1. Calculate the expected value, variance, and standard deviation for the following possible asset values. "(a,b) for prob. (p,q)" means X takes on value a and b with probabilities p and q.
(a) (1,2,3) and (0,2,4) for prob. (1/3, 1/3, 1/3) [2 cases].
(b) (9,10,11), (8,10,12), and (0,10,20) for prob. (1/4, 1/2, 1/4) [3 cases].
(c) (1,2,3) and (0,2,4) for prob. (1/10, 8/10, 1/10) [2 cases].
(d) (0,1), (0,4), (9,11), (8,12), (0,20) for prob. (1/2, 1/2) [5 cases].
2. What general propositions do each of the parts (a) - (d) of the previous problem illustrate?
3. Suppose a junk bond is worth $1000 with probability 0.90 and $200 with probability 0.10. What would an investor for square root utility be willing to pay for this bond? Please explain why that figure is less that the mathematical expected value of the dollar payouts.
4. Suppose the risk-free interest rate of 10% can be obtained by buying government bonds. Stock in Company A, on the hand, sells today for $20 per share and has an expected value next year of $25, which is an increase of 25%. Unfortunately, the standard deviation of the stock's value next year is 8 so it is not without risk. What are the mean and standard deviation (for next year) of a portfolio consisting of $500 in government bonds and $500 in stock? How about the portfolio of $1000 in bonds and no stock and the portfolio of no bonds and $1000 in stock?
5. Using the information in the previous question, what are the mean and variance (for next year) of a portfolio consisting of $2000 in stock financed by $1000 of your money and $1000 borrowed from a bank at 10%?
6. Graph the results for the two previous questions using expected dollars on the vertical axis and standard deviation on the horizontal axis. (A sketch will be fine.) Should you get a straight line? Why or why not?
7. Prove or disprove the following conjecture. Allowing for various proportions of your wealth invested in two assets, a portfolio containing those two assets can be no safer than the safer asset of the two.
8. Draw two intertemporal substitution diagrams, each at least 4 inches by 4 inches. On one illustrate the income and substitution effects of an increase in interest rates for a saver. On the other illustrate the income and substitution effects of an increase in interest rates for a borrower.
9. Draw a production possibility frontier diagram that illustrates the motivation for saving for your retirement when you are in the 30-60 age range.
10. Suppose the one-year interest rate this year is 5%, but that rate is expected to increase to 7% next year? What current two-year rate would satisfy the expectations hypothesis?
11. Suppose the current one-year interest rate is 8% and the current two-year interest rate is 12%. You are going to receive $100 at the end of the fist year and you want to invest that sum over year two. Explain how you can lock in the forward interest rate implicit in the current term structure. What is that forward rate?
12. Suppose the difference between the one-year and two-year rates in the previous question is interpreted as a risk premium. What is the risk in locking in the forward rate?
13. Draw a diagram comparing the current yield curve to the ones on the handout.