Financial Subject: Sharpe Ratio and Jensen’s alpha
To rank the portfolios according to their performance, finance specialists rely on 3 different
ratios: the Sharpe Ratio, the Treynor Ratio and Jensen’s alpha.
Sharpe Ratio
The Sharpe ratio is a measure of stock or fund performance, it measures the reward per unit of
risk. By definition, it is the ratio of an asset’s excess return to its volatility. It is also known as
the reward-to-variability ratio. The Sharpe ratio can be computed either ex-ante or ex-post.
Mathematically, the Sharpe ratio, in its ex-post expression (based on realized returns) is as
follows:
𝑆𝑆𝑆𝑆 =
𝑟𝑟𝑖𝑖 − 𝑟𝑟
𝜎𝜎𝑖𝑖
Where r i is the return on asset i, r is the return on the benchmark asset, and σ i is the standard
deviation of r i .
Treynor ratio
The Treynor ratio is a measure of stock performance. It measures the performance of an asset
i compared to a risk free asset (typically Treasury bills) per unit of assumed market risk.
Mathematically, the Treynor ratio is as follows:
𝑇𝑇𝑇𝑇 =
𝑟𝑟𝑖𝑖 − 𝑟𝑟𝑓𝑓
𝛽𝛽𝑖𝑖
Where r i is the return on asset i, r f is the return on the risk free asset, and β i is the beta of
asset i.
The Treynor ratio can also be computed for a portfolio; the higher is the Treynor ratio, the
better is the performance of the fund. Treynor ratios are commonly used to rank fund
managers. The ranking given by Treynor ratios is usually the same as the ranking obtained via
Jensen’s alpha because both of them rely on systematic risk. Theoretically, the Treynor ratio
can be negative, but in this case it is very difficult to interpret. A negative Treynor ratio may
imply that the fund manager has outperformed the risk free rate while reducing systematic
risk (negative Beta) which is a favorable situation. The opposite situation with a positive beta
but having underperformed the risk free return is very unfavorable.
Jensen’s alpha
Jensen’s alpha is a measure of a security’s excess return with respect to the expected return
given by the Capital Asset Pricing Model. Investors are looking for assets or portfolios with
positives alphas, as it signals positive abnormal return. An asset with a positive alpha has a
higher return than the risk adjusted return estimated by the CAPM. Computations of Jensen’s
alpha are based on realized returns. For a stock or portfolio i, we thus have:
𝛼𝛼𝐽𝐽 = 𝑟𝑟𝑖𝑖 − �𝑟𝑟𝑓𝑓 + 𝛽𝛽𝑖𝑖 �𝑟𝑟𝑀𝑀 − 𝑟𝑟𝑓𝑓 ��
where r i is the return on asset i, r f is the return on the risk free asset, β i is the beta of asset i,
and r m is the market return.