FuturesHedging.PortfolioBeta Method

Estimates the beta of a portfolio in the sense of the Capital Asset Pricing model (CAPM).

public double PortfolioBeta(
   double[] portfolioReturns,
   double[] indexReturns,
   double[] riskFreeReturns
);

Parameters

portfolioReturns: An array where the k-th element is the return of the portfolio over the k-th period considered given in decimal format (i.e. 1 percent = 0.01). Please see the example above for further details.
indexReturns: An array where the k-th element is the return of the index considered over the k-th period considered given in decimal format (i.e. 1 percent = 0.01). Please see the example above for further details.
riskFreeReturns: An array where the k-th elements is the risk free rate of return over the k-th period considered given in decimal format (i.e. 1 percent = 0.01). Please see the example above for further details.

Return Value

The beta of a portfolio with respect to a given index.

Remarks

Beta according to the CAPM

There are two slightly differing notions of beta used widely within finance, namely:

Portfolio view of Beta: The contribution of an equity's risk to the systematic (non-diversifiable) risk of the portfolio, where the risk is taken as being standard deviation of the portfolio.
CAPM view of Beta: The slope of the linear relationship between the expected excess return (over the risk free interest rate) of an equity and the expected excess return of the market.

Note: By excess returns in this context we are referring to the return over and above the risk free interest rate.

Here we will estimate the beta in the sense of the CAPM using the standard least squares approach. In particular, we will find the slope of the line of best fit with respect to the least squares approach for a scatter plot of the portfolios excess returns against the markets excess returns.

Example

If we wished to find the beta of a portfolio of FTSE stocks which had a return over three years of 8, 10, 14 percent when the return of the FTSE over the same three year period was 12, 13, 25 percent and where the risk free rate paid on cash over the same three years was 6, 6, 5 percent then we would need to pass the following parameters:

indexReturns = { 0.12, 0.13, 0.25 }
portfolioReturns = { 0.08, 0.10, 0.14 }
riskFreeReturns = { 0.06, 0.06, 0.05 }

In this case the beta (in accordance with the CAPM) will be approximately 0.4508.

Application of the beta evaluated with respect to the CAPM

Here we detail some of the consequences of the beta evaluated in accordance with the CAPM. From the example mentioned above the beta was 0.4508, or equivalently you need a weighting of 1/0.4508 approx 2.218, in order to obtain the expected return of the market over the period in question. What this means is that by gearing the portfolio by 221.8 percent by borrowing from the market at the risk free rate we would expect to obtain the return of the index. In particular:

1st Year: Index Return = 12 percent, Portfolio Return = 8 percent, Risk Free Rate = 6 percent
Return of the Portfolio after gearing by 221.8 percent = 2.218.(Portfolio Return) - (cost of gearing) = 2.218(0.08) - (1.1218)(0.06) = 11.01 percent
2nd Year: Index Return = 13 percent, Portfolio Return = 14 percent, Risk Free Rate = 5 percent
Return of the Portfolio after gearing by 221.8 percent = 2.218.(Portfolio Return) - (cost of gearing) = 2.218(0.10) - (1.1218)(0.06) = 15.45 percent
3rd Year: Index Return = 25 percent, Portfolio Return = 14 percent, Risk Free Rate = 5 percent
Return of the Portfolio after gearing by 221.8 percent = 2.218.(Portfolio Return) - (cost of gearing) = 2.218(0.14) - (1.1218)(0.05) = 24.96 percent

Note that in each of the three years the rate obtained from the geared portfolio is not exactly equal to the return on the index. However, over the three year period the beta found allows the best fit (in accordance with the least squares approach) for the geared portfolio detailed above to match the return of the index.