Sharpe vs Treynor ratio
Same excess return on top, different risk underneath: total risk or market risk.
A return means little until you know the risk taken to earn it. The Sharpe ratio divides a portfolio's excess return over the risk-free rate by its total risk (standard deviation). The Treynor ratio divides the same excess return by its market risk (beta).
The only difference is the risk in the denominator, and that difference decides which one to use. XXI-A asks you to calculate both, and to say which fits a given client.
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The formulas
- Sharpe ratio
- (Rp − Rf) ÷ σp, where Rp is portfolio return, Rf the risk-free rate and σp the portfolio's standard deviation. Called reward to variability: excess return per unit of total risk.
- Treynor ratio
- (Rp − Rf) ÷ βp, where βp is the portfolio's beta against the market. Called reward to volatility: excess return per unit of systematic (market) risk.
- Jensen's alpha
- Rp − [Rf + βp × (Rm − Rf)], the return above what CAPM predicts for the beta taken. Positive alpha means the manager added value for the market risk carried.
- Information ratio
- (Rp − Rb) ÷ tracking error, where Rb is the benchmark return and tracking error is the standard deviation of the return differences over several periods. It measures consistency of beating the benchmark.
- Sortino ratio
- Like Sharpe, but divides by downside deviation, so only harmful volatility counts as risk.
A worked comparison where they disagree
Two equity PMS approaches both returned 15% last year, with a risk-free rate of 7%, so each has 8% excess return. Approach A holds 15 concentrated stocks: standard deviation 24%, beta 0.8. Approach B is diversified across 45 stocks: standard deviation 16%, beta 1.0.
Sharpe: A = 8 ÷ 24 = 0.33; B = 8 ÷ 16 = 0.50. B wins. Treynor: A = 8 ÷ 0.8 = 10.0; B = 8 ÷ 1.0 = 8.0. A wins.
Why: A carries a lot of stock-specific risk that beta does not see. Sharpe counts it; Treynor ignores it, on the assumption that it will be diversified away elsewhere.
Which one fits the client
Risk measured
Sharpe
Total risk (standard deviation)
Treynor
Systematic risk (beta)
Assumes
Sharpe
Nothing about the rest of the client's wealth
Treynor
The portfolio is one part of a well-diversified whole
Use when
Sharpe
The PMS is most or all of the client's investments
Treynor
The PMS is a slice of a larger diversified portfolio
Gives the same ranking when
Sharpe
Portfolios are well diversified, so total risk is close to market risk
Treynor
Same
| Sharpe | Treynor | |
|---|---|---|
| Risk measured | Total risk (standard deviation) | Systematic risk (beta) |
| Assumes | Nothing about the rest of the client's wealth | The portfolio is one part of a well-diversified whole |
| Use when | The PMS is most or all of the client's investments | The PMS is a slice of a larger diversified portfolio |
| Gives the same ranking when | Portfolios are well diversified, so total risk is close to market risk | Same |
Free account, this exam preselected.
Reading the numbers correctly
- check_circleHigher is better for Sharpe, Treynor and the information ratio.
- check_circleCompare like with like: same period, same risk-free rate, similar strategies. A Sharpe of 0.6 for a debt approach and 0.6 for an equity approach say different things.
- check_circleComparing a portfolio's Sharpe with the market's Sharpe is valid on its own: a higher figure means better reward per unit of total risk than the market.
- check_circleOne year of data is thin. Tracking error, for example, cannot be computed from a single period's difference.
- check_circleThese ratios describe the past; they are not a forecast and not a reason to promise a return.
How XXI-A tests this
Most questions are calculations: Sharpe from return, standard deviation and risk-free rate; Treynor from beta; actual return from alpha and CAPM. The trap options divide raw return by risk without subtracting the risk-free rate, swap beta and standard deviation, or put the risk-free rate in the denominator. Conceptual questions ask which ratio uses total risk (Sharpe) and which suits a diversified investor (Treynor).
FAQs
What is the difference between the Sharpe ratio and the Treynor ratio?expand_more
Both divide excess return over the risk-free rate by a measure of risk. Sharpe uses standard deviation (total risk); Treynor uses beta (systematic risk only).
How do you calculate the Sharpe ratio?expand_more
Subtract the risk-free rate from the portfolio return and divide by the portfolio's standard deviation. A 12.5% return, 6.5% risk-free rate and 8% standard deviation gives (12.5 − 6.5) ÷ 8 = 0.75.
When should the Treynor ratio be used instead of Sharpe?expand_more
When the portfolio is one part of an investor's well-diversified holdings, so stock-specific risk is diversified away and only market risk (beta) matters.
Is a higher Sharpe ratio always better?expand_more
For the same period and comparable strategies, a higher Sharpe means more excess return per unit of total risk. It is a measure of past results, not a guarantee of future returns.