Calculating alpha, beta and sharpe ratio

Alpha, beta and sharpe ratios are the most commonly used performance and risk metrics in the world of investing.

These metrics can be calculated for any portfolio or instrument. We will not go into the details of what these metrics are. Rather, we will discuss the methodology used by us for calculating these metrics.

Time period and sampling frequency

When calculating alpha, beta and sharpe, it is important to define the period over which the metric is to be calculated.

For stocks, we provide alpha, beta and sharpe in the last 1 year, 3 years and 5 years. For mutual funds and ETFs, due to lack of availability of benchmark data, we have restricted coverage of alpha, beta and sharpe ratio – only available for last 3 years.

These metrics are calculated using the daily return of the stocks and the benchmark (for alpha and beta). For mutual funds and ETFs, the calculation is done using monthly returns. Both alpha and sharpe ratio are annualized.

Prices used for calculations are adjusted for splits, bonuses, rights issues, and dividends. Nifty 50 Total Return Index (TRI) is used as the benchmark for stocks.

Geometric mean for alpha calculation

For calculating beta and sharpe ratio, we stick to the traditional formulae:

Beta = Covariance (Return of the stock, Return of the benchmark) / Variance (Return of the benchmark)

Sharpe Ratio = (Return of the stock - Risk-Free Rate) / Standard Deviation of the stock's return

Note that returns used in the above sharpe ratio formula are “arithmetic mean” return. This is the correct approach in case of sharpe ratio.

Now here is the traditional formula for calculating stock’s alpha:

Alpha = (Return of the stock) - (Risk-Free Rate + Beta x (Return of the benchmark - Risk-Free Rate))

Again, traditionally, “arithmetic mean” return of the stock (and the benchmark) is used. But unlike sharpe ratio, using arithmetic mean for alpha calculation can create problems.

Let’s take an example. We will calculate alpha for 20MICRONS in the last 5 years (14th March 2018 to 14th March 2023).

Using daily returns of 20MICRONS and Nifty 50 TRI, stock beta comes out to be 1.098. If we use arithmetic mean, annualized alpha of the stock was 7.04%.

Now let’s revisit the definition of alpha - alpha refers to the excess return earned by a stock relative to the beta-adjusted benchmark return.

By this definition, one would conclude that 20MICRONS has delivered an annualized excess return of 7.04%.

Now comes the twist!

In our sample 5-year period, 20MICRONS delivered a CAGR of 8.3% as against Nifty 50 CAGR of 12.0%. Investors who chose 20MICRONS over Nifty 50 would have lost 3.7% of CAGR. But alpha value doesn’t reflect this.

The problem is with the usage of arithmetic mean. For highly volatile stocks (as is the case in our example), arithmetic mean of stock returns can be higher than the arithmetic mean of benchmark returns. But it is possible that CAGR (geometric mean) of the stock is lower than that of the benchmark (this happens due to a phenomenon called “variance drain”).

Therefore, we make a slight adjustment in alpha calculation. We use CAGR for both the stock and the benchmark instead of arithmetic mean.

Note that the traditional formula of using arithmetic mean is not wrong. Technically, alpha (Jensen’s alpha) is the excess return delivered over “expected return”. But using this leads to unintuitive values, especially for highly volatile stocks. Hence, we make this switch from arithmetic mean to geometric mean.