In this question sigma (sigma) is used to denote the standard deviation of asset

Question

In this question sigma (sigma) is used to denote the standard deviation of asset or portfolio returns. It is also a measure of volatility. The Japanese stock market has a sigma of 18%, when computed in yen. The U.S. stock market has a sigma of 17% in US$ and the US$/Yen exchange rate has a sigma of 6%. The correlation between the Japanese stock market and US$/Yen currency movements is -0.1. The correlation between the Japanese and U.S. stock market is equal to 0.4, measured either in local currency or in dollars. What is the sigma of the Japanese market when expressed in dollars? Using this number, calculate the sigma (in dollars) of a portfolio made up of 50% of Japanese stocks and 50% of US stocks.  Suppose the dollar volatility of the Japanese stock market is 18.97%, what can you conclude about the correlation between the Japanese stock market movements and exchange rate movements? Assume that the domestic volatility (standard deviation in yen) of the Japanese bond market is 8%. The volatility of the yen against the U.S. dollar is 6%. Suppose the dollar volatility of the bond market is 11.35%, what can you conclude about the correlation between the Japanese bond market movements and exchange rate movements?  Assume that you have a portfolio made up of 50% of Japanese stocks, 20% of Japanese bonds, and 30% of U.S. stocks. Formulate an expression for the sigma (in dollars) of your portfolio in terms of sigma_JS^2, sigma_USS^2, sigma_JB^2, rho_USS_JS, rho_USS_JB, rho_JS_JB where the subscripts JS, USS and JB denote Japanese stocks, US stocks and Japanese bonds respectively.

Explanation / Answer

a.Total risk of Japanese stocks in $: 2 2 2 in $ in $/ in $/ in 2 338.4 18.4%

b.Risk of a diversified portfolio 50 US/50 Japan, measured in $: 2 portfolio (0.5 US ) 2 (0.5 japan ) 2 2 0.5 0.5 0.4 US japan 219.45 14.8%

c.Because the actual dollar volatility is 18.4%, we conclude that the correlation between Japanese stock market returns and exchange rate movements is negative. The actual correlation is 0.1. This can be explained by the idea that a weak currency is associated with rising stock prices; a depreciation of the yen is good for Japanese corporations.

If the correlation between stock market returns and exchange rate movements were equal to zero, the dollar volatility of the Japanese stock market would be: 2 2 2 2 2 in $ in $/ in $/ 2 in $ 2 (18) (6) (2)(0)(18)(6) 360 18.97%.

d.Because the actual dollar volatility is 11.35%, we conclude that the correlation between Japanese bond market returns and exchange rate movements is positive.

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