- 30 Marks
FM – L2 – Q23 – Portfolio theory and CAPM
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:
Answer
(a). EV of return (x̄)
= (0.25 × 15) + (0.60 × 10) + (0.15 × 2) = 10.05.
| Probability | Return | x – x̄ | p(x – x̄)² |
|---|---|---|---|
| p | x | ||
| 0.25 | 15 | 4.95 | 6.1256 |
| 0.60 | 10 | (0.05) | 0.0015 |
| 0.15 | 2 | (8.05) | 9.7204 |
| Variance σ² | 15.8475 |
Standard deviation of return σ_x = √15.8475 = 3.98.
(b). EV of return (ȳ)
= (0.25 × 20) + (0.60 × 8) + (0.15 × (-6)) = 8.90
| Probability | Return | y – ȳ | p(y – ȳ)² |
|---|---|---|---|
| 0.25 | 20 | 11.10 | 30.8025 |
| 0.60 | 8 | (0.90) | 0.4860 |
| 0.15 | (6) | (14.90) | 33.3015 |
| Variance σ² | 64.59 |
Standard deviation of return σ_y = √64.59 = 8.04
(c). Probability x – x̄ y – ȳ p(x – x̄)(y – ȳ) 0.25 4.95 11.10 13.7363 0.60 (0.05) (0.90) 0.0270 0.15 (8.05) (14.90) 17.9917 cov_xy 31.7550
(d). Correlation coefficient
ρ_xy = 31.7550 / (3.98 × 8.04) = +0.992
This shows a high level of positive correlation between the returns from Security X and the returns from Security Y
(e). The EV of the return from a portfolio consisting of 50% Security X and 50% Security Y
= (0.50 × 10.05) + (0.50 × 8.90) = 9.475%.
The variance of the returns from this portfolio would be:
[(0.50)² × 15.8475] + [(0.50)² × 64.5900] + [2 × 0.50 × 0.50 × 31.7550]
= 3.9619 + 16.1475 + 15.8775 = 35.9869.
The standard deviation of the portfolio returns = √35.9869 = 6.0%.
(f). The EV of the return
= (0.80 × 10.05) + (0.20 × 8.90) = 9.82%.
The variance of the returns from this portfolio would be:
[(0.80)² × 15.8475] + [(0.20)² × 64.5900] + [2 × 0.80 × 0.20 × 31.7550]
= 10.1424 + 2.5836 + 10.1616 = 22.8876.
The standard deviation of the portfolio returns = √22.8876 = 4.78%.
Note: In this example, since Security Y has a lower expected return than Security X and a higher standard deviation, expected returns will be highest and risk lowest with a ‘portfolio’ consisting of Security X only, and none of Security Y.
- Tags: expected return, Portfolio theory, Security X, Security Y, Standard deviation, Variance
- Level: Level 2
- Uploader: Samuel Duah