Questions: Risk-Adjusted Return Measures: Sharpe and Treynor Ratios

Question 1

Fund A has a Sharpe ratio of 0.6 and a Treynor ratio of 0.9. Fund B has a Sharpe ratio of 0.8 and a Treynor ratio of 0.5. A pension fund manager is selecting one fund to add to an already well-diversified multi-manager portfolio. Which fund should she prefer, and why?

• A. Fund B, because its higher Sharpe ratio indicates superior risk-adjusted performance
• B. Fund A, because its higher Treynor ratio means it delivers more excess return per unit of systematic risk — the relevant risk for a component of a diversified portfolio
• C. Either fund — both metrics should produce the same ranking for diversified investors
• D. Fund A, because a lower Sharpe ratio signals lower total volatility, which is always desirable

Answer: (B)

For a portfolio that is one component of a larger diversified portfolio, idiosyncratic risk washes out at the total portfolio level. Only systematic risk (beta) matters. The Treynor ratio measures excess return per unit of beta — exactly the right metric here. Fund A's higher Treynor (0.9 vs. 0.5) means it contributes more return per unit of market risk it adds. Fund B's higher Sharpe reflects better total-risk-adjusted performance — relevant only if Fund B is the investor's entire portfolio.

Question 2

An investor holds only a single hedge fund as her entire investment. Which ratio should she use to evaluate it, and why?

• A. Treynor ratio, because systematic risk is what hedge funds specialize in managing
• B. Both ratios equally — they are interchangeable for concentrated investors
• C. Sharpe ratio, because she cannot diversify away the fund's idiosyncratic risk — total volatility is the relevant cost
• D. Neither — hedge funds should be evaluated by absolute return only

Answer: (C)

When a fund represents your entire investment, you bear all of its risk — both idiosyncratic (company-specific) and systematic (market-wide). The Sharpe ratio uses standard deviation (total volatility) as the risk measure, making it appropriate for stand-alone or concentrated allocations. The Treynor ratio uses beta (systematic risk only), which understates the true risk faced by an undiversified investor whose idiosyncratic component cannot be cancelled out.

Question 3

Two portfolios with identical Sharpe ratios can have different Treynor ratios.

• T. True
• F. False

Answer: True

Yes — the ratios measure different things. Sharpe uses total standard deviation; Treynor uses beta. A portfolio with high idiosyncratic volatility (high standard deviation) but low market sensitivity (low beta) could have a low Sharpe but a high Treynor. Conversely, a portfolio highly correlated with the market might have high beta relative to its standard deviation, giving a high Sharpe but a low Treynor. The two ratios can produce any relative ranking.

Question 4

A portfolio with a higher Sharpe ratio is generally a better investment than one with a lower Sharpe ratio.

• T. True
• F. False

Answer: False

The Sharpe ratio is only the right metric for investors who hold the portfolio as their entire investment. For a diversified institutional investor evaluating one sleeve of a multi-asset portfolio, the Treynor ratio is more appropriate — and the Sharpe-superior fund may have a worse Treynor ratio, making it the inferior choice in that context. 'Better' depends on the investor's situation: whether they can diversify away the idiosyncratic risk determines which measure is relevant.

Question 5

Explain when the Treynor ratio is more appropriate than the Sharpe ratio, and why the two measures can rank the same set of portfolios differently.

Model answer: The Treynor ratio is appropriate when a portfolio is one component of a larger diversified portfolio, so its idiosyncratic risk washes out at the total portfolio level. Only systematic risk (beta) is then relevant. The Sharpe ratio is appropriate when the portfolio represents the investor's entire investment, so total volatility (including idiosyncratic) is the relevant cost. They rank portfolios differently because a fund can have high idiosyncratic risk (raising standard deviation and lowering Sharpe) while having low beta (raising Treynor) — or vice versa.

The core insight is that 'risk' is context-dependent: what risk you bear depends on what else you hold. For a concentrated investor, idiosyncratic risk is real and costly. For a diversified investor, it's irrelevant. Using the wrong metric in the wrong context leads to systematically incorrect portfolio selection — preferring funds that hurt diversified investors or dismissing funds that would complement a diversified portfolio well.