The document summarizes William Sharpe's single index model from 1963, which simplified Harry Markowitz's earlier portfolio selection model. The single index model assumes that only one macroeconomic factor, represented by a market index like the S&P 500, influences the systematic risk of stock returns. It expresses the return of a security as the sum of its expected excess return, its sensitivity to market movements, and random error. This allows estimating portfolio variance and minimum variance portfolios based only on market risk rather than the full covariance matrix.

1. SHARPE’S
SINGLE INDEX MODEL
BY
SAKINA AHMED ALI

2. In 1952, “Harry Markowitz” published a PORTFOLIO SELECTION MODEL
that maximized a portfolio's return for a given level of risk.
This model required the estimation of :
• Expected returns for each security
• Variances for each security and
• A covariance matrix
(that calculated the covariance between each possible pair of securities
within the portfolio based on historical data or through scenario
analysis.)

3. Sharpe’s SINGLE INDEX MODEL
The model has been generated by “WILLIAM SHARPE” in 1963.
The Single Index Model is a simplified analysis of
“PORTFOLIO SELECTION MODEL”
To measure both Risk and Return on the stock.
• The SINGLE INDEX MODEL greatly reduces the number of calculations that
would otherwise have to be made for a large portfolio of thousands of
securities.

4. ASSUMPTIONS OF SINGLE INDEX MODEL
• There is only 1 macroeconomic factor that causes the systematic
risk affecting all stock returns and this factor can be represented by the
rate of return on a market index, such as the S&P 500.
• The return of any stock can be decomposed into the expected excess
return of the individual stock due to firm-specific factors, commonly
denoted by its alpha coefficient (α), which is the return that exceeds
the risk-free rate.
• The return due to macroeconomic events that affect the market, and
the unexpected microeconomic events that affect only the firm.

5. The single index model can be expressed by the following equation.
• Ri= the return on security i
• RM=the return on the market index
• αi=that part of security i’s return independent of market performance.
• Βi= a constant measuring the expected change in the dependent
variable, Ri, given a change in the independent variable RM
• ei= random residual error.
iMiii eRR  
SINGLE INDEX MODEL-RETURNS

6. In Single Index Model, the total risk of a security, as measured by its
variance, consists of two components: market risk and unique risk.
= Market risk+ company specific risk
SINGLE INDEX MODEL-RISK
2222
eiMi i
 
• In the single index model, the covariance between two stocks
depends only on the market risk Therefore covariance between two
securities can be written as
• TOTAL PORTFOLIO VARIANCE=PORTFOLIO MARKET RISK+ PORTFOLIO RESIDUAL VARIANCE
This single security variance can be extrapolated for finding the minimum
variance set of portfolios.
2
Miij  

7. • We can vary the amount invested in each type of asset and get an idea of
the relation between portfolio expected return and beta:
• It estimates the expected risk premium per unit of risk.
• We can also calculate the reward to risk ratio for all individual securities.
P
fP RRE



)(
RatioRisk-to-Reward