Sharpe, Sortino and information ratio: return per unit of risk

The Sharpe ratio is the most common way to judge a fund or a trading strategy. The Information ratio is how a manager hired to beat an index gets judged, and the Sortino ratio answers a fair complaint about Sharpe. This post covers all three.

All three build directly on mean and standard deviation.

Why you need a ratio at all

Over 2020 to 2025, GS returned 28.7% a year and GLD (gold) 17.5%. Is GS better? Not necessarily. GS swung with a 33.0% yearly standard deviation, gold with 16.3%, so gold gave you more return per unit of risk (0.95 against 0.81, as Table 6 shows). Every ratio here has the same shape:

the ratioreturnsrisk

$$\key{\text{ratio}} = \frac{\good{\text{extra return}}}{\risk{\text{risk}}}$$

They differ only in what “extra” and “risk” mean.

Sharpe ratio

Plain meaning: how much return you got above a risk-free investment, for each unit of total volatility. This is the form in Romero and Balch’s What Hedge Funds Really Do (chapter 2), with the symbols in Table 1:

$$\key{\text{Sharpe}} = \frac{\good{r_p} - r_f}{\risk{\sigma_p}}$$

The book’s own example: the S&P 500 at 10% a year, bills at 2.5%, volatility 15% gives $(10 - 2.5)/15 = 0.5$.

With daily data, you annualise by multiplying by $\sqrt{252}$:

$$\key{\text{Sharpe}_{\text{annual}}} = \sqrt{252} \times \frac{\text{mean}(\good{r_t} - r_{f,t})}{\risk{\text{std}}(\good{r_t})}$$

where $\good{r_t}$ are the daily returns. (When the risk-free rate is the same every day, the standard deviation of $\good{r_t}$ and of $\good{r_t} - r_{f,t}$ are identical, so you’ll see both versions.)

Why $\sqrt{252}$ and not 252? The top of the fraction (a mean) grows with time, so $\times 252$. The bottom (a standard deviation) grows with the square root of time, so multiply by $\sqrt{252}$. Divide one by the other and you’re left with $252/\sqrt{252} = \sqrt{252}$. (The square-root rule is explained in the standard deviation post.) For monthly data, use $\sqrt{12}$. For weekly, $\sqrt{52}$.

Worked example

SPY’s first five daily returns of 2020, from the mean and standard deviation post: −0.8%, 0.4%, −0.3%, 0.5%, 0.7%. Risk-free rate 0, to keep it simple. The mean is 0.1% and the standard deviation 0.628%.

$$\key{\text{daily Sharpe}} = \frac{0.1}{0.628}$$ $$\key{\text{daily Sharpe}} = 0.159$$ $$\key{\text{annual Sharpe}} = \sqrt{252} \times 0.159$$ $$\key{\text{annual Sharpe}} = 2.53$$

Romero and Balch call a Sharpe of 1.0 “very loosely” fair pay for the risk taken, and note that ratios at or above 1.0 are very uncommon. A Sharpe of 2.53 is what you get from annualising one week: five days is far too short to trust any of these numbers. Over the full 2020 to 2025 run SPY’s Sharpe is 0.69.

The risk-free rate per day

If the risk-free rate is 4% a year, the daily rate is the 252nd root of the yearly growth:

$$r_f^{\text{daily}} = (1 + 0.04)^{1/252} - 1 = 0.015565\%$$

The shortcut $4\% / 252 = 0.015873\%$ is close enough for most uses. Many people simply use 0 when rates are very low.

What Sharpe gets wrong

The standard deviation punishes all swings, up and down. A fund with a few huge gains gets penalised exactly like a fund with a few huge losses. Investors don’t mind upside surprises.

Sortino ratio

Plain meaning: like Sharpe, but only the bad days count as risk (Table 2 names the symbols).

$$\key{\text{Sortino}} = \frac{E[\good{R_p} - T]}{\risk{\text{DD}}}$$ $$\risk{\text{DD}} = \sqrt{\frac{1}{n}\sum_{t=1}^{n} \min(\good{r_t} - T,\ 0)^2}$$

Good days count as zero shortfall, so they don’t add to the risk. Note that the average is over all $n$ days, good and bad. That’s the common convention; some references divide by the number of bad days only, which gives a different number, so always check which one a report uses. Annualise with $\sqrt{252}$, as before.

Worked example

Same five SPY days, target 0 (Table 3):

$$\risk{\text{DD}} = \sqrt{0.146}$$ $$\risk{\text{DD}} = 0.382$$ $$\key{\text{daily Sortino}} = \frac{0.1}{0.382}$$ $$\key{\text{daily Sortino}} = 0.262$$ $$\key{\text{annual}} = \sqrt{252} \times 0.262$$ $$\key{\text{annual}} = 4.15$$

Higher than the Sharpe of 2.53, because the three up days no longer count against the fund; only the two bad days do.

Where Sortino earns its keep

I made up three funds with exactly the same mean and standard deviation, so their Sharpe ratios are identical. Only the shape differs (Figure 1 shows the two lopsided ones):

(Risk-free rate 4% a year.) Sharpe can’t tell the three in Table 4 apart. Sortino ranks the “big losses” fund clearly worst, which is what an investor would say too. Strategies that sell insurance-like options often look like that last fund: steady small gains, then a rare large loss. Their Sharpe can look excellent right up until the bad day.

Information ratio

Plain meaning: how much a manager beat their benchmark, for each unit of straying from it.

A fund manager is usually hired to beat an index, not cash. So “extra return” becomes the return above the index (the active return), and “risk” becomes how much the fund’s return differs from the index’s (the tracking error). This is the version in Grinold and Kahn’s Active Portfolio Management:

$$\key{\text{IR}} = \frac{E[\good{R_p} - R_b]}{\risk{\sigma}(\good{R_p} - R_b)}$$ $$\key{\text{IR}} = \frac{\good{\text{active return}}}{\risk{\text{tracking error}}}$$

where $\good{R_p}$ is the fund’s return and $R_b$ the benchmark’s.

Romero and Balch (chapter 9) measure skill the same way, but on alpha: the part of the return left after removing what the market explains, using the CAPM (the capital asset pricing model):

$$\key{\text{IR}}(\good{\alpha}) = \frac{\text{mean}(\good{\alpha})}{\risk{\text{s.d.}}(\good{\alpha})}$$

For a fund that moves one-for-one with its index (a beta of 1), alpha is simply fund minus index, and the two versions agree.

Worked example

Treat NVDA as the “fund” and SPY as its benchmark, over the same five days (Table 5):

Active returns: mean 0.16%, standard deviation 0.868%.

$$\key{\text{daily IR}} = \frac{0.16}{0.868}$$ $$\key{\text{daily IR}} = 0.184$$ $$\key{\text{annual IR}} = \sqrt{252} \times 0.184$$ $$\key{\text{annual IR}} = 2.93$$

One good day against the index (day 3), and the week’s IR is 2.93, twice what the full record gives. Over 2020 to 2025 NVDA beat SPY by 58.5% a year with a tracking error of 41.6%, an IR of $\key{1.41}$: the same stock, a different number, and only the longer horizon means something.

2020 to 2025, five tickers

Table 6 has all three ratios on the 2020 to 2025 daily data, risk-free 2% a year, SPY as the benchmark:

Read across a row: Sortino is above Sharpe in every row (the up days stop counting), and the ranking does not change, because these daily returns are fairly symmetric. Read down the IR column: XOM and GLD beat the index a little but wandered far from it, so their IR is near zero; GS earns 0.55; only NVDA’s extra return was large next to its wandering.

Two managers

In Table 7, B earned three times more extra return, but took six times more risk relative to the index. Per unit of risk, A is twice as good. If you like A’s skill but want more return, you can give A more money to work with, or let A take bigger positions. B’s extra return came with a lot of noise.

The IR leads to the Fundamental Law of Active Portfolio Management, from Grinold and Kahn, as Romero and Balch write it in chapter 9:

$$\key{\text{IR}} = \text{IC} \times \sqrt{\text{BR}}$$

In plain words: a manager’s IR comes from how good each bet is (the information coefficient, IC, the correlation between forecasts and what actually happened) times the square root of how many investment decisions they make each year (the breadth, BR). Small skill applied many times can beat big skill applied rarely. That’s a post of its own.

Drawdown: the fall from the peak

None of the three ratios says how bad the worst stretch was. Drawdown does: it is the fall from the highest value so far to a later low, measured from the peak, not from the start. The maximum drawdown is the deepest such fall in the record, and it has a second dimension, duration: the longest time spent below a previous peak before getting back to it.

Two things Table 8 says that the Sharpe column cannot. XOM, with the lowest Sharpe in the table, fell 55.0%, which needs a 122% gain to undo and took almost a year and a half; NVDA, with the best Sharpe in the table at 1.36, fell further still, 66.3% between November 2021 and October 2022. And gold, with a mild 22% drawdown, spent 897 trading days, more than three years, below its 2020 peak: depth and duration are different things and often come from different episodes.

A historical maximum drawdown, like SPY’s 33.7% in Figure 2, is a sample, not a ceiling: the next one can be deeper. Excluding a bad period from the record understates it. And Sharpe is unchanged by leverage (holding more than you own, with borrowed money), because doubling every position doubles the excess return and the volatility together, as long as the borrowed money costs the risk-free rate. Drawdown is not: doubling the positions makes the worst fall about twice as deep. That is why a fund report shows both.

The code

All three ratios and the maximum drawdown for the five tickers. It reads the blog’s price file from the examples repo, so it runs as pasted (pip install numpy pandas). r holds the daily returns, one column per ticker; rf the risk-free rate as a daily number. Each function is one formula from above.

import numpy as np
import pandas as pd

K = np.sqrt(252)                                 # daily to yearly, for every ratio

def sharpe(r, rf=0.0):
    ex = r - rf                                  # excess return over the risk-free rate
    return K * ex.mean() / ex.std(ddof=1)

def sortino(r, target=0.0):
    downside = np.sqrt(np.mean(np.minimum(r - target, 0) ** 2))   # only shortfalls count
    return K * (r - target).mean() / downside

def info_ratio(r, benchmark):
    active = r - benchmark                       # how much it beat the benchmark each day
    return K * active.mean() / active.std(ddof=1)

def max_drawdown(r):
    value = (1 + r).cumprod()                    # what $1 grew to
    return (value / value.cummax() - 1).min()    # the deepest fall below a previous peak

URL = "https://raw.githubusercontent.com/ankurtrapasiya/examples/main/data/prices.csv"
prices = pd.read_csv(URL, index_col="date", parse_dates=True)
r = (prices / prices.shift(1) - 1).dropna()     # daily returns, one column per ticker
rf = (1 + 0.02) ** (1 / 252) - 1                 # 2% a year as a daily rate

for t in r.columns:
    print(f"{t:5} Sharpe {sharpe(r[t], rf):.2f}  Sortino {sortino(r[t], rf):.2f}  "
          f"IR vs SPY {info_ratio(r[t], r['SPY']) if t != 'SPY' else float('nan'):5.2f}  "
          f"max drawdown {max_drawdown(r[t]):.1%}")
dd = max_drawdown(r["XOM"])
print(f"XOM needs {1 / (1 + dd) - 1:.0%} to recover")

It prints one row per ticker, the Sharpe, Sortino, IR and maximum drawdown columns of Tables 6 and 8, then XOM’s 122%. np.minimum(r - target, 0) keeps each day’s shortfall and turns every good day into 0; value.cummax() is the highest value so far on each day.

The scripts behind all the numbers

The tables and the made-up skewed funds come from examples/risk-adjusted-ratios/ratios.py and examples/risk-adjusted-ratios/drawdown.py. To run either, save it next to the shared helpers market_data.py and chartstyle.py and the price file data/prices.csv, keeping the folders as in the links, then run it with uv run.

Summary

Table 9 puts the three side by side.

All three: annualise daily figures with $\sqrt{252}$, and never trust them on a short history.

References

• Philip J. Romero and Tucker Balch, What Hedge Funds Really Do: An Introduction to Portfolio Management (Business Expert Press, 2014), chapter 2 (the Sharpe ratio, with the S&P 500 example of 0.5 and the note that ratios at or above 1.0 are very uncommon) and chapter 9 (the information ratio on alpha, and the Fundamental Law, “performance = skill $\times \sqrt{\text{breadth}}$”, after Grinold and Kahn)
• Richard C. Grinold and Ronald N. Kahn, Active Portfolio Management (2nd ed., McGraw-Hill, 2000)
• William F. Sharpe, “The Sharpe Ratio”, The Journal of Portfolio Management 21(1), 1994
• Frank A. Sortino and Lee N. Price, “Performance Measurement in a Downside Risk Framework”, The Journal of Investing 3(3), 1994 (Romero and Balch name the Sortino ratio in chapter 2 but give no formula; the formula here is from this paper)