# Alpha

In finance, Alpha (also called Jensen’s alpha) is a measure of an investment portfolio’s excess return. It is determined as the difference between the actual return on a portfolio and the required return given the portfolio’s systematic risk. A higher alpha is better, and an investment manager’s skill is demonstrated by sustained alpha-generation.

There are two approaches to investment management: active management and passive management. Active managers overweight/underweight different sectors and/or stocks in their investment portfolio relative to their benchmark to beat the market. A positive alpha means they succeeded in beating the market.

## Formula

$$\alpha\ =\ \text{R} _ \text{p}-\ (\text{r} _ \text{f}+\beta\times(\text{r} _ \text{m}-\text{r} _ \text{f}))$$

Where α is the Jensen’s alpha, Rp is the portfolio return, β is the portfolio beta coefficient, rf is the risk-free rate and rm is the return on the benchmark, let’s say S&P 500.

The portfolio return is the weighted average holding period return of the individual investments. The holding period return is calculated as the percentage of difference between the beginning and closing investment value and portfolio income with reference to the initial investment value:

$$\text{Portfolio Return}=\frac{\text{P} _ \text{1}-\text{P} _ \text{0}+\text{I}}{\text{P} _ \text{0}}$$

Over multiple periods, geometric average return and time-weighted rate of return are appropriate approaches to determine portfolio’s actual return.

Portfolio beta equals the weighted average beta coefficient of the constituent investments.

The second term on the right-hand side is the equation for the capital asset pricing model. It is because we are attempting to measure the return generated over and above the return required given the portfolio’s risk. In a well-diversified portfolio, only the systematic risk is relevant because unsystematic risk can be diversified and hence shouldn’t be compensated.

## Example

Calculate alpha for the following portfolio

Stock Shares Stock price as at Dividend
per Share
Beta
01 Jan 2017 31 Dec 2017
A 2000 30 28 1 1.5
B 1000 55 65 2 1.2
C 500 125 140 5 0.8

The relevant risk-free rate is 5% and the actual return on the applicable broad market index (benchmark) is 9.5%.

First, we need to find out individual holding period returns:

$$\text{Holding period return}\ (\text{A})=\frac{\text{28}-\text{30}+\text{1}}{\text{30}}=-\text{3.33%}\$$

$$\text{Holding period return}\ (\text{B})=\frac{\text{65}-\text{55}+\text{2}}{\text{55}}=\text{21.82%}\$$

$$\text{Holding period return}\ (\text{C})=\frac{\text{140}-\text{125}+\text{5}}{\text{125}}=\text{16%}\$$

Next, we must find out relative weights of each stock in the portfolio:

$$\text{Weight of Stock A}\ \\=\frac{\text{500}\times\text{\140}}{\text{2,000}\times\text{\28}+\text{1,000}\times\text{\65}+\text{500}\times\text{\140}}\\=\text{29.32%}$$

Similarly, weights of Stock B and C are 34.03% and 36.65%

We can work out the portfolio return by finding the products of stock weight and return and them summing them up:

$$\text{Portfolio return}\\=\text{29.32%}\times-\text{3.33%}+\text{34.03%}\times\text{21.82%}+\text{36.65%}\times\text{16%}\\=\text{12.31%}$$

The same applies to portfolio beta:

$$\text{Portfolio beta}\\=\text{29.32%}\times\text{1.5}+\text{34.03%}\times\text{1.2}+\text{36.65%}\times\text{0.8}\\=\text{1.14}$$

Portfolio alpha can be calculated as follows:

$$\text{Portfolio alpha}\\=\text{12.31%}-\ (\text{5%}+\text{1.14}\times(\text{9.5%}-\text{5%}))\\=\text{12.31%}-\text{10.14%}=\text{2.18%}$$

The portfolio generated excess return of 2.18%.