Key Performance Measures (FRM Part 1, Part 2)

1. Context

In this swatch, we look at key risk adjusted performance measures or performance indicators used to measure the performance of portfolios, specifically Sharpe Ratio, Treyner’s Measure and Jensen’s Alpha. We summarize and compare these measures and look at their respective usage. This topic appears in the curriculum of both FRM Part 1 and FRM Part 2. The details of the related readings are given below:

Area	Foundations of Risk Management
Reading	Applying the CAPM to Performance Measurement
Reference	Noel Amenc and Veronique Le Sourd, Ch10. Applying the CAPM to Performance Measurement: Single-Index Performance Measurement Indicators. In Portfolio Theory and Performance Analysis, West Sussex, England: John Wiley & Sons, 2003.

Area	Investment Management
Reading	Portfolio Performance Evaluation
Reference	Zvi Bodie, Alex Kane, and Alan J. Marcus, Ch24. Portfolio Performance Evaluation. In Investments, 10th Edition, New York: McGraw-Hill, 2013.

2. Terminology

We will use the following symbols in the sections below:

$P,M,F$	Any Portfolio, Market Portfolio, Risk-free Portfolio
$R$	Random (future) return
$r$	Constant or realized return
$E()$	Expectation operator
$\sigma$	Standard deviation or volatility
$\beta$	Measure of systematic risk, with respect to a benchmark (index or market portfolio)

3. Sharpe Ratio

Definition	Average (or expected) return in excess of risk-free return, per unit of total risk (measured by standard deviation). Ex-ante calculation uses expected returns while ex-post estimation uses realized returns.
Calculation	$$\begin{align}S_P&=\frac{E(R_P)-r_F}{\sigma_P}&\mbox{(ex-ante)}\\ S_P&=\frac{\bar{r}_P-r_F}{\sigma_P}&\mbox{(ex-post)} \end{align}$$NB: If $P$ is fully invested in risk-free asset, it’s $S_P=0$.
Risk Used	Total Risk
Source	Portfolio Theory
Market Portfolio	Sharpe Ratio for Market Portfolio ($S_M$) is the slope of Capital Market Line (CML): $$S_M=\frac{E(R_M)-r_F}{\sigma_M}$$ Computed value of $S_P$ can tell if the portfolio plots above ($S_P>S_M$) or below ($S_P>S_M$) the CML.
Limitations	Uses total risk as denominator, while only systematic risk is priced. As a stand-alone number, the ratio is not very informative.
Usage	Higher the portfolio’s Sharpe Ratio, higher it’s risk-adjusted performance, where risk is proxied by standard deviation. Can be applied to compare portfolios of different risk, portfolios that are not very well diversified and/or that represent an individual’s total personal wealth. Sharpe Ratio works very well for performance evaluation of past period (ex-post).

4. Treynor’s Measure

Definition	Expected return of portfolio in excess of risk-free return per unit of systematic risk taken (proxied by $\beta$ of portfolio). It only penalizes the systematic component of risk as it cannot be diversifed away.
Calculation	$$T_P=\frac{E(R_P)-r_F}{\beta_P}$$
Risk Used	Systematic
Source	CAPM
Market Portfolio	Treynor’s Measure for Market Portfolio ($\beta_M=1$) is: $$T_M=E(R_M)-r_F$$or, the expected market risk premium.
Limitations	Treynor’s Measure (since it involves the $\beta$), is heavily dependent on the choice of index.
Usage	Applicable to portfolios of different risk, well-diversified portfolios or sub-portfolios of larger fully diversified portfolio that represent an investor’s total personal wealth.

5. Jensen’s Alpha

Definition	Denotes the portfolio’s excess return, over and above the fair return commanded by the portfolio given it’s systematic risk. The fair return is calculated using a model like the CAPM.
Calculation	$$\alpha_P=E(R_P)-r_F-\beta_P\left(E(R_M)-r_F)\right)$$ It can be calculated as the intercept from regressing excess return of portfolio (over risk-free rate) vs excess return of market (over risk-free rate) i.e. $$R_{P,t}-R_{F,t}=\alpha_P+\beta_P(R_{M,t}-R_{F,t})+\varepsilon_t$$
Risk Used	Systematic
Source	CAPM
Market Portfolio	Jensen’s Alpha for market portfolio ($\alpha_M$) is zero.
Limitations	Jensen’s Alpha (since it involves the $\beta$), is heavily dependent on the choice of index. Also, it cannot be used to rank/compare portfolios of very different (systematic) risk.
Usage	Used to rank portfolios with same beta, to rank managers in their peer groups. A high $\alpha_P$ denotes a measure of a manager’s skill in constructing a portfolio that achieves a return in excess of it’s fair or reference return (as given by CAPM).