Financial Risk Manager Part 1
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An analyst has attempted to get some insight into the relationship between the return on stock A (R_A,t) and the return on the Nasdaq Composite index (R_NC,t). The analyst gathers historical data and comes up with the following estimates:
| Expected mean return for A | 10% |
|---|---|
| Annual mean return for Nasdaq Composite | 6% |
| Annual volatility for Nasdaq Composite | 15% |
| Covariance between the returns of A and Nasdaq Composite | 5% |
The analyst goes ahead and formulates the following regression model using the data: [ R_{A,t} = \alpha + \beta R_{NC,t} + \epsilon_t ]
Using the ordinary least squares technique, which of the following models will the analyst obtain?_
Explanation:
Explanation
The regression coefficients for a model specified by ( Y = \hat{\alpha} + \hat{\beta} X + \varepsilon ), where ( \varepsilon ) represents the error term, are obtained using the formula:
[ \hat{\beta} = \frac{\text{Cov}(X,Y)}{\text{Var}(X)} = \frac{0.05}{0.15^2} = 2.2222 ]
[ \hat{\alpha} = E(Y) - \hat{\beta} E(X) = 0.1 - 2.2222(0.06) = -0.03333 ]
Where:
- ( \text{Cov}(X,Y) = 0.05 ) (covariance between returns)
- ( \text{Var}(X) = (0.15)^2 = 0.0225 ) (variance of Nasdaq Composite returns)
- ( E(Y) = 0.10 ) (expected mean return for stock A)
- ( E(X) = 0.06 ) (annual mean return for Nasdaq Composite)
Therefore, the correct regression model is: [ R_{A,t} = -0.03333 + 2.2222 R_{NC,t} + \epsilon_t ]_