tag:blogger.com,1999:blog-59941116582486558302024-03-29T00:32:32.879-07:00Retirement Income ScenariosWilliam Sharpehttp://www.blogger.com/profile/05491169505008130323noreply@blogger.comBlogger15125tag:blogger.com,1999:blog-5994111658248655830.post-64460347064976155112017-07-08T15:25:00.000-07:002018-04-01T15:08:41.885-07:00Table of Contents<div style="text-align: center;">
<b>Table of Contents</b><br />
<br />
<b>Links to posts in chronological order -- use browser back arrow to return to this post</b></div>
<br />
<a href="http://www.retirementincomescenarios.blogspot.com/2013/08/plans-for-this-blog.html">Plans for this blog</a><br />
<br />
<a href="http://www.retirementincomescenarios.blogspot.com/2013/09/why-scratch.html">Why Scratch?</a><br />
<br />
<a href="http://www.retirementincomescenarios.blogspot.com/2013/09/longevity-graphs.html">Longevity Graphs</a><br />
<br />
<a href="http://www.retirementincomescenarios.blogspot.com/2013/09/retirement-income-scenarios.html">Retirement Income Scenarios</a><br />
<br />
<a href="http://www.retirementincomescenarios.blogspot.com/2013/11/video-on-longevity.html">Video on Longevity</a><br />
<br />
<a href="http://www.retirementincomescenarios.blogspot.com/2013/12/savings-and-income-scenario-settings.html">Savings and Income Scenario Settings</a><br />
<br />
<a href="http://www.retirementincomescenarios.blogspot.com/2013/12/the-x-rule.html">The X% Rule</a><br />
<br />
<a href="http://www.retirementincomescenarios.blogspot.com/2013/12/investment-returns-and-inflation.html">Investment Returns and Inflation</a><br />
<br />
<a href="http://www.retirementincomescenarios.blogspot.com/2014/01/rmd-accounts.html">RMD Accounts</a> <br />
<br />
<a href="http://www.retirementincomescenarios.blogspot.com/2014/01/analyzing-multiple-scenarios.html">Analyzing Multiple Scenarios</a><br />
<br />
<a href="http://www.retirementincomescenarios.blogspot.com/2014/04/yearyear-incomes.html">Year/Year Incomes</a><br />
<br />
<a href="http://www.retirementincomescenarios.blogspot.com/2014/04/present-values.html">Present Values</a><br />
<br />
<a href="http://www.retirementincomescenarios.blogspot.com/2014/04/pricing-kernels.html">Pricing Kernels</a> <br />
<br />
<a href="http://www.retirementincomescenarios.blogspot.com/2017/07/the-rismat-e-book-and-software-astute.html" target="_blank">The RISMAT E-book and Software</a><br />
<br />
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<div class="post-title entry-title" itemprop="name" style="text-align: center;">
<b>Using the Scratch Software</b></div>
<div class="post-header">
</div>
<h2 style="text-align: center;">
</h2>
<br />
Go to <a href="http://scratch.mit.edu/projects/20867413" target="_blank">scratch.mit.edu/projects/20867413</a><br />
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<br />
Notes:<br />
To put Scratch in turbo mode, hold the shift key then click the green flag<br />
To see the program full-screen, click the icon in the upper left corner<br />
To select an item from the main menu, click the corresponding screen button <br />
To turn context-sensitive help on or off, press the keyboard up arrow key<br />
To return to the main menu, press the keyboard left arrow key<br />
The release date of the software will be included in the title (RIS-yyyymmdd)<br />
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<br />William Sharpehttp://www.blogger.com/profile/05491169505008130323noreply@blogger.com39tag:blogger.com,1999:blog-5994111658248655830.post-9275792331707859042017-07-08T14:56:00.001-07:002018-03-20T11:07:55.202-07:00The RISMAT E-book and Software<div style="text-align: center;">
<h2 style="text-align: left;">
</h2>
</div>
The astute reader of this blog will note that this post was
written more than three years after its predecessor. This was not due
to lethargy on the part of the author. Rather, for the last three
years I have devoted my research and writing to a project titled:<br />
<br />
<br />
<i><b> R</b></i><i>etirement </i><i><b>I</b></i><i>ncome </i><i><b>S</b></i><i>cenario
</i><i><b>M</b></i><i>atrices</i> (<b>RISMAT)</b>
<br />
<br />
<br />
<br />
<br />
The project has produced both an e-book and a suite of software
written in the MATLAB language. Both are available at:<br />
<br />
<br />
<span style="color: blue;"><i> www.stanford.edu/~wfsharpe/RISMAT</i></span><br />
<br />
<br />
<span style="color: black;"><span style="font-style: normal;">A compressed (.zip) file with the matlab software is available at:</span></span><br />
<span style="color: black;"><span style="font-style: normal;"><br /></span></span>
<span style="color: black;"><span style="color: blue; font-style: normal;">https://drive.google.com/file/d/1GTrNOkDrKnbE4kdmVJ95GKrhjUKD44FM/view?usp=sharing </span></span><br />
<br />
A compressed (.zip) file with the ebook chapters is also available at:<br />
<br />
<span style="color: blue;">https://drive.google.com/file/d/14sTIRbzfCF0ChyXo8lQJeJTs8DX361cB/view?usp=sharing</span><br />
<br />
<span style="color: black;"><span style="font-style: normal;">The
material may be used for any purpose without the payment of any fees,
subject to the terms of the Create Commons Attribution 4.0 License,
for which a link is provided on the RISMAT table of contents page.</span></span><br />
<br />
<br />
<br />
<span style="color: black;"><span style="font-style: normal;">The overall
approach is similar to that taken in the previous chapters of this
blog, which can be considered a precursor for RISMAT. However, the
MATLAB software can handle many more scenarios at speeds that are
orders of magnitude faster than those of the Scratch software used
for this blog. Moreover, the RISMAT e-book covers many more topics,
each in considerable depth. </span></span>
<br />
<br />
<br />
<br />
<span style="color: black;"><span style="font-style: normal;">My hope is
that this new material and the accompanying programs will be utilized
by financial engineers and financial advisors who offer their
services to retirees. I believe the RISMAT e-book and software can
provide a useful base for comprehensive analyses of alternative
approaches for providing retirement income. </span></span>
<br />
<br />
<br />
<br />
<span style="color: black;"><span style="font-style: normal;">It seems
fitting to conclude this blog with the last sentence from the RISMAT
e-book:</span></span><br />
<br />
<br />
<br />
<span style="color: black;"><i>Following tradition, we conclude with the
admonition that more research is needed (and probably more
programming) -- tasks that the author, having reviewed actuarial
tables, chooses to leave to others</i></span><span style="color: black;"><span style="font-style: normal;">.</span></span><br />
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</style>William Sharpehttp://www.blogger.com/profile/05491169505008130323noreply@blogger.com27tag:blogger.com,1999:blog-5994111658248655830.post-593999512115439022014-04-22T14:45:00.000-07:002014-04-22T18:17:49.273-07:00Pricing Kernels<div class="separator" style="clear: both; text-align: center;">
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<span style="font-size: small;"><span style="font-family: inherit;">In June of 2011, I was invited to give the keynote speech at the annual meeting of the European Financial Management Association in Braga, Portugal. The talk was to be followed by an article for the EFMA journal. I of course chose to speak about my favorite topic. Here is the reference to the published version:</span></span><br />
<span style="font-size: small;"><span style="font-family: inherit;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: inherit;">"Post-retirement Financial Strategies: Forecasts and
Valuation", <span style="font-style: italic;">European
Financial Management</span>, Vol. 18, No. 3, 2012, pp.
324-351.</span></span><br />
<span style="font-size: small;"><span style="font-family: inherit;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: inherit;">A pre-publication version is available <span style="color: blue;"><a href="http://www.stanford.edu/~wfsharpe/retecon/PRS_20111001.pdf">here</a></span>.</span></span><br />
<span style="font-size: small;"><span style="font-family: inherit;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: inherit;">In the paper I tried to succinctly and somewhat formally summarize the approach to asset pricing that I and my coauthors Jason Scott and John Watson took in our earlier publications, that I have taken in my subsequent research and that I have used for the present value calculations in the RIS software. Here I'll use portions of the EFMA paper with added comments to provide a semi-formal description. </span></span><br />
<span style="font-size: small;"><span style="font-family: inherit;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: inherit;">Warning: some of this will be technical. Feel free to skim or ignore it, as needed.</span></span><br />
<span style="font-size: small;"><span style="font-family: inherit;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: inherit;">Here goes. </span></span><br />
<br />
<span style="font-size: small;"><span style="font-family: inherit;">------------ </span></span><br />
<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;">Building on the approach of [Arrow 1952] and [Debreu 1959], I consider first a one-
period setting in which there are <span style="font-style: italic;">n </span>possible states of the world. Assume that securities are
priced based on their payoffs in the states of the world, using a set of state prices, where
p<sub>s</sub> is the price today of $1 if and only if state s occurs at the end of the period. If the probability of state s is π<span style="font-weight: normal;"><sub>s</sub></span>, define the <span style="font-style: italic;">pricing kernel value </span>for state s at time t as
m<sub>st</sub>≡ p<sub>st</sub>/π<sub>st</sub>, a value that I called in [Sharpe, 2007] the <span style="font-style: italic;">price per chance </span>(PPC). The set
of n values of m<sub>st</sub> is the pricing kernel for time t. To say that only the market portfolio is priced in a single period is to assert that all states with the same market return have the
same price per chance. Moreover, societal risk-aversion implies that the higher a state’s
market return, the lower should be its price per chance. More fundamentally, if markets
are to clear, prices must adjust so that income in a state of scarcity (low market return)
costs more than income in a state of plenty (high market return).</span></span><br />
<div class="column">
<span style="font-family: 'Times'; font-size: 12.000000pt;"> -------------</span><br />
<span style="font-size: small;"><span style="font-family: inherit;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: inherit;">The references are as follows:</span></span><br />
<span style="font-family: 'Times'; font-size: 12.000000pt;"><span style="font-size: small;"><span style="font-family: inherit;"><br /></span></span></span>
<span style="font-family: 'Times'; font-size: 12.000000pt;">-------------- </span><br />
<span style="font-family: 'Times'; font-size: 12.000000pt;">Arrow, Kenneth J., 1952, “Le Role de valeurs boursiers pour la repartition le meillure des
risques,” Econometrie, Colloques Internationaux du Centre National de la Recherche
Scientifique 11, pp. 41-47.</span><br />
<br />
<span style="font-family: 'Times'; font-size: 12.000000pt;">
<span style="font-family: 'Times'; font-size: 12.000000pt;">Debreu, Gerard, 1959, </span><span style="font-family: 'Times'; font-size: 12.000000pt; font-style: italic;">The Theory of Value, </span><span style="font-family: 'Times'; font-size: 12.000000pt;">Wiley and Sons, New York.</span></span><br />
<br />
<span style="font-family: 'Times'; font-size: 12.000000pt;"><span style="font-family: 'Times'; font-size: 12.000000pt;">
</span></span><br />
<div class="column">
<span style="font-family: 'Times'; font-size: 12.000000pt;">Sharpe, William F., 2007, </span><span style="font-family: 'Times'; font-size: 12.000000pt; font-weight: 700;">Investors and Markets: Portfolio Choices, Asset Prices
and Investment Advice, </span><span style="font-family: 'Times'; font-size: 12.000000pt;">Princeton University Press, 2007. </span><br />
<span style="font-family: 'Times'; font-size: 12.000000pt;">--------------- </span><br />
<br /></div>
<span style="font-family: 'Times'; font-size: 12.000000pt;"><span style="font-size: small;"><span style="font-family: inherit;"> </span></span></span><br />
<span style="font-size: small;"><span style="font-family: inherit;">There are two major approaches to asset pricing in Financial Economics. One, starting with the Capital Asset Pricing Model (CAPM), builds on the mean/variance portfolio theory of Harry Markowitz. The other, using Pricing Kernels, builds on the state/preference theory developed by Kenneth Arrow and Gerard Debreu. It may surprise some that I favor the latter for applications such the analysis of retirement income strategies; after all, I was awarded a Nobel Prize in Economics for my work on the CAPM. But over the years I have found the state/preference approach better suited to the task of modeling the determination of asset prices in a one-period setting and even moreso in settings involving many periods, as with retirement income analysis. </span></span><br />
<span style="font-size: small;"><span style="font-family: inherit;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: inherit;">I labored long and hard to justify this preference in a series of lectures given at Princeton and the subsequent 2007 book referenced above. However, the focus of the book was on a traditional one-period setting in which investment is made at the present time based on estimates of the probabilities of various returns over a single period such as a year. As I showed there, those who have been brought up on the CAPM need not fear obsolescence, since its major qualitative conclusions hold in the state/preference setting. The market portfolio is still an efficient investment for the average investor. Moreover, security and portfolio expected returns are linearly related to a measure of return sensitivity to the returns on the market. There is of course a difference. In the CAPM the appropriate measure (beta) is based on the covariance of a security or portfolio's return with that of the market portfolio. In the Pricing Kernel version, the measure is based on the covariance of a security or portfolio's return with a function of the return on the market portfolio, a result that I called its<i> kernel beta</i>. But the main qualitative message holds in both approaches: securities and portfolios are priced so that higher expected returns are associated with greater probabilities of doing badly in bad times. </span></span><br />
<span style="font-size: small;"><span style="font-family: inherit;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: inherit;">The CAPM can be considered a special case of the more general pricing kernel approach in which the kernel for a single investment period is a linear function of the return on the market portfolio over that period. In the book I argued that it is much more plausible to assume that the relationship is non-linear, with a positive price for every possible future state of the world. </span></span><br />
<span style="font-size: small;"><span style="font-family: inherit;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: inherit;">The case for such a relationship is far stronger when one attempts to model the determination of asset prices in a multi-period world -- a necessity for analysis of retirement income strategies. This requires some assumption about the efficiency of alternative strategies for investors with different horizons. Moreover, the assumption must be consistent with market clearing -- that is, the collective demands for available securities must equal the supplies. To return to the EFMA paper:</span></span><br />
<br />
------------<br />
<br />
<span style="font-size: small;"><span style="font-family: 'Times';">Consistent with the focus on strategies that utilize only the market portfolio and a riskless
asset, I assume that only market returns are priced, both in any single period and also for
any multi-period horizon. This could be consistent with a model of multi-period
equilibrium in the capital markets, although I have no aspirations to develop one here (or
elsewhere). In any event, as I will show, the assumption greatly restricts the
characteristics of the pricing function. </span></span><br />
<span style="font-family: 'Times'; font-size: 12.000000pt;">------------</span><br />
<br />
<span style="font-size: small;"><span style="font-family: inherit;">This may be far too simple a characterization of equilibrium in capital markets. But to value different combinations of income over future years one must make some assumptions and to be credible they should be consistent with market clearing. The approach taken in the EFMA paper seems reasonable. And some such model is better than none. Interestingly, in a world of this sort, it is quite simple to determine the manner in which assets should be priced. Here is the formal analysis from the paper:</span></span><br />
<br />
------------</div>
<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;">Now, consider a two-period case. Letting <span style="font-weight: 700;">m<sub>t</sub> </span>and <span style="font-weight: 700;">r<sub>t</sub> </span>represent vectors of pricing kernel
values and market total returns (value-relatives) respectively for time t, the pricing kernels for periods 1 and 2 can be written as: </span></span><br />
<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-weight: 700;"> m<sub>1</sub> </span>= <span style="font-style: italic;">f<sub>1</sub></span>(<b>r</b><sub>1</sub> ) </span></span><br />
<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-weight: 700;"> m<sub>2</sub> </span>= <span style="font-style: italic;">f<sub>2</sub>(<b>r</b><sub>2</sub>)</span></span></span><br />
<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-style: italic;"><br />
</span>The pricing kernel for a horizon that includes both periods 1 and 2 will be the (dot) product of the two kernels. Thus: </span></span><br />
<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-weight: 700;"> m<sub>1</sub></span>⋅ <span style="font-weight: 700;"><span style="font-weight: 700;">m<sub>2</sub> </span> </span>= <span style="font-style: italic;">f<sub>1</sub></span>(<b>r</b><sub>1</sub> )⋅ <span style="font-style: italic;">f<sub>2</sub>(<b>r</b><sub>2</sub>)</span></span></span><br />
<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-weight: 700;"> </span></span></span><br />
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<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;">In order for (1) the market to be priced in the same manner for each period and (2) for
only the market to be priced for any multi-year horizon, it must be the case that: </span></span><br />
<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"> <span style="font-style: italic;">f</span>(<b>r</b><sub>1</sub> ) ⋅ <span style="font-style: italic;">f(<b>r</b><sub>2</sub>) = </span><span style="font-style: italic;"> <span style="font-style: italic;">g(</span></span></span></span><span style="font-style: italic;"><span style="font-style: italic;"><b>r</b><sub>1</sub>⋅ </span></span><b>r</b><sub>2</sub>)<br />
<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;">A necessary and sufficient condition for this to be the case is that the one-period pricing
function be isoelastic: </span></span><br />
<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-weight: 700;"> m</span></span></span><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><sub>t</sub></span></span><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><sub><sub> </sub> </sub>= A<b>r</b></span></span><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><sub>t</sub></span></span><sup>-b</sup> </span></span><br />
<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;">More generally, if <span style="font-weight: 700;">M</span></span></span><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><sub>t</sub></span></span></span></span><span style="font-weight: 700;"> </span>represents the pricing kernel for payments t periods hence and <span style="font-weight: 700;">V</span><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><sub>t</sub></span></span><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-weight: 700;">
</span>the cumulative market return over that horizon: </span></span><br />
<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-weight: 700;"> M</span></span></span><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><sub>t</sub></span></span><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"></span></span><span style="font-weight: 700;"><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><sub><sub> </sub></sub></span></span></span></span></span><sub></sub>= A<sup>t</sup> <b>V</b><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><b><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"></span></span></b><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><sub>t</sub></span></span></span></span><b><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"></span></span><span style="font-weight: 700;"><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><sub><sub> </sub></sub></span></span></span></span></span></b></span></span><sup>-b</sup> <br />
------------<br />
<br />
<span style="font-size: small;"><span style="font-family: inherit;">While this may look formidable, it is quite sensible. Back to the paper:</span></span><br />
<br />
------------<br />
<span style="font-family: 'Times'; font-size: small;">Taking the logarithms of both sides of the equation:</span><br />
<span style="font-size: small;"><br /></span>
<span style="font-family: 'Times'; font-size: small;"> log(<b>M</b></span><span style="font-family: 'Times'; font-size: small;"><span style="font-family: 'Times';"><span style="font-family: 'Symbol';"><span style="font-family: 'Symbol';"><span style="font-family: 'Symbol';"><sub><span style="font-family: 'Symbol';"><sub><span style="font-family: 'Times';"><sub>t</sub></span></sub></span></sub></span></span></span></span></span><span style="font-size: small;"> ) <span style="font-family: 'Symbol';">= </span><span style="font-family: 'Times';">log(A</span><span style="font-family: 'Times';"><span style="font-family: 'Times';"><span style="font-family: 'Symbol';"><sup>t</sup></span></span> ) </span><span style="font-family: 'Symbol';">− </span><span style="font-family: 'Times'; font-style: italic;">b</span><span style="font-family: 'Times'; font-style: italic;"><span style="font-family: 'Times';"><span style="font-family: 'Times';"><span style="font-family: 'Symbol';">⋅</span></span></span>log(<b>V</b></span><span style="font-family: 'Times'; font-style: italic;"><span style="font-family: 'Times';"><span style="font-family: 'Symbol';"><span style="font-family: 'Symbol';"><span style="font-family: 'Symbol';"><sub><span style="font-family: 'Symbol';"><sub><span style="font-family: 'Times';"><sub>t</sub></span></sub></span></sub></span></span></span></span></span></span> <span style="font-family: 'Times';">)
</span><br />
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<span style="font-family: 'Times'; font-size: small;"><br /></span>
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<span style="font-family: 'Times'; font-size: small;">Clearly, the b coefficient indicates the elasticity of the pricing kernel with respect to
cumulative market return – for every one percent increase in the latter, the pricing kernel
decreases by approximately </span><span style="font-family: 'Times'; font-size: small; font-style: italic;">b </span><span style="font-family: 'Times'; font-size: small;">percent. As is well known, this can be interpreted as
indicating that a “representative investor” who holds the market portfolio has a utility
function with a constant relative risk-aversion coefficient of </span><span style="font-family: 'Times'; font-size: small; font-style: italic;">b (</span>for a further discussion, see <span style="font-family: 'Times'; font-size: small; font-style: italic;">Sharpe, 2007).</span><br />
<span style="font-family: 'Times'; font-size: 12.000000pt; font-style: italic;">------------</span><br />
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<span style="font-size: small;"><span style="font-family: inherit;">The relationship is log-linear. Among other things, this means that every state price is positive, no matter how large or small the return on the market portfolio. This is not the case if the kernel is linear, which can be if one assumes that all investors have quadratic utility functions (and hence care only about the mean and variance of portfolio returns). </span></span><br />
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<span style="font-size: small;"><span style="font-family: inherit;">This is not the place to pick a fight between proponents of traditional mean/variance portfolio theory with a strict interpretation of the of the CAPM and those (including me) who prefer the state/preference pricing kernel approach. The key point is that we now have a pricing kernel with desirable properties that could be consistent with a multi-period equilibrium. For any period in the future, we can use it to determine the price per chance (PPC) for $1 to be received at that future time if and only if the state occurs. PPC is a function of the cumulative compounded return on the market portfolio from the present to that future time and the function is of the form shown above. And clearly, the price for $1 to be received at a future time and in a given state is equal to the PPC times the chance that the state will occur at the time in question.</span></span></div>
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<span style="font-size: small;"><span style="font-family: inherit;">To return to the paper: </span></span><br />
------------<br />
<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;">I will use the pricing kernel primarily to estimate the values of the distributions of
spending (also called “payments” or “paychecks”) provided by a strategy. This requires
a set of state prices – each of which represents the cost today of obtaining $1 at a given
future time and state. Since the pricing kernel is simply a set of ratios of state prices to
state probabilities: </span></span><br />
<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-weight: 700; vertical-align: 2pt;"> P</span></span></span><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-weight: 700; vertical-align: 2pt;"><span style="font-family: 'Times'; font-size: small;"><span style="font-family: 'Times';"><span style="font-family: 'Symbol';"><span style="font-family: 'Symbol';"><span style="font-family: 'Symbol';"><sub><span style="font-family: 'Symbol';"><sub><span style="font-family: 'Times';"><sub>t</sub></span></sub></span></sub></span></span></span></span></span></span></span></span> <span style="vertical-align: 2pt;">= </span><span style="vertical-align: 2pt;"><b>M</b></span><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="vertical-align: 2pt;"><b><span style="font-family: 'Times'; font-size: small;"><span style="font-family: 'Times';"><span style="font-family: 'Symbol';"><span style="font-family: 'Symbol';"><span style="font-family: 'Symbol';"><sub><span style="font-family: 'Symbol';"><sub><span style="font-family: 'Times';"><sub>t</sub></span></sub></span></sub></span></span></span></span></span></b></span></span></span>⋅<b><span style="vertical-align: 2pt;"><b>∏</b></span></b><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><b><span style="vertical-align: 2pt;"><b><span style="font-family: 'Times'; font-size: small;"><span style="font-family: 'Times';"><span style="font-family: 'Symbol';"><span style="font-family: 'Symbol';"><span style="font-family: 'Symbol';"><sub><span style="font-family: 'Symbol';"><sub><span style="font-family: 'Times';"><sub>t</sub></span></sub></span></sub></span></span></span></span></span></b></span></b></span></span><br />
<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="vertical-align: 2pt;"><br />
</span>Where <span style="font-weight: 700;">P</span></span></span><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-weight: 700;"><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"></span></span><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-weight: 700; vertical-align: 2pt;"><span style="font-family: 'Times'; font-size: small;"><span style="font-family: 'Times';"><span style="font-family: 'Symbol';"><span style="font-family: 'Symbol';"><span style="font-family: 'Symbol';"><sub><span style="font-family: 'Symbol';"><sub><span style="font-family: 'Times';"><sub>t</sub></span></sub></span></sub></span></span></span></span></span></span></span></span></span></span></span><span style="font-weight: 700;"> </span>is a vector of state prices for payments at a future time t, and <b>M</b><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"></span></span><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="vertical-align: 2pt;"><b><span style="font-family: 'Times'; font-size: small;"><span style="font-family: 'Times';"><span style="font-family: 'Symbol';"><span style="font-family: 'Symbol';"><span style="font-family: 'Symbol';"><sub><span style="font-family: 'Symbol';"><sub><span style="font-family: 'Times';"><sub>t</sub></span></sub></span></sub></span></span></span></span></span></b></span></span></span><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><sub><span style="font-family: 'Times';"><b> </b></span></sub>and <span style="vertical-align: 2pt;"><b>∏</b></span></span></span><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="vertical-align: 2pt;"><b><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"></span></span><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="vertical-align: 2pt;"><b><span style="font-family: 'Times'; font-size: small;"><span style="font-family: 'Times';"><span style="font-family: 'Symbol';"><span style="font-family: 'Symbol';"><span style="font-family: 'Symbol';"><sub><span style="font-family: 'Symbol';"><sub><span style="font-family: 'Times';"><sub>t</sub></span></sub></span></sub></span></span></span></span></span></b></span></span></span></b></span></span></span><sub><span style="font-family: 'Times';"><b> </b></span></sub>are, respectively, vectors for the pricing kernel and probabilities of the states for that time. In
the simulations ... market returns were drawn randomly from the underlying
probability distributions for <span style="font-style: italic;">n </span>multi-year scenarios. Considering each scenario as a state,
the probabilities all equal 1/n so that:<br />
<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-weight: 700; vertical-align: 2pt;">P</span></span></span><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"></span></span><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-weight: 700; vertical-align: 2pt;"><span style="font-family: 'Times'; font-size: small;"><span style="font-family: 'Times';"><span style="font-family: 'Symbol';"><span style="font-family: 'Symbol';"><span style="font-family: 'Symbol';"><sub><span style="font-family: 'Symbol';"><sub><span style="font-family: 'Times';"><sub>t</sub></span></sub></span></sub></span></span></span></span></span></span></span></span><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="vertical-align: 2pt;"><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="vertical-align: 2pt;"><b><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"></span></span></b></span></span></span><sub> </sub>= </span><span style="vertical-align: 2pt;"><b>M</b></span></span></span><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="vertical-align: 2pt;"><b><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"></span></span><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="vertical-align: 2pt;"><b><span style="font-family: 'Times'; font-size: small;"><span style="font-family: 'Times';"><span style="font-family: 'Symbol';"><span style="font-family: 'Symbol';"><span style="font-family: 'Symbol';"><sub><span style="font-family: 'Symbol';"><sub><span style="font-family: 'Times';"><sub>t</sub></span></sub></span></sub></span></span></span></span></span></b></span></span></span></b></span></span></span><sub> </sub>/n<br />
---------- <br />
<br />
<span style="font-size: small;"><span style="font-family: inherit;">But how to determine the key parameters in the equation (<i>A</i> and<i> b</i>)? In the simulations for the EFMA paper, I ran a million simulations for each year for every strategy analyzed. Moreover, my Matlab program made it possible to run all the results for each year at one time, then find the pricing kernel that best fit the resulting million simulated compound market returns. For each future year, I found values for the two variables (A and<i> b</i>) that would be consistent with valuations of the market portfolio and the riskless asset for that horizon. As I indicated in a footnote in the paper:</span></span><br />
<br />
------------<br />
<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;">In the simulations ... to reduce sampling errors of draws
from the underlying return distribution, the values of A and b used for each horizon were found iteratively
based on the requirement that the implied present values of both the cumulative market returns and the
cumulative risk-free return were both within a very small distance from 1.0. </span></span><br />
<span style="font-family: 'Times'; font-size: small;"><span style="font-size: x-small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-family: 'Times';">------------</span></span></span></span><br />
<br />
<span style="font-size: small;"><span style="font-family: inherit;">John Watson is currently doing research on the efficacy of this and other possible approaches to valuation when using a simulated sample to approximate the universe of all possible scenarios. This procedure produces pricing kernels that may fit the simulated scenarios but may violate the underlying assumption about the multi-period equilibrium. I shall have more to say about this in subsequent posts. But the discussion is not particularly germane for those using the RIS software for two reasons. First, it is infeasible to run a million scenarios. And second, since the software generates results a scenario at a time, this fitting procedure is not feasible since it would require storing at least 40 or 50 million results (a matrix with a row for each scenario and a column for each year). </span></span><br />
<span style="font-size: small;"><span style="font-family: inherit;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: inherit;">Fortunately, there is another way, using a formula derived by John Watson in our earlier paper:</span></span><br />
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<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;">Scott, Jason S. with William F. Sharpe and John G. Watson, 2009, "The 4% Rule -- At
What Price?", <span style="font-style: italic;">Journal of Investment Management, </span>Vol. 7, No. 3, Third Quarter 2009, pp.
31-48 </span></span></div>
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<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"> </span></span><span style="font-size: small;"><span style="font-family: inherit;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: inherit;">It is available<a href="http://www.stanford.edu/~wfsharpe/retecon/4percent.pdf"> <span style="color: black;">here</span></a>.</span></span><br />
<br />
<span style="font-size: small;"><span style="font-family: inherit;">As derived in the paper, for the special case in which the return on the market portfolio is assumed to be lognormally distributed in each period (year), the coefficients of the asset pricing equation can be calculated directly. Here is the EFMA paper's description:</span></span><br />
<br />
<span style="font-family: 'Times'; font-size: 12.000000pt;">----------- </span><br />
<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;">The values of the coefficients in the pricing equation can be computed directly from the
parameters of the assumed distributions of annual returns as follows:</span></span><br />
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<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2DIB4iq5ZTemxoie7dUqxmkUTqXXxp-wXcq3nQsb-W0eL891w5hh-dsuMlbFmGd9Ik2i0bA6iTU8KUO8ldhkSpMfwB9bam-SYJldPxrUCpn-FNq9rAkkilsPyAJ4gu2VXx7n1xji-UeQ/s1600/Screen+Shot+2014-04-18+at+12.08.52+PM.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="" border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2DIB4iq5ZTemxoie7dUqxmkUTqXXxp-wXcq3nQsb-W0eL891w5hh-dsuMlbFmGd9Ik2i0bA6iTU8KUO8ldhkSpMfwB9bam-SYJldPxrUCpn-FNq9rAkkilsPyAJ4gu2VXx7n1xji-UeQ/s1600/Screen+Shot+2014-04-18+at+12.08.52+PM.png" height="118" title="" width="200" /></a></span></span></div>
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<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;">In the above equations, R</span></span><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="vertical-align: 2pt;"><b><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="vertical-align: 2pt;"><b><span style="font-family: 'Times'; font-size: small;"><span style="font-family: 'Times';"><span style="font-family: 'Symbol';"><span style="font-family: 'Symbol';"><span style="font-family: 'Symbol';"><sub><span style="font-family: 'Symbol';"><sub><span style="font-family: 'Times';"><sub>f</sub></span></sub></span></sub></span></span></span></span></span></b></span></span></span></b></span></span></span></span></span> is the total risk-free return, <span style="font-family: 'Times';">E</span><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"></span></span><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="vertical-align: 2pt;"><b><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="vertical-align: 2pt;"><b><span style="font-family: 'Times'; font-size: small;"><span style="font-family: 'Times';"><span style="font-family: 'Symbol';"><span style="font-family: 'Symbol';"><span style="font-family: 'Symbol';"><sub><span style="font-family: 'Symbol';"><sub><span style="font-family: 'Times';"><sub>m</sub></span></sub></span></sub></span></span></span></span></span></b></span></span></span></b></span></span></span></span></span></span></span> = E[R<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"></span></span><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"></span></span><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"></span></span><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="vertical-align: 2pt;"><b><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="vertical-align: 2pt;"><b><span style="font-family: 'Times'; font-size: small;"><span style="font-family: 'Times';"><span style="font-family: 'Symbol';"><span style="font-family: 'Symbol';"><span style="font-family: 'Symbol';"><sub><span style="font-family: 'Symbol';"><sub><span style="font-family: 'Times';"><sub>t</sub></span></sub></span></sub></span></span></span></span></span></b></span></span></span></b></span></span></span></span></span></span></span></span></span></span></span>] is the yearly expected
total market return, and S<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"></span></span><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="vertical-align: 2pt;"><b><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="vertical-align: 2pt;"><b><span style="font-family: 'Times'; font-size: small;"><span style="font-family: 'Times';"><span style="font-family: 'Symbol';"><span style="font-family: 'Symbol';"><span style="font-family: 'Symbol';"><sub><span style="font-family: 'Symbol';"><sub><span style="font-family: 'Times';"><sub>m</sub></span></sub></span></sub></span></span></span></span></span></b></span></span></span></b></span></span></span></span></span></span></span> = (Var[R<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"></span></span><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"></span></span><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="vertical-align: 2pt;"><b><span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="vertical-align: 2pt;"><b><span style="font-family: 'Times'; font-size: small;"><span style="font-family: 'Times';"><span style="font-family: 'Symbol';"><span style="font-family: 'Symbol';"><span style="font-family: 'Symbol';"><sub><span style="font-family: 'Symbol';"><sub><span style="font-family: 'Times';"><sub>t</sub></span></sub></span></sub></span></span></span></span></span></b></span></span></span></b></span></span></span></span></span></span></span></span></span>])<span style="font-size: small;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: small;"><span style="font-family: 'Times';"><span style="font-family: 'Times';"><span style="font-family: 'Symbol';"><sup>1/2</sup></span></span></span></span> is the annual market volatility.</span></span></div>
------------<br />
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<span style="font-size: small;"><span style="font-family: inherit;">The RIS software uses these formulas to compute the parameters for the pricing kernel (computing <i>b</i> first, then <i>A</i>). While producing multiple scenarios, the formula is used to compute the state price for each income and fee payment, obtain the resulting present value and update the appropriate cumulative sum.</span></span><br />
<span style="font-size: small;"><span style="font-family: inherit;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: inherit;">Unfortunately, with feasible numbers of scenarios the results will be imperfect since the sample scenarios will not be fully representative of the universe of possible future scenarios. As indicated in the previous post, this is reflected in the design of the RIS software. First, present values are shown only when 5,000 or more scenarios have been analyzed. Second, the computed values are not shown, only the percentages of the total value associated with the prospects for three recipients (the household, the estate and those who receive fees). Third, the actual percentages are not shown, only a pie chart indicating their magnitudes. And finally, the user is encouraged to generate additional sets of multiple scenarios to determine the variation in the allocation of the present values from case to case. </span></span><br />
<span style="font-size: small;"><span style="font-family: inherit;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: inherit;">Enough caveats. Despite these limitations, the breakdown of present values in the RIS software should provide valuable information in most cases. And some of the analyses that I will describe in subsequent posts will be obtained using Matlab software, employing large numbers of scenarios and, in some cases using different and hopefully more accurate approximations of present values. </span></span><br />
<br />
<span style="font-size: small;"><span style="font-family: inherit;">I will have much to say about the present values of future possible outcomes in the future analyses of alternative retirement income strategies. You should expect nothing less from an economist.</span></span><br />
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William Sharpehttp://www.blogger.com/profile/05491169505008130323noreply@blogger.com17tag:blogger.com,1999:blog-5994111658248655830.post-50208673490898171792014-04-19T11:15:00.001-07:002014-04-19T11:15:13.616-07:00Present Values<div class="separator" style="clear: both; text-align: center;">
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A retirement income strategy is designed to provide incomes at different times to different recipients. However, the outcomes will depend on future events -- in our scenarios, longevity, investment returns and inflation. But it is helpful to consider the present value of prospective incomes and how it is distributed among the key recipients. </div>
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In this post I'll describe the basic approach used for the Present Value Analysis included in the RIS software; more details will follow in the next post.</div>
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To start, consider the problem of valuation of an investment in a share of Apple common stock. This is a claim that provides the owner with any future dividends that Apple might pay as well as the ability to sell the share at any future date. Since shares are traded publicly it is relatively straightforward to determine the market price, which represents a consensus opinion of the present value of Apple's future prospects. If you think the share is worth more, you may well want to buy more shares; if less, you may want to sell any shares you hold. But the market price reflects the overall evaluation of other investors. Economists define this as its present value and consider it a highly relevant measure.</div>
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Note that the price of an Apple share is based on its uncertain future prospects. In some possible scenario, future earnings and dividends will be disappointing and the prior price will turn out to have been too high. In another scenario, the future earnings and dividends will be spectacular and the prior price will turn out to have been too low. But the price today reflects investors' assessment of many different possible scenarios.</div>
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The RIS software takes the same approach when calculating the present values of prospective incomes. In particular, it generates a number of scenarios, each covering many years. </div>
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Now, imagine a large matrix (table) with 5,000 rows (one for each scenario) and 40 columns (one for each future year). In each cell of the matrix there is an income payment and an indication of the recipient (you and your partner, you alone, your partner alone, or your estate). </div>
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Next, imagine that for every cell in this matrix there is a number representing the present value of $1. For example, in row (scenario) 123, column (year) 10 there is a number representing the amount that investors would pay today to receive $1 in year 10 if and only if scenario 123 (and no other) happened. What would determine this value? First, the probability that the scenario would take place. And second, the state of the world in that scenario.</div>
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The probability is easily determined. In our simulation there are 5,000 scenarios so the probability that any one will occur is 1/5000. But there is good reason to believe that the prices will differ across scenarios.</div>
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To approach this in a more general context, economists use a concept that I like to term the <i>price per chance</i> (PPC) for payment at a given future time in a given future state of the world. And there is good reason to assume that this value will depend on (1) the number of years before the state occurs and (2) the economic health of the economy in that state. In particular, it makes sense to assume that other things equal, the present (market) value of a future unit of purchasing power is greater for states of scarcity than for states of plenty. In our simulations the best available measure of the state of the economy is the future level of the real value of the market portfolio of world bonds and stocks. So we assume that the present value of $1 in a future state divided by the probability of that outcome depends on the number of years before the state occurs and on the cumulative real value of $1 invested in the market portfolio today and held until that future year. In sum, other things equal, the present value of $1 to be received in a given state divided by the probability is lower, the farther in the future the state and the smaller the cumulative return on the market over the period.</div>
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The<i> pricing kernel</i> formula reflects these assumptions. It indicates the price (present value) per chance (probability) of $1 for a given state in the future. To return to our matrix with 5,000 rows (scenarios) and 40 columns (years), imagine that there is present value of $1 in each cell, computed by multiplying the corresponding price per chance by the probability (in this case, 1/5,000). Then we can value each cell's payment by multiplying the amount paid by the present value. Moreover we can sum up the present values of all the payments made to various recipients. In effect, this is what the RIS software does (although it actually updates sums as it creates scenarios in order to avoid the need to store large matrices). In addition, the software provides similar computations for any fees paid to financial advisors or others (as if they were stored in a second matrix).</div>
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Here is a result (based on a standard RMD account with 1% fees), obtained by clicking the <i>Present Values</i> button under <i>Analyses</i>.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQ9XuuZf7RUlXSJLgOpLsLXfQMpH0onj2A3elJlDX7m6tHuqcYxaHi9m16qloRM4FuFOMsBPX1HjOUECKk_Tehn9F7vEbuSaIXPVyJTQe4OgLwT0zqyHKt5eb8tEK8YNrlqung5oiom58/s1600/Screen+Shot+2014-04-18+at+11.59.43+AM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQ9XuuZf7RUlXSJLgOpLsLXfQMpH0onj2A3elJlDX7m6tHuqcYxaHi9m16qloRM4FuFOMsBPX1HjOUECKk_Tehn9F7vEbuSaIXPVyJTQe4OgLwT0zqyHKt5eb8tEK8YNrlqung5oiom58/s1600/Screen+Shot+2014-04-18+at+11.59.43+AM.png" height="279" width="320" /></a></div>
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In this case the present value of the household's prospective incomes is roughly 70% of the total value of the future prospective payments. The present value of the estate's prospects are somewhat more than half of the remainder, with the present value of the financial advisor's prospects taking the rest.</div>
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I willl have more to say about these pie charts in future posts analyzing alternative strategies. But here are two generic observations.</div>
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First, only annuities offer reasonable income prospects for a household without providing valuable prospects for an estate. Moreover, different variations on a theme can substantially change the division of present value between the household and its estate, as can be seen by changing settings, then generating another set of scenarios.</div>
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Second, fees that may seem small may offer very valuable prospects for the person or organization charging the fees, with an associated decrease in the value of the prospects for the household and/or its estate. Note that in the figure above, a 1% fee is worth almost 1/8th of the total present value -- a cautionary tale.</div>
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Finally, some mechanics. </div>
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Present value calculations using a limited number of scenarios are subject to some error. I'll explore this more in subsequent posts, but suffice it to say that it is important to have at least 5,000 scenarios before making such a calculation. You will see that if the scenario settings called for fewer than 5,000 scenarios when you produced multiple scenarios, the Present Values analysis will politely refuse to give you any results. The solution is, of course, to change the scenario settings, then produce a new set of multiple scenarios.</div>
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You will notice also that there are no numbers given for the percentages of the various recipients' prospects. This was not an oversight. I chose to show only the pie chart in deference to the imprecision of the estimates. You can test the dependence of the results on the scenarios generated by pressing the multiple scenarios button again, generating a new set of scenarios, then seeing the resulting present value pie chart. With luck, the variation may be relatively slight. And the larger the number of scenarios, the smaller it should be. </div>
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<br />Here's the bottom line. Any Monte Carlo analyses is at best an approximation of reality. It is a complex world out there. At best we can only hope that our models and simulations shed useful light on the relative prospects of different strategies.</div>
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<br />William Sharpehttp://www.blogger.com/profile/05491169505008130323noreply@blogger.com12tag:blogger.com,1999:blog-5994111658248655830.post-22548134190318620032014-04-16T14:21:00.003-07:002014-04-17T14:37:39.254-07:00Year/Year Incomes<div class="separator" style="clear: both; text-align: center;">
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The previous post ("Analyzing Multiple Scenarios") showed how to generate multiple scenarios using the RIS software, then view the yearly income ranges using the Analysis routine titled (appropriately enough) "Yearly Incomes". The resulting graph shows the probabilities of exceeding alternative levels of income in each of the future years in the selected range. </div>
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In economic analyses of multiple years' incomes, it is often assumed that people have "time separable utility functions." In principle, a person with such preferences can evaluate the probabilities of alternative incomes in any given year, then calculate the "expected utility" of that range of possibilities. If in this manner the results for each of the future years are expressed in terms of present-day happiness, the expected utilities for each of the years can be added together to get a single measure of the overall desirability of the prospective future incomes. An individual preferences of this type can in theory determine the desirability of a strategy by studying only a graph showing yearly incomes with the accompanying information about the probability of being alive in each year. To be sure, even in this simplified setting, a couple would have to consider the probabilities of being alive and most people would want to include information about the possible amounts that could be left for an estate. But information on the changes in income from each year to the next would not be needed.</div>
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However, many people are in fact concerned with the extent to which their income might change from year to year. To accommodate them the RIS software also includes graphs showing at least some relevant information. On the Analysis page click the button labelled "Yr/Yr Incomes"; you will get a graph something like this:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiLUxRj_D0SIIGG6SCKm0VkdpKu3iK7dn1pP7iho4sWxd2yDDYq_QhU83zNqklonlZ_zTkcojkt9ZiGO3cZWCpFJTdUtETEHFVBhhWeX6qvuBPcYamXmU_JJfLkc6cZLeVqPcb9lQxQqp8/s1600/Screen+Shot+2014-04-15+at+9.18.56+AM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhIFnJdkZ1MRK-WlvP2V9hv_LyOckAgr5QC6OWOe0JXhv4zPPPMTMbhZQlTq4Ly2XIq4KCmHI1SkMkFvEmvoMMj-TIKmEZ2yvdOCPL_iwqajCH8dgdvPRnAwAjqFvxNbp_CizW0gDAz-ac/s1600/Screen+Shot+2014-04-15+at+9.18.19+AM.png" height="297" width="400" /></a></div>
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<span style="color: #0000ee;"><span style="color: black;">In this case, each curve shows (on the vertical axis) the probability of exceeding</span> a </span>given value of a ratio of income divided by the prior year's income (X), shown on the horizontal axis. The ratios shown run from 0 to 2.0, with a ratio of 1.0 indicated by a green vertical line. As with the Yearly Income graph, the chance that one or both of you and your partner will be alive in a given year is shown in the upper right. Also, as with that graph, there are two variants. In the first, as in the case shown above, each graph starts at 100% and shows the ratios for cases in which at least one of the members of the household is alive. The other, shown below, plots the probabilities that the ratios will (1) exceed various levels and (2) that one or both of the members of the household will be alive.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiLUxRj_D0SIIGG6SCKm0VkdpKu3iK7dn1pP7iho4sWxd2yDDYq_QhU83zNqklonlZ_zTkcojkt9ZiGO3cZWCpFJTdUtETEHFVBhhWeX6qvuBPcYamXmU_JJfLkc6cZLeVqPcb9lQxQqp8/s1600/Screen+Shot+2014-04-15+at+9.18.56+AM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiLUxRj_D0SIIGG6SCKm0VkdpKu3iK7dn1pP7iho4sWxd2yDDYq_QhU83zNqklonlZ_zTkcojkt9ZiGO3cZWCpFJTdUtETEHFVBhhWeX6qvuBPcYamXmU_JJfLkc6cZLeVqPcb9lQxQqp8/s1600/Screen+Shot+2014-04-15+at+9.18.56+AM.png" height="305" width="400" /></a></div>
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The first item in the Analysis Settings determines which graph will be shown -- C(ontingent), as in our first example, or A(ctual), as in the second. As with the yearly income graph, the number of seconds of delay between years will be that indicated in the Analysis Settings.<br />
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It is important to understand how these graphs are constructed. With some exceptions, as each one of the multiple scenarios is generated, the ratio of each year's income to that of the prior year is calculated, with the result added to tables to be used to produce the graphs. Excluded are cases in which the recipient in a year differs from that in the prior year. Thus the final year in which any remaining savings goes to the estate is excluded as well as any year in which the household changes from two people to one (since some strategies call for changes in income in such instances that are not related to investment performance). <br />
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As with the yearly income graph, the year/year income graph should be interpreted as showing information available at the present time about the range of outcomes that could take place future years. For example, the range of possible year/year income ratios for years 19 and 20 is based on information available today. When year 19 actually arrives, the range of possible ratios of year 20 income to year 19 income will undoubtedly be very different, since many possible scenarios concerning the first 19 years will have failed to take place.<br />
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This distinction helps illustrate the difficulty of assessing the range of possible outcomes for any multi-year income strategy. There are simply too many income combinations to consider in detail. To emphasize the point, look at the graph below, showing the likelihoods of different combinations of income one and two years hence for a market-based strategy.<br />
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It is possible that someone with training and patience could study such a graph, compare it with another showing the likelihoods of incomes from some other strategy, then choose the preferred strategy. But most retirement income strategies have many more dimensions (years of income) than two. It is simply impossible to portray and assess alternative 40-dimensional probability distributions of income. One needs rather to concentrate on a manageable number of attributes. For some investors our yearly income graphs will suffice. Others may wish to also consider the year/year income graphs. More than that is beyond this project.<br />
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<br />William Sharpehttp://www.blogger.com/profile/05491169505008130323noreply@blogger.com7tag:blogger.com,1999:blog-5994111658248655830.post-61981492437076368512014-01-18T10:54:00.001-08:002014-02-20T16:13:44.941-08:00Analyzing Multiple Scenarios<h2>
Multiple Scenarios </h2>
While it is very useful to see possible scenarios for future income and savings one at a time, there is merit in getting a view of the range of possible outcomes over many possible scenarios. Starting with RIS-20120120, my software on the Scratch site allows for the analyses of multiple scenarios. This post will describe the required procedures and show some examples, but will be short on analysis. Future posts will discuss the relevant economics and analyze alternative retirement income strategies.<br />
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In a previous post I discussed the scenario settings. There is now one additional setting that indicates the number of scenarios that you wish to create for multiple scenario analyses. It is the last one shown in the figure below. To obtain meaningful results you will need to analyze a great many scenarios so the setting is stated in thousands. The default is 5 thousand. I allow as few as one thousand but strongly recommend at least five thousand and more if you are willing to take the time.<br />
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The Main Page</h2>
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The main page now contains the ten buttons shown below. The new ones provide for <i>Analysis Settings</i>, <i>Multiple Scenarios</i> and <i>Analyses</i>. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEifS9D7AMR-YY8VFp73qtzsJH4gNwe8qA9GQrkwWYv0oC0zuVYuKHjTM9eUG9dkPG93iL7Q_-Ls7c67AzP6ikynL5nhOKu9KzkAif8nXw8mDiPcJjSMy4IIS2zWTf8oQIbtt7EszxaMi9M/s1600/Screen+Shot+2014-01-17+at+4.16.20+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEifS9D7AMR-YY8VFp73qtzsJH4gNwe8qA9GQrkwWYv0oC0zuVYuKHjTM9eUG9dkPG93iL7Q_-Ls7c67AzP6ikynL5nhOKu9KzkAif8nXw8mDiPcJjSMy4IIS2zWTf8oQIbtt7EszxaMi9M/s1600/Screen+Shot+2014-01-17+at+4.16.20+PM.png" height="243" width="320" /></a></div>
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The Multiple Scenarios Button</h2>
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To create multiple scenarios, you need only click on the Multiple Scenarios button. After a short while you will be given an estimate of the time required to complete the process and asked whether you wish to proceed. If you say<i> no</i>, you will return to the main page but there will be no scenario statistics available to be used for any subsequent analyses. If you say <i>yes</i>, the desired number of scenarios will be generated and statistics gathered. You will see the progress on the screen and it is very important that you do not interrupt the process. After it is completed, you may click the Analyses button at any time to see various properties of the scenarios. </div>
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Analysis Settings</h2>
The Analysis Settings button allows you to change the default settings for various analyses. At present there are only two settings, although more will be added. They are shown below.<br />
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The first setting indicates whether you want to see the <i>actual probabilities</i> of receiving income (A) or the <i>contingent probabilities</i> (C). As shown, the default is actual. I'll describe the two alternatives below.<br />
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The second setting indicates the time that the software should wait between years when plotting multiple yearly outcomes. The default is 0.25 seconds, which makes the plots come rather fast. You may want to use a larger value to slow down the display, although you can always stop temporarily by pressing and holding the '<i>s</i>' key (then resuming by pressing the space bar).<br />
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The Analysis Page</h2>
When you press the <i>Analysis</i> button on the main page, you will be transferred to the Analysis Page. The figure below shows its current state.<br />
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At present, only the first button is operative. The other two will be programmed in the future, and more may be added as well. Note that all analyses will use the multiple scenarios that you generated most recently. As usual, you may return to the main page by pressing the left arrow key.<br />
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Yearly Income Analyses</h2>
Now to the good part --what happens when you press the <i>Yearly Income</i> button. <br />
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I strongly suggest that you start by doing this using the software initial defaults settings (which include 5,000 multiple scenarios previously produced) to see the results in their full animated glory. <br />
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I'll start with graphs produced using all the default settings (based on an RMD account). In this case the Analysis Setting calls for <i>A</i>ctual probabilities. The figure below shows the graph after 18 years.<br />
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To produce this figure, I froze the display by pressing and holding the <i>'s'</i> key after the 18th year was shown. To produce the next figure, I simply pressed the space bar. (As usual, you can find context-sensitive help instructions by pressing the up arrow key to get a help message).<br />
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Let's look at the results. As shown at the top of the graph, the red curve is for year 18 (18 years in the future since the current year is year 0). The chance that you, your partner or both will be alive in that year is 89.7%. The horizontal axis shows levels of income from 0 to 80 $ thousand (the upper limit, taken from your scenario settings). There are twenty vertical grid lines, so in this case, each covers 4 ($thousand). Here the values shown on the horizontal axis are for real income, also taken from your scenario settings. You may change any of the <i>Scenario Settings</i> to produce different graphs, then producing a new set of multiple scenarios by pressing the <i>Multiple Scenario</i> button.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj7HWgkwLI863WPlr_FwVIX6E1YgjiwGtWI5ZmvwLlZSEQvFvj9_87lZzdThoyQmMWfOCk8xu53UpZQD8NjQRzWe-1Lp-WBO9ifa4jpi04OO_ENOl0XmWA18ZTnqlJwVdJiF2MMA6ZDTVs/s1600/Screen+Shot+2014-01-18+at+8.08.58+AM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj7HWgkwLI863WPlr_FwVIX6E1YgjiwGtWI5ZmvwLlZSEQvFvj9_87lZzdThoyQmMWfOCk8xu53UpZQD8NjQRzWe-1Lp-WBO9ifa4jpi04OO_ENOl0XmWA18ZTnqlJwVdJiF2MMA6ZDTVs/s1600/Screen+Shot+2014-01-18+at+8.08.58+AM.png" height="239" width="320" /> </a></div>
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As indicated, the vertical axis shows the chance that (1) income will exceed the value shown on the horizontal axis <b>and</b> (2) that one or both of you will be alive. Values range from 0% to 100% (or, for those of you who think in probability terms, from 0 to 1.0). Each horizontal grid line covers 5%, and the 50% (median) line is indicated as well. </div>
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You can read this graph in either of two ways. </div>
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You could pick a real income goal, say $40,000, find it on the horizontal axis, then go up to the curve and look over to the vertical axis to see your chances of doing that well or better -- in other words, beating that goal. Clearly, the better your chances, the happier you will be. Thus higher curves are better than lower ones.</div>
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Or you could pick a chance, say 50%, find it on the vertical axis, then go to the curve and look down to the horizontal axis to see the goal that you have a 50% chance of beating. The higher that goal, the happier you will be. Thus curves farther to the right are better than ones to the left.</div>
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If you have a statistical background, you may recognize this graph as similar to a cumulative probability distribution, but with one key difference. The typical statistical graph shows the probability of falling below the value on the horizontal axis, not the probability of exceeding it. I think this is not the way most human beings think about attaining goals and strongly prefer the approach I've employed in my prior research and incorporated in the RIS software. I'll probably have more to say about this "goal/chance" approach in future blogs.</div>
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Most of those who analyze retirement income strategies pick one, two or three probabilities (chances), then show the incomes associated with each of them in each future year. I feel that it is far better to show the entire ranges, as does the RIS software. I'll undoubtedly have more to say about this as well in the future.</div>
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To return to the figure, note that when income exceeds the maximum shown on the horizontal axis, the plot is just to the right of the vertical right edge of the graph box. Here, the actual income values are greater than 80 $thousand maximum plotted, but there is no way to tell how much greater they may be. If this is of concern you may want to change the <i>Scenario Settings</i> to provide higher maximum incomes, then run a new set of multiple scenarios, and analyze the results using the appropriate Analysis tools.</div>
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Now, back to the case at hand. The figure below shows the graph after all the years specified in the Scenario Settings have been shown. Not surprisingly, as time goes on, the chance of any income diminishes as mortality takes its toll. Moreover, there is a wide range of possible incomes in all but the initial year, and the range tends to be larger for later years. In future posts I'll discuss such matters at length when analyzing specific retirement income strategies.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUgYVV6BMlrcXyQTzy3GiVKMmxIWosl3PXGnG5ZmM_ZRTxXYmxbUord_wJpkosu9gG1wJPEVPZeDjCPqeMvbV4zrIvhyphenhyphenJAqF8qc5FmUxbR90fz1aEH2tObBQbMjgZ6irjCdhtzK-HmZMM/s1600/Screen+Shot+2014-01-18+at+8.09.28+AM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUgYVV6BMlrcXyQTzy3GiVKMmxIWosl3PXGnG5ZmM_ZRTxXYmxbUord_wJpkosu9gG1wJPEVPZeDjCPqeMvbV4zrIvhyphenhyphenJAqF8qc5FmUxbR90fz1aEH2tObBQbMjgZ6irjCdhtzK-HmZMM/s1600/Screen+Shot+2014-01-18+at+8.09.28+AM.png" height="240" width="320" /></a></div>
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I'll finish this post with graphs produced using the <i>C</i>ontingent Probabilit<i>y</i> setting in the <i>Analysis Settings</i>. (Happily, you do not have to run a new set of Multiple Scenarios to change between <i>A</i>ctual and <i>C</i>ontingent probabilities).<br />
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The figure below shows years 0 through 14 in yellow and year 15 in red. For the first few years the graphs are virtually the same is in the previous case, since the probability that one or both of you will be alive in the near future close to or equal to 100%. The only difference is the heading for the vertical axis, which shows that the results indicate the chance that income will exceed the amount on the horizontal axis <b>if</b> one or both is alive. (In that sense, it is contingent). Note that this shows that for at least the next 15 years the median (50%) real income is larger in future years, the low-probability worst cases (90% and above) are somewhat worse, and the rosier low-probability cases (say, 25% and below) are considerably better.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhETFb2rh9yX7Df-01ULjtD2S84rUpqJE93HUZ3Rz_tzXQzAz-Ya4LKED5vE6YzDaE8HaNWqfVXEaalHJBYrEFUP1auA6BLGzcEuzU7Y_zP6oYz6J2vmBUT6I3tP6Oa2yyBJ2-MHqybYr0/s1600/Screen+Shot+2014-01-18+at+9.38.13+AM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhETFb2rh9yX7Df-01ULjtD2S84rUpqJE93HUZ3Rz_tzXQzAz-Ya4LKED5vE6YzDaE8HaNWqfVXEaalHJBYrEFUP1auA6BLGzcEuzU7Y_zP6oYz6J2vmBUT6I3tP6Oa2yyBJ2-MHqybYr0/s1600/Screen+Shot+2014-01-18+at+9.38.13+AM.png" height="241" width="320" /></a></div>
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The figure below completes the picture, including all the future years through year 49. As can be seen, the prospects for the very distant years become quite dismal. But of course the chances that anyone will be alive at the time are small.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjsl7E40dBu9CDSGJgjzDUmBmIeGLi9fi0iu8DkywLf_3SDRgC3HIXRZ2BtFOJtnWztxJuRLoN932fur0gvdHdRYVNHowW7OHXHx2uA5kcQk6jwe0m3ie2hu-ScyR0ooit9AdvctYnyu2Y/s1600/Screen+Shot+2014-01-18+at+9.37.13+AM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjsl7E40dBu9CDSGJgjzDUmBmIeGLi9fi0iu8DkywLf_3SDRgC3HIXRZ2BtFOJtnWztxJuRLoN932fur0gvdHdRYVNHowW7OHXHx2uA5kcQk6jwe0m3ie2hu-ScyR0ooit9AdvctYnyu2Y/s1600/Screen+Shot+2014-01-18+at+9.37.13+AM.png" height="320" width="320" /></a></div>
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Note also that in this case the curves for distant years are far from smooth. This reflects the fact that while the results were based on 5,000 scenarios, there are very few scenarios in later years in which anyone is alive, so the sample sizes are insufficient to provide good indications of the overall range of possible future outcomes. For example, in year 49 (shown in red), there were only 5 scenarios (0.1% of 5,000) -- far from sufficient to make well-informed estimates of the whole range of possibilities. Unfortunately, the only way to improve the reliability of distant forecasts is to take the (considerable) time required to run many more scenarios. But with at least 5,000 you should be able to get a rough idea of possible prospects.<br />
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There are profound differences between viewing future retirement income prospects using actual probabilities and conditional probabilities, as these figures show. Indeed, there is considerable debate about the extent to which people should weigh each of these two views when choosing among alternative retirement income strategies. I'll have more to say about this anon. Meanwhile, please do use the software to experiment with these new features. <br />
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As indicated in previous posts, an analysis using the RIS software
can employ one or more <i>accounts</i>, each of which provides retirement
income. Earlier I described the X% Rule account. This post covers
another type, based on the Required Minimum Distribution requirements
specified by the U.S. Internal Revenue Service for those older than
70 ½ holding investments in tax-deferred accounts such as Individual
Retirement Accounts and 401(k)s.
<br />
<br />
The IRS rules are provided in IRS Publication 590. Required
distributions each year are determined by dividing the value of an
account by a life expectancy. Equivalently, the required distribution
is equal to a percentage of the value of the account, with the
percentage equal to the reciprocal of the life expectancy. For
example, if the life expectancy is 20 years, the required withdrawal
percentage is 1/20, or 5%. The required distribution amount each year
must be moved from tax-deferred accounts and declared as income
subject to regular income tax rates; otherwise a prohibitive tax is
levied.<br />
<br />
<br />
Life expectancies are given in three tables, each of which is
applicable for taxpayers in a particular category. The simplest and most
widely applicable is Table III, which is required for use by:
“Unmarried Owners, Married Owners whose Spouses are Not More than
10 Years Younger, and Married Owners Whose Spouses are Not the Sole
Beneficiaries of their IRAs” (IRS Publication 590, p. 109).
<br />
<br />
<br />
The first two columns of the table below are taken directly from
publication 590. “Dist Period” is the Distribution Period (Life
Expectancy). I have added the final column, which shows the
percentage of an account that must be distributed at each age.<br />
<br />
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<table border="0" cellspacing="0" cols="3" frame="VOID" rules="NONE">
<colgroup><col width="98"></col><col width="98"></col><col width="98"></col></colgroup>
<tbody>
<tr>
<td align="CENTER" height="18" width="98"> Age</td>
<td align="CENTER" width="98"> Dist Period</td>
<td align="CENTER" width="98"> Percent</td>
</tr>
<tr>
<td align="RIGHT" height="18">70</td>
<td align="RIGHT">27.4</td>
<td align="RIGHT">3.65%</td>
</tr>
<tr>
<td align="RIGHT" height="18">71</td>
<td align="RIGHT">26.5</td>
<td align="RIGHT">3.77%</td>
</tr>
<tr>
<td align="RIGHT" height="18">72</td>
<td align="RIGHT">25.6</td>
<td align="RIGHT">3.91%</td>
</tr>
<tr>
<td align="RIGHT" height="18">73</td>
<td align="RIGHT">24.7</td>
<td align="RIGHT">4.05%</td>
</tr>
<tr>
<td align="RIGHT" height="18">74</td>
<td align="RIGHT">23.8</td>
<td align="RIGHT">4.20%</td>
</tr>
<tr>
<td align="RIGHT" height="18">75</td>
<td align="RIGHT">22.9</td>
<td align="RIGHT">4.37%</td>
</tr>
<tr>
<td align="RIGHT" height="18">76</td>
<td align="RIGHT">22.0</td>
<td align="RIGHT">4.55%</td>
</tr>
<tr>
<td align="RIGHT" height="18">77</td>
<td align="RIGHT">21.2</td>
<td align="RIGHT">4.72%</td>
</tr>
<tr>
<td align="RIGHT" height="18">78</td>
<td align="RIGHT">20.3</td>
<td align="RIGHT">4.93%</td>
</tr>
<tr>
<td align="RIGHT" height="18">79</td>
<td align="RIGHT">19.5</td>
<td align="RIGHT">5.13%</td>
</tr>
<tr>
<td align="RIGHT" height="18">80</td>
<td align="RIGHT">18.7</td>
<td align="RIGHT">5.35%</td>
</tr>
<tr>
<td align="RIGHT" height="18">81</td>
<td align="RIGHT">17.9</td>
<td align="RIGHT">5.59%</td>
</tr>
<tr>
<td align="RIGHT" height="18">82</td>
<td align="RIGHT">17.1</td>
<td align="RIGHT">5.85%</td>
</tr>
<tr>
<td align="RIGHT" height="18">83</td>
<td align="RIGHT">16.3</td>
<td align="RIGHT">6.13%</td>
</tr>
<tr>
<td align="RIGHT" height="18">84</td>
<td align="RIGHT">15.5</td>
<td align="RIGHT">6.45%</td>
</tr>
<tr>
<td align="RIGHT" height="18">85</td>
<td align="RIGHT">14.8</td>
<td align="RIGHT">6.76%</td>
</tr>
<tr>
<td align="RIGHT" height="18">86</td>
<td align="RIGHT">14.1</td>
<td align="RIGHT">7.09%</td>
</tr>
<tr>
<td align="RIGHT" height="18">87</td>
<td align="RIGHT">13.4</td>
<td align="RIGHT">7.46%</td>
</tr>
<tr>
<td align="RIGHT" height="18">88</td>
<td align="RIGHT">12.7</td>
<td align="RIGHT">7.87%</td>
</tr>
<tr>
<td align="RIGHT" height="18">89</td>
<td align="RIGHT">12.0</td>
<td align="RIGHT">8.33%</td>
</tr>
<tr>
<td align="RIGHT" height="18">90</td>
<td align="RIGHT">11.4</td>
<td align="RIGHT">8.77%</td>
</tr>
<tr>
<td align="RIGHT" height="18">91</td>
<td align="RIGHT">10.8</td>
<td align="RIGHT">9.26%</td>
</tr>
<tr>
<td align="RIGHT" height="18">92</td>
<td align="RIGHT">10.2</td>
<td align="RIGHT">9.80%</td>
</tr>
<tr>
<td align="RIGHT" height="18">93</td>
<td align="RIGHT">9.6</td>
<td align="RIGHT">10.42%</td>
</tr>
<tr>
<td align="RIGHT" height="18">94</td>
<td align="RIGHT">9.1</td>
<td align="RIGHT">10.99%</td>
</tr>
<tr>
<td align="RIGHT" height="18">95</td>
<td align="RIGHT">8.6</td>
<td align="RIGHT">11.63%</td>
</tr>
<tr>
<td align="RIGHT" height="18">96</td>
<td align="RIGHT">8.1</td>
<td align="RIGHT">12.35%</td>
</tr>
<tr>
<td align="RIGHT" height="18">97</td>
<td align="RIGHT">7.6</td>
<td align="RIGHT">13.16%</td>
</tr>
<tr>
<td align="RIGHT" height="18">98</td>
<td align="RIGHT">7.1</td>
<td align="RIGHT">14.08%</td>
</tr>
<tr>
<td align="RIGHT" height="18">99</td>
<td align="RIGHT">6.7</td>
<td align="RIGHT">14.93%</td>
</tr>
<tr>
<td align="RIGHT" height="18">100</td>
<td align="RIGHT">6.3</td>
<td align="RIGHT">15.87%</td>
</tr>
<tr>
<td align="RIGHT" height="18">101</td>
<td align="RIGHT">5.9</td>
<td align="RIGHT">16.95%</td>
</tr>
<tr>
<td align="RIGHT" height="18">102</td>
<td align="RIGHT">5.5</td>
<td align="RIGHT">18.18%</td>
</tr>
<tr>
<td align="RIGHT" height="18">103</td>
<td align="RIGHT">5.2</td>
<td align="RIGHT">19.23%</td>
</tr>
<tr>
<td align="RIGHT" height="18">104</td>
<td align="RIGHT">4.9</td>
<td align="RIGHT">20.41%</td>
</tr>
<tr>
<td align="RIGHT" height="18">105</td>
<td align="RIGHT">4.5</td>
<td align="RIGHT">22.22%</td>
</tr>
<tr>
<td align="RIGHT" height="18">106</td>
<td align="RIGHT">4.2</td>
<td align="RIGHT">23.81%</td>
</tr>
<tr>
<td align="RIGHT" height="18">107</td>
<td align="RIGHT">3.9</td>
<td align="RIGHT">25.64%</td>
</tr>
<tr>
<td align="RIGHT" height="18">108</td>
<td align="RIGHT">3.7</td>
<td align="RIGHT">27.03%</td>
</tr>
<tr>
<td align="RIGHT" height="18">109</td>
<td align="RIGHT">3.4</td>
<td align="RIGHT">29.41%</td>
</tr>
<tr>
<td align="RIGHT" height="18">110</td>
<td align="RIGHT">3.1</td>
<td align="RIGHT">32.26%</td>
</tr>
<tr>
<td align="RIGHT" height="18">111</td>
<td align="RIGHT">2.9</td>
<td align="RIGHT">34.48%</td>
</tr>
<tr>
<td align="RIGHT" height="18">112</td>
<td align="RIGHT">2.6</td>
<td align="RIGHT">38.46%</td>
</tr>
<tr>
<td align="RIGHT" height="18">113</td>
<td align="RIGHT">2.4</td>
<td align="RIGHT">41.67%</td>
</tr>
<tr>
<td align="RIGHT" height="18">114</td>
<td align="RIGHT">2.1</td>
<td align="RIGHT">47.62%</td>
</tr>
<tr>
<td align="RIGHT" height="18">115 and over</td>
<td align="RIGHT">1.9</td>
<td align="RIGHT">52.63%</td></tr>
<tr><td align="RIGHT" height="18"><br /></td><td align="RIGHT"><br /></td><td align="RIGHT"><br /></td></tr>
<tr><td align="RIGHT" height="18"><br /></td><td style="text-align: left;"><br /></td><td align="RIGHT"><br /></td></tr>
<tr><td align="RIGHT" height="18"><br /></td><td align="RIGHT"><br /></td><td align="RIGHT"><br /></td></tr>
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</tbody></table>
The calculations made by the IRS to generate this table are not
specified. Presumably, mortality tables were utilized, with some sort
of averaging across possible combinations of unmarried investors of
both sexes and those married with spouses of both sexes and with
differing ages.
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<br />
<br />
There is no presumption that the owner of a tax-deferred account
must spend the amount on which taxes must be paid. And many investors
have additional sources of retirement income. This said, it has
occurred to some investors and analysts that it might be desirable to
adopt a retirement income strategy with a policy of spending
the percentages of overall savings given in the final column above.
Prominent studies of the efficacy of such an approach include:<br />
<br />
<br />
<div style="text-align: left;">
Sun, Wei and Anthony Webb, 2012, “Should Households base
Asset Decumulation Strategies onRequired Minimum Distribution
Tables?” Center for Retirement Research at Boston College Working Paper (available <a href="http://crr.bc.edu/working-papers/should-households-base-asset-decumulation-strategies-on-required-minimum-distribution-tables/">here).</a></div>
<br />Blanchett, David, Maciej Kowara and Peng Chen, 2012, “Optimal
Withdrawal Strategy for Retirement-Income Portfolios,” Retirement
Management Journal, 2(3): 7-20<br />
<br />Blanchett, David M. 2013. “Simple Formulas to Implement Complex
Withdrawal Strategies.” <em>Journal of Financial Planning</em> 26
(9): 40–48, available <a href="http://www.fpanet.org/journal/SimpleFormulastoImplementWithdrawalStrategies/">here</a> .<br />
<br />In their 2012 paper, Sun and Webb concluded that the RMD
approach was preferable to the 4% rule. In his 2013
paper, Blanchett found that “the RMD approach works well for
periods less than 15 years...” and advocated the use of a more
complex approach for subsequent years. I remain agnostic on the issue
but feel that the approach is worthy of investigation.<br />
<br />
Now, to the details of the RMD account.<br />
<br />
<br />
To cover ages not shown in the IRS table, I have made the
assumption that the life expectancy for any age younger than 70 will
be (70 – age) years longer than that for age 70. Thus for a 65-year
old the assumed life expectancy is 27.4 + 5, or 32.5 years. Moreover,
when there are two participants (you and your partner), I base the
withdrawal percentage each year on the age of the older participant
in that year.<br />
<br />
<br />
The settings for an RMD account are shown below.<br />
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<br />
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<br />
<br />
The initial setting (line 2) is the usual one that determines
whether or not the account is active. The second (line 4) indicates
the initial balance – here, a million dollars (1,000 $ thousand).
The next setting allows for a variation of the strategy in which the
RMD longevity numbers are altered by adding or subtracting a constant
number of years. For example, if the adjustment is 2, the life
expectancy at age 70 will be 29.4 (27.4 + 2) years, and every other
life expectancy will be adjusted by adding 2 years to the amount
shown in the table. Lengthening the life expectancies in this manner
will reduce the percentages of savings paid, lowering retirement
income payments and increasing possible estate values. If desired,
you may enter a negative number for this setting. This will reduce
the life expectancies and increase the percentage payments. (Not to worry -- if this
would result in any expectancies less than one, they are replaced with
1.0).<br />
<br />
<br />
The remaining settings are the same as those for the X% Rule
settings. The fee indicates the annual percentage of the account
value charged as fees to third parties. The three settings for the
investment strategy are, as for the X% Rule, the initial proportion
of the account invested in the market portfolio, the number of years
for any glide path, and the proportion of the account invested in the
market portfolio at and after the end of the glide path period. As
with the X% Rule, the default settings provide for a constant
investment solely in the market portfolio in each year.
<br />
<br />
The RMD approach is a special case of a more general class that I
have called Proportional Payout (PPO) strategies, in which a
pre-specified proportion of an investment account is paid out to
provide retirement income in each year. In previous research, I have
used a set of proportions specified for the Fidelity Income
Replacement 2042 Fund, which is designed to pay out all the assets in
the portfolio by the end of 2042. For a detailed analysis, see my
paper in the European Financial Management Journal, a version of
which is <a href="http://www.wsharpe.com/retecon/FinancingRetirement.pdf">here</a>.
While the Fidelity Funds are designed specifically for producing
retirement income, they will pay out all assets by a target date no
more than 30 years in the future. In contrast, the RMD approach as
implemented in the RIS software will provide some income until the
both participants are gone, leaving at least some funds for an
estate. For this reason, and because it uses non-proprietary data, I
chose to include the RMD method in the software. However, it would be
a simple matter for a user to alter the longevity table used for the
calculations to produce different results.<br />
<br />
Do try the RMD account. Unlike the X% Rule, it conforms with two sensible criteria in each year:<br />
<br />
<b>The amount you spend should depend on<br /> 1. How much money you have, and<br /> 2. How long you are likely to need it</b><br />
<br />
This doesn't mean it is the best approach for you. But, combined with a sensible investment policy, it might provide a desirable component for your overall retirement income strategy.<br />
<br />
<br />
<br />
<br />
William Sharpehttp://www.blogger.com/profile/05491169505008130323noreply@blogger.com76tag:blogger.com,1999:blog-5994111658248655830.post-20493915895116361242013-12-29T10:51:00.001-08:002013-12-29T10:53:44.913-08:00Savings and Income Scenario Settings<style type="text/css"></style><br />
<br />
This is about the scenarios that can be generated and shown in the
RIS software. I assume that you have dealt with the <i>client settings</i>
and the <i>market settings</i> and have also set up one or more <i>accounts</i> and made at least one of them active. At this point you are
almost ready to generate and plot scenarios, But you will probably
first need to alter the initial <i>scenario settings</i>. The figure below
shows the six settings and their default values in the RIS-20140101
version.<br />
<br />
<br />
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<br />
<br />
<br />
<br />
The first setting (line 2) indicates whether you want to project
real (inflation-adjusted) or nominal values. I strongly suggest that
you focus on real values for both savings and income, since these are
far more relevant in estimating possible consumption of goods and
services. However, it may be instructive in some cases to look at the
nominal values, since some retirement income strategies focus on
them. That said, it is important not to be fooled by such displays. For
example, it is not enough for a strategy to provide constant or
slightly increasing nominal income if the increases are not
sufficient to cover inflation. Here is a simple bromide to keep in mind:<br />
<br />
<b>Real people should care about real income</b><br />
<br />
<br />
By all means, feel free to look at the nominal values of savings
and income for scenarios, but then examine the real values in order to seriously
evaluatie a strategy.<br />
<br />
<br />
The second setting (line 4) indicates the number of future years
that you wish to display on the savings and income graphs. Any values for
subsequent years will be shown just outside the right-hand border of
the graph. This conforms with a general rule: <i>If a value falls outside the range of a graph, it is shown
outside the border, using the closest x and/or y value </i><br />
<br />
<br />
<div style="font-weight: normal;">
The remaining settings indicate the
maximum values shown inside the borders for the four possible
graphs (real savings, real income, nominal savings and nominal
income). You may have to experiment a bit to find the most
satisfactory values for these settings. A useful rule of thumb for
strategies that rely on an initial investment is to set the maximum
real savings at twice the initial investment and the maximum real
income at twice the initial income. For nominal values it is useful
to set maximum values at four times the initial amounts, since with typical settings for expected
inflation, nominal values in the later years can be twice as large as real values.</div>
<div style="font-weight: normal;">
<br /></div>
<div style="font-weight: normal;">
I suggest that you use make rough
estimates for all these settings, then generate some scenarios to see
what happens. If too many values fall outside the borders, increase
the corresponding setting. If there is too much empty space within
the graph, decrease the corresponding setting. It should not take
much time to find settings that provide a reasonable balance, excluding few values and using the space within the graph
efficiently.</div>
<div style="font-weight: normal;">
<br /></div>
<div style="font-weight: normal;">
Once you have adjusted these scenario
settings, you are ready to see the results of your handiwork by
clicking either the <b>savings scenario</b> or <b>income scenario</b>
button. I'll cover these in the next post.</div>
William Sharpehttp://www.blogger.com/profile/05491169505008130323noreply@blogger.com13tag:blogger.com,1999:blog-5994111658248655830.post-72927051735918776782013-12-17T16:28:00.000-08:002013-12-17T16:28:28.562-08:00The X% Rule<style type="text/css"></style><br />
<h2>
Accounts
</h2>
<br />
The RIS software allows the user to specify one or more sources of
retirement income. Each is described in an <i>account</i>. And each
such account has a number of settings.
<br />
<br />
An account might be a bank account, an
account managed by a financial advisor, an annuity in which an
insurance company provides monthly payments, etc.. All accounts
provide annual payments that sum to equal retirement income. Many
also have balances that sum to equal total savings. An account will
make payments that may depend on the mortality of the recipients.
Thus an account might pay more if both you and your partner are alive
than if only one is alive. And in the first year in which you and your partner are both dead, any account with a balance will pay
the total amount remaining to your estate.<br />
<br />
In this post I will describe the first type of account, which I
have called, generically, the X% Rule. This is a generalization of an
approach widely recommended by financial advisors, based on a
strategy initially termed the “4% Rule”.
<br />
<br />
<br />
<h2>
The 4% rule</h2>
<br />
The 4% rule, first advocated by William Bengen in “Determining
Withdrawal Rates Using Historical Data”, <i>Journal of Financial
Planning</i>, vol. 7, no. 4, October 1994, pp. 171-180, is widely used by
financial advisors. Bengen initially analyzed annual returns on bonds
and stocks in the United States over every possible 30-year sub-period
within a 90-year period . For each sub-period, he calculated the outcomes
of a policy of withdrawing a constant real amount equal to 4% of an
initial investment value, assuming that funds were invested in a
constant mix split evenly between stocks and bonds. He found that in
almost all of the 30-year periods, such a policy would not “run out
of money” and suggest that in this sense it “should be safe”. In a recent <a href="http://www.spreecast.com/events/does-the-4-rule-still-work">webinar</a> he
advocated following the policy with a withdrawal amount equal to 4.5% of the
initial value.<br />
<br />
Much has been written about the 4% rule. Jason Scott, John Watson
and I analyzed it at considerable length in (“The 4% Rule: At What
Price?”, Journal of Investment Management, vol. 7, no.3 (Third
Quarter) 2009, pp. 1-18"), available <a href="http://www.wsharpe.com/retecon/4percent.pdf">here</a>, in which we pointed out a number of its
shortcomings. In a recent paper in the September/October issue of the<i> Financial Analysts Journal</i>, Jason and John document subsequent
studies of variants of the 4% rule and advocate a very different
approach.<br />
<br />
I won't go into details here, but my view is that the 4% rule and
the variants that I have allowed in the x% account are sorely
lacking. It seems to me that first principles dictate that any rule
for spending out of a retirement account should at the very least
adhere to the following principle:<br />
<br />
<b>The amount you spend should depend on<br /> 1. How much money you have, and<br /> 2. How long you are likely to need it</b><br />
<br />
<ol><ol></ol>
</ol>
The x% rule can meet both criteria in the first year, since the
amount spent is x% of the initial value and the value of x can be set
taking into account the ages of the recipients (in practice, some advisors do modify the initial payment percentage, making it larger for older clients and smaller
for younger ones). But after the first year, the rule fails on both
counts. The amount paid is completely divorced from the value of the
account. And no account is taken of changes in life expectancy, death
of a principal, etc..
<br />
<br />
In a quest for simplicity, the x% approach to providing retirement
income comes up very short on first principles. However, since it persists as a kind of
standard in much of the practice of financial advice, I
have included it in the software. I'll have more to say about this in
subsequent posts. In the meantime, I encourage you to experiment with
different settings of the account, the market and the client to
better understand the properties of this rule.
<br />
<br />
<br />
Now to the details of the software.<br />
<br />
<br />
<h2>
The X% account and its settings</h2>
<br />
This section will cover details of the implementation of the
account in the RIS software. I'll probably belabor some aspects that
may be obvious. For many users it will suffice to look at the settings
using the software, make any desired changes, and see the
implications for savings and income in alternative possible future
scenarios.
<br />
<br />
The figures below shows the settings for the X% account. These may be reviewed and/or changed by clicking the "Account Settings" button on the RIS home page, then clicking the "X% Rule" button on the account settings page. When the settings have been reviewed or changed, simply press the keyboard left arrow to return to the home page.<br />
<br />
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<br />
<br />
The initial setting (line 2) indicates whether this account is
<i>active</i> or not. If an account type is not be be used, this
setting must be N (no). When reviewing the settings for the account if
the answer is N, the dialogue will be terminated and the account
button will be dimmed. If the setting is Y the account button will be
shown fully and the remaining settings will be reviewed and may be
changed.<br />
<br />
The second setting (line 4) concerns the amount of money initially
placed in the account. In general, dollar amounts in RIS are stated
in thousands. In this default case the account starts with a million
dollars (1000 $ thousand).
<br />
<br />
The next setting (line 6) indicates the amount to be paid out
initially, also stated in thousands of dollars. This will be paid
immediately and is thus not subject to any uncertainty. In the
default case this is equal to 40 ($ thousand), which is 4% of the initial investment, as in the
classic 4% rule.<br />
<br />
The original formulation of the 4% rule did not take mortality
into account. The assumption was made that an amount with the same
purchasing power would be paid in each subsequent year unless there
were insufficient funds, in which case the remaining funds would be
paid out and subsequently nothing would be available<br />
<br />
In RIS I have generalized the approach somewhat to allow different amounts forthree possible conditions: both are alive (line 8), only you are
alive (line 10) and only your partner is alive (line 12). In the
default settings, all three equal the initial amount, giving the
original 4% rule.<br />
<br />
The remaining settings concern fees and the investment strategy to
be followed when implementing the rule.<br />
<br />
Line 14 shows the annual fee as a percentage of the account
value. In the default case, 1% of the value of the account will be
deducted for fees each year, just before payment is made to the
beneficiaries. In practice a smaller fee (for example, 1/12 of 1%) is likely
to be deducted each month but since RIS uses only annual returns and
valuations, a single deduction is utilized. Financial institutions and
advisors often charge lower percentage fees for larger accounts, but
1% is not atypical for accounts of a million dollars. You should
adjust this setting to reflect the likely cost of such services in
your case. Of course, you could follow an x% rule without an
intermediary, saving a considerable amount of money. In such an instance
you would set this amount to 0.
<br />
<br />
The remaining settings describe the investment strategy to be
followed. The original versions of the rule assumed that funds would
be invested in a relatively constant mix of stocks and bonds –
typically with 50 or 60% invested in stocks and the remainder in
bonds. More recently, some have advocated the use of a “glide path”
in which the proportions of bonds and stocks vary from year to year.
To generalize, I have allowed for limited versions of either
approach.
<br />
<br />
To focus on real returns, RIS has only two major types of
investments – a market portfolio and a riskless real security (as discussed in a previous post). Any
investment strategy can thus described by the proportion of funds (by
value) in the market. For example, if the proportion in the market is 0.60
(60%), the remainder (0.40 or 40%) will be invested in the riskless
real security. The settings allow the proportion in the market to be
as low as 0 and as high as 5. As I discussed in my earlier post, values greater than 1.0 assume that it is possible to
“lever up” the market portfolio by borrowing at the riskless real
rate of interest (or equivalently, that some sort of investment with
the equivalent expected return and risk of such a levered position can be obtained) – an
assumption that may be inappropriate in some cases.<br />
<br />
Three settings determine the investment policy. The first
(line 16) indicates the initial proportion of funds in the market
portfolio. This specifies the investment mix that will be used to determine
the return at the end of the first year. In the default case it is
1.00, reflecting investment totally in the market portfolio. The
next two settings determine the length of the “glide path” and
the proportion invested in the market portfolio when it ends. For
example, if the glide path were to last 20 years with the proportion
in the market at that point equal to .50, line 18 would be set at
20 and line 20 at 0.50. The RIS software assumes that after the glide
path ends, the proportion invested in the market remains constant.
Thus if the glide path lasts 20 years and ends at 0.50, the
proportion in the market portfolio will equal 0.50 in years 21, 22,
and thereafter.<br />
<br />
The default settings specify the special case in which there is, in
effect, no glide path. The market proportion starts at 1.0, the glide
path lasts for 0 years and then the proportion continues at 1.0
thereafter.<br />
<br />
While the use of these three settings is somewhat forced in the default
case, it seemed desirable to allow for a glide path in order to allow
the analysis of approaches advocated by some practitioners. I have
limited the possibilities by requiring the glide path to be linear,
but this should capture at least the key attributes of approaches
recommended by some analysts. A similar approach will be used for some of the additional types of account to be added to the software in the future.<br />
<br />
Once you have arranged the settings of an X% account to your satisfaction and made it active, you need only return to the RIS home page (via the keyboard left arrow) and click either the "Income Scenarios" or "Savings Scenarios" to see plots of possible future outcomes. It's that easy.<br />
<br />
<br />William Sharpehttp://www.blogger.com/profile/05491169505008130323noreply@blogger.com3tag:blogger.com,1999:blog-5994111658248655830.post-57872526773936916672013-12-03T08:15:00.003-08:002013-12-03T08:15:59.225-08:00Investment Returns and Inflation<style type="text/css">P { margin-bottom: 0.08in; }</style><br />
<div style="margin-bottom: 0in;">
With the exception of some social
insurance programs, most sources of retirement income depend to a
greater or lesser extent on the returns on one or more investment
vehicles. The RIS software simulates returns on two types
of such investments as well as changes in the overall cost of living.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
There are five key settings for this
process, shown, along with default values, in the market settings
list below.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
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<br />
<br />To see the current values of these settings and make any desired changes, simply click the Market
Settings button on the RIS home page.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
This post will discuss the procedures
used to provide simulated returns and changes in the cost of living
as well as the reasons why I have chosen them for the initial releases of
the software. I'll deal with the three key aspects in turn. </div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
Warning!
Of necessity, some of this discussion is going to be lengthy and rather
technical. Feel free to skim it as needed.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<h3 style="margin-bottom: 0in; text-align: center;">
Risk-free real returns</h3>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
There are, at present, only two
possible types of investment in the RIS software. The first is a
risk-free instrument that promises a constant real return in each
future year. An example of a (hopefully) risk-free real return
instrument would be the Treasury Inflation-Protected Securities
(TIPS) issued by the U.S. Government. The coupon and principal
payments of such bonds are adjusted using a Consumer Price Index with
the intention of providing fixed amounts of purchasing power. Of
course the “cost of living” of any given individual or household
will depend on the prices of a particular basket of goods and
services and will, at best, only be approximated by a standard price
index. There have been attempts in the U.S. To produce an index that
more closely reflects the goods and services purchased by retirees,
but the results were mixed and some concluded that the
resulting index was not demonstrably superior to the
standard CPI for this purpose.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
As you will see, in the scenario projections I have chosen to focus
on real returns, real investment values and real retirement income since it is far more important to consider the
spending power of future incomes than the nominal amounts. Unhappily,
some vendors of retirement strategies market their products by focusing on seemingly
desirable prospects for nominal income, leading retirees to fail to
consider the erosion of purchasing power that inflation may cause.
Many retirees will find that in the last years of their lives a
dollar (or other currency) buys less than half the goods and services
that it did when they retired. Those who ignore the possibility of
inflation do so at their own great peril.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
As shown above, the default setting is for a
real (inflation-adjusted) risk-free return of 1% per year in each
future year. This “flat yield curve” assumption is at variance
with both history and current yields on TIPS. For example, here are
the TIPS real yields for bonds of various maturities at the end of
November, 2013:</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
Maturity Annual Yield</div>
<div style="margin-bottom: 0in;">
5 years - 0.32 %</div>
<div style="margin-bottom: 0in;">
10 years 0.60 %</div>
<div style="margin-bottom: 0in;">
20 years 1.23 %</div>
<div style="margin-bottom: 0in;">
30 years 1.53 %</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
Such a “rising yield curve” is not
uncommon. This may reflect a preference for more liquidity or
possibly a prediction that shorter-term rates are likely to increase
in the future. But it adds a complexity that is not currently included in the RIS software.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
Note also that at the time the real return on
(relatively) riskless investments for a 5-year period was actually
negative, suggesting that you could have invested your money for a promise to
obtain fewer future goods and services than you sacrificed initially. To
some extent this may have been due to a feature of TIPS that precludes
reductions in payments below certain levels if there is deflation.
But more likely it was an artifact of the “quantitative easing”
program, in place at the time, in which the Federal Reserve Bank
was purchasing huge amounts of government bonds each month in an attempt to
hold interest rates down, due to the still tepid recovery from the
recession of 2007-2009. Whatever the reason, low
yields on both inflation-protected and traditional bonds imposes a
huge burden on those attempting to finance their retirement – a
result infrequently noted in the popular press and political discourse.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
In any event, the RIS system assumes that there is a
constant riskless real rate of interest. The user is free to chose
any desired value for this setting. The default of 1% is roughly
equal to the average across all maturities in the latter part of 2013 and lower than the rates provided in previous years by inflation-protected securities in
the U.S. and some other countries. While the
lack of a complete term structure of interest rates in our simulations may omit
important aspects of some possible investment policies, it may be an acceptable simplifying assumption for broad-based comparisons of alternative approaches.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
<br /></div>
<h3 style="margin-bottom: 0in; text-align: center;">
Market bond/stock portfolio returns</h3>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
The other possible type of investment
in the RIS software is a portfolio of bonds and stocks. In the
settings I call this the “market bond/stock portfolio” although I
often refer to it simply as the “market portfolio”. Ideally,
this should include all bonds and stocks traded relatively
actively around the globe, with each represented in proportion to its
outstanding value. In practice you may want to favor bonds from your home country.<br />
<br />
A key assumption is that the market portfolio includes securities with values proportional to the total outstanding values. Thus if the total outstanding value of Apple
shares is $A and the total outstanding value of Microsoft shares is
$M, the relative values of the two shares in the market portfolio will
be $A/$M. More simply put, if the portfolio has x% of the total
shares issued by Apple, it will have x% of the total shares issued by
Microsoft and every other issuer. It will also have x% of the bonds issued by each of the included firms or governments. Importantly, changes in the relative
prices of Apple and Microsoft will not require the purchase or sale of either. Actual trades will be required only to
deal with dividends on stocks, coupon payments on bonds, share
repurchases, bonds that are redeemed, new issues and the like. In this sense our market portfolio is a low-turnover fund and it should be possible to obtain an index fund with similar returns and low overall expenses.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
The market portfolio plays a central
role in may theories of the pricing of capital assets and resultant prescriptions concerning the relative desirability of different
investment strategies. Importantly, it represents the portfolio held
by the sum of all those who invest in traded bonds and stocks. Any
investor holding a different combination of such securities must, in a
sense, be offset by one or more investors holding a complementary
portfolio. Thus if I underweight Microsoft and overweight Apple
relative to the market portfolio, one or more investors must
overweight Microsoft and underweight Apple. Only the market portfolio can
be said to be “macro-consistent” – that is, everyone could hold it and
markets would clear.
</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
Note that our market portfolio is not
the often-used market value weighted portfolio of equity securities,
represented by some popular stock index. Rather it is intended to
represent all relatively liquid bonds and stocks and therefore conform more
closely to the “market portfolio” construct of academic theories
about the pricing of capital assets.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
But how to predict the return on such a
portfolio? Anyone with experience in security markets knows that it
is impossible to predict the total return on the market in any given year. One can, at best, aspire to specify a range of
possible outcomes and the likely probability of each one. Both
academics and practitioners assume that this can be done, with the
actual return considered a draw from a pre-specified probability
distribution of possible returns. But what is the shape of the
distribution? And what are its parameters: the central return, the
range of possible returns, etc.? It would be nice if we could reasonably assume
that every year in history was a drawn from an unchanging probability
distribution and if we had many centuries or such draws. But it is
implausible that the return in 1865 was drawn from
the same distribution as that in 2013. The world changes, the financial system varies, and the sources of uncertainty change as well. Despite decades of
sophisticated statistical analyses, there is little agreement among academics and practitioners about "the" probability distribution of the return on the market portfolio.
</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
My opinion is that predicting the
possible range of returns on the market is ultimately the responsibility of the
investor with the aid of a financial advisor whom he or she trusts. I am not that advisor. The RIS
software makes some assumptions about the type of probability
distributions from which market returns and inflation will be drawn, but it is up to
the user to choose the specific inputs. I have provided defaults that
are similar to those used by some institutional investors, but you
should feel free to change them. That said, the current software does
employ a particular type of probability distribution and has
additional built-in assumptions. If you believe these are not appropriate, you or someone else may create a version
of the software with different computations. The code is available at the Scratch site and you are free to make any desired modifications.</div>
<div style="margin-bottom: 0in;">
<br />
In the current version of the software I have assumed that
each year's total market return is drawn from a lognormal
probability distribution. By “total market return” I mean the
ratio of the year-end value to the value at the beginning of the year. Thus if the return is 10%, the total return is 1.10. Equivalently, I assume that the logarithm of the total market return is drawn
from a normal distribution (the bell-shaped symmetric version that
you undoubtedly studied in school). </div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
Why this assumption? Here is a possible justification. The annual total return on a portfolio will equal the product of
the daily total returns. From this it follows that the logarithm of
the annual total return will equal the sum of the logarithms of the
daily total returns. Now, as you may have learned in class, if you
repeatedly add up a set of values each of which is drawn randomly
from a distribution, the distribution of the sums will be close to a
normal distribution, and the more the numbers you sum each time, the
closer this will be to such a distribution (this is the famous
“central limit theorem”). So if you think about the total return
on the market over a year as the product of the total returns for each of the trading days in the year and assume that the daily
returns are drawn independently, you will conclude that the
distribution of annual total returns will be very close to lognormal. Moreover, the central limit theorem holds approximately in many cases where these assumptions are not met in every detail. In any event, such relationships can provide some justification for the assumption that annual returns are lognormally distributed. But there may be occasional "perfect storms" and the software does not take such a possibility into account; that said, the long run effects on retirement income for at least some strategies may not be radically different from those produced in the simulations.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
For good or ill, the RIS software draws each annual market
return from an unchanging lognormal probability distribution. You (or
I at some future date) could of course create a version with a
different set of assumptions to accommodate, for example, a “fat left tailed”
distribution or some other set of assumptions, but the current
version doesn't allow for such an alternative.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
Two settings are used to fix the
parameters of the market return distribution. The first is the
expected annual return premium over the riskless real rate. For
example, given the default value of 4% per year along with the
riskless real return of 1% , the expected real return of the market
portfolio would equal 5% (the sum). Note that this is the <i>expected
return</i>, defined as the value obtained by weighting each possible
value by its probability. The corresponding risk measure is the
standard deviation of the annual real return, obtained by squaring
the deviation of each possible real return from the expected value,
weighting each such value by its probability, then taking the square
root of the resulting sum. The default value is 12% per year. Note
that, following convention, both these measures relate to the annual
return, not its logarithm.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
An important relationship is given by
the ratio of the market expected return premium to its standard
deviation -- usually called the Sharpe Ratio (although I originally
termed it the <i>Reward to Variability Ratio</i>). In this case it is 4/12,
or 1/3. This is a commonly made assumption, reflecting a plausible
relationship between risk and the additional expected return required
for investors to bear the risk of a market portfolio.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
In traditional models of asset pricing
such as the Capital Asset Pricing Model, the market portfolio
provides the highest possible Sharpe Ratio. Combinations of the
market portfolio and the riskless asset will provide the same Sharpe
Ratio, assuming that investors can borrow or lend at the riskless
rate. In the RIS software, all asset mixes are combinations of the
riskless asset and the market portfolio, so this condition is met.
However, substantial amounts of borrowing (negative positions in the
riskless asset) at the same riskless rate may not in fact be feasible
in the real world. Fortunately, most retirement income strategies involve investment risks equal to or lower than that of our market portfolio.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
In more general models of asset pricing such as those employing pricing kernels, the market portfolio also plays a central role. I have written about this in a 2007 book and utilized the approach in papers analyzing alternative retirement income strategies. For more information, see <a href="http://www.wsharpe.com/">my web site</a>.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
It is important to note that when total returns are lognormally distributed, the median (50/50) return will
be smaller than the expected return, since the distribution of total
returns will be skewed to the right. This is an important aspect of
the RIS assumption. My view is that the expected return is a
non-intuitive concept and that ordinary human beings relate far
better to the median -- that is, the outcome for which there is roughly a 50%
chance that the actual return will larger and a 50% chance that
the return will be smaller. The distinction between the expected value and the median is important when
returns are drawn from asymmetric distributions, as they are in
the software.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
A final assumption about market returns concerns the
relationship between the return on the market in one year and that in
the next. The software assumes that each annual return is drawn
independently from a given lognormal distribution, so that there is
no predictable relationship between one year's return and that of any
other. In academic-speak, annual returns are independent and identically
distributed (<i>iid</i>). An interesting aspect of
such returns is that, regardless of the nature of the distribution of
annual returns, the distribution of possible cumulative return over a
period of many years will be close to lognormal, and hence skewed to
the right (due to the the central limit theorem). In our case, however, the return over <i>any</i> period of years (from 1
to many) will be lognormally distributed.
</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
You may well wonder why only two
possible investments are included explicitly in the RIS software. The
reason is that in a simple setting, all efficient investment
strategies should be constructed using the most efficient risky
portfolio and a riskless security. We assume that people care about
real, not nominal, returns and so the market portfolio is assumed to
be the most efficient risky portfolio in real terms. Accordingly, the
only real risk that is rewarded with greater expected real returns is the risk
borne by investing in the market portfolio; moreover, this will be
the case for any single year or multi-year holding period. No
additional source of risk is rewarded with higher expected real
returns.
</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
In the real world, many investment
strategies recommended for retirees employ mixes of stocks and
bonds. An investment in our market portfolio could be considered
as roughly equal to a portfolio with 60% of its value in
stocks and 40% in bonds. However, the relative values of stocks and
bonds in our market portfolio will change as the relative values of
outstanding stocks and bonds vary. The investor holding our market
portfolio will not have to sell bonds and buy stocks when the stock
market falls more than the bond market. Nor will he or she have to
sell stocks and buy bonds when the stock market rises more than the
bond market. As I have discussed elsewhere, strategies that call for
specified proportions of value invested in stocks and bonds require
“contrarian” behavior – selling relative winner and buying
relative losers, and only a minority of investors can do this.
As always, for every seller there must be a buyer. This
calls into serious question the desirability of any investment
strategy that requires rebalancing to maintain specific proportions of values of different asset classes, especially when trading costs
are considered.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
My paper on these issues and a helpful
calculator with historic data on the relative values of world bonds
and stocks can be found <a href="http://www.wsharpe.com/aaap">here</a>.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
Unfortunately, at present there is no
low-cost mutual fund or ETF that provides returns similar to those of
a world bond/stock portfolio. It is possible to find low-cost index
funds or ETFs that cover the major components -- world stocks, U.S. Bonds and
non-U.S. Bonds. But the investor holding such funds would have to
monitor the relative values of these components periodically to adjust for new issues, bond maturities and the like. While this might not be too arduous, I
continue to hope that the financial industry will provide a
single fund for those who wish to invest in a truly representative
world bond/stock portfolio. In the meantime, relatively low-turnover
mixes of broad-based bond and stock funds will probably suffice.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
If you wish to analyze strategies in which some alternative risky portfolio is utilized, you may of course adjust the assumptions about the market portfolio's expected return premium and standard deviation of return accordingly. However, any changes in asset allocation will have to rely on combinations of this portfolio and the riskless real security.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
A final issue in this area concerns our
assumption that the expected return on the market is constant from
year to year. Some evidence suggests that stock returns
are not independently distributed from year to year. Instead, the stock
expected returns may be higher after stocks have declined and
lower after they have risen. Formally, the return on the stock market may have negative serial correlation. Importantly, this is not inconsistent with our
assumption that the returns on the overall bond/stock market portfolio returns are
independent from year to year. For example, assume that stocks have
fallen in value and that the value of the stocks has changed from 60%
of the total to 50%. If bond expected returns are
unchanged, for the overall market expected return to be the same,
stock returns will have to be greater. More generally, our assumption
that the returns on the broad market of bonds and stocks are
independent from year to year is not incconsistent with negative
serial correlation in stock returns.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<h3 style="margin-bottom: 0in; text-align: center;">
Inflation
</h3>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
The last two market settings relate to
inflation. They are the expected annual inflation, for which the
default value is 2.5% per year and the standard deviation of
inflation, with a default value of 1.0%. While these are the
parameters for annual inflation, the amounts generated for
simulations are drawn from a lognormal distribution. Thus if annual
inflation is 2.5%, the comparable relative value of purchasing power
is 1.025 and in the simulations, the logarithms of such relative values
are drawn from a normal distribution.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
For simplicity, the annual rates of
inflation are assumed to be independently and identically
distributed (in academic-speak, they are said to be <i>iid</i>). Moreover, they are assumed to be uncorrelated with the
real returns on the market portfolio. These assumptions are somewhat inconsistent
with much of the empirical evidence. Annual inflation appears to be
positively serially correlated, with abnormally high periods of
inflation likely to be followed by periods of smaller but still
above-average values and with abnormally low periods of inflation
likely to be followed by periods of higher but still below-average
inflation. Moreover, in some countries there tends to be a
negative relationship between real returns on equity and inflation,
due perhaps to the fact that firms' taxes are based on nominal rather than real returns, so an increase in inflation may
lower after-tax profits.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
While these empirical results raise
relevant questions about our inflation assumptions, it is not likely
that changing them would greatly affect the key simulation
results. If the variation in inflation from year to year is
relatively small, the impact on retirement income strategies may be
minor. And, as is well known, central banks in most large countries
and regions make every attempt to keep inflation within narrow
ranges such as those assumed in our default settings. Our default assumption
for the standard deviation of inflation (1%) may reflect an overly
optimistic view of such banks' abilities to control inflation, but it
can of course be easily changed.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
Turning to expected inflation, the
target set by many if not most central banks is 2 % per year.
Historically, many countries have experienced somewhat greater levels
(and many advocate the central banks encourage this, at least when
economies are sluggish). Some observers believe that expected
inflation can be inferred from the spread between the yield on a
nominal treasury security and that on a real security. For example,
at the end of November, 2013 the real yield on a 20-year U.S. Treasury TIPS
was 1.23% while the nominal yield on a regular U.S. Treasury bond with the same maturity was
3.54%. The difference (2.31%) could be a consensus of investors'
estimates of future inflation over that period, although it might also
reflect other considerations. In any event, our estimate of 2.5% may
be reasonable, although it too can be easily changed.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
With simulated real returns on the
market portfolio and inflation, it is easy to determine the
associated nominal returns on the market. Rather than adding the two
amounts I use the more precise relationship:<br />
</div>
<div style="margin-bottom: 0in;">
</div>
<div style="margin-bottom: 0in;">
(1+n) = (1+r)*(1+i)<br />
</div>
<div style="margin-bottom: 0in;">
</div>
<div style="margin-bottom: 0in;">
Thus if the market return is 8% and
inflation is 2%:</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
(1+n) = 1.08*1.02</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
so (1+n) is 1.1016 and the nominal
return is 10.16%. Given this relationship, the nominal returns on
the market portfolio will be lognormally distributed, since both the
market real return and inflation are lognormally distributed and the product of two
variables drawn from such distributions will be as well.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in; text-align: center;">
<h3>
<b>Summary</b></h3>
</div>
<div style="margin-bottom: 0in;">
Here are some key points concerning the
market and inflation assumptions utilized in the RIS software.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
First, investors are assumed to
diversify their risky asset holdings not only within asset classes
but more broadly, across asset classes, focusing on a highly
diversified portfolio of bonds and stocks. Second, the investment
world is assumed to focus on real returns, with investors avoiding
any “money illusion”. Third, the only risk that is rewarded with
higher expected returns in any year or multi-year period is that
associated with the broad overall market, represented ideally by the portfolio
of all traded world bonds and stocks.
</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
The relative merits of
different retirement income strategies may change if these
assumptions are changed substantially, either by modifying the market settings or
by changing the code to produce a system with qualitatively different investment
and/or inflation assumptions. Outputs may well depend
on inputs. My goal is to provide a base that others can use or modify
as desired.</div>
William Sharpehttp://www.blogger.com/profile/05491169505008130323noreply@blogger.com6tag:blogger.com,1999:blog-5994111658248655830.post-13021419501756546462013-11-19T14:50:00.002-08:002013-11-23T12:44:55.370-08:00Video on Longevity<div class="separator" style="clear: both; text-align: center;">
<iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/P9LQETnB4m0?feature=player_embedded' frameborder='0'></iframe></div>
<br />
<br />
Here is a video showing how to use the RIS system to obtain a longevity graph. It also provides an introduction to the general use of the software.<br />
<br />
The first few seconds may appear blurry, but don't give up -- the video will become clear very quickly. William Sharpehttp://www.blogger.com/profile/05491169505008130323noreply@blogger.com2tag:blogger.com,1999:blog-5994111658248655830.post-13222628072139933742013-09-23T15:03:00.001-07:002013-09-23T15:03:22.999-07:00Longevity Graphs
<style type="text/css">A:link { }</style>
<br />
There are many uncertainties associated with one's retirement
years. This project is about retirement income – how much income
will be available in each year and for how many years will income be
needed. There will be much to say about the generation of income but
a key aspect of any strategy for producing retirement income is the
question of how long it will be needed.
<br />
<br />
To put it crassly – how long will the primary recipients of
income live?<br />
<br />
<br />
Very few want to address this question. But it is a key component
of the process required to make sensible financial decisions in
retirement. The programs that I will develop will rely heavily on
longevity probability estimates. Hence it seems suitable to start the
Retirement Income Scenarios (RIS) software with such estimates.<br />
<br />
<br />
Consider Bob and Sue Smith, whom we will call “The Client”.
Bob is a 66 year old male and Sue a 63year old female. How long will
they live? The answer is almost certainly that no one really knows.
To approach this question in any rational manner one must deal with
<i>probabilities</i>. Let's say that the probability that Bob will
die next year is 1.3%. This means that out of cohort of 1,000 men of
Bob's age, 13 are likely to die in the next twelve months. To put it
more positively, 987 of them are likely to survive to reach 67.
<br />
<br />
<br />
Mortality tables contain estimates of the probabilities of death
at various ages for people in a particular segment of society. There
are usually different tables for male and female members of that
segment. Based on the numbers in the tables, estimates can be made of
the probabilities that an individual or a pair of individuals will
live to various ages. These are the probabilities shown in the RIS
longevity graphs.<br />
<br />
<br />
For the RIS software I used tables provided by the United States
Society of Actuaries based on statistics about the longevity of
participants in a number of retirement plans in the U.S. The basic
RP-2000 tables give estimated mortality probabilities for the year
2000 for (1) males and (2) females of various ages. The original data
concerned mortality rates for employees up to age 70 and “healthy
annuitants” (those receiving retirement benefits) from ages 50
through 120. Then these were combined to provide “combined healthy”
mortality rates for all ages through 120 – the numbers that I used.
For details, see the
<a href="http://www.soa.org/research/experience-study/pension/research-rp-2000-mortality-tables.aspx">RP-2000 mortality tables.</a><br />
<br />
<br />
The Society of Actuaries has also produced two mortality
improvement tables (BB) – one for males, the other for females.
These provide estimates of the annual improvement in mortality for
each age (although the annual improvements are zero for the lowest
and highest ages). By applying these each year, it is possible to
produce, in effect, a table for 2001, 2002, …, 2013, 2014 ,... and
so on. The client settings in the RIS program include the current
year to allow for updating to the present. Then the factors are used
to compute estimates of future mortality. Thus Bob's mortality next
year when he is 67 will be given by this year's mortality for a
67-year old plus the one year's mortality improvement using the
factor for that age. The mortality for Bob the next year when he is
68, will be given by this year's mortality for a 68 year old male
plus two times the annual mortality improvement factor for that age.
And so on. For more on scale BB, see
<a href="http://www.soa.org/research/experience-study/pension/research-mortality-improve-bb.aspx">Mortality Improvement Scale BB.</a><br />
<br />
<br />
Of course, no one really knows the percentage of males or females
of a given age that will die in each future year. There is thus
uncertainty about the probabilities computed in this manner. I like
to call this “table risk”. We don't really know what the future
statistics will be. What if there is a cure for a major type of
cancer? What if there is a nuclear holocaust? What if an
antibiotic-resistant virus spreads around the world? The retirement
income scenarios in my software will ignore this additional source of
uncertainty. But insurance companies that sell annuities quite
rightly worry about it a great deal, charging higher prices than
would be dictated by standard annuity tables in order to
provide a cushion if mortality rates increase (for life insurance) or
decrease (for annuities). To some extent, this danger can be
mitigated by issuing both types of policies, but life insurance is
(appropriately) purchased mostly by younger people and annuities
(appropriately) by older folks, so any offsets will be, at best,
imperfect. I'll have more to say about the pricing of annuities and
possible societal approaches for dealing with such table risk in
later blogs.<br />
<br />
<br />
In practice there are many different mortality tables. Insurance
companies use tables with higher mortalities when computing prices
for life insurance policies, to reflect the likelihood that the pool
of applicants will be less healthy than the average person (adverse
selection) and that people with such policies might take more risk
(moral hazard). Conversely, insurance companies use tables with lower
mortalities for annuity policies, on the assumption that the pool of
purchasers will be more healthy than the average person and likely to
take better care of themselves.<br />
<br />
<br />
The U.S. Internal Revenue services requires corporate pension
plans under its jurisdiction to use the RP-2000 tables with mortality
improvement. A number of academic studies have utilized the tables as well.
Hence my choice.<br />
<br />
<br />
Now, to the Longevity Graph feature of the RIS software.
<br />
<br />
<br />
To access the current version of the software, go to
<a href="http://scratch.mit.edu/">scratch.mit.edu</a> and type <i>wfsharpe</i> in the search box. Then
click on the latest RIS version shown. Click the green flag. You will
see two buttons. Click the “Client Settings” button to see the
current settings and to put in your own information. To leave a
setting as is, simply press the Return key. When you are finished you
will return to the main menu.
<br />
<br />
<br />
At any time (except when you are entering inputs), you may press
the keyboard up arrow to turn context-sensitive help on or off. You
may also return to the main menu by pressing the keyboard left arrow.<br />
<br />
<br />
To make the software run faster at any time hold down the Shift
key and click the green flag at the top of the window to place Scratch in<i> turbo mode</i>. To show the
information in full-screen mode, click the icon at the top left of
the window. To return to the smaller version, click it again.<br />
<br />
<br />
If you sign up for a free Scratch account, you may look at and, if
desired, modify the software and save the revised version as a
project in your own account at the Scratch site or on your own
computer. If you have modified the client settings, the latest
information will be saved with the software and will be available
when you re-load it.<br />
<br />
<br />
So much for logistics – back to substance.<br />
<br />
<br />
If you feel that you or your partner are more or less healthy than
the average healthy person of your age, you may want to put in a
different age than your actual physical age. There are web sites that will give you an estimate
for this purpose. I have looked at several and found them wanting.
Some are blatant attempts to get your health information in order to
sell you something (a magic elixir, anyone?). Others seem quite crude. I
tried one that provides your “death date”. It took my
health information, then told me that my death date had passed and parted by telling me to have a nice day (I'm not making this up). My friends confirmed that I am not
yet dead (Monty Python fans will recognize the phrase). Perhaps the
best approach is to ask your family doctor for his or her estimate of your effective age, health-wise.<br />
<br />
<br />
Once you have changed the client settings, click the “Longevity
Graph” button and you will see your personal Longevity Graph. Here
is the one for the Smiths:<br />
<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKPe28-iCDrfMcu6or25SVlOqS-9fMtnHlTUeGnEw_gc6Vds7vBLn5twVl_ZupniMMSDt__L9Ti967Dp3hV0tp5twNt2KlXzokDCK8qC0uFHqvaVxY7iC_0Lulyo_-w1YRpfoWSw5Qipc/s1600/Screen+Shot+2013-09-20+at+8.21.11+AM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="255" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKPe28-iCDrfMcu6or25SVlOqS-9fMtnHlTUeGnEw_gc6Vds7vBLn5twVl_ZupniMMSDt__L9Ti967Dp3hV0tp5twNt2KlXzokDCK8qC0uFHqvaVxY7iC_0Lulyo_-w1YRpfoWSw5Qipc/s320/Screen+Shot+2013-09-20+at+8.21.11+AM.png" width="320" /></a></div>
<br />Each bar shows the probability that (1) both you and your partner
will be alive in a future year, (2) only you will be alive in that
year, or (3) only your partner will be alive.
<br />
<br />
<br />
If you do not have a partner, change the client settings to make
your fictional partner 120 years old. This rather crude approach will
insure that he or she is not around in any future year.<br />
<br />
<br />
Of course in any specific projected future scenario you will live some specific number
of years, as will your partner, This will be evident with alternative
possible scenarios are shown in future versions of the RIS system.
But the probabilities shown in the longevity graph provide some
context as you think about alternative retirement income strategies.
<br />
<br />
<br />
You may find all this terribly depressing. It is not pleasant to
even think about dying and to consider the chances that you and/or
your partner might not be around in some near or distant future year.
On the other hand it may be depressing to think about the possibility
that you will need income for decades in the future. I share your
pain. But longevity is truly a fact of life, and it is one of three
major uncertainties that must be faced when making plans for
retirement income (the other two are investment returns and health
issues). Forewarned is forearmed.
<br />
William Sharpehttp://www.blogger.com/profile/05491169505008130323noreply@blogger.com7tag:blogger.com,1999:blog-5994111658248655830.post-30455546516919595552013-09-17T10:34:00.002-07:002013-09-17T10:54:27.453-07:00Why Scratch?<style type="text/css">P { margin-bottom: 0.08in; }</style>As indicated in the previous blog, I am
developing a suite of software dealing with retirement income
scenarios using the Scratch programming language. Those who know
something about Scratch may consider this a strange choice. Here I'll
try to show why I consider it well suited for this project.
<br />
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
I have been writing computer programs
for over fifty years. My PhD dissertation included (in addition to an
early version of the Capital Asset Pricing Model) the description of
an algorithm for solving a special class of portfolio optimization
problems and a program for implementing it. Since then I have written
programs in a variety of languages. I published the first commercial
book on the BASIC language and wrote an interpretive compiler to
implement it when I was at the University of Washington. For my own
research I now use Matlab, a scientific programming language. For
years I used the standard Matlab constructs but now rely on the more
recently added object-oriented capabilities. I love to program –
there is much gratification when a program does what you intended it
to do -- more than enough to offset the frustration when it doesn't.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
I also feel very strongly that everyone
should be exposed to programming as part of the curriculum in Junior
High School and/or High School. The benefits are many. Students can
learn to think logically, divide complex tasks into a series of
sub-tasks, test ideas rigorously, and explore aspects of mathematics,
statistics and many other fields by doing experiments. They can also
gain a deeper understanding of the ways in which computers, tablets,
phones, televisions, movies and many things we encounter in our daily
lives do what they do. Most people now spend hours every day
interacting with technology but in an important sense they are
interacting with programs. One hears “the computer did such and so”
but it would be more accurate to say that a program made the computer
do it.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
Most important, as the Scratch team
emphasizes, one can experiment and be creative when writing programs
– far more so than when using programs written by others.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
Unfortunately, programming is included
in the required public curriculum in only a minority of public
schools in most countries. There are groups trying to fill this need
– see, for example,<a href="http://www.computerclubhouse.org/%E2%80%8E">Computer Clubhouse</a>, <a href="http://coderdojo.com/">Coder Dojo</a>, and
<a href="http://www.code.org/%E2%80%8E">Code.org</a>. But far more is needed.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
Since I had never taught pre-college
students, I thought it would be a good idea to understand more about
the benefits and challenges associated with including programming in
the curriculum. I began by researching languages that would be suitable
for doing so. I very shortly narrowed my list to one – the Scratch
Programming Language developed at the Massachusetts Institute of
Technology (MIT) ( <a href="http://scratch.mit.edu/">scratch.mit.edu</a>)– for reasons that I'll give shortly. I spent some
time learning the rudiments of the language and then volunteered to
teach it to a small group of middle school students in a summer
program sponsored by the Community Partnership for Youth in Seaside
California, near my home in Carmel. I had a great time, as did the
students. We wrote programs to create designs using geometric
figures, to administer arithmetic tests, to run a horse race and to
allow people to play pong. The students learned key aspects of
logical thinking, how to break tasks into key components, and some
applied mathematics. They also gained a better understanding of how
much of the world of technology works. I learned as much from them as
they did from me. Most importantly, it was great fun for us all.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
More than ever, I am convinced that the
school curriculum needs to include programming. And that the best
language, at least for the first course, is Scratch.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
Scratch was developed and is supported
by the Lifetime Kindergarten research group at the MIT Media Lab.
Work began in 2003 and the first version was launched publicly in
2007. At present there are over one million members of the Scratch
Community and over three million projects have been posted on the
Scratch web site.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
Here is a description of the choice of
the name from a 2009 article by the members of the team ( <a href="http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CDAQFjAA&url=http%3A%2F%2Fweb.media.mit.edu%2F~mres%2Fpapers%2FScratch-CACM-final.pdf&ei=rJU4Up7nPIbliAK_oYGoCw&usg=AFQjCNHhCIBu2ZhRva6I9NZ2-qwBHO-u5w&sig2=9LxBD-ffO6gjAXrF3Bua4A">Scratch: Programming for All</a>).</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
“The name 'Scratch' itself highlights
the idea of tinkering, as it comes from the scratching technique used
by hip-hop disc jockeys, who tinker with music by spinning vinyl
records back and forth with their hands, mixing music clips together
in creative ways. In Scratch Programming, the activity is similar,
mixing graphics, animations, photos, music, and sound.”</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
The most recent version, Scratch 2.0,
became public in May, 2013. It allows users to write, edit and run
programs using only a browser. Programs may also be downloaded to the
user's computer. A downloadable version of the language editor and
processor is also available (in a beta version as I write this).
</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
Anyone may join the Scratch community,
create programs, and, if desired, make them available on the Scratch
website. Any program made public by its author may be used by anyone.
It is also possible to “look inside” to see a public program's
code. Anyone may adapt such a program for other uses, subject only to
the terms of a Creative Commons attribution and sharing license.
As indicated earlier, users are encouraged to “remix” existing programs in
order to create new capabilities (with attribution, of course) .
</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
There are no charges. The Lifetime
Kindergarten group has received support from the likes of the
National Science Foundation, the Intel and Microsoft Foundations, the
MacArthur Foundation, Google and many others. Your support is also
welcome but not required.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
It should not be surprising that an
undertaking of this importance and quality comes from MIT. A
legendary pioneer in the use of computers by people of all ages was
Seymour Paper, who developed the Logo programming language. Indeed,
Mitchel Resnick, the head of the Lifetime Kindergarten research
group, recently published a description of the genesis of Scratch
under the title <a href="http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CC4QFjAA&url=http%3A%2F%2Fwww.media.mit.edu%2F~mres%2Fpapers%2Feducational-technology-2012.pdf&ei=7pY4UsmdGqnViwK0x4CYBw&usg=AFQjCNEfuaOi3bh9I3JAWTWHtQOQbdR7lw&sig2=k8X4gRoyzm-2EKXVRLjn3w&bvm=bv.52164340,d.cGE">Reviving Papert's Dream</a>.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
This is not the place for a detailed
description of Scratch. Resnik's recent paper is an excellent introduction,
as is the formerly cited 2009 paper by the entire Scratch team. Here
I'll give just a flavor of why it is different from most conventional
programming languages.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
First, there are no error messages
because it is very difficult, if not impossible to make a syntactic
error. The grammar is based on a set of graphical programming blocks
and items that are “snapped together” to create a program. And
the items have shapes and colors that indicate their nature. If a
something doesn't fit in a location, it can't be used in that manner.
This avoids myriad errors, at the relatively small cost of requiring
more grabbing, moving and assembling than required in most
programming languages.
</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
Scratch has many attributes of a modern
object-oriented programming language. Objects (called sprites) can
have local variables and methods. Sprites communicate by broadcasting
and receiving messages, which allows for more modular programming and
event-driven execution. As indicated earlier, there are features that
facilitate animation, graphic user input, graphic output, sound,
inclusion of photos, external material and much more. If desired,
programs can even be written to process input and output from some
external devices.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
All these features make Scratch ideal
for its intended purpose. The 2009 paper states: “The core audience
on the site is between the ages of eight and 16 (peaking at 12),
though a sizeable group of adults participates as well.” The
students that I taught were between 11 and 13 and I can attest to the
suitability of Scratch for that demographic. But next year I will be
five times as old as the upper limit of the range for the core
audience. Is Scratch right for me and for my work on retirement
income? I think so.</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
As I have learned more about Scratch
and used it for complex projects, I have realized that the underlying
structure is truly brilliant. The structure has been carefully
crafted to allow great generality but with consistent and highly
logical underpinnings. To be sure, there are limitations, but one can
get around most of them or adapt as needed. At present I have not
stressed the system by attempting very large simulations with sizable
intermediate data, but early experiments indicate that Scratch can
accomplish rather complex tasks quite efficiently.
</div>
<div style="margin-bottom: 0in;">
<br /></div>
<div style="margin-bottom: 0in;">
So, here is my plan. I will start with
an overall structure that allows me to add features as items on a
menu. The first release will have only one such feature (a “longevity
graph”). Subsequent releases will add other features, all related
in some manner to the forecasting and analysis of retirement income
scenarios. I invite you to try the programs. Together we will see how
far this undertaking can go.</div>
William Sharpehttp://www.blogger.com/profile/05491169505008130323noreply@blogger.com6tag:blogger.com,1999:blog-5994111658248655830.post-17541557657157687002013-09-16T15:33:00.002-07:002013-09-16T15:35:41.929-07:00Retirement Income Scenarios<style type="text/css"></style><br />
This blog will be devoted to discussions of issues surrounding the
provision of income for a person or couple during their retirement
years. Much of the analysis will be conducted by forecasting a number
of possible future scenarios, then analyzing the properties of chosen
strategies for producing retirement income across the scenarios. I
call this approach “retirement income scenario analysis”. It uses
the method of Monte Carlo simulation with an underlying set of
assumptions about the behavior of capital market and macro-economic
variables as well as an assumed basis for valuations of possible
future cash flows.<br />
<br />
Since it is important to generate sufficient scenarios to provide
a representative set of possible future outcomes, computer programs
play a central role in the analyses. I have developed a series of
routines for large-scale projections using the Matlab programming
language, taking particular advantage of its object-oriented
capabilities. I have been using Matlab for decades and find it an
excellent language for scientific analysis. However, there is no
simple way to make Matlab programs available for use by those who
have not purchased the software or have access to it through colleges
and universities. Hence I do not plan to try to make these programs
available for use by others. Instead I will use the Matlab programs
to illustrate and illuminate some of the fundamental relationships
involved in retirement income planning.<br />
<br />
Fortunately, there is a programming language that can freely be
used by anyone, and a supporting system that allows programs to be
made available for use, study and modification by others. Moreover,
only a standard web browser is required to use the system or to run
programs written in the language. Its name? Scratch. I am in the
process of preparing a series of programs written in Scratch that
will be available for anyone to use, study or modify.
<br />
<br />
I'll discuss Scratch and the reason why I chose it in some detail
in the next blog. Subsequent blogs will describe the components and
capabilities of the Scratch programs as I complete them and make them
available. I call the overall system RIS, which stands for Retirement
Income Scenarios. As befits the subject of a series of blogs, this is
an ongoing undertaking, with capabilities that will grow over time.<br />
<br />
Of course there will be more in the blogs than discussions of
programs. Much of the material will deal more generally with key
aspects of the economics of retirement income..
<br />
<br />
There is much to cover. Please join me in trying to understand and
explore the many relevant aspects of this crucially important topic.<br />
<br />
<br />William Sharpehttp://www.blogger.com/profile/05491169505008130323noreply@blogger.com52tag:blogger.com,1999:blog-5994111658248655830.post-31080755173118495302013-08-04T13:08:00.003-07:002013-08-04T13:08:57.153-07:00Plans for this blogThis is a new blog on which I plan to post material on creating and analyzing ranges of scenarios for retirement income using different strategies for investing, spending and annuitizing retirement savings.<br />
<br />
With luck, I'll have new material here relatively soon.<br />
<br />
wfsWilliam Sharpehttp://www.blogger.com/profile/05491169505008130323noreply@blogger.com3