Question

Suppose you can invest in only the market index fund and 12-month Treasury bill because of some regulation. E[r]=11%, SD[r]=20%, and rf=3% per year. You are a portfolio manager at Fidelity. Suppose, based on your price of risk and the portfolio theory, you invest today $0.4m in the market index fund and $0.6m in the 12-month treasury bill whose maturity is one year from now. One year later, the annual net return of the market index fund turns out 30%. If E[r], SD[r], rf, and your price of risk remain the same, how should you rebalance your portfolio? That is, do you have to sell or buy the market index fund after you observe its 30% return per year, based on the portfolio theory?

Answer 1

Given the fact that you had allocated 30% in market fund and 70% in risk-free asset, your expected returns were

E(R) = 30% x 11% + 70% x 3% = 5.4% and SD = 30% x 20% = 6%

However, your realized returns = 30% x 40% + 70% x 3% = 14.1%

In other words, your allocation to market fund = 0.4 x 1.3 = 0.52, while that to risk-free asset = 0.6 x 1.03 = 0.618

=> weight of market fund = 0.52 / (0.52 + 0.618) = 45.69% and weight of treasuries = 54.31%

Hence, you would have to rebalance your portfolio by selling 15.69%(45.69%-30%) of total assets from market fund and buy treasuries for the same amount to bring the asset allocation back to 30/70.

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