Am I a Good Investor? How Long Should a Performance Track Record Be To Know For Sure
The following post focuses on using performance data to evaluate the quality of a trader or investment manager. To be sure, qualitative assessments are also valuable but are not considered here explicitly.
The Sharpe Ratio
The most commonly used tool to judge traders (and hedge funds) is the Sharpe Ratio (SR), defined as the ratio of the excess expected returns to the standard deviation of returns i.e.,
where the excess expected return is usually computed relative to the risk-free rate, Rf. Given a daily or weekly performance track record measured over N periods, the SR’s parameters can be estimated with the familiar estimators i.e.,
where Rᵢ is the return in each period i.
Note that mean and variance both grow linearly with N, assuming each period is uncorrelated with prior periods. Therefore, the SR grows with the square root of N. To see this, observe that
For purposes of comparison, therefore, the Sharpe Ratio is almost always annualized. For example, we can estimate the daily SR and then multiply it by √365 (or √252 if we only use stock market trading days).
Estimation Error of the Sharpe Ratio
Lo (2002) shows how we can compute the standard errors for SR. A standard result from Statistics is that
(Note that these equations hold asymptotically i.e., for “large” N.)
Now say that we have a new statistic (b) that is a function g() of other statistics a1, a2, etc. With a lot of fancy math, it can be shown that if
(Note that this assumes that estimators for aⱼ and aₖ are uncorrelated.)
Hence, for the SR, we can write
With this result, we can compute the 95% confidence interval for SR as
Annualization of the Sharpe Ratio
As mentioned earlier, Sharpe Ratios are typically expressed in annualized terms with
If SR is measured with daily returns data, then q = 365 (or 252 if we only include trading days).
To estimate the annualized version of the Sharpe Ratio, we compute
Where T is the number of years in the return time series. Note that
Typically, daily or even weekly returns are adequate to use this version of the formula.
Finally, we can write
This result is amazing because it shows that the precision of SR estimation depends only on the number of years.
So how many years is enough? If we want to know the SR within 1.0, then
T = 3.84 years! In other words, assuming a “good” Sharpe Ratio is over 1.0, an investor’s track record should be at least 3.84 years old.
Sharpe Ratios Can Be Misleading
There are a couple of problems with this procedure.
One problem is that it assumes that the “true” SR remains constant over time. Consider an investor with a 4-year track record. Say that the investor manages risk wisely in years 1 and 2 (i.e., low σ), but then gets overconfident and reckless in year 3 (i.e., high σ). After 2 years, we’d estimate a high SR, but the “true” SR for year 3 is lower.
In other words, investors’ performance quality does not remain constant over time. Hence, measuring Sharpe Ratio over 20 years isn’t necessarily more meaningful than measuring it over 5 years.
There is a trade-off here: more data reduces the statistical error, but it increases overall estimation error since recent data is probably more informative than older data. One solution to this problem is simply not to use more than ~5 years of data.
The second problem is that multiplication by √N only works if returns are not correlated over time. This is especially relevant at the aggregate market level. For example, the stock market seems to have periods lasting 5–15 years where short-term returns are correlated.
Consider a 3-decade period in the stock market; each decade has SRs of -1.0, -1.0, 2.0, respectively. Say an investor does nothing but invest in the stock market during the last decade. Based on the SR estimation procedure, we’d conclude that this investor’s SR is 2.0. However, this would be misleading because the true SR over 3 decades is actually net zero.
We can largely address this latter problem with the Marginal Sharpe Ratio (see below). For more fun facts about the problem of correlated returns, see Lo (2002).
Estimation Error of the Marginal Sharpe Ratio (MSR)
I introduced the MSR in a prior post. It solves the market correlation problem because it directly benchmarks the SR against market returns. An ancillary benefit, as shown below, is that it almost always has a standard error less than that of the ordinary SR.
As discussed in the earlier post,
To derive the standard error, we can use the same procedure as above. First note that α is the constant term of a simple regression, where y is the return of p and x is the return of m. From elementary Statistics, we know that
where σₑ is the standard error of the regression. We can rewrite this as
To simplify further, we can write
Now, we can apply the same method as above.
Finally, we can annualize this formula.
In other words, the MSR has an ancillary benefit over the ordinary SR: for large q, it always has standard errors less than those of the SR.
α vs MSR
To be sure, one might ask “why not just compute standard errors on alpha directly?” Of course, we could do so. However, the result is less elegant.
It can be shown that
Note that, unlike before, σ² doesn’t cancel out.
Finally, we can offer the following heuristic to measure performance quality.
If this condition is true, then an investor is highly likely to have a positive MSR and hence a positive alpha. That said, for the reasons discussed above, data more than ~5 years old should be viewed with suspicion.
Lo, A. W. (2002). The statistics of Sharpe ratios. Financial Analysts Journal, 58(4):36–52.