(Originally William F. Sharpe, Beta, and the Paradox of Risk-Adjusted Returns)
Money managers like to take into account not only the potential returns of an investment but also the amount of risk involved. They use all sorts of formulae to quantify this balance: alpha, beta, the Sharpe ratio, the Sortino ratio, value at risk, the gain/loss ratio, and so on.
(Just to be clear, I’m talking about the returns of funds—mutual, closed-end, exchange-traded, hedge; strategies; and portfolios. In other words, bundles. It’s hard to sensibly apply these measures to the returns of individual securities—stocks, bonds, options, and so on.)
Most money managers use the Sharpe ratio, which is a very simple and elegant measure: you take the average monthly return of your investments, subtract the risk-free return (inflation, ten-year treasury bonds, ten-year CD rate, what have you), and divide by the standard deviation of the monthly returns.
Unfortunately, the results are frequently nonsense. Let’s say you invest in MINT, the largest actively managed exchange-traded fund, which is comprised of short-duration investment-grade debt securities. MINT has an annualized variability of 0.53% (we’ll use the five-year figures here). Compare that to SPY (an S&P 500 ETF), with a variability of 12.13%. Now let’s look at their returns. Over the last five years, SPY has returned a total of about 80% (not annualized, but total). MINT has returned less than 5%. Inflation over the last five years is about 5.8%. In other words, if you invest in MINT, you don’t keep up with inflation.
Now which fund has a higher Sharpe ratio? MINT (its five-year Sharpe is 1.63; SPY’s is 0.96). Yes, MINT is less risky than SPY, but how can its “risk-adjusted returns” be higher when its actual returns are so piddly? (The same goes for the Sortino ratio, a variation of the Sharpe ratio—MINT’s Sortino is about double that of SPY’s.)
There are similar problems with all of the other measurements of risk. I’ll discuss alpha and beta more fully later—I think they are truly valuable measurements, but they have to be used in the right way. Alpha is essentially the expected return on an investment when the market as a whole is earning 0%, so SPY has an alpha of 0% while MINT has an alpha of 0.77%, making it a (marginally) better risk-adjusted investment, according to the usual theories.
There are, of course, other ways to measure risk. Value at risk gives you interesting numbers along the lines of “you have a 5% chance of losing 25% of your investment in one year.” Once again, these figures will show that MINT is far less risky than SPY. Then there’s the gain/loss ratio, the ratio of positive monthly returns to negative monthly returns multiplied by the ratio of the average gain during the positive months to the average loss during the negative months. Once again, MINT beats SPY, 3.83 to 1.94. There’s even an “ulcer index” based on the square root of the sum of the squares of the drawdowns. Once again, MINT beats SPY handily.
None of this makes sense to me. I can’t imagine any justification for putting your money into MINT instead of SPY unless you believe that future returns are going to be the inverse of past ones.
So where’s the flaw?
We’re looking at risk in terms of the day-to-day, week-to-week, or month-to-month prices of the securities in a portfolio. In fact, there are only two times when the price of a security matters: the day you buy it and the day you sell it. The times in between are irrelevant. All this discussion of variability, monthly gains and losses, and drawdowns assumes that your portfolio’s value varies from moment to moment depending on the values of the securities you own. But that’s purely an illusion, one made possible by your ability to follow the day-to-day (or minute-to-minute) price variations of the securities it’s composed of. In some neighborhoods, it’s riskier to put your money under your mattress (standard deviation: 0%) than to invest it in VIX (standard deviation: 64%). Yes, that’s a joke, but it makes a serious point: variability is a rather limited way of thinking about risk.
If the price of a security that you’re proud to own goes down, it’s a great time to buy more of that security. It’s not a “loss” or a “drawdown”—it’s an opportunity. In fact, the more variable the security’s price, the more opportunities there are to make money off of it by buying low and selling high. That’s one reason why the formulae I’m discussing are ill-suited for individual securities.
This brings us to how we define financial risk. If you ask someone who’s not in the investment world, he or she will tell you that it’s the probability of a permanent loss of capital. MINT is certainly less risky than SPY in the short term, but after investing in SPY for five years, the chances of losing that 80% that you’ve gained are pretty damn low no matter what you do. By making lots of money, you’ve eliminated lots of risk. In other words, the best way to ensure that any losses you incur are only temporary is to make as much money as quickly as possible. The higher the return, the less risk of permanent loss.
Now this definition is, admittedly, not the one that money managers use. For them, risk is the probability of not meeting investors’ expectations. But even using this metric, the Sharpe ratio and similar measurements have some other flaws:
- They put the risk factor in the denominator. As a result, non-equity securities can have comparably huge ratios because their variability is so low. This is not as much of a problem for equity-based funds.
- They’re all calculated on the basis of average monthly rather than compounded returns. Take, for example, a fund that gains 25% in value one month and loses 20% in value the next. Its average return is 2.5%; its actual return is 0%. This would be less of a problem if you used annual returns.
- They ignore the order of returns, so that a fund that gains 5% a month for six months and then loses 5% a month for six months has the same risk-adjusted return as a fund that alternates its 5% gains and losses, even though the second fund displays far less overall variability. Again, this would be less of a problem if you used annual returns.
Now there’s a third definition of financial risk, which nobody ever states outright, but is inherent when talking about drawdowns or the gain/loss ratio or an “ulcer index.” A drawdown is the difference between the high value of a portfolio and its subsequent low value. Its only significance is as a measure of a temporary loss of capital, or a bump in the road. This has become an implicit definition of risk. The obvious question, then, is: how is a loss that’s only temporary risky? It only becomes so if it has a chance of becoming permanent. I propose that we ignore this definition and focus on the other two. A purely temporary loss is unpleasant but not risky.
So, given all this, how can we get a formula for a risk-adjusted return that will show that SPY is clearly a better investment than MINT, despite its risk?
The simple answer is that this goal appears, at first, to be logically impossible. It’s what I call the paradox of risk-adjusted returns.
We’ve seen that financial risk can be defined in two very different ways. If we define it as the failure to meet expectations, the essential component of risk is unpredictability. Logically, then, no measure of risk will exhibit any persistence, since persistence is the opposite of unpredictability. Therefore, a measurement of ex-post risk will have no correlation with ex-ante risk. On the other hand, if we define risk as the permanent loss of capital, then there’s no logical way to adjust returns for risk, since positive returns are the sole guarantee against the permanent loss of capital, and negative returns are the sole guarantee of the permanent loss of capital. The concept of risk-adjusted returns is a chimera.
Now this is not a result that will make anyone happy. Do we just rely on historical returns without adjusting them for the risk involved? Are all funds that have delivered the same mean annual excess return equally risky?
Although I now realize I was engaged in a logically fruitless quest, I spent months trying to solve this conundrum, doing experiment after experiment. I came up with an elaborate formula for something I called “noisefree returns,” but like the other formulas I investigated, it had low persistence. There seemed to me to be no way to create a risk-adjusted performance measure that was more predictive than simple mean annual returns, which really isn’t terribly predictive.
So I propose we start solving this paradox by going back to origins. After all, the people who came up with all these risk-adjusted return measures were not idiots. William Sharpe, for instance, won the Nobel Prize. So let’s take another look at the 1966 paper in which he invented the Sharpe ratio (he didn’t call it that), “Mutual Fund Performance.” I’m not going to provide a summary of the paper, which is pretty wide-ranging. But I want to point out a few extremely important elements.
First, Sharpe defines risk as money managers still do: basically, risk is the standard deviation of returns. In an earlier paper, “Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk,” Sharpe justifies this definition as follows: “Assume that an individual views the outcome of any investment in probabilistic terms; that is, he thinks of the possible results in terms of some probability distribution.” Sharpe focuses on two aspects of that probability distribution: expected return and standard deviation from that return. In other words, risk is defined as the component of return that deviates from the expected return, or the probability that you won’t get what you expect. (To be snarky, one could say that Sharpe therefore defines risk as the probability of the improbable.)
Second, Sharpe’s paper analyzes annual mutual fund returns, not monthly fixed-income returns. This obviates all three of the objections I raised to the use of the Sharpe ratio. Only fixed-income securities give devastatingly low variability for the denominator, and annual returns aren’t as susceptible to the other two objections as monthly returns.
Third, Sharpe’s study found a strong and linear correlation between returns and risk. In other words, the greater the risk, the greater the return.
Fourth, Sharpe ranked mutual funds on the basis of his ratio during a recent decade, and then ranked the same funds on the same basis during the previous decade. He found a low correlation between the two ranks. He then ranked the same funds according to what he called the Treynor index (mean annual return divided by beta), a ratio he named after the man who formulated financial beta, and the correlation was significantly higher. In other words, Sharpe himself admitted that his ratio was a poor predictor—that funds with a high Sharpe ratio were not much more likely to continue to have high Sharpe ratios than similar funds with a low Sharpe ratio. And every subsequent study since then has borne this out.
Now the Treynor index is a quite different risk-adjusted return measure; Sharpe invented it in this paper, but didn’t say much about its usefulness, perhaps because it has none. Basically, it’s just like the Sharpe ratio except with volatility (beta) rather than variability (standard deviation) in the denominator. But after noting that it has stronger predictive value than the Sharpe ratio, Sharpe did not ask why this was the case.
The answer is that variability, in essence, varies from one period to another much more than volatility, which tends to be consistent across long periods. In other words, it actually has some predictive power. (I’m using the words variability and volatility in the sense that they were used in the 1960s by researchers like Sharpe and Treynor, not the way they’re used today, which is pretty interchangeably. Variability is variation in returns; volatility is the variation in a fund’s returns that is correlated with the variation in the market’s returns.)
Now I’m not trying to claim that variability, or standard deviation, and unpredictability are exactly the same, or that standard deviation has absolutely no persistence. Obviously, a three-stock portfolio will have a much greater standard deviation than a fifty-stock portfolio (and its returns will be much harder to predict), and that difference will persist, making it predictable. Standard deviation is far from a useless measure. But Sharpe was looking at mutual funds with similar numbers of holdings, not imaginary three-stock portfolios. When compared to similar funds or portfolios, variability is not terribly consistent over different time periods; on the other hand, volatility is.
So this leads us to ask what, precisely, is volatility (as distinct from variability)—in other words, what is beta (and, of course, alpha)?
The best way to think of this is to take the returns of a fund and the returns of a benchmark over the same period. Now create a scatter-point graph mapping the two returns, the benchmark on the x axis and the fund’s return on the y. You’ll get something that looks like this (I’ve used the annual returns, measured monthly for twenty years, of PCVAX [AllianzGI NFJ Small-Cap Value Fund A], adjusted for dividends; the benchmark is the annual returns of SPY, also adjusted for dividends):
You’ll see I’ve drawn a linear regression line through the points. The slope of the line is the fund’s beta (in this case 0.5127); the y-intercept is the fund’s alpha (in this case 6.23%); the y point on the line where x (the benchmark) is average is the mean annual return. (In this case, the mean annual benchmark return is 8.24% and the mean annual fund return is 10.46%. And 10.46% equals alpha [6.23%] + 8.24% times beta [0.5127].)
In other words, beta measures how much the fund’s returns vary from the benchmark’s, and a higher beta corresponds to a steeper slope; alpha is the fund’s return when the benchmark is 0%. (I’m going back here to Jack Treynor’s original concept of alpha and beta; the way that, say, Morningstar measures these is a bit more complicated, and I’ll get to that later. In addition, Treynor and subsequent analysts subtract the risk-free rate from both the portfolio returns and from the benchmark; since in the past few years the risk-free rate has been very close to zero—or negative, if one takes inflation into account—I’ve refrained from doing this for simplicity’s sake.)
It’s important to realize that the slope of a linear regression will have a “pivot point” at the average benchmark return. In other words, if you have two strategies with similar returns when the benchmark is above average, the one with the worse return when the benchmark is below average will have a higher slope. Higher slope does not mean higher performance—it means higher performance above the pivot point and worse performance below.
It turns out that the most predictable thing about mutual fund returns—or for that matter strategy returns—is beta. I did an experiment comparing returns of thirty different broad-based equity mutual funds over two different seven-year time periods (2001–2007 and 2010–2016), and beta was by far the single most correlative factor. The correlation of beta was 0.65, while that of CAGR, Sharpe ratio, Treynor index, alpha, and the information ratio were all between –0.16 to 0.12 (in other words, there was no significant correlation). I then repeated the experiment using thirty different strategies as backtested on Portfolio 123. These strategies all consisted of holding the top 50 stocks out of the Russell 3000 based on ranking according to five randomly selected factors for a certain number of weeks, then rebalancing. (For example, one strategy consisted of the top fifty stocks ranked on the basis of their free cash flow growth, operating income growth, asset turnover, debt-to-EBITDA ratio, and return on capital, with a yearly rebalance; another ranked on the basis of their asset-to-equity ratio, balance-sheet accruals, Piotrowski F-score, net profit margin, and forward earnings yield, rebalanced every four weeks.) The results were similar: the correlation of beta was 0.65, once again; there was no significant correlation of alpha, Treynor index, or information ratio; and there was a low correlation of CAGR (0.24) and Sharpe ratio (0.28).
These results confirm earlier studies by William Sharpe and his colleagues. If the beta of a fund/strategy is high, it tends to remain high; if beta is low, it tends to remain low. And, precisely as one would expect, during bear markets, low-beta strategies significantly outperform, and during bull markets, high-beta strategies significantly outperform. (Once again, I must emphasize that this applies to general stock-market funds and investing strategies but not to individual securities or sector-specific funds and strategies; it also applies only if there is a decent correlation between the return and the benchmark.) Of course, there is no crystal ball to tell you if a bear or bull market is in the cards.
Now William Sharpe won the Nobel Prize not for his ratio, but for his work on beta. Sharpe, however, made some fundamental errors after he wrote this paper. He applied the result of his study of mutual funds to individual securities, ignored the pivot-point principle, and concluded that investing in stocks with high beta would get greater returns than stocks with low beta since the market in general trends upwards and has an alpha of zero percent. When, in 1992, Eugene Fama and Kenneth French challenged Sharpe’s results and found no link between relative volatility (beta) and long-term returns, Fama told the New York Times, “beta as the sole variable explaining returns on stocks is dead.” Sharpe responded, “I am not willing to make investment decisions based on the theory that there is no relationship between beta, properly measured, and expected returns.” Sharpe’s faith that high beta produced high returns was unshakeable. But there is an enormous difference between the past returns of a strategy for choosing stocks (high price-to-sales ratio and sustainable growth, for instance) and the past returns of the stocks themselves. The former have a built-in tendency to persist; the latter have a built-in tendency to be cyclical. Both Sharpe and Fama were wrong—Sharpe was wrong to apply the theory of beta to individual securities; Fama was wrong in wanting to throw out the theory altogether.
In 1973, Sharpe had summarized his theory of beta as follows: “In equilibrium the expected excess return on a security over and above the pure interest rate will equal some constant times its ex ante risk, measured by the security’s so-called ‘beta coefficient.’ . . . The expected excess return of any security or portfolio over and above the pure interest rate, will equal the expected excess return of the market portfolio, times the security’s beta relative to the market.” This has been called the CAPM equation in financial literature.
This is a rather drastic oversimplification of the concept of slope. It’s true only when alpha equals zero and only if you change the word security to portfolio, strategy, or fund. (The slope of a linear regression is only a simple multiplier when the intercept is zero, which is rarely the case.) In addition, the implication here is that a higher beta increases excess returns in general, rather than only during a bull market.
I have come to the conclusion that of all the notions about risk floating around, the ideas of alpha and beta make the most sense. Go back to Jack Treynor’s original conception: if you regress the returns of an equity investment strategy or fund against a suitable benchmark, beta is the slope and alpha is the intercept. Given those two numbers, one can quickly arrive at a range of expectations for the future return of the strategy, expectations that will have a high degree of correlation with the strategy’s future returns.
Now the conventional measures of alpha and beta are based on monthly rather than annual returns. Sharpe and Treynor might have preferred that measure themselves, if only monthly returns had been less hard to come by in the mid-1960s.
It is conventional nowadays to annualize alpha (measured monthly) while leaving beta (which, if measured monthly, can’t be annualized) alone. So if you look up the alpha and beta of a fund on any website, you’ll get two numbers that have only a tenuous relationship to each other.
To arrive at annual alpha and beta, take a list of the past returns of the fund or the backtested returns of the strategy along with the returns of the benchmark over the same period. For every price, calculate the increase since the same date last year as a percentage. If you’re using Excel, you’ll now have four columns: one with the price of the fund, one with the price of the benchmark, one with the annual growth of the fund at every point, and one with the annual growth of the benchmark. Now perform a linear regression on the latter two columns by using the “slope” and the “intercept” commands, with the benchmark return as the x values and the fund’s return as the y. The slope will be beta and the intercept alpha. To calculate monthly alpha and beta, do the same but calculating the increase since the same date last month (or 21 trading days ago).
If you prefer to use monthly alpha and beta rather than annual, be sure to de-annualize any published monthly alpha by adding one, taking the twelfth root, and subtracting one. Make sure that you then use monthly benchmark returns (substantially lower than annual) when considering future results.
The practical way to apply alpha and beta is to come up with a range of possible returns depending on future benchmark returns. Let’s call the future benchmark return x. Then y, the expected return of your fund or strategy, will be α + βx. So if you have a strategy like the one I illustrated in the above scatter plot, with a 6% alpha and a beta of 0.5, then you’d expect an annual return of 11% when the benchmark returns 10% and a return of 1% when the benchmark returns –10%. It’s that simple.
Given two funds with the same alpha and different betas, you’d want to invest in the high-beta fund in a bull market and a low-beta fund in a bear market. Now, just as alpha is uncorrelated with the market, beta is uncorrelated with performance. If, like most investors, you have no idea whether a bear or a bull market is in the cards, a high-alpha fund is always a better investment than a low-alpha fund, as long as you have some measure of confidence that the alpha will persist.
Now, since beta, unlike alpha, is persistent, why disregard it when faced with two funds with very different alphas? Why not base your investing choices on α + βx, where x is the mean benchmark return? And, since this is exactly equal to the mean portfolio return, aren’t we back at where we started—at unadjusted returns?
By taking beta out of the equation and focusing only on alpha, we’re taking the most predictable part of the return equation out and focusing on the least predictable part of the return. In other words, we’re removing a constant to arrive at a variable. Alpha, as Michael Jensen argued when he came up with the idea in 1968, is the part of the return attributable to the skill of the fund manager (or of the strategy designer). Yes, alpha has no more persistence than total returns or the Sharpe ratio, but it’s a better measure of past performance because it eliminates the vagaries of the market.
You can now try to find funds with high alphas and high betas for bull markets and high alphas and low betas for bear markets. Just keep in mind that beta is a lot more predictable than alpha, and adjust your expectations accordingly.
I believe that considering annual alpha and beta when investing lowers risk because the calculation addresses both of the definitions of risk we’ve considered. First, it addresses the potential loss of capital since it considers what will happen to the strategy during a bear market; it also addresses drawdowns through beta: the higher the beta, the more severe the drawdown will be. Second, it addresses predictability—because of the persistence of beta, it is the most predictable of all of the measures of returns we’ve considered.
Let’s now go back to MINT vs. SPY. If you look at just the last five years, MINT has an annual alpha of 0.77% and a beta of 0.011. So now you know that almost no matter what the market does, you should expect a return of between 0.5% and 1.5% per annum. SPY, on the other hand, has an alpha of 0% and a beta of 1. Which means that its return will match the market exactly. Considering that the US equities market has a mean annual average return of about 10.5% (including dividends) since 1871 (this is higher than its CAGR, since mean returns are usually higher than compounded returns), that that hasn’t changed significantly in recent years, and that given the current supply of investable securities it’s unlikely to change considerably in the future, I think you can tell which fund gives a better risk-adjusted return.