Introduction

In the realm of modern portfolio management, the assessment of investment performance goes beyond mere returns. It involves measuring returns relative to the risks undertaken. This chapter delves into a range of performance measures that shed light on how well a portfolio or investment strategy has performed, considering the inherent risks. These metrics offer insights into risk-adjusted returns, portfolio efficiency, and the effectiveness of active management.

Sharpe Ratio

The Sharpe ratio, named after Nobel laureate William F. Sharpe, evaluates the excess return of an investment per unit of its risk (volatility). Mathematically, it’s calculated as: $$\text{Sharpe Ratio }=\frac{(R_p-R_f)}{\sigma_p}$$

Where:

• $R_p=$ Portfolio’s Expected Return
• $R_f=$ Risk-Free Rate
• $\sigma_p=$ Portfolio Standard Deviation

A higher Sharpe ratio indicates better risk-adjusted returns. For instance, if Portfolio A has a Sharpe ratio of 1.5 and Portfolio $B$ has a ratio of 1.0, Portfolio A generates superior returns relative to its risk exposure.

Treynor Ratio

Developed by Jack Treynor, this index gauges the reward per unit of systematic risk (beta). It’s calculated as:

$$\text{Treynor Ratio = }\frac{(R_p-R_f)}{\beta_p}$$

Where:

• $R_p=$ Portfolio’s Expected Return
• $R_f=$ Risk-Free Rate
• $\beta_p=$ Portfolio Beta

The Treynor ratio helps investors assess how efficiently a portfolio’s returns compensate for market risk. A higher ratio signifies better compensation for systematic risk.

Jensen’s Alpha

Named after Michael Jensen, the Jensen index evaluates a portfolio’s risk-adjusted return relative to its Capital Market Line (CML) or Security Market Line (SML). It’s calculated as:

$$\text{Jensen’s Alpha = } R_p- (R_f+\beta_p \times (R_m-R_f) )$$

Where:

• $R_p$ = Portfolio’s Expected Return
• $R_f=$ Risk-Free Rate
• $\beta_p=$ Portfolio Beta
• $R_m=$ Market Return

A positive alpha implies that the portfolio outperformed the expected return based on its risk.

Tracking Error

The Tracking Error quantifies the standard deviation of the difference between the returns of a portfolio and its benchmark over a specific period. It measures how closely the portfolio tracks its benchmark.

$$\text{Tracking Error =}\sqrt{\frac{\sum_{i=1}^n(R_{p, i}-R_{b, i})^2}{n-1}}$$

Where:

• $n=$ Number of periods
• $R_{p, i}=$ Portfolio’s return in period $i$
• $R_{b, i}=$ Benchmark’s return in period $i$

Information Ratio

The Information Ratio assesses the portfolio manager’s ability to generate excess return compared to a benchmark, relative to the tracking error. It is calculated by dividing the portfolio’s excess return by its tracking error.

$$\text { Information Ratio }=\frac{(R_p-R_b)}{\text { Tracking Error }}$$

Where:

• $R_p=$ Portfolio’s return
• $R_b=$ Benchmark’s return

The Information Ratio helps investors determine whether a portfolio’s excess return is a result of superior management skill or simply due to increased risk.

Sortino Ratio

The Sortino ratio is a performance measure that focuses on downside risk by considering only the volatility of negative returns. It is particularly useful for assessing the risk-adjusted return of an investment strategy while giving more weight to downside volatility. The formula for the Sortino ratio is as follows:

$$\text{Sortino Ratio =}\frac{(R_p-R_f)}{\sigma_d}$$

Where:

• $R_p=$ Portfolio’s Expected Return
• $R_f=$ Risk-Free Rate
• $\sigma_d=$ Downside Deviation (Standard Deviation of Negative Returns)

The Sortino ratio formula is similar to the Sharpe ratio formula, but instead of using the total standard deviation of returns, it uses the downside deviation. The downside deviation is calculated using only the negative returns of the portfolio.

$$\sigma_d=\sqrt{\frac{1}{N} \sum_{i=1}^N\left(\min \left(0, R_{p, i}-R_f\right)\right)^2}$$

Where:

• $N=$ Number of observations (periods)
• $R_{p, i}=$ Portfolio return for the ith observation
• $R_f=$ Risk-Free Rate

The Sortino ratio provides investors with a more nuanced perspective of risk-adjusted returns by focusing on the risk of losing money rather than the overall volatility of returns. A higher Sortino ratio indicates better risk-adjusted returns, particularly during downward market movements.