## Sorting the Sheep from the Goats

The determination of good from bad investments using quantitative measures.

"Past Performance is Not an Indication of Future Results": like most boilerplate text, this warning is often consigned to the margins of investors’ consciousness, its ability to pack a punch eroded by constant repetition. We occasionally need to be reminded that the truth contained in this statement is both profound and complex.

Often the most accessible means of assessing an investment comes in the form of past performance, and some valuable information may be gleaned from it through cautious assessment. However, potential investors need to remain aware of the limitation and pitfalls of some of the most commonly used assessment techniques.

Here we will be taking a fresh look at some of the most frequently used quantitative measures, exposing some of their commonly overlooked logical weaknesses and suggesting some ways of rehabilitating them into the investor’s toolbox.

## Forecasting return

Most investments that investors are called upon to evaluate can be assessed using the representation of their history in the form of a time series. Both amateurs and professionals are inclined to evaluate certain quantitative characteristics of such time series for the purpose of judging whether the underlying investment is a “good” one or not; that is, whether it will produce positive returns over future years at a level of risk they can understand and accept.

The statistics that they choose to calculate and evaluate need to be as well suited to this purpose as possible; they need to not only capture information about the past but also to be capable of providing a means for forecasting the salient features of the future of the time series, a point that is often forgotten, overlooked or wilfully misrepresented.

Let us say we would like to compare the investment potential of two venerable investments, the US stock market and gold, based on their historical performance (Figure 1).

## Figure 1: Which is the better investment?

The return of each time series can be measured in many different ways – the past year, two years, three, etc, or the average rate of return (compound or otherwise) over the investment’s life – which give different results (Table 1).

## Table 1: What measures to use?

## Return, error and risk

For historical return to be a useful statistic for forecasting future return, the time series should, as a rule, be as long as possible. That is because return as a statistic often has a high error associated with it; this error is what is often described as the risk. High returns in particular are usually inextricably linked with higher risks and, ironically, it is higher risks that are more predictable from high returns than are future high returns.

This problem is particularly great with shorter-term returns. The error in forecasting the future return is proportional to the inverse of √*t* where *t* is the length of the time series. Thus, a time series history of four times the length will halve the error in the return forecast. For typical investments ‒ stock funds, for example ‒ a reasonable long-run estimate of expected return and risk might be 10% annually with a standard deviation of 20%.

However, the last year’s return from a particular fund might be +80%, with an annualised monthly standard deviation of 100%. A four-year series would reduce that standard deviation to 50%, a nine-year series to 33%, and a sixteen-year series to 25%; starting to approach the reasonable long run estimate. Clearly, the 80% return is the last statistic that one should reasonably extrapolate into the future, yet both professional and amateur investors regularly make this crude mistake.

Let us take the case of an investor tempted to buy tech stocks at the end of 1998 on the basis of recent returns (Figure 2).

## Figure 2: Point of assessment

Figures 3 to 4 and Table 2 show the kind of information they would have derived from analysing different time windows, and, in retrospect, the bearing it might have had on the fate of the investment.

## Figure 3: The longer the time series, the better the estimate (a)

## Figure 4: The longer the time series, the better the estimate (b)

## Table 2: The longer the time series, the better the estimate (c)

It is also worth bearing in mind that an estimate of the error on returns is likely to be more accurate than an estimate of the returns themselves (Figure 5). The error associated with a return forecast coming from a historical time series remains substantial even when the return is estimated from 10 to 20 years of historical time series data.

## Figure 5: It is easier to estimate error than return

In sum, it is safe to assume that large positive historical returns imply large negative future returns as well as large positive ones ‒ that is, one can expect high volatility in both directions, rather than exclusively large positive future returns.

## Risk-adjusted return

All of this is known to professional money managers which is why they have devised better time series statistics for measuring investment quality. The key step forward was to calculate the amount of return per unit of risk taken, or, put differently, to standardise the return in risk units.

The most popular derivation of this concept is the Sharpe ratio. The Sharpe ratio is calculated for a time series by dividing the mean period return (daily, monthly, yearly), in excess of the risk free rate, by the standard deviation of such returns. The Sharpe ratio overcomes some of the problems inherent in the pure return statistic.

The Sharpe ratio, however, suffers from a number of drawbacks as a statistic, of which it is wise to be aware [3]. Firstly, the denominator is standard deviation, which is only a reliable and meaningful statistic for time series where the distribution of the first differences (price changes) is both parametric and stationary. “Parametric” implies that the distribution can be characterised by a known and meaningful distribution (for example: normal, binomial, T, etc) with finite variance.

Some financial time series do not satisfy this criterion ‒ as an example, option granting strategies, which produce lots of small profits and occasional large losses. In such cases, the Sharpe ratio would not give an accurate representation of the investment’s risk/return profile (see Figure 6).

## Figure 6: These distributions have the same Sharpe ratio

Other return distributions might be bi- or multi-model, and their standard deviation may give a very misleading impression of the probability of certain events.

“Stationary” implies that the time series volatility remain constant through time. This criterion would typically not be satisfied where the investment strategy or assets underlying the time series have changed too much through time.

A common example would be a hedge fund starting with a high leverage in order to produce impressive returns, then gearing down on maturity to ease liquidity constraints and collect management fees (Figure 7).

## Figure 7: This is not a stationary return process

A simple test for stationarity is to ensure that there is no major trend in the rolling volatility. A related problem is that the underlying form of many distributions is unknown. Lower credits hedged with higher ones will tend to pick up a steady excess over the risk-free rate, resulting in very high Sharpe ratio, presumably at the expense of occasional very large losses, such as incurred by Long-Term Capital Management.

Another problem for Sharpe ratio is that it is symmetric regarding upside and downside risk. High returns have the effect of increasing the value of the denominator (standard deviation), and lowering the value of the ratio. Conversely, for a positively skewed return distribution such as that of a managed futures strategy, the Sharpe ratio can be increased by removing the largest positive months. This is patently absurd.

Finally, as with return, there is an overwhelming issue of data bias. Bias can be introduced into a time series in all sorts of innocuous ways as well as deliberately; but regardless of its source, it has the effect of undermining the value of any statistic as a forecasting tool.

Thus, all retrospectively constructed time series, portfolio backtests, and so on must be viewed askance from the point of view of calculating forecast statistics. Similarly, short time series are more vulnerable to statistical bias; this is the well-known point about coin tossing chimpanzee-like fund managers.

## Drawdown

One statistic that is used to try to overcome the parametricity issue is drawdown. The drawdown is the maximum peak to subsequent trough fall in a time series; the maximum loss the investor has experienced from a previous high (which, in the worst instance could have been when he invested and thus represent his worst possible loss). This does not rely on the return process having any particular form and does have intuitive physical appeal.

It also, however, has important weaknesses of which it is necessary to be aware [4]. First, for two time series with otherwise equal characteristics, the longer will tend to have the greater drawdown. For most, investable time series longevity would be presupposed to be a good thing implying survival, robustness, experience, etc. All other things being equal however, a longer track record implies a larger worst drawdown.

## Figure 8: The longer the track record, the deeper the drawdown

Secondly, maximum drawdown is a single number and will therefore have a large and uncertain error distribution. Thus we cannot be at all sure that a time series with a larger worst drawdown is being produced by a return generating process that will tend to produce worse drawdown! Essentially, by using a single number as the denominator, we are balancing too much inferential weight on too slender a quantity of data.

Another statistic that is used to address the skewness problem is that of Sortino ratio. This is the mean period return divided by the semi-standard. This statistic does not demand symmetry of profits and losses and is a better measure than Sharpe ratio for time series resulting from dynamic investment strategies such as managed futures.

## Back to return

Even solidly built risk-adjusted statistics are not a panacea. They still leave us with the problem of underestimating the value of decent returns compared to strategies that consistently produce just a little over the risk-free rate.

We have recently suggested some amendments to traditional measures to remedy this problem. In the first instance, the Sortino ratio can be adjusted to incorporate a minimum acceptable return. This ratio utilises the mean return in excess of the minimum acceptable hurdle divided by the semivariance with respect to that (minimum acceptable) return. This is a more useful statistic for real life situations, such as those faced by the promoter of an investment product bearing several percentage points per annum of fixed fees or the trustee of a pension fund that requires a certain level of returns in order to meet its liabilities.

A mathematical statistic named Omega tackles the issues of parametricity in a robust manner, but involves more complex computations [5, 6]. Both Omega and the modified Sortino ratio redirect much-needed attention to pure return, but in a more constrained fashion. In the end, the market recognises the value of the very long-term compound annual average rate of return (Warren Buffet’s 26% for 40 years, the US stock market’s 5% in real terms for 200 years (Figure 9) or so in forming judgments.

## Figure 9: Long-term returns from US equities

To produce 5% per annum for 20 years is not that exceptional or extraordinary; to produce 15% is very good. Yet the former process can easily produce much higher Sharpe and Sortino ratios and lower drawdown. Without an understanding of the statistical concepts of populations and samples, parametric distributions, statistical moments and their errors, investors are doomed to carry on being misled by their intuition into making one mistake after another.

### References

[1] M. Getmansky, A. W. Lo & I. Makarov, An Econometric Model of Serial Correlation and Illiquidity in Hedge Fund Returns, Journal of Financial Economics, 2004.

[2] W. Goetzman, J. Ingersoll, M. Spiegel & I. Welch, Sharpening Sharpe Ratios, Working Paper, Yale School of Management, International Center for Finance, 2002.

[3] D. W. Harding, Sharpe Justification?, Hedge Funds Review, July 2003.

[4] D. W. Harding, G. Nakou & A. Nejjar 2003, The Pros and Cons of “Drawdown” as a Statistical Measure of Risk for Investments, AIMA Journal, April 2003.

[5] C. Keating & W. Shadwick, A Universal Performance Measure, The Finance Development Centre Limited, 2002.

[6] C. Keating & W. Shadwick, An Introduction to Omega, The Finance Development Centre Limited, 2002.

[7] A. W. Lo, The Statistics of Sharpe Ratios, forthcoming in Financial Analysts Journal 58.4:36-52, 2001.

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