by Performance evaluation is the systematic process of assessing how effectively an investment portfolio has performed over a specific period. It examines the return generated in relation to the risk undertaken and compares actual results with predetermined objectives or appropriate benchmarks. The purpose is to determine whether investment decisions have added value and whether the portfolio continues to meet the investor’s financial goals. Performance evaluation is therefore an important part of portfolio management and helps support future investment decisions.
1. Problem on Sharpe Ratio
Suppose a portfolio has earned a return of 16%, the risk-free rate is 6%, and the portfolio’s standard deviation is 12%. Calculate the Sharpe Ratio.
Formula:
Sharpe Ratio = (Rp − Rf) ÷ σp
= (16% − 6%) ÷ 12%
= 10% ÷ 12%
= 0.833
Therefore, the Sharpe Ratio = 0.83. This means the portfolio generated approximately 0.83 units of excess return for every unit of total risk undertaken. A higher Sharpe Ratio generally indicates better risk-adjusted performance when comparing similar portfolios.
2. Problem on Treynor Ratio
Suppose Portfolio B generated a return of 18%, the risk-free rate is 7%, and the portfolio beta is 1.10. Calculate the Treynor Ratio.
Formula:
Treynor Ratio = (Rp − Rf) ÷ βp
= (18% − 7%) ÷ 1.10
= 11% ÷ 1.10
= 10%
Therefore, the Treynor Ratio = 10%. The result indicates that Portfolio B generated 10 percentage points of excess return per unit of systematic market risk. The measure is particularly appropriate for evaluating well-diversified portfolios where unsystematic risk is relatively less important.
3. Problem on Jensen’s Alpha
Suppose a portfolio earned 17%, the risk-free rate is 6%, the portfolio beta is 1.20, and the market return is 14%. Calculate Jensen’s Alpha.
Formula:
Jensen’s Alpha = Rp − [Rf + βp(Rm − Rf)]
First, calculate the expected return:
= 6% + [1.20 × (14% − 6%)]
= 6% + (1.20 × 8%)
= 15.6%
Now:
Alpha = 17% − 15.6%
= +1.4%
Therefore, Jensen’s Alpha = +1.4%. The positive alpha indicates that the portfolio earned 1.4 percentage points more than the return expected for its systematic risk according to CAPM.
Comparison of Two Portfolios Using Sharpe Ratio
Consider two portfolios:
| Particular | Portfolio A | Portfolio B |
|---|---|---|
| Return | 15% | 17% |
| Risk-Free Rate | 5% | 5% |
| Standard Deviation | 10% | 16% |
For Portfolio A:
Sharpe Ratio = (15 − 5) ÷ 10 = 1.00
For Portfolio B:
Sharpe Ratio = (17 − 5) ÷ 16 = 0.75
Although Portfolio B has the higher absolute return, Portfolio A has the higher Sharpe Ratio. Therefore, Portfolio A generated better risk-adjusted performance because it provided more excess return for each unit of total risk.
Comparison of Two Portfolios Using Treynor Ratio
Suppose two portfolios have the following characteristics:
| Particular | Portfolio A | Portfolio B |
|---|---|---|
| Return | 14% | 17% |
| Risk-Free Rate | 6% | 6% |
| Beta | 0.80 | 1.40 |
For Portfolio A:
Treynor Ratio = (14 − 6) ÷ 0.80 = 10%
For Portfolio B:
Treynor Ratio = (17 − 6) ÷ 1.40 ≈ 7.86%
Therefore, Portfolio A has the higher Treynor Ratio, despite Portfolio B having a higher absolute return. This suggests that Portfolio A provided better compensation for each unit of systematic risk.
Comparison Using Jensen’s Alpha
Suppose the market return is 12% and the risk-free rate is 5%.
Portfolio A has a return of 14% and beta of 0.90.
Expected Return = 5% + [0.90 × (12% − 5%)]
= 11.3%
Jensen’s Alpha = 14% − 11.3% = +2.7%
Portfolio B has a return of 16% and beta of 1.50.
Expected Return = 5% + [1.50 × (12% − 5%)]
= 15.5%
Jensen’s Alpha = 16% − 15.5% = +0.5%
Although Portfolio B generated a higher actual return, Portfolio A generated a higher Jensen’s Alpha and therefore performed better relative to the systematic risk it assumed.
Interpretation of Sharpe, Treynor and Jensen Measures
These three measures evaluate portfolio performance from different perspectives. Sharpe Ratio considers total portfolio risk through standard deviation. Treynor Ratio considers only systematic risk through beta. Jensen’s Alpha measures abnormal return after accounting for systematic risk according to CAPM. Therefore, the same portfolio can receive different evaluations under the three measures. Investors should select the measure according to the portfolio’s diversification and the specific purpose of performance evaluation.
Comprehensive Performance Evaluation Problem
Suppose Portfolio X has a return of 18%, beta of 1.20, standard deviation of 15%, market return of 14%, and risk-free rate of 6%.
Sharpe Ratio:
(18 − 6) ÷ 15 = 0.80
Treynor Ratio:
(18 − 6) ÷ 1.20 = 10%
Jensen’s Alpha:
18% − [6% + 1.20(14% − 6%)]
= 18% − 15.6% = +2.4%
Thus, the portfolio has a Sharpe Ratio of 0.80, Treynor Ratio of 10%, and Jensen’s Alpha of +2.4%. These results indicate positive risk-adjusted performance, although final evaluation should involve comparison with other portfolios or an appropriate benchmark.
