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Plate 5 of 19 · Finance

Return, risk and diversification

About 7 minutes to read, then three questions that count toward your mastery.

A share is judged by the return it earns for the risk it carries. Combining shares that don’t move together cuts risk faster than it cuts return.

The idea

Risk here means how much returns swing: volatility, the standard deviation of returns, quoted per year (daily volatility × √252). The Sharpe ratio divides the return above the risk-free rate by volatility, so shares of different riskiness can be compared. The Sortino ratio does the same but counts only the downside swings.

Beta is the slope of a share’s excess returns against the market’s: a beta of 1.5 means the share tends to move 1.5% when the market moves 1%. The intercept of that regression, alpha, is return the market doesn’t explain. Daily returns are noisy, so an alpha needs a low p-value before it means skill rather than luck.

A portfolio’s return is the weighted average of its shares’ returns, but its volatility is not, because the shares’ bad days don’t all coincide. The lower the correlation, the more the swings cancel. The efficient frontier traces the best mixes, though the weights it suggests move a lot when the noisy average returns are re-estimated.

The formula

Sharpe = R − rfσ, σP2 = wA2σA2 + wB2σB2 + 2 wAwB ρ σAσB

R
expected (or average) return per year
rf
risk-free rate per year
σ
volatility: the standard deviation of returns per year
w
the share of the money in each holding; the weights add up to 1
ρ
correlation between the two shares’ returns, from −1 to +1

A worked example

Plate 5

Share A is expected to return 10% a year with 20% volatility; share B 6% with 12%. Their correlation is 0.2 and the risk-free rate is 4%. Is a 50:50 mix better than either?

  1. Sharpe ratios alone: A (10% − 4%) ÷ 20% = 0.30; B (6% − 4%) ÷ 12% = 0.17.
  2. The mix returns 0.5 × 10% + 0.5 × 6% = 8%.
  3. Its variance is 0.25 × 0.20² + 0.25 × 0.12² + 2 × 0.25 × 0.2 × 0.20 × 0.12 = 0.0100 + 0.0036 + 0.0024 = 0.0160, so its volatility is √0.0160 = 12.6%.
  4. Its Sharpe ratio is (8% − 4%) ÷ 12.6% = 0.32.

The mix beats both shares on return per unit of risk. Its volatility, 12.6%, is barely above B’s alone and well below the 16% average of the two: that gap is the gain from diversification.

Try it

Shares A and B from the example: 10% return and 20% volatility, against 6% and 12%.

50%
0.2
4.0%

Mix two shares

0.32Sharpe ratio of the mix
Expected return
8.0%
Volatility
12.6%
Risk saved against the average volatility
3.4 points
Best Sharpe ratio of either share alone
0.30

This mix beats both shares on return per unit of risk. The best mix here puts about 60% in A.

The common mistake

Averaging the volatilities

Taking 0.5 × 20% + 0.5 × 12% = 16% as the mix’s risk assumes the shares always move together (a correlation of 1). It understates the Sharpe ratio, 4% ÷ 16% = 0.25 instead of 0.32, and hides the reason to diversify.

Remember

  • Sharpe ratio = (return − risk-free rate) ÷ volatility. Forgetting to subtract the risk-free rate flatters every share.
  • Beta scales the market’s move: beta 1.5 and a 2% market day means about 3%. Alpha needs a small p-value to mean anything.
  • Portfolio return averages; portfolio risk doesn’t. Low correlation is what makes diversification pay.

Check yourself

Three questions from the Learn room’s bank. Your answers update your mastery of this skill, the same as in Practise.

Use it on real numbers

Open Return and risk, in Finance › Stock analysis and try the idea on your own numbers.