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Analysing the output from stochastic investment models

A great deal of research has been done by actuaries on developing suitable stochastic investment models for projecting the growth in value of a portfolio of assets, but less has been published on how to use the output from such models. This output typically consists of many thousands of random simulations for the fund at a particular time horizon (or the projected surplus/deficit if liabilities are being projected as well as assets). The simulations can be used to derive probability distributions for the projected fund, but how should these be compared? If the aim of the exercise is to compare different investment strategies, each one will give rise to its own distinct distribution.

Stochastic dominance
A method of comparison familiar to actuarial students who have taken the new course in financial economics (subject 109) uses the concept of stochastic dominance. A probability distribution for future wealth, W, may be represented by its distribution function F(X):
F(X) = Probability {W<=X}
If we have two distributions, FA(X) and FB(X), we say that distribution A exhibits first-order stochastic dominance over distribution B if:
FA(X)<=FB(X) for all X, with the inequality
holding for some values of X.
This means that for distribution A, the probability of falling below any level of wealth X is never greater than the corresponding probability for distribution B. This is a very powerful relationship and implies that distribution A should be preferred by any investor who prefers more wealth to less.

Stochastic dominance and utility
Stochastic dominance is related to utility theory. The utility function for wealth, U(W), is a measure of the satisfaction gained from that level of wealth. Although this function varies from person to person, there are some general properties it should have for all investors. For example, it should be strictly increasing because people prefer more wealth to less, thus:
U’(W)>0 for all W
Under certain assumptions, it can be shown that utility functions can be used to rank probability distributions for wealth by calculating the expected utility under each distribution. Thus if distribution A is preferred to distribution B we must have:
U(X).dFA>U(X).dFB
It is not difficult to show that if distribution A exhibits first-order stochastic dominance over distribution B, the above inequality will hold for all strictly increasing utility functions.

Relative risk aversion
First-order stochastic dominance gives us a very easy method of ranking probability distributions, but we would generally have to select and apply a suitable utility function to rank the distributions arising from different investment strategies. A utility function often used by financial theorists is the power utility function, given by:
U(X)=
The parameter g is equal to the investor’s relative risk aversion, for which the general formula is:

The use of a utility function with constant relative risk aversion implies that the individual will invest the same fraction of total wealth in risky assets, whatever the total amount of wealth happens to be. And the value of the parameter g, which will vary from one investor to another, is inversely proportional to the fraction of total wealth invested in risky assets. The Society of Actuaries’ textbook on financial economics suggests that setting g=2 gives a reasonable description of the behaviour of an average investor.

Empirical distributions from Barclays Capital data
We will now apply these ideas to a very simple stochastic model for the real annual returns on equities and gilts, respectively. In this model, we randomly select one of the 100 real annual returns from the last century to get our simulated return, as provided by the Barclays Capital Equity/Gilt Study 2000. It follows that cumulative frequency graphs for this data will give distribution functions for real annual returns on equities and gilts, respectively.
These graphs are shown in figure 1 above left. Note that the line for real equity returns is nearly always below the line for real gilt returns, implying that we are very close to first-order stochastic dominance of equities over gilts. If we use a power utility function to calculate the expected utility under each distribution, we find that this is greater for equities when g=2 and roughly equal for equities and gilts when g=4.
So, based on the experience of the last century, a typical investor should prefer equities to gilts, even over a timescale as short as one year.
Simulated distributions from Wilkie model
The Wilkie model (1995) was based on data taken from the period 192394, using the yield on Consols as a proxy for the gilt-edged market. By using the model to simulate real annual returns on equities and gilts, we obtain distributions which can be compared with those derived from the Barclays Capital returns. Figures 2 and 3 (across) show cumulative frequency graphs for the projected real fund accumulated over one-year and five-year time periods, respectively. This time we have included a 50/50 portfolio in our comparison as well the equity and gilt funds.
Figure 2 demonstrates that the 50/50 portfolio exhibits first-order stochastic dominance over gilts when comparing real annual returns. The same is not true for the 100% equity portfolio, although the expected utility of the gilt fund only exceeds that from the equity fund when g=5. Thus, a typical investor would again prefer equities to gilts over a one-year timescale. Figure 3 shows that over a five-year time period both equities and the 50/50 portfolio exhibit first-order stochastic dominance over gilts. The difference between the one-year and five-year comparisons occurs because the Wilkie model allows for autocorrelation between variables over adjacent periods. Thus, the assumption that the equity dividend yield reverts to a long-term average increases the relative attractiveness of equities by making annualised returns less variable over longer time periods.

Why the past may not be the best guide to the future
The last century has been a good one for equities relative to gilts, but should we assume that this experience is the most appropriate basis on which to make future projections? If we have good reasons to believe that the risk premium on equities may not match the experience of the last century, it would not be prudent to base our projections solely on this data. Below we give two plausible reasons for why the future may differ from the past.

The inflationary period 194979
During the period 194979, the UK underwent a period of sustained and increasing price inflation that had a negative impact on real investment returns. The consequence for real gilt returns was more serious than for real equity returns. According to the Barclays Capital study, the real return on equities over 194979 was 4.4% pa, as compared with 5.7% pa over the whole century. For gilts, the real return over this 30-year period was 3.4% pa, as compared with 1.0% pa over the century.
But why have we selected the period 194979, given that significant rates of inflation continued into the 1990s? The reason is that over this 30-year period short-term interest rates did not appear to compensate investors for the high inflation rates. According to the study, a fund invested in Treasury Bills at the end of 1949 would have lost 23% of its real value by the end of 1979. Either the market was persistently underestimating the future inflation rate, or government controls over monetary policy and foreign exchange forced lenders to accept negative real returns.
The negative real short-term interest rates over this period must have affected the gilt-edged market by keeping gilt yields lower than they otherwise would have been. Had gilt yields risen to a higher level, this would have initially meant greater capital losses for investors, but the re-investment of income at higher yields is the dominant effect over longer periods. Thus the negative real gilt returns over 194979 may have been caused by factors which are unlikely to recur. This is illustrated by what happened later during 1988-1991, when there was a brief spurt of inflation averaging 7% pa. The real return on Treasury Bills over this four-year period was 5.8% per annum, and even the Barclays Gilt Fund managed a respectable real return of 2.6% pa.

Is the equity market currently overvalued?
Equities have recently been priced at historically high levels and traditional actuarial wisdom suggests that future returns will be lower as a result. This idea is certainly built into the Wilkie model, which assumes that the dividend yield reverts to a long-term average. Although such reasoning is difficult to reconcile with the efficient markets hypothesis (EMH), the latter has been seriously challenged by a number of financial economists, and the growing field of behavioural finance is an indication that the EMH is no longer the orthodoxy that it once was.
If we want to build a lower expected return into our stochastic projection of equity returns, the Wilkie model will automatically do this if the initial dividend yield is set to the current (low) dividend yield. The problem arises, however, in choosing the long-term average yield where the past may not be the best guide to the future. A reasonable approach might be to adjust the parameters of the Wilkie model so that the expected return on shares is reduced by the extent to which the earnings yield is below its long-term average, or use an alternative model based on earnings rather than dividends.

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