**Are investment returns lognormally distributed?**

There is an excellent theoretical basis for the proposition that investment returns are distributed according to a Lognormal distribution.* A random variable is usually thought of as being Lognormally distributed if it can be expressed as the product of other factors. This is exactly the case for compounded investment returns which are computed by expressing the individual period percentage returns as fractions and adding 1.0 to form the

*relative return*expression. For example, a 12% return expressed as a relative return would be given by 1+.12 = 1.12. Thus, a compounded overall return over n number of periods (months, years, etc.) is the product of the relative returns expressed like this: (1+r1)(1+r2)...(1+rn). The data can always be thought of as the result of combining data from shorter intervals, daily data combined to form monthly data, monthly data combined to form annual data, etc.

Further, a Lognormal distribution can never be negative in the same way that with conventional investments the most you can lose is all the money you have invested. Even the vaunted Ibbotson & Associates SBBI Annual reports that the Large Stocks asset class is not exactly Normally distributed.

But let's see how the distributions for the various asset classes actually look. Using monthly total return data from 1927-2007 for Large Stocks the distribution looks like the following histogram. It contains 81 years x 12 months = 972 data items. On this graph are superimposed the best fit Normal distribution as a red line and the best fit three-parameter Lognormal distribution as a blue line.

As you can see, the Normal distribution in red is completely covered by the Lognormal line indicating that the best fit Lognormal distribution is in actuality a Normal distribution. What this means is that the assumption made by MPT that investment return data is Normally distributed is vindicated by the data and that MPT's use of standard deviation is an excellent measure of risk even by PMPT standards. At least for the Large Stock asset class anyway. Let's see if the coincident plots of Normal versus Lognormal distributions continues for the other asset classes. Here is the histogram for the Small Stock asset class.

The two distributions appear coincident on this graph as well. Let's look at the histogram for Long-Term Corporate Bonds.

Here the two plots diverge only minimally as the red line begins to appear in a few spots. Next is the graph for Long-Term Government Bonds.

Now we begin to see some distribution divergence as the histogram demonstrates a noticeable amount of positive skewness. The assumption of a normal distribution is still quite appropriate however. Lastly, let's look at the histogram for Intermediate-Term Government Bonds.

Again, here we see a similar amount of divergence between the Normal and Lognormal distribution fits as a visually perceptible amount of positive skewness appears. The Normal distribution is still an excellent approximation to the Lognormal for this asset class however.

No histogram is presented for the U.S. Treasury Bills asset class as that data in monthly total return fractional form is available to only four places to the right of the decimal point. The lack of sufficient data resolution results in an almost random uniform distribution rendering it useless for this analysis.

The previous three graphs above for the various Bond asset classes exhibit an excess of "peakiness" (known as

*kurtosis*) relative to the histograms for the Large and Small Stock asset classes. The high kurtosis of these distributions suggests that neither Normal nor Lognormal distributions completely explain the behavior of these debt based asset classes. A high kurtosis in a distribution usually means more of the variation is due to infrequent larger deviations as opposed to frequent smaller sized deviations. This sounds very much like the behavior of bonds where a fixed interest rate provides much of the return with interest rate market fluctuations providing a smaller relative capital appreciation/loss portion.

**Conclusion & PMPT Evaluation**

We have shown that the MPT assumption that investment returns are Normally distributed is quite workable and serves to confirm the validity of the portfolio allocations presented in earlier installments of this article.

To summarize this evaluation of PMPT:

1. PMPT's criticisms of MPT are basically theoretical and have been shown here to have little practical validity.

2. The tools that PMPT provides run contrary to the basic tenets of MPT, those being efficient risk minimization through broad diversification over many asset classes and the employment of Index funds to represent those asset classes as asserted by the Efficient Market Hypotheses (EMH).

3. Investment returns appear to very closely follow a Normal distribution particularly the stock asset classes in contradiction to the assertions of PMPT. The bond asset classes do exhibit some skewness but a Normal distribution nevertheless provides a very close fit. As such, the standard deviation, derided by PMPT as a measure of risk, is in fact an excellent objective measure.

4. PMPT is silent on correlations of asset class behavior and is thus useless for determining optimum portfolio allocations among various asset classes.

I hope you have enjoyed my series on MPT and PMPT and I am grateful to Dr. Kris for allowing me the space to present it.

*A lognormal distribution is the probability distribution of a random variable whose logarithm is normally distributed.

*Posted by Prof. Pat*

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