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 or standard deviation of a portfolio, we need to take into account the individual volatilities of the investments and their correlation. In this case, we need to consider the volatility of firms of type S and type I, as well as their correlation.
(a) Volatility of type S firms:
Given that there are 20 firms of type S, and each firm has a 60% probability of a 15% return and a 40% probability of a -10% return, we can calculate the volatility using the formula:
Volatility_S = √[w1 * σ1^2 + w2 * σ2^2 + ... + wn * σn^2],
where w is the weight or proportion of each investment, and σ is the standard deviation of each investment.
In this case, the weights are equal, as there are 20 firms and an equal investment in each. Let's assume w = 1/20 for each firm.
Given that the standard deviation (σ) of a 15% return is 15%, and the standard deviation of a -10% return is -10%, we can calculate the volatility as follows:
Volatility_S = √[20 * (1/20) * (0.6 * 0.15^2 + 0.4 * (-0.1)^2)]
= √[(0.6 * 0.0225 + 0.4 * 0.01)]
= √[(0.0135 + 0.004)]
= √[0.0175]
≈ 0.1321 (or 13.21%)
Therefore, the volatility of a portfolio consisting of an equal investment in 20 firms of type S is approximately 13.21%.
(b) Volatility of type I firms:
Given that type I firms move independently, the calculation of volatility remains the same. We continue using the formula:
Volatility_I = √[20 * (1/20) * (0.6 * 0.15^2 + 0.4 * (-0.1)^2)]
= √[(0.0135 + 0.004)]
= √[0.0175]
≈ 0.1321 (or 13.21%)
Therefore, the volatility of a portfolio consisting of an equal investment in 20 firms of type I is also approximately 13.21%.
In both cases, the volatility or standard deviation of the portfolio is approximately 13.21%.