ECO 342 Fall 2011 Chris SimsEXERCISE DUE TUESDAY 11/22 AND DISCUSSION QUESTIONS FOR PRECEPT(1) There is an investment opportunity that will pay a rate of return 6% with probability .5 anda rate of return -3% with probability .5. You have $1000 of your own money to invest.(a) If you simply invest your own money, what is your expected rate of return? the vari-ance of your rate of return?(b) Suppose you can borrow an additional $1000, achieving a leverage ratio of 1 to 1, pay-ing an interest rate of one per cent. What would be the distribution of rates of returnon the leveraged investment? What would the mean and variance of rates of return beon the leveraged investment?(c) For each of the following combinations of borrowing rates of interest and leverageratios calculate the distribution of rates of return and its mean and variance:(i) borrowing rate 1%, leverage ratio 20 to 1.(ii) borrowing rate 2%, leverage ratio 1 to 1.(iii) borrowing rate 2%, leverage ratio 20 to 1.(d) Can you state a general rule for determining from the distribution of rates of return onthe unleveraged investment and from the borrowing rate, whether increasing leverageincreases expected return?(2) Suppose the economy is in a steady state in which real balances Mt/Ptare constant and themoney stock is growing at the constant gross growth rate G, so Mt= GMt−1. Suppose thecentral bank has an old-fashioned central bank balance sheet, so that its liabilities are justMtand its assets Btare government bonds paying a gross interest rate Rt, with Bt= Mtalways (because the bank turns over seignorage earnings to the treasury). Assume thatthere is a constant gross real interest rate ρ that must match the real return on governmentdebt so thatRt≡ ρPt+1Pt.The central bank’s profits at date t in real terms are then (Rt−1−1)Bt−1/Pt.(a) Show that in this steady state these profits move one-for-one with the other measureof seignorage, (Mt− Mt−1)/Pt.(b) Suppose that the demand for real balances in steady state decreases with the nominalinterest rate, according to the money demand equationMtPt=1√Rt.Determine what steady state growth rate G for money maximizes seignorage revenue.Discussion questions(i) Under a pure gold standard (almost never seen) the central bank holds actual gold to backevery issue of currency. Its assets are all gold and its liabilities are all currency. Does itgenerate seignorage revenue by expanding the amount of currency? Why or why not?c2011 by Christopher A. Sims. This document is licensed under theCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License2 EXERCISE DUE TUESDAY 11/22 AND DISCUSSION QUESTIONS FOR PRECEPT(ii) Discuss the Stella article: What is a “revaluation account” and why do they appear onmany central bank balance sheets? What are examples of central banks that have issuedlarge amounts of debt on their own account? Why do they do that and what problemsdoes it create? Stella gives examples of central bank “recapitalizations” that do not actuallyimprove the bank’s situation, because the assets provided by the treasury are of little value.What are examples of these assets and why are they of little value? Stella documents adownward trend in central bank profits in recent years. Why does this raise problems andwhat might explain the