School Launch

Risk, Return and the Sharpe Ratio

Markowitz’s insight was that risk belongs to the whole portfolio, not to each holding alone. Mix two or three investments, set how closely they move together, and watch the portfolio’s risk fall below the average of its parts. Then measure return per unit of risk with the Sharpe and Treynor ratios, and judge a fund against CAPM.

Series 65 Lessons 8.1 and 8.5 · also Lessons 3.4, 3.5 and 8.3 · about 25 minutes

Across: risk (standard deviation). Up: expected return. The curve shows every mix of A and B. Click or drag along it to choose a mix; with the chart focused, the left and right arrow keys move 1% of the money between A and B.

Portfolio expected return
Portfolio standard deviation
Sharpe ratio

What diversification bought

Judge a fund: CAPM, alpha, Sharpe and Treynor

CAPM expected return
Alpha (actual − CAPM)
Sharpe ratio
Treynor ratio
Market’s Sharpe and Treynor

The same thing in a spreadsheet

From a column of past returns: =AVERAGE(B2:B11) for the mean, =STDEV.S(B2:B11) for the standard deviation, =CORREL(B2:B11, C2:C11) for the correlation of two columns, and =SLOPE(fund column, market column) for beta.

The investments

The mix

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The fund to judge

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Try an example

From the Series 65 lessons; numbers the lessons leave open are made up for the example.

Mixes

Funds to judge

Return and risk of a mix

A portfolio’s expected return is simply the weighted average of its parts:

Ep = wAEA + wBEB

Its standard deviation is not. For two investments,

σp2 = wA2σA2 + wB2σB2 + 2wAwBρσAσB

The correlation ρ runs from −1 to +1. At +1 the two always move together and the risk is just the weighted average: no benefit. Anything less than +1 makes the last term smaller, so the mix is less risky than the average of its parts. At −1 the right mix can remove the risk entirely. That is why the exam’s answer to “which asset diversifies best?” is always the one with the lowest correlation.

The efficient frontier

Every mix of A and B lies on the curve. The left-most point is the least-risk mix. Mixes below it are inefficient: another mix has the same risk and more return. With many investments the upper edge of all possible mixes is Markowitz’s efficient frontier.

Return per unit of risk

The Sharpe ratio divides the return above the risk-free rate by total risk:

Sharpe = (rp − rf) ÷ σp

It is the slope of the green line from the risk-free rate to the portfolio; the steeper, the better. The Treynor ratio divides by beta instead, counting only market risk, so it suits a portfolio that is already well diversified:

Treynor = (rp − rf) ÷ β

CAPM and alpha

The capital asset pricing model says a security should earn the risk-free rate plus beta times the market’s risk premium:

E(r) = rf + β(rm − rf)

Alpha is the actual return minus that CAPM return: what the manager earned beyond what the fund’s beta explains. On the chart, it is how far the fund sits above (positive alpha) or below (negative alpha) the security market line.

Check yourself

  1. The risk-free rate is 2%, the market is expected to return 8%, and a stock’s beta is 1.5. Find its CAPM return.
    Answer

    2% + 1.5 × (8% − 2%) = 11%. The trap, 1.5 × 8% + 2% = 14%, forgets to subtract the risk-free rate.

  2. A fund returned 12% with a beta of 0.9. T-bills paid 3% and the market returned 11%. Find its alpha and Treynor ratio.
    Answer

    CAPM: 3% + 0.9 × 8% = 10.2%, so alpha = 12% − 10.2% = +1.8%. Treynor: (12 − 3) ÷ 0.9 = 10, against the market’s (11 − 3) ÷ 1 = 8.

  3. Fund P returned 10% with a 10% standard deviation; Fund Q returned 16% with 28%. T-bills paid 2%. Which did better for its risk?
    Answer

    Sharpe of P = (10 − 2) ÷ 10 = 0.80; Q = (16 − 2) ÷ 28 = 0.50. P earned more per unit of risk. Load the example and slide the mix: the best blend of the two (correlation 0.3 in this example) beats both.

  4. A 60/40 mix of a stock fund (9%, σ 16%) and a bond fund (4.5%, σ 6%) has correlation 0.2. Find its expected return and standard deviation.
    Answer

    Return 0.6(9) + 0.4(4.5) = 7.2%. Variance 0.36(256) + 0.16(36) + 2(0.6)(0.4)(0.2)(16)(6) = 107.136, so σ = 10.35%: below the 12% weighted average of the two standard deviations.

A bit of history: Harry Markowitz published “Portfolio Selection” in 1952; William Sharpe built CAPM on it in 1964, and the two shared the 1990 Nobel prize in economics with Merton Miller.

“Give a portion to seven, and also to eight; for thou knowest not what evil shall be upon the earth.” — Ecclesiastes 11:2