Looking beyond return
If you asked most investors whether they wanted a 10-percent return or an 8-percent return, in addition to a puzzled expression you might also receive a hand on your forehead in an attempt to determine if you were running a fever. Two percent extra is two percent extra.
In casual conversation, any investor would naturally assume “All else being equal” is implied in the question. But little is ever equal with different investment opportunities, and the primary inequality comes in the amount of risk or uncertainty you have to accept to achieve your investment goal.
For example, if you asked investors whether they wanted a 10-percent annual return with a high probability of being down more than 65 percent at some point over the next year or an 8-percent return with a 90-percent probability of never being down more than 15 percent at any time over the next year, they might change their tune. These are extreme examples, of course, but they illustrate that the key aspect of the question is not return, but risk-adjusted return.
Consider Figure 1, which shows monthly results for two hypothetical commodity trading advisors (CTAs — see “Key Concepts”). Starting with $100,000 initial investments, both advisors end the year with $124,575 — a healthy 24.58-percent return. If you couldn’t see the month-to-month results, you might think the decision to invest in one over the other was best determined by a coin toss.
The monthly returns detailed in Table 1 convey a great deal about each manager’s trading style and what you might experience as a potential investor. Although Advisor A actually had one more losing month than Advisor B (four vs. three), Advisor B’s monthly swings are much wider than Advisor A’s. Although Advisor B’s average winning month was almost four times the size of Advisor A’s (10.06 percent vs. 2.51 percent), B’s average losing month was also more than 20 times larger than Advisor A’s (–0.04 percent vs. –10.66 percent).
The higher volatility of Advisor B’s monthly returns — represented by a standard deviation more than 10 times the size of Advisor A’s — implies Advisor B assumed much more risk to achieve the same return as Advisor A. “Variance and standard deviation” (below) has a detailed explanation of these concepts, but no math is required to understand that Advisor B’s huge month-to-month jumps would be a roller-coaster ride many investors would find difficult to stomach. The fund lost more than 40 percent of its value in February, and subsequent months featured many more large percentage swings. An exceptionally strong finish brought Advisor B even with Advisor A at the end of the year, but Figure 1 illustrates how rocky the overall road was.
If Advisor B had produced a total return that was at least 10 times bigger than Advisor A’s, there might be room for discussion about which program to invest in, but based on this information, most investors would select Advisor A over Advisor B.
Adjusting for reality
There are many ways to calculate risk-adjusted return (RAR), and the formulas can be a bit confusing. But at their core, such measures simply place an investment’s return in the context of the risk required to achieve it, with the assumption that more risk should translate into higher returns.
For example, we could define risk as the historical maximumdrawdown (see “Key Concepts”) an investment has experienced. The logic is that maximum drawdown is a proven, real risk — the investment has already realized it — so it is fair to expect a similar-sized loss to occur in the future. In this context, it makes sense to measure the investment’s return relative to this risk. In Table 1, neither advisor has back-to-back losing months, so their maximum drawdowns are their biggest monthly losses: -1.87 percent (in February) for Advisor A and -43.46 percent (also in February) for Advisor B.
As we know, both advisors ended the year up 24.58 percent. As a result, a simple RAR measure would be to divide this return by the funds’ respective maximum drawdowns. The higher the resulting number, the better the investment’s risk-adjusted performance:
Return / drawdown = risk-adjusted return
Advisor A: 24.58 / 1.87 = 13.13
Advisor B: 24.58 / 43.46 = 0.57
Notice the minus signs are removed from the maximum drawdowns to produce an easy-to-work-with positive number. The important consideration is the proportion of return relative to risk — Advisor A comes out well ahead of Advisor B.
One of the best-known RAR measures in the financial industry is the Sharpe ratio, developed by Nobel Laureate William Sharpe in the 1960s. The Sharpe ratio adjusts an investment’s return by 1) subtracting a “risk-free” rate of return from the investment’s return and 2) dividing that result by the standard deviation of the investment’s returns. The first step lets you know what you’re earning above and beyond a theoretically risk-free investment such as Treasury bills. The second step produces a number that is higher for an investment with returns that are less volatile from period to period.
The Sharpe ratio formula is:
Sharpe ratio = IR – RF
StD
where
IR = Investment’s return (typically, the annualized return)
RF = the risk-free return
StD = Standard deviation of the investment’s returns (annualized)
Table 1 also includes the Sharpe ratio for both advisors, and shows Advisor A (2.54) again ranking far ahead of Advisor B (0.25). It should be noted the hypothetical monthly returns used here — for both advisors — are fairly extreme. For example, the top-10 performing CTAs managing more than $10 million for February 2010 tracked by BarclayHedge (www.barclayhedge.com) had an average Sharpe ratio a little above 1.00; these funds returned an average of 10.97 percent for the month of February.
No such thing as perfect
There are many critics of risk-adjusted return measures such as the Sharpe ratio, noting that, among other shortcomings, they don’t lend themselves to universal application across different investments, and they are not necessarily a good indicators of future performance.
These final two points were raised in a study titled “The myth of persistent Sharpe ratio” by SiewLing Lay of Singapore-based hedge fund research and consultancy firm GFIA (www.gfia.com.sg) that analyzed the Sharpe ratios of more than 300 hedge funds between 2007 and 2009. Although the study was conducted on Asian hedge funds, its findings provide food for thought regarding the usefulness of Sharpe ratios and similar performance metrics in determining the future performance of any actively managed portfolio.
The study ranked managed funds according to their Sharpe ratios during an initial observation period and segregated them in four percentile groups: the top 25 percent of Sharpe ratios, the 25 to 50th percent of Sharpe ratios, and so on. It then analyzed the funds’ Sharpe ratios for the next year to see if funds in each group tended to remain in the same quartile — that is, whether funds with high or low Sharpe ratios tended to have high or low Sharpe ratios in the next period. In analyzing the entire group of funds, as well as funds in different strategy sub-groups, Lay found there was no correlation in Sharpe ratios from one period to the next. In each test, fewer than 50 percent of the funds in a particular Sharpe-ratio percentile group appeared in the same group the following year.
However, if the Sharpe ratio doesn’t have any predictive value, what use does it have for investors who might be trying to decide where to allocate money? Lay says the Sharpe ratio simply isn’t designed to aid in such decisions, even though many people believe it is. One problem is that the Sharpe ratio doesn’t distinguish between positive volatility (i.e., equity gains) and negative volatility (losses, or drawdowns).
“Investors only complain about losing money, whereas having a very volatile positive return with no downside at all is probably not an issue to them,” Lay says. “In many cases, two funds can have the same volatility despite one having much more downside than the other.”
The study ultimately concluded the Sharpe ratio was mostly useful in providing insight regarding a fund’s historical performance. But this problem isn’t unique to the Sharpe ratio. No single calculation can provide the “best” answer about a trading program’s future performance and risk.
“Decision making should probably never be based on just one of them,” Lay says. “A combination of complementary metrics probably will give a better picture of any investment. For example, if you’re looking to analyze the downside risks of funds, you should probably also look at the Sortino ratio.”
The Sortino ratio is designed specifically to address the difference between positive and negative volatility in an investment’s returns. The formula is:
Sortino ratio = IR – RF
StD(NR)
IR = Investment’s return (typically, the annualized return)
RF = the risk-free return
StD(NR) = Standard deviation of the investment’s negative returns (typically annualized)
Although there are no easy answers when attempting to categorize risk-adjusted returns, more information is better than less. A trading program with consistently favorable risk-adjusted returns — especially using different measures — over a much longer time period will provide greater confidence (but no guarantee) of repeating that performance in the future.
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Variance and standard deviation Standard deviation measures the degree to which the values in a group (say, monthly returns) vary from their mean (average) value. Consider the following series of numbers: 10, 10, 9, 11, 10 12, 8, 10, 14, 9.5, 10.5, 6, 10… If you had to guess what the next number in the series might be, a good choice probably would be 10. With two exceptions – 14 and 6 – the numbers stay very close to 10, fluctuating slightly above and below it. In other words, the individual numbers in the series deviate very little from the mean value of approximately 10. Now look at this series of numbers: 10, .02, 557, 12, 54, 3000, .0099, 20, 92, 20001, 1208, 443… In stark contrast to the first series, these numbers deviate dramatically from the average value, which is around 2116. This is an example of a high-standard deviation series of numbers. Here’s a quick review of the math involved in standard deviation. In statistics-speak, standard deviation is the square root of variance, which simply measures how spread out a group of values are. Mathematically, variance is the “average squared deviation” of each number in the group from the group’s mean value. For instance, for the numbers 8, 9 and 10, the mean value is 9 and the variance is:
3 3 3 The standard deviation of these numbers is their square root: sqrt(.667) = .817 The higher the standard deviation, the greater the volatility. Calculating the standard deviation of the monthly returns for Advisor A in the main story is done the same way, as shown in the accompanying table. The average return for all 12 months (1.88 percent) is subtracted from each of the monthly returns. The result for each month is squared (raised to the second power), and the average of all 12 calculations is 0.06 percent. The standard deviation is the square root of this number, or 2.37 percent. For more information on variance, standard deviation, and other statistical topics, visit http://davidmlane.com/hyperstat/index.html. |
(8-9)2+ (9-9)2+ (10-9)2 = (1 + 0 + 1) = 2 = .667