# Portfolio Performance: Comparing Portfolio Returns using the Sharpe Ratio, Treynor Ratio, and Jensen's Alpha

Portfolios contain groups of securities that are selected to achieve the highest return for a given level of risk. How well this is achieved depends on how well the portfolio manager or investor is able to forecast economic conditions and the future prospects of companies, and to accurately assess the risk of each security under consideration. Many investors and some portfolio managers adopt a **passive portfolio management** strategy by simply holding a basket of securities that is weighted to reflect a market index, or by buying securities based on a market index, such as most exchange-traded funds.

Some investors and portfolio managers think they can do better than the market, and so engage in **active portfolio management**, buying and selling securities as conditions change. Most active portfolio managers use sophisticated financial models to base their investment decisions.

However, most studies have shown that few portfolio managers outperform the market, especially over a long time, and because there are thousands of portfolio managers selling their services, the fact that some outperform the market over an extended period may be due to luck. For instance, if thousands of portfolios were constructed by choosing securities at random, and buying and selling them at random, some would do much better than all the rest—simply by chance. Another factor weighing on the performance of active portfolios are the fees charged by their managers, and the trading costs of frequently buying and selling. There are some managers who seem to outperform the market consistently, such as Warren Buffett of Berkshire Hathaway; however, Warren Buffet uses a buy-and-hold strategy rather than active trading. Another factor that individual investors doing their own active portfolio management should consider is whether any gains are worth the amount of time necessary to actively manage their portfolio.

## Measuring Portfolio Returns

Portfolio returns come in the form of current income and capital gains. **Current income** includes dividends on stocks and interest payments on bonds. A **capital gain** or **capital loss** results when a security is sold, and is equal to the amount of the sale price minus the purchase price. The return of the portfolio is equal to the net of the capital gains or losses plus the current income for the holding period. Unrealized capital gains or losses on securities still held are also added to the return to evaluate the **holding period return** of the portfolio. The portfolio return is adjusted for the addition of funds and the withdrawal of funds to the portfolio, and is time-weighted according to the number of months that the funds were in the portfolio. Below is the formula for calculating the portfolio return for 1 year:

Time Weighted Adjustment of the Portfolio = (Added Funds × Number of Months in Portfolio / 12) - (Withdrawn Funds × Number of Months Withdrawn from Portfolio / 12)

Portfolio Return | = | Dividends + Interest + Realized Gains or Losses + Unrealized Gains or Losses Initial Investment + Time Weighted Adjustment of the Portfolio Return |

**Realized gains** (or **losses**) are gains or losses actualized by the selling of the securities, whereas **unrealized gains or losses** are securities that are still owned but are marked to market to determine the portfolio's return.

## Comparing Portfolio Returns

There are several ways of comparing portfolio returns with each other and with the market in general. A simple comparison is to simply compare their returns. However, returns by themselves do not account for the risk taken. If 2 portfolios have the same return, but one has lower risk, then that would be the preferable, more efficient portfolio.

There are 3 common ratios that measure a portfolio's risk-return tradeoff: Sharpe's ratio, Treynor's ratio, and Jensen's Alpha.

### Sharpe ratio

The **Sharpe ratio** (aka **Sharpe's measure**), developed by **William F. Sharpe**, is the ratio of a portfolio's total return minus the risk-free rate divided by the standard deviation of the portfolio, which is a measure of its risk. The Sharpe ratio is simply the risk premium per unit of risk, which is quantified by the standard deviation of the portfolio.

Risk Premium = Total Portfolio Return – Risk-free Rate

Sharpe Ratio = Risk Premium / Standard Deviation of Portfolio Return

The risk-free rate is subtracted from the portfolio return because a risk-free asset, often exemplified by the T-bill, has no risk premium since the return of a risk-free asset is certain. Therefore, if a portfolio's return is equal to or less than the risk-free rate, then it makes no sense to invest in the risky assets.

Hence, the Sharpe ratio measures the performance of the portfolio compared to the risk taken—the higher the Sharpe ratio, the better the performance and the greater the profits for taking on additional risk.

#### Example—Calculating the Sharpe Ratio

If a fund has a return of 12% and a standard deviation of 15%, and if the risk-free rate is 2%, then what is its Sharpe ratio?

Solution:

**Sharpe Ratio** = (12% – 2%) / 15% = 10% / 15% = **66.7%** (rounded)

### Treynor Ratio

While the Sharpe ratio measures the risk premium of the portfolio over the portfolio risk, or its standard deviation, Treynor's ratio, popularized by Jack L. Treynor, compares the portfolio risk premium to the systematic risk of the portfolio as measured by its beta.

Treynor Ratio | = | Total Portfolio Return – Risk-Free Rate Portfolio Beta |

Note that since the beta of the general market is defined to be 1, the Treynor Ratio of the market equals its return minus the risk-free rate. The Treynor ratio measures the return per unit risk: it is higher with either higher portfolio returns or lower portfolio betas.

#### Example—Calculating the Treynor Ratio

*Case 1:* If a portfolio has a return of 12% and a beta of 1.4, and if the risk-free rate is 2%, then what is its Treynor ratio?

**Treynor Ratio** = (12 – 2) / 1.4 = 10 / 1.4 = **7.14** (rounded)

*Case 2:* Same as Case 1, except that the portfolio Beta equals the market beta of 1:

**Treynor Ratio** = (12 – 2) / 1.0 = 10 / 1.0 = 10 (rounded)

Note that here we used whole numbers instead of percentages for the return and risk-free rate because it simplifies the math and because it makes no difference when comparing portfolios if the same method is used consistently. Also note that, in Case 2, the same return was achieved with lower risk, so that, therefore, would be a more desirable portfolio.

### Jensen's Alpha (aka Jensen Index)

**Alpha** is a coefficient that is proportional to the **excess return** of a portfolio over its required return, or its expected return, for its expected risk as measured by its beta. Hence, alpha is determined by the fundamental values of the companies in the portfolio in contrast to beta, which measures the portfolio's return due to its volatility. **Jensen's alpha** (aka **Jensen index**), developed by Michael C. Jensen, uses the capital asset pricing model (CAPM) to determine the amount of the return that is firm-specific over that which is due to market volatility as measured by the firm's beta in relation to the market beta.

Jensen's Alpha = Total Portfolio Return – Risk-Free Rate – [Portfolio Beta × (Market Return – Risk-Free Rate)]

Jensen's alpha can be positive, negative, or zero. Note that, by definition, Jensen's alpha of the market is zero. If the alpha is negative, then the portfolio is underperforming the market; thus, higher alphas are more desirable.

#### Example—Calculating Jensen's Alpha

*Case 1:*

- portfolio return = 12%
- market return rate = 8%
- beta = 1.4
- risk-free rate = 2%

**Jensen's alpha** = 12 – 2 – 1.4 × (8 – 2) = 10 – 1.4 × 6 = 10 – 8.4 = **1.6**

*Case 2:*

- beta = 1.0

**Jensen's alpha** = 12 – 2 – 1.0 × (8 – 2) = 10 – 1.0 × 6 = 10 – 6 = **4**

*Case 3:*

- beta = 0.8

**Jensen's alpha** = 12 – 2 – 0.8 × (8 – 2) = 10 – 0.8 × 6 = 10 – 4.8 = **5.2**

The 3 portfolios have the same return, but Case 3 achieves the return with the lowest volatility, and, therefore, with lowest market risk among the 3 portfolios. Note also that in Case 2, a portfolio return that is 4% higher than the market return was achieved with no more volatility due to systematic risk than the general market.