Risk-Adjusted Return

A risk-adjusted return is a measure of investment performance that takes into account how much risk was taken to achieve a return. Two funds that both gained 10% in a year are not equally impressive if one did so with steady, modest swings and the other with wild ups and downs or heavy borrowing. Risk-adjusted measures put such results on a common footing, and they are used by fund investors, pension trustees, portfolio managers, banks and regulators.

The most widely used measures are ratios that divide some form of excess return by some measure of risk. They differ in which risk they count: total variability, market sensitivity, downside losses or capital at risk. All of them build on the risk–return trade-off, and several come directly from the Capital Asset Pricing Model (CAPM).

Think of two delivery drivers who each finish their route in the same time. One drove at a sensible speed; the other ran red lights and nearly crashed twice. Their results look the same on paper, but nobody would call them equally good drivers, because one took far more danger to get there.

Investing is similar. A raw return tells you how fast the money grew, not how hair-raising the trip was. A risk-adjusted return asks how much reward was earned for each unit of risk taken, so that a careful steady performer and a lucky daredevil can be told apart.

The best-known is the Sharpe ratio, introduced by William Sharpe in 1966 as the "reward-to-variability" ratio:

• $R_p$ is the portfolio's return over the period.
• $R_f$ is the risk-free rate over the same period.
• $\sigma_p$ is the standard deviation of the portfolio's returns, its total volatility.

The Treynor ratio, published by Jack Treynor in 1965, replaces total volatility with beta, counting only systematic risk:

Jensen's alpha, proposed by Michael Jensen in 1968, measures the return above what CAPM predicts for the fund's beta:

Here $\beta_p$ is the portfolio's beta and $R_m$ is the market return. Worked example: a fund returned 10% with a standard deviation of 15% and a beta of 1.2, while the risk-free rate was 4% and the market returned 9%. Its Sharpe ratio is $(10 - 4) \div 15 = 0.40$; its Treynor ratio is $6 \div 1.2 = 5.0$; and its alpha is $10 - [4 + 1.2 \times 5] = 0$. The fund earned exactly what its market exposure would predict, so it added no alpha despite a respectable raw return.

• Sortino ratio. Like the Sharpe ratio but uses only downside deviation, on the view that investors do not mind upside surprises.
• Information ratio. Excess return over a benchmark divided by tracking error, the volatility of that difference; widely used to judge active managers.
• M-squared. Proposed by Franco and Leah Modigliani in 1997, it rescales a portfolio to the market's volatility so the result is a percentage return that is easy to compare.
• Calmar and drawdown ratios. Return divided by the largest peak-to-trough fall, or drawdown, popular among hedge fund and managed-futures investors.
• RAROC. Risk-adjusted return on capital, developed at Bankers Trust in the late 1970s, divides a business line's risk-adjusted profit by the economic capital needed to support it. Banks use it to price loans and allocate bank capital, often with capital estimated from value at risk or similar models.

Which measure fits depends on context. For an investor's whole portfolio, total volatility matters, so the Sharpe ratio is appropriate. For one fund held among many, much of its own volatility is removed by diversification, so beta-based measures such as Treynor and Jensen's alpha are more relevant. Comparisons should use the same period, the same frequency of data and annualized figures, and should start from total return including dividends and fees. Most mutual fund fact sheets and research services publish Sharpe ratios and standard deviations alongside raw returns.

• Volatility is not all of risk. Standard deviation treats upside and downside alike and misses the chance of rare large losses in assets with fat tails or option-like payoffs.
• Smoothed prices. Assets valued by appraisal, such as private equity and property, show artificially low volatility, inflating their ratios.
• Short records. Ratios estimated from a few years of data have wide error margins, and strategies that sell insurance against crashes can show excellent ratios until the crash arrives.
• Benchmark choice. Alpha and information ratios depend heavily on the benchmark, which managers may choose to flatter results.

• William F. Sharpe, Stanford University: The Sharpe Ratio