Question

The Japanese market is on the verge of rising again, and the UK market is doing better than the US market in the current economic recovery. Therefore, as a US investor, John is contemplating investing in the Japanese and UK stock markets. Using the historical data, John has estimated the means, volatilities, and correlation of the US, UK, and Japanese stock markets as the following:
US UK Japan
Means 0.120 0.150 0.084
St. Dev. 0.150 0.240 0.220
Correlation matrix:
US UK Japan
US 1.000 0.500 0.266
UK 0.500 1.000 0.358
Japan 0.266 0.358 1.000
Assume that the risk-free rate is 4%. .
(a) Given the data above, what are the Sharpe ratios on the US portfolio and the UK portfolio? Given the correlation between US and UK above, should John add the UK portfolio to his US only portfolio?
(b) To see how robust his conclusion is on the issue of adding UK stocks to his portfolio, John wants to know the lowest expected return on UK stock can be in order to improve his Sharpe ratio from holding the two country stocks, given the correlations, volatilities and US Sharpe ratio. What is it? Show your reasoning and computations.
(c) What is the Sharpe ratio for the Japanese equity? Similar to (b), compute the lowest expected return for the Japanese equity. Are the differences between your answers to (b) and (c) surprising? Why or why not?

Answer 1

a) Sharpe ratio = (Return on portfolio - Risk free rate)/ Standard deviation of portfolio

Sharpe ratio of US Stock = (0.120-0.04)/0.150 = 0.5333

Sharpe ratio of UK Stock = (0.150-0.04)/0.240 = 0.4583

Correlation between US and UK Stock is 0.5.This means both the stock are not perfectly correlated which means john can add UK stocks to US Stock only portfolio to get the advantage of diversification to some extent.

b) Adding 50% of UK Stock to US portfolio , expected return of portfolio would be = 0.120(0.5) + 0.150(0.5)

Expected return of the portfolio would be  = 0.06+0.0750 = 0.1350

c) Sharpe ratio of japense Stock = 0.084-0.04/0.220 = 0.2