Measuring Performance: Sharpe, Sortino, Alpha and Factor Regression

“Our fund returned 12% last year.” Is that good? The honest answer is: you can’t tell from that. How much risk was taken? What did the market do over the same period? Did investors actually get those 12%? And was it skill or luck? This lesson shows the tools professionals use to answer such questions.

Whose return? Time-weighted and money-weighted#

Fund companies publish the time-weighted return. It chains the returns of individual periods and ignores how much money was in the fund at which time. That way it measures the performance of the manager, who has no control over inflows and outflows.

What matters to you as an investor, however, is what happened to your money. That’s measured by the money-weighted return (internal rate of return): it takes into account that more money was invested in good or in bad phases.

This isn’t a thought experiment. Morningstar studies regularly find that fund investors earn one to two percentage points a year less than the funds themselves report because of poor timing – often much more in volatile thematic funds. They buy after strong runs and sell after losses.

Return per unit of risk: Sharpe, Sortino and Calmar#

From portfolio theory you know the Sharpe ratio: excess return over the risk-free rate divided by volatility. A fund with an 8% return and 15% volatility at a 2% rate has a Sharpe ratio of (8 − 2) ÷ 15 = 0.4. Broad equity markets have often been in the 0.3 to 0.5 range over the long run.

The Sharpe ratio has two weaknesses. It treats upside volatility the same as downside volatility. And it assumes returns are roughly normally distributed. Strategies that collect small gains for a long time and rarely lose big – such as selling options – therefore look better on Sharpe than they are.

• The Sortino ratio divides excess return only by downside deviation – the standard deviation of returns below a target. If our fund has a downside deviation of 8%, its Sortino ratio is (8 − 2) ÷ 8 = 0.75.
• The Calmar ratio divides the annual return by the largest drawdown. It answers the question that keeps investors awake at night: how much did I have to endure along the way to get this return?

Against the index: tracking error and information ratio#

An active equity fund shouldn’t be measured against the risk-free rate but against its benchmark index – otherwise it would be praised just because stocks rise over time. Two measures are central:

• Active return: fund return minus index return.
• Tracking error: the volatility of that difference. It shows how far the manager strays from the index. An index fund has a tracking error near zero, a concentrated stock picker often 5 to 10%.

A fund that beats its index by 1% a year with 4% tracking error has an information ratio of 0.25. Values above 0.5 are considered very good and are rare over long periods. Incidentally, an ETF’s tracking difference is nothing other than the (usually negative) active return of an index fund.

Alpha by regression#

Where did a fund’s excess return come from? To find out, its monthly returns are compared with the market’s by regression. The result is an equation:

Beta measures how strongly the fund swings with the market; alpha is what remains. A fund with a 10% excess return over the rate, a beta of 1.1 and a market premium of 6% has an alpha of 10 − 1.1 × 6 = 3.4% a year. That looks like skill.

Factors in disguise#

Since the work of Eugene Fama and Kenneth French in the 1990s, factors are added: value (cheap stocks), size (small companies), and later momentum, profitability and investment. The regression then gets more terms:

The coefficient of determination R² shows how much of the fund’s movement the factors explain. If it is 0.98, the fund is essentially an expensive index fund (a “closet indexer”) – however active it calls itself.

Skill or luck?#

Even a genuinely positive alpha could be chance. Statistics tell you how confident you can be: the so-called t-statistic of alpha roughly equals the information ratio times the square root of the number of years. To speak of skill with about 95% confidence you need a value of around 2.

With a very good information ratio of 0.5 that’s 16 years. With a realistic 0.25 it’s 64 years – longer than most managers’ careers. On top of that there is survivorship bias: bad funds are closed or merged and vanish from the statistics.

Summary#

Time-weighted returns measure the fund, money-weighted returns the investor. Sharpe, Sortino and Calmar relate return to risk; tracking error and information ratio to deviation from the index. Regression separates alpha from market and factor risk – often revealing that little alpha remains. And even then it takes decades to tell skill from luck. That’s the sober reason why broad, low-cost index funds are the best choice for most investors.