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.


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