21. Consider an economy with two types of firms, S and I. S firms all move together. I firms move independently. For both types of firms, there is a 60% probability that the firms will have a 15% return and a 40% probability that the firms will have a -10% return. What is the volatility (standard deviation) of a portfolio that consists of an equal investment in 20 firms of (a) type S, and (b) type I?
To calculate the volatility (standard deviation) of a portfolio, we need to understand the characteristics of each type of firm and their returns.
In this case, we have two types of firms: S and I. S firms all move together, meaning that they have the same returns, while I firms move independently, indicating that each I firm can have different returns.
For both types of firms, there is a 60% probability of a 15% return and a 40% probability of a -10% return.
To calculate the volatility of the portfolio that consists of an equal investment in 20 firms of each type, we need to consider the risk and return characteristics of both types of firms.
Calculating for Type S firms:
Since S firms move together, the 20 firms will have the same returns. To calculate the volatility of the portfolio for type S firms, we need to calculate the standard deviation of their returns.
Let's denote the 15% return as R1 and the -10% return as R2. The probabilities of these returns are P(R1) = 0.6 and P(R2) = 0.4.
We can use the formula for calculating the standard deviation of a portfolio:
σs = √[w1^2 * σ1^2 + w2^2 * σ2^2 + 2 * w1 * w2 * ρ * σ1 * σ2]
Where:
- σs is the standard deviation of the portfolio
- w1 and w2 are the weights (equal for each firm in this case, so both are 1/20)
- σ1 and σ2 are the standard deviations of the returns (15% and -10% respectively)
- ρ is the correlation coefficient between the firm returns (since S firms move together, the correlation is 1)
Plugging in the values:
σs = √[(1/20)^2 * 0.15^2 + (1/20)^2 * (-0.10)^2 + 2 * (1/20) * (1/20) * 1 * 0.15 * (-0.10)]
Calculate this expression to find the standard deviation of the portfolio's returns for type S firms.
Now let's calculate for Type I firms:
Since I firms move independently, each firm can have different returns. However, we know the probabilities of the returns. Here, we have the same weights for all 20 firms, which are 1/20.
Apply the formula for standard deviation again with the values of the returns (15% and -10%) for type I firms.
In this case, the correlation coefficient (ρ) is not specified, so we assume it to be zero as there is no indication of correlation between I firms.
So the formula becomes:
σi = √[(1/20)^2 * 0.15^2 + (1/20)^2 * (-0.10)^2]
Calculate this expression to find the standard deviation of the portfolio's returns for type I firms.
The calculated σs and σi represent the volatility (standard deviation) of the portfolio for type S and type I firms, respectively.