Math 5621Financial Mathematics IIMathematics of Corporate FinanceFall 2010Final ExaminationDecember 10 - 15, 2010This is a take-home examination due back to me by 5 PM Wednesday, De-cember 15, in my department mail box, under my o¢ ce door, or by email. Youmay consult any written source, including textbooks, notes, solution manuals,websites, or anything else written. You may NOT consult with any other per-son, which would result in your failing the course. Be sure to put your nameon all papers submitted. Please show all of your work and give all reasoningand calculations associated with your answers; give me a chance to give partialcredit on an incorrect answser. The seven questions will be equally weightedin the grading.1. Consider a put option with an exercise price of 50, expiring four yearsfrom today, on an underlying asset whch pays no dividends, has a marketvalue of 40 today, and a standard deviation of return equal to 0:40. Usea binomial model with N = 8 steps and probabilities qu= qd=12ateach step. (BE CAREFUL HERE: do not just copy formulas from thetextbook, there will be n o partial credit for use of formulas inappropriateto these instructions.) Use a risk-free annual rate of return of 2%. (BECAREFUL HERE: an annual rate of return is not the same thing as acontinuously compounded annual rate of return, a force of interest.)(a) What would be wrong with using the formulas for u and d in thetextbook?(b) What is the value of the put option today if it is an American option?(c) Why is this value greater than 10, which I could get for exercisingright away?(d) What is the earliest time it might possibly be best to exercise thisAmerican put option?(e) At time t = 1, if you are at the up-down node of the tree, willthe value of the risk-free bonds in the replicating portfolio for a putoption, after rebalancing the portfolio, be larger for the Americanput option or for the corresponding European put option? By howmuch?(f) Why are the two values in (e. ) di¤erent?2. Suppose that the c urrent price in the market for blank silicon wafers usedas raw material for chip manufacturing is $0:50 per wafer. Your engi-neering sta¤ tell you that their best and most reliable consultants forecast1that the price of blank silicon wafers will rise an average rate of 2% peryear for the next 3 years, 6% per year for the following 2 years, and reachlong run equilibrium at 3% per year thereafter forever. You think thatthe forecast makes a lot of sense. You expect to be using 400; 000 blanksilicon wafer per year in your manufacturing operation for each of the next20 years. Assume that blank silicon wafers have a  = 0, that the riskfree rate is 1% for the next three years and 3% thereafter forever, andthat any excess stock of silicon wafers from year to year can be stored fora negligible cost. For each of the the next 20 years you have purchaseda European call option expiring at the end of that year on 400; 000 blanksilicon wafers with a strike price of $0:75 per wafer to hedge your exposureto a rise in price. For each of th e next 20 years you have sold short aEuropean put option expiring at the end of that year on 400; 000 blanksilicon wafers with a strike price of $0:75 p e r wafer in order to help …nancethe call option purchase. What is the value today of your net positionin all of these options?3. The risk free rate is 2%. Portfolio A has an expected return of 11% and astandard deviation of returns of 64%. Portfolio B has an expected returnof 8% and a standard deviation of returns of 25%.(a) From a risk/reward perspective which of the two portfolios is supe-rior? Why?(b) Suppose the returns from the two portfolios have a correlation coef-…cient of 0:4. What is the optimal allocation ratio to each of A andB in a new portfolio to be constructed as a combination of a portionof A and a portion of B?4. Gimmel Inc. has a beta of 0:5 on its equity, 40% debt in its capitalstructure, with the debt being valued by the market as essentially risk-freeat a 5% pre-tax annual yield. The expected return on the entire market is13%. Gimmel is considering a project called Gamma to develop a chain ofhigh-end urban retail outlets for its products that it expects will producea cash yield of 25% annually on an afte r tax basis. The main competitorwill be Himmel Inc., which is thought to have similar risk characteristicsas project Gamma. Himmel’s equity beta is 2:2 and it has 25% debt in itscapital structure. Assume that the marginal tax rate for both companiesis 45% and that Gamma will be funded with 40% debt and 60% fromretained earnings. From a purely …nancial perspective should Gimmelproceed with the Gamma project? Give a speci…c …nancial an alysis andreason.5. The Black-Scholes formula for the price of a European call option isc = S (d1)  erTK (d2)where d1and d2are certain expressions that you can evaluate. Onceyou know d1, the value  (d1) can be obtained from a table of normal2probability distribution values (or a computer algorithm to obtain thosevalues). Therefore,  (d1) must be the probability of some event occuringor some state of a¤airs being realized. Correct? Explain exactly whatthat event or state of a¤airs is.6. A stock has a dividend yield of 2% and the company pays 7:5% intereston its long term debt. The ROE based on beginning of year equity is16%. The are 10 million shares outstanding. The market to book ratiois 1:25 and the share price is $40. The interest payments on the long termdebt amount to $2:50 per share. What is the maximum possible growthrate the company can …nance without using any new external sources of…nancing of any kind?7. For years, a company has plowed back 60% of earnings while making a 20%return on equity and maintaining a 3% dividend yield. The debt ratio hasremained constant. The market has priced the shares as if the growth ratecorresponding to this …nancial performance could continue forever. Bywhat % and in what direction will the share price change if the companysuddenly annou nce s, in a complete surprise to the market, that is hasno further opportunities for pro…table growth beyond its current scale ofoperations, it now plans no further growth at all, and will begin to payout all of its earnings as dividends every