1. Derivation of the Beta Risk Factor
Calculate portfolio variance
Split into market proportional variance and firm specific variance
m: number of assets in portfolio
M: ‘market’

2. Derivation of the Beta Risk Factor

3. Derivation of the Beta Risk Factor
Split
Firm specific covariance is assumed zero. Split the variances and covariances
Market proportional Firm specific
variance covariance variance covariance

4. Derivation of the Beta Risk Factor
Firm specific covariance is assumed zero

5. Derivation of the Beta Risk Factor
Systemic risk contribution only
Portfolio covariance is a linear function of beta
Systemic and
non-systemic
(firm specific)
Contribution to variance for asset i

6. Derivation of the Beta Risk Factor
Substitute into CAPM formula
The two assumptions in the derivation are:
A market proportional covariance matrix can be constructed and
Covariances between firm specific risks are zero
Return other than from taking market driven risk is captured by α which in the CAPM model is a normally distributed random variable with an expected value of zero

7. Portfolio Beta Derivation