Sharpe, Sortino and drawdowns

Key takeaways

• The Sharpe ratio is excess return over the risk free rate divided by the volatility of returns. Annualise it by multiplying a monthly Sharpe by sqrt(12).
• The Sortino ratio replaces volatility with downside deviation, which counts only returns below a target, so it does not penalise upside volatility.
• The information ratio is active return over a benchmark divided by tracking error, and measures how consistently a manager adds value against that benchmark.
• Maximum drawdown is the largest peak to trough fall in cumulative value; the Calmar ratio divides annual return by it.
• Ratios from short track records carry large estimation error, and smoothed or option like return streams can produce high Sharpe ratios with hidden tail risk.

The Sharpe ratio

The Sharpe ratio, introduced by William Sharpe in 1966, divides a portfolio's excess return (its return above the risk free rate) by the standard deviation of its returns. The excess return is the reward for taking risk at all, since the risk free rate is available to anyone. Dividing by volatility expresses that reward per unit of total risk. A Sharpe of 0.5 means the portfolio earned half a percentage point of excess return for each percentage point of annual volatility.

The ratio is computed at the data's frequency and then annualised. Because the numerator scales with time and the denominator with the square root of time, an annual Sharpe equals the periodic Sharpe multiplied by the square root of the number of periods per year: sqrt(12) for monthly data and sqrt(252) for daily. That annualisation inherits every assumption of the square root rule, so it is only as reliable as the independence of the returns.

As rough reference points, a broad equity index has historically delivered a long run Sharpe somewhere around 0.3 to 0.5, and a sustained Sharpe above 1.0 after fees is rare for a single manager. One property matters for allocators: the Sharpe ratio is unaffected by leverage at the risk free rate. Doubling both the excess return and the volatility leaves it unchanged, which is why it is the natural measure for comparing strategies that can be scaled up or down.

The Sortino ratio and downside deviation

The Sharpe ratio treats a surprise gain exactly like a surprise loss, because the standard deviation squares both. Investors do not. The Sortino ratio keeps the same numerator but replaces volatility with downside deviation: the root mean square of returns that fall short of a target, usually the risk free rate or zero. Months above the target contribute nothing to the risk measure.

Two conventions need checking whenever a Sortino figure is quoted. The first is the target, which should match the numerator: if the numerator is return above the risk free rate, the shortfalls should be measured against the risk free rate too. The second is the divisor. The standard definition divides the sum of squared shortfalls by the total number of observations, not by the number of shortfall months only. Dividing by the shortfall count alone inflates downside deviation for funds with few down months, and some tear sheets use whichever convention flatters.

For a fund with roughly symmetric returns, the Sortino ratio is typically somewhat above the Sharpe ratio. The gap widens for strategies with positively skewed returns, such as trend following, and narrows or reverses for negatively skewed ones. A Sortino that is far above the Sharpe deserves a look at the return distribution: it is either a genuinely right skewed strategy or a short series with few losing months.

Information ratio and tracking error

Many managers are hired to beat a benchmark rather than cash: a long only equity manager against an index, or an equity long/short fund against a blend. For them the relevant reward is active return, the portfolio return minus the benchmark return, and the relevant risk is tracking error, the standard deviation of active returns. The information ratio (IR) divides one by the other. It measures how consistently the manager adds value relative to the benchmark.

Tracking error describes how far the portfolio strays from the benchmark, not how risky it is in absolute terms. An index fund has a tracking error close to zero. A concentrated stock picker might run 6 to 10 percent. An information ratio of 0.5 sustained over many years is generally regarded as good for an active manager, and above 1.0 is exceptional. Because the IR depends on the benchmark chosen, the benchmark should be agreed before the track record is examined, not selected afterwards.

The illustrative benchmark below is built to move with Fund X but with smaller amplitude. Fund X beats it by more in good months than it trails it in bad ones, so the active returns are small, steady and positive on average. The result is an information ratio far above the Sharpe ratio, which is a useful pattern to recognise: a manager can be excellent at the job of beating a benchmark while the benchmark itself offers little reward for its risk.

Maximum drawdown and the Calmar ratio

Volatility is an abstraction. Drawdown is what an investor feels. A drawdown is the percentage fall in cumulative value from its previous peak; the maximum drawdown is the largest such fall over the record. It captures something volatility misses: the sequence of returns. Three moderate losing months in a row produce a deeper hole than the same three months scattered among gains, even though the volatility is identical.

To compute it, build a wealth index by compounding the returns, track the running maximum of that index, and at each date measure how far the index sits below its running maximum. The maximum drawdown is the most negative of those readings. The time from peak to recovery, sometimes called the drawdown duration or time under water, matters just as much to an allocator explaining a loss to an investment committee.

The Calmar ratio divides the annualised compound return by the absolute value of the maximum drawdown. It was designed for managed futures, where it is conventionally computed over the trailing 36 months. A close cousin, the MAR ratio, uses the whole track record. Both inherit a weakness of maximum drawdown: it is a single worst event, and it can only grow as the record lengthens, so a longer track record mechanically looks worse.

Reading the ratios together

Each ratio answers a narrow question, and the useful reading comes from comparing them. A Sortino well above the Sharpe says the volatility is mostly upside. An information ratio well above the Sharpe says the manager is good at beating a benchmark that itself pays little for its risk. A Calmar ratio that is low relative to the Sharpe says losses arrive in clusters, which volatility hides.

All of these ratios are estimates, and short records make them noisy. A common approximation for the standard error of an annual Sharpe ratio estimated from T years of data is roughly sqrt((1 + Sharpe^2 / 2) / T). With three years of data and a true Sharpe of 0.5, that is about 0.6, so the 95 percent confidence band runs from below zero to well above one. Twelve months, as in the example here, are enough to teach the arithmetic and nowhere near enough to judge a manager.

Finally, ratios that divide by volatility can be gamed by strategies whose risk does not show up in volatility. Selling out of the money options produces a steady stream of small gains, a low volatility and a high Sharpe ratio, until the event the options insured against arrives. Smoothed marks do the same for illiquid funds. A high ratio with negative skew, positive autocorrelation or very few losing months is a reason to dig, not to allocate.

import numpy as np
import pandas as pd

r = pd.Series([0.021, -0.014, 0.030, 0.008, -0.026, 0.019,
               0.024, -0.007, -0.012, -0.031, 0.028, 0.016])
b = pd.Series([0.015, -0.010, 0.022, 0.005, -0.020, 0.014,
               0.018, -0.003, -0.015, -0.025, 0.020, 0.011])
rf = 0.003  # monthly risk free rate

ex = r - rf
sharpe = ex.mean() / ex.std() * np.sqrt(12)

downside = np.sqrt((np.minimum(ex, 0) ** 2).mean())  # divides by all n
sortino = ex.mean() / downside * np.sqrt(12)

active = r - b
ir = active.mean() / active.std() * np.sqrt(12)

wealth = (1 + r).cumprod()
mdd = (wealth / wealth.cummax() - 1).min()
calmar = (wealth.iloc[-1] - 1) / -mdd

print(f"Sharpe        {sharpe:.2f}")
print(f"Sortino       {sortino:.2f}")
print(f"Info ratio    {ir:.2f}")
print(f"Max drawdown  {mdd:.2%}")
print(f"Calmar        {calmar:.2f}")

Sharpe        0.27
Sortino       0.39
Info ratio    1.28
Max drawdown  -4.93%
Calmar        1.11

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