Returns, volatility and annualisation

Key takeaways

• A simple return is the percentage change in value over a period; a log return is the natural log of one plus the simple return. Simple returns add across assets, log returns add across time.
• The arithmetic mean of returns is always at least the geometric mean, and the gap (volatility drag) is roughly half the variance. Investors earn the geometric mean.
• Volatility is the standard deviation of returns. Practitioners use the sample version, dividing by n minus 1.
• Annualise a mean by multiplying by the number of periods per year and a volatility by the square root of that number. The square root rule assumes returns are independent from one period to the next.
• Smoothed or autocorrelated returns, common in private and illiquid funds, make the square root rule understate true annual volatility.
• Real return distributions have fatter tails than the normal distribution, so a move a normal model calls once in decades happens far more often.

Simple returns and log returns

The simple return (also called the arithmetic or discrete return) over one period is the change in value plus any income received, divided by the starting value. A share bought at $100 that pays a $2 dividend and ends the period at $105 has a simple return of 7 percent. Simple returns are what investors experience and what fund reports quote. They also have one property that makes them indispensable for portfolios: the return of a portfolio over a period is exactly the weighted average of the simple returns of its holdings.

The log return (also called the continuously compounded return) is the natural logarithm of one plus the simple return. For small moves the two are almost identical: a 1 percent simple return is a 0.995 percent log return. For large moves they diverge, and log returns are always lower than simple returns for gains and more negative for losses. The property that makes log returns useful is that they add across time. The log return over a year is exactly the sum of the twelve monthly log returns, which makes multi period statistics and many models far simpler.

The two kinds of return therefore have complementary jobs. Use simple returns when combining assets into a portfolio at a point in time. Use log returns when chaining one asset's returns through time or fitting a statistical model. Mixing them up is a common source of small but persistent errors, such as averaging log returns across holdings or adding simple returns across months.

Compounding and the geometric mean

Returns compound: each period's return is earned on the value left by the previous one. The growth of $1 over n periods is the product of one plus each period's simple return, and the compound (or cumulative) return is that product minus one. The geometric mean return is the constant per period return that would produce the same final value, which is why it is also called the compound annual growth rate (CAGR) when the periods are years.

The arithmetic mean is the ordinary average of the period returns. It answers a different question: what is the expected return in a single, randomly chosen period? It is the right input for forecasting one period ahead and for portfolio optimisation. It is the wrong number for describing what an investor actually earned over many periods, because it ignores the fact that losses are taken on a larger base after gains and on a smaller base after losses.

Compounding frequency matters too. A rate of 1 percent per month compounds to (1.01)^12 minus 1, or 12.68 percent per year, not 12 percent. With continuous compounding, a 12 percent annual log return corresponds to a simple annual return of e^0.12 minus 1, or 12.75 percent. When a manager quotes an annual figure, check whether it is twelve monthly returns multiplied by twelve, compounded, or a log figure.

Standard deviation and volatility drag

Volatility, in the language of the buyside, is the standard deviation of returns. It measures how widely returns scatter around their mean, with large deviations counted heavily because they are squared. Practitioners almost always use the sample standard deviation, which divides the sum of squared deviations by n minus 1 rather than by n. The adjustment corrects for the fact that the mean was itself estimated from the same data, and it matters in short series: with twelve observations it raises the estimate by about 4.4 percent.

Volatility is not only a risk measure. It directly reduces compound growth. The gap between the arithmetic and geometric means is called volatility drag, and a good approximation is half the variance (the squared volatility). Two portfolios with the same arithmetic mean do not deliver the same wealth: the more volatile one ends up with less. A strategy that alternates +40 percent and minus 20 percent years has an arithmetic mean of 10 percent but grows at only sqrt(1.40 x 0.80) minus 1, about 5.83 percent per year.

The same arithmetic explains why large losses are so expensive. A 50 percent loss needs a 100 percent gain to recover, and a 75 percent loss needs a 300 percent gain. This asymmetry is why drawdown control features so heavily in how hedge funds and multi asset allocators describe their risk management, and it is the subject of the next topic.

Annualising with the square root of time

Returns and volatilities are measured at whatever frequency the data comes in, daily for a listed portfolio and monthly for most hedge funds, but they are compared annually. The mean scales with time: a monthly arithmetic mean is multiplied by 12, a daily mean by the number of trading days, conventionally 252. Volatility scales with the square root of time: monthly volatility is multiplied by sqrt(12), about 3.464, and daily volatility by sqrt(252), about 15.87. A portfolio that moves 1 percent on a typical day therefore has an annual volatility of about 15.9 percent.

The square root rule is not a convention; it is a consequence of an assumption. If returns in successive periods are independent and identically distributed, the variance of their sum is the sum of their variances, so variance grows linearly with time and standard deviation grows with its square root. When that assumption fails, the rule fails with it.

The most important failure for an allocator is autocorrelation, where this month's return tends to resemble last month's. It is pervasive in funds that hold illiquid assets, because marks are smoothed: appraisals lag the market, and a manager who reprices slowly spreads one real move across several reporting periods. Positive autocorrelation makes the monthly volatility look small and the square root rule then makes the annual volatility look small too. Smoothing flatters every ratio that divides by volatility, which is why the Sharpe ratios of private credit and real estate funds computed from reported returns cannot be compared directly with those of listed assets.

Fat tails and skew

Much of risk arithmetic quietly assumes returns follow a normal distribution, which is fully described by its mean and standard deviation. Real returns are not normal. Two departures matter most. Skewness measures asymmetry: negative skew means the large moves are more often losses than gains, a common pattern in equity indices and in strategies that sell insurance, such as short volatility or merger arbitrage. Kurtosis measures how much of the variance comes from extreme observations; excess kurtosis above zero means fatter tails than the normal distribution.

The table below shows how rare large moves are under a normal distribution. A 4 standard deviation daily move should happen roughly once every 63 years of trading days, and a 5 standard deviation move once in about 6,900 years. In equity markets, moves of that size relative to recent volatility have occurred several times in living memory: on 19 October 1987 the S&P 500 fell about 20 percent in a day. When a manager says an event was a '6 sigma' surprise, the usual explanation is not bad luck but a model that assumed thin tails.

Higher moments are hard to estimate. Fund X's twelve returns have a sample skewness of minus 0.39 and an excess kurtosis of minus 1.32, which would suggest thin tails, but twelve observations say almost nothing about the tails, since by construction the tails are where observations are rare. Reliable estimates of skew and kurtosis need long daily histories, and even then a single crisis can change them. Treat any tail statistic from a short track record as a hypothesis, not a measurement.

import numpy as np
import pandas as pd

# Illustrative monthly returns for Fund X, as decimals
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])

arith = r.mean()
growth = (1 + r).prod()
geo = growth ** (1 / len(r)) - 1
vol = r.std()  # pandas uses the sample version, ddof=1

print(f"arithmetic mean   {arith:.2%}")
print(f"geometric mean    {geo:.2%}")
print(f"monthly vol       {vol:.2%}")
print(f"annual vol        {vol * np.sqrt(12):.2%}")
print(f"compound return   {growth - 1:.2%}")
print(f"sum of log rets   {np.log1p(r).sum():.2%}")

arithmetic mean   0.47%
geometric mean    0.45%
monthly vol       2.16%
annual vol        7.49%
compound return   5.48%
sum of log rets   5.33%

Templates for this lesson

Included with a Premieur membership.

• NotebookPython analysis notebookPrice tables, simple and log returns, rolling volatility, monthly resampling and a fund against its benchmark, in pandas.Download