Solutions to Exam 1
1. (Chapter 1, slides, Q3) Suppose there are two investments.The expected returns from the investments are 10% and 15%, the standard deviation of the returns are 16% and 24%, and the correlation between returns is 0.2.Let w1 be the proportion of wealth put into the first investment. (a). Calculate the expected return and the standard deviation for portfolio w1=0,0.2,0.4,0.6. (b). Draw a picture of these risk and returns for w1. (c). What is it called? (d). Draw the picture of (c) when there is a riskless asset. (e). How should an investor choose the optimal investment? Solutions: (a)
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(c) It is called Efficient Frontier. An efficient frontier represents the limit of how far we can move in a northwest direction and is illustrated in the following figure. There is no investment that dominates a point on the efficient frontier, in the sense that it has both a higher expected return and a lower standard deviation of return. (d)
(e) Every possible combination of the risky assets, without including any holdings of the risk-free asset, can be plotted in risk-expected return space, and the collection of all such possible portfolios defines a region in this space. The left boundary of this region is a hyperbola, and the upper edge of this region is the efficient frontier in the absence of a risk-free asset.
2. (Chapter 1, Practice Questions 14) The return from the market last year was 10% and the risk-free rate was 5%. A hedge fund manager with a beta of 0.6 has an alpha of 4%. What return did the hedge fund manager earn? Solutions:
3. (Chapter 2, Q3)
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(a). How many banks are there in the United States? (b). What is the size of the assets held by these banks? Solutions: (a)&(b)
4. (Chapter 2, slides, Q8) Suppose that there is a severe recession and as a result the bank's loan losses rise by 3.2%of assets, to 4% next year. We assume that other items on the income statement in Table 2.3 are unaffected. (a) Calculate the pre-tax net operating loss. (b) Assuming a tax rate of 30%. Calculate the after-tax loss of about 1.8% of assets. (c) Calculate the reduction of the equity capital. (d) Determine whether the bank can survive. (e) What is the lesson of this exercise?
Solutions: (a) The pretax net operating loss is
pre-tax operating income
increase in loan loss
(b) The after-tax loss is
after tax payment
(c) The reduction in equity capital is
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(d) An arter-tax loss equal to 1.8% of assets can be absorbed by the equity capital. It would result in a reduction of the equity capital to 3.2% of assets. Even a second bad year similar to the first would not totally wipe out the equity. (e) Equity capital is important. 5. (Chapter 3, slides, Q7) Consider a mortality table. Assume that interest rates for all maturities are 4% and premiums are paid once a year at the beginning of the year. (a). What is an insurance company's break-even premium for $100,000 of oneyear term life insurance for a man of average health aged 90? (b). What is the premium payment?
Solutions: (a) & (b) If the term insurance lasts one year, the expected payout is 0.181789 x 100,000, or $18,179. Assume that the payout occurs halfway through the year. The premium is $18,179 discounted for six months at 4%. Assuming semiannual compounding, this is 18,179/1.02, or $17,822. 6. (Chapter 4, slides, Q2) What is the size of the assets in mutual funds industry in 2008? Solutions: Mutual funds have grown very fast in since the Second World War. These estimates of assets managed by mutual funds were over $10 trillion by 2008. About 50% of US households own mutual funds.
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7 (Chapter 4, slides, Q18) Suppose that a hedge fund manager is presented with an opportunity where there is a 0.4 probability of a 60% profit and a 0.6 probability of a 60% loss, with the fees earned by the hedge fund manager being 2 plus 20%....
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