Question

You have 3 billion dollars in the fund, which you can invest in any combination of Australian stocks, US stocks, and Australian Treasury. The idea is to use your knowledge of portfolio theory to make an argument for having an internationally diversified portfolio, rather than just holding domestic assets. The data are monthly returns and the relevant sample statistics are summarized in the following table:

Stock	E[R]	Var(R]	Cov(Aus, US)
Aus Index	0.00959	0.00222	0.00088
US Index	0.00727	0.00348
Aus Treasury	0.00300	0.00000

1. Using the results of portfolio theory and the estimates above, compute the tangency mutual fund (portfolio) between Australian and US stocks (i.e., the optimal split between Australian and US stocks). Find the tangency portfolio using the Solver in Excel. Paste the table used with Solver to your Word document and discuss your findings.

2. Suppose you would like to achieve an average return of 0.5% per month in excess of the T-bill rate with the smallest possible risk. What is the optimal split between Australian stocks, US stocks, and T-bills? That is, how much of the $3 billion should you invest in each country and how much should you borrow or lend? What is the standard deviation of this portfolio?

3. After a bad year on the US stock market, some people try to influence you to divest (i.e., sell all of) the holdings of US stocks. How much should you invest in Australian stocks and T-bills alone to obtain the same level of risk as you obtained in part 2.? (Hint: you want the standard deviation of the divested portfolio to be the same as the nondivested portfolio.)

4. What would be the cost in terms of expected monthly return from divesting in the US stocks? What would be the cost in terms of annual return (note: the returns are continuously compounded)? What would be the cost in dollar terms on the $3 billion portfolio each year?

Answer 1

1) Risk free rate of return = 0.00300 = 0.3 %

Stock	E(r)	Var.(r)	Std, Devn.(r)	Cov.(Aus,US)
Aus Index	0.00959	0.00222	0.04711	0.00088
US Index	0.00727	0.00348	0.05899
Aus Treasury	0.00300	0.00000

	Weight
Aus Index	0.839075
US Index	0.160925
Sum	1
Portfolio return	0.0092167
Portfolio Variance	0.0018903
Portfolio SD	0.0434775
Sharpe ratio	0.1429855

Optimal portfolio consists of 83.9075% investment in Australian index and 16.0925% investment in US Index.

Portfolio return in this case is 0.92167% per month and standard deviation is 0.189%. Sharpe ratio is 0.1429855

2) Target return = 0.5% + 0.3% = 0.8 %

Investment targets: Aus Index = 65.55264% of $ 3 billion = $ 1,966,579,200

US Index = 15.95038% of $ 3 billion = $ 478,511,400

Aus Treasury = 18.49698% of $ 3 billion = 554,909,400

Portfolio Standard deviation = 0.03148151 = 3.502%

3) Investment in Aus stocks = 74.33192% of $ 3 billion = 2,229,957,600

Investment in Aus treasury = 25.66808% of $3 billion = 770,042,400

4) Portfolio return (after divestment of US stocks) = 0.78985 %

Cost in terms of returns = 0.8 - 0.78985 = 0.01015 %

Cost in terms of annual returns = e^0.0001015 - 1 = 0.000101505 = 0.0101505 % per annum

Cost in terms of monthly returns = 0.0101505 / 12 = 0.000845875 % per month

Cost in dollar terms = 0.0101505 % * $ 3 billion = $ 304,515