Sunday, December 29, 2013

Savings and Income Scenario Settings

This is about the scenarios that can be generated and shown in the RIS software. I assume that you have dealt with the client settings and the market settings and have also set up one or more accounts 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 scenario settings. The figure below shows the six settings and their default values in the RIS-20140101 version.

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:

         Real people should care about real income

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.

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: If a value falls outside the range of a graph, it is shown outside the border, using the closest x and/or y value

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.

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.

Once you have adjusted these scenario settings, you are ready to see the results of your handiwork by clicking either the savings scenario or income scenario button. I'll cover these in the next post.

Tuesday, December 17, 2013

The X% Rule


The RIS software allows the user to specify one or more sources of retirement income. Each is described in an account. And each such account has a number of settings.

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.

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”.

The 4% rule

The 4% rule, first advocated by William Bengen in “Determining Withdrawal Rates Using Historical Data”, Journal of Financial Planning, 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 webinar he advocated following the policy with a withdrawal amount equal to 4.5% of the initial value.

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 here, in which we pointed out a number of its shortcomings. In a recent paper in the September/October issue of the Financial Analysts Journal,  Jason and John document subsequent studies of variants of the 4% rule and advocate a very different approach.

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:

        The amount you spend should depend on
              1. How much money you have, and
              2. How long you are likely to need it

    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..

    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.

    Now to the details of the software.

    The X% account and its settings

    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.

    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.

    The initial setting (line 2) indicates whether this account is active 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.

    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).

    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.

    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

    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.

    The remaining settings concern fees and the investment strategy to be followed when implementing the rule.

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    Tuesday, December 3, 2013

    Investment Returns and Inflation

    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.

    There are five key settings for this process, shown, along with default values, in the market settings list below.

    To see the current values of these settings and make any desired changes, simply click the Market Settings button on the RIS home page.

    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. 

    Warning! Of necessity, some of this discussion is going to be lengthy and rather technical. Feel free to skim it as needed.

    Risk-free real returns

    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.

    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.

    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:

            Maturity         Annual Yield
                5 years           - 0.32 %
             10 years              0.60 %
             20 years              1.23 %
             30 years              1.53 %

    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.

    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.

    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.

    Market bond/stock portfolio returns

    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.

    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.

    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.

    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.

    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.

    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.

    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). 

    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.

    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.

    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 expected return, 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.

    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 Reward to Variability Ratio). 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.

    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.

    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 my web site.

    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.

    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 (iid). 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 any period of years (from 1 to many) will be lognormally distributed.

    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.

    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.

    My paper on these issues and a helpful calculator with historic data on the relative values of world bonds and stocks can be found here.

    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.

    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.

    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.


    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.

    For simplicity, the annual rates of inflation are assumed to be independently and identically distributed (in academic-speak, they are said to be iid). 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.

    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.

    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.

    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:
             (1+n) = (1+r)*(1+i)
    Thus if the market return is 8% and inflation is 2%:

            (1+n) = 1.08*1.02

    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.


    Here are some key points concerning the market and inflation assumptions utilized in the RIS software.

    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.

    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.