William Sharpe’s Single Index Model | CA Final AFM

Determining Portfolio Risk Using William Sharpe’s Approach

According to William Sharpe, portfolio risk is the aggregate of systematic and unsystematic risk.

For determining portfolio risk using Sharpe’s approach, first determine systematic risk of portfolio, using variance approach. Then, independently compute unsystematic risk of portfolio as explained below for n stocks:

Sharpe’s Single Index Model

Constructing Optimal Portfolio:

This model generates cut off rate and only those securities which have higher excess return to beta ratio than cut off rate are included in optimal portfolio. The single Index model formulates cutoff rate based on data inputs and selects only those securities which have higher excess return to beta ratio as compare to cut-off rate. Then based on residual variance (unsystematic risk) of the security, excess return to beta ratio, beta of the security and cut-off rate, proportion or weightage of the investment of the selected security is computed. Various financial and statistical methodologies are used for implementation of the model.

With the help of Single index model, portfolio managers and security analysts can easily identify based on security’s excess return to beta ratio, whether security should be included as part of optimal portfolio or not. Single index model gives ‘single value’ which explains the desirability of including any security in the portfolio. This ‘single value’ is excess return to beta ratio of that security. This excess return to beta ratio shows how much additional return for the security is generated for every unit of systematic risk (non-diversifiable risk). Symbolically, excess return to beta ratio can be shown as below:

Where,

R_i = Expected Return on security i

R_f = Risk free rate

β_i = Beta of the security i

Securities are ranked based on excess return to beta ratio from Highest to lowest, this ranking represent the desirability of including that security in portfolio. So, if security with particular ranking is included in portfolio, all the securities with ranking above will be included as well.

Same way if security with particular ranking is not part of the portfolio all securities below that security in terms of ranking will also be excluded from portfolio. Selection of the security is done based on cutoff rate. So, all the securities having

(*C = cut-off rate)

The steps to be followed for which security and in what proportion it is included in optimal portfolio by applying Single index model are;

Step 1

Determine “excess return to β” ratio for each stock and rank them on that basis from highest to lowest

Step 2

Determine cut-off rate (C)

Step 3

Find the optimal cut-off point (the highest of all) and select all securities upto such cut-off point from Rank 1 onwards.

Step 4

Determine the proportions of each security (w) to be included in the portfolio by using the following formula:

Step 5

Convert the proportions into appropriate weights such that aggregate of weights should be equal to 1.00 or 100%.

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