- 30 Marks
FM – L2 – Q23 – Portfolio theory and CAPM
Calculate mean and standard deviation of expected return for Security X based on economic states.
Question
An investor is planning to invest in two securities, Security X and Security Y. The expected return from each security will depend on the state of the economy, as follows:
| State of the economy | Probability | Return from Security X | Return from Security Y |
|---|---|---|---|
| Strong | 0.25 | 15% | 20% |
| Fair | 0.60 | 10% | 8% |
| Weak | 0.15 | 2% | (6%) |
Required:
(a) Calculate the mean and standard deviation of the expected return from Security X.
(b) Calculate the mean and standard deviation of the expected return from Security Y.
(c) Calculate the covariance of the returns from Security X and Security Y. The formula for a covariance is:
Cov_x,y = Σ p (x – x̄)(y – ȳ)
(d) Calculate the correlation coefficient for returns from Security X and Security Y, for a portfolio consisting of 50% of the funds invested in Security X and 50% of the funds invested in Security Y. The formula for correlation coefficient is:
ρ_XY = Covariance_XY / (σ_X σ_Y)
where:
σ_x = the standard deviation of returns from Security X
σ_y = the standard deviation of returns from Security Y
Cov_x,y = Covariance of X and Y
Comment on the correlation coefficient.
(e) Calculate expected return, the variance and standard deviation of a portfolio consisting of 50% of the funds invested in Security X and 50% of the funds invested in Security Y. The formula for correlation coefficient is: a²(Variance X)² + (1-a)²(Variance Y)² + 2a(1-a)Cov_x,y
where:
a = the proportion of the portfolio invested in Security X
(1-a) = the proportion of the portfolio invested in Security Y
Variance X = the variance of the returns from Security X
Variance Y = the variance of the returns from Security Y
(f) Calculate expected return, the variance and standard deviation of a portfolio consisting of 80% of the funds invested in Security X and 20% of the funds invested in Security Y.
Answer:
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