Fnal Exam Fall 4 Econ 8-67 Closed Book. Formula Sheet Provded. Calculators OK. Tme Allowed: hours Please wrte your answers on the page below each queston. (5 ponts) Assume that the rsk-free nterest rate s % and that gold s costless to store. The spot prce of gold s $, per ounce. Accordng to spot-futures party, what must the futures prce for delvery of gold one year hence be?. (5 ponts) A factory costs $ mllon to buld. To fnance the factory, the frm borrows $5 mllon at a % nterest rate. The profts from the factory are $5, per year. What s the return on equty?. ( ponts) Mergenthaler Enterprses stock sells for $8. After one year, t wll ether go up to $94 (wth probablty.75) or down to $7 (wth probablty.5). The rsk-free rate s %. Fnd the prce of a European call opton to buy one share n Mergenthaler Enterprses for $8 one year from today. 4. (5 ponts) Suppose that the rsk-free nterest rate s percent. Consder the followng possble confguraton of prces of European optons and current share prces on a non-dvdend payng stock wth a maturty of one year hence. Say f t s possble or mpossble. If t s mpossble, please say why. Strke Prce Value Call Opton 6 Put Opton 6 Stock - 5. ( ponts) What s the duraton of a bond that sells at par, has a coupon rate of 8% per annum (pad annually) and a remanng tme to maturty of years? 6. ( ponts) Suppose that there are two potental rsky assets, A and B. The expected return on A s 6% and t s standard devaton s 5%. The expected return on B s 8% and t s standard devaton s %. The covarance (not correlaton) between the two assets s -.5. The rsk-free rate s 4%. You want to form a portfolo of assets A and B alone so as to maxmze the Sharpe rato. What weght wll you put n asset A? 7. ( ponts) Please state whether each of the followng statements s true or false. No explanaton needed. () The prce of a call opton on the S&P 5 ndex wth an exercse prce of, s greater than the prce of a call opton on the S&P 5 ndex wth an exercse prce of,4. () Callable bonds have a hgher yeld to maturty than otherwse dentcal bonds that are not callable. () In fall 8, the ol futures curve was n normal backwardaton. (v) Fxed rate mortgages had hgher delnquency rates than adjustable rate mortgages durng the fnancal crss.
(v) Amercan optons cannot be worth less than European optons. 8. (5 ponts) Suppose that the rsk-free rates n the Unted States and n the Unted Kngdom are % and %, respectvely. The spot exchange rate between the dollar and the pound s pound = $.5. What should the futures prce of the pound for a one-year contract be to prevent arbtrage opportuntes? 9. ( ponts) Suppose that a stock s tradng for $. Its volatlty s σ =.. The stock pays no dvdends and the rsk-free nterest rate s zero. You buy a European call opton on ths securty at a strke prce of $ wth an expraton of year hence. (a) Accordng to the Black-Scholes formula, what should ths call opton be worth? (b) What s the delta of ths opton (.e. how many stocks would you need to buy to hedge t)?. ( ponts) The Japanese one-year nterest rate s, and the Australan one-year nterest rate s 4%. The current exchange rate s that Australan Dollar = 9 Yen. Suppose that n one years tme, you thnk that there are three possble values for the Australan Dollar-Japanese Yen nterest rate as n the table below Exchange Rate Probablty AUD = 75 Yen % AUD = 85 Yen % AUD = 95 Yen 7% Consder the carry trade strategy of startng wth a fxed amount of yen, convertng to Australan dollars, nvestng n Australa at the one-year nterest rate, and convertng back at the end of the year. What s the Sharpe rato on ths strategy from the perspectve of a Japanese yen nvestor?. ( ponts) Consder the followng data for a one-factor economy. All portfolos are well-dversfed. Portfolo Expected Return Beta A % B F 6% (a) Accordng to APT, what s the expected return on portfolo B? (b) Suppose that another portfolo, portfolo E, s well-dversfed wth a beta of 4 and an expected return of %. Would an arbtrage opportunty exst? If so, what would be the arbtrage strategy?. ( ponts) (a) To test the CAPM, researchers have run cross-sectonal regressons of the return for stock ( r ) onto the book-to-market rato of that stock ( BM ), the sze of that stock ( S ) and the beta of that stock ( β ): r = γ + γbm + γs + γβ + ε (a) Accordng to the CAPM, what should the coeffcents γ, γ, γ be? (b) When the regresson n (a) was run by Fama and French and other researchers, what results dd they fnd? Please be very specfc about the sgns of the coeffcents that they found for γ.
Solutons. $,*.=$, per ounce.. Equty s $5,,. Profts net of nterest costs are $5,-$5,=$5,. So return on equty 5, s 7% 5,, =.. If I buy the call opton, the payoff wll be $ on the upper branch and $ on the lower branch. Suppose I buy the stock and borrow $6.64. Ths costs me $6.6. On the lower branch, the payoff wll be $7-$7=$. On the upper branch, the payoff wll be $94-$7=$4. So ths portfolo s effectvely call optons. The prce of the call opton should be $8.8. 4. Impossble. It volates the arbtrage condton C S. 5. The prce of the bond s $ and the yeld s 8%. The duraton s 8 8 8 ( *) + ( *) + ( *) =.78 years..8.8.8 6. The expected excess returns on A and B are % and 4%, respectvely. So the weght on asset A s.*..4*(.5). = =..*. +.4*.5 (. +.4)*(.5). 7. () True. () True. () False. (v) False. (v) True. 8. By covered nterest party, the futures prce F satsfes.5*. = F *. Solvng ths gves the futures prce of pound=$.5.. 9. (a) In the Black-Scholes formula, d = (ln( ) + ) /. =.5 and d =.5. =.5. From the normal dstrbuton tables, N (.5) =.5596 and N(.5) =.444. Hence the call prce should be C = (*.5596) (*.444) = $.58. (b) The hedge rato s N (.5) =.5596.. The returns n the three scenaros are: Exchange Rate Return Probablty AUD = 75 Yen -. % AUD = 85 Yen -.778 % AUD = 95 Yen.97778 7% The mean return s.556 and the varance s.5875, so the Sharpe rato s.67.
. (a) The key equaton s Er ( ) = r + βλ. Because the expected return on F s 6%, we know that the f rsk-free rate must be 6%. Because the expected return on A s %, we know that λ =.. Hence the expected return on B must be 9%. (b) Wth a beta of 4, the expected return on E ought to be 8%. So here s the strategy. Invest $ n E. Payoff s $*(.+4F). Invest -$4 n B. Payoff s $-4*(.9+F) Invest $ at the rsk-free rate. Payoff s $*.6 Addng the peces up, there s no cost today, but my payoff s $. Ths s an arbtrage.. (a) γ should be the rsk-free rate, γ should be zero, should be the expected market excess return. (b) γ s postve, γ s negatve, s smaller than the average market excess return.
Gradng Rubrc for Fnal. 5 ponts for correct answer. ponts for correct answer apart from vsble algebra error.. 5 ponts for correct answer. ponts for correct answer apart from vsble algebra error.. 6 ponts for constructng the replcatng portfolo ncludng workng out the cost of the portfolo and payoffs on the two branches. ponts for complete answer. ponts off for any algebra error. 4. No partal credt. 5. 7 ponts for wrtng down the formula. ponts for the correct answer (wth or wthout formula). 6. ponts off for usng returns nstead of excess returns; ponts off for mxng up standard devaton wth varance; ponts off for algebra mstake. No credt f does not have answer and does not wrte down the correct formula (apart from errors above). 7. No partal credt. 8 5 ponts for correct answer. ponts wth algebra error. No ponts for an answer of.5/.. 9. (a) 6 ponts. ponts off for algebra mstake, falure to look up normal dstrbuton tables correctly, mxng up varance and standard devaton or other specfc ms-applcatons of the Black-Scholes formula. Must have the elements of the Black-Scholes formula for any credt. (b) 4 ponts. ponts for dentfyng that the hedge rato s N(d) and more ponts for workng ths out. Full credt for ths part f got the wrong d n (a) but worked out N(d) correctly.. ponts for computng returns, ponts for computng the mean, ponts for computng the varance and ponts for computng the Sharpe rato. No ponts on any part that contans an algebra error. But an algebra error on one part should not affect mark on other parts. For example, f student computes returns ncorrectly, but gets everythng else rght wth those ncorrect returns, then (s)he would get 8 ponts. (a). 4 ponts for correct answer. ponts for wrtng down formula correctly but makng an algebra error. (b). 6 ponts for correct answer. If student has the rght arbtrage strategy, there s no need to dentfy what the expected return on E should be. If student does not have the rght arbtrage strategy, then ponts for dentfyng that the expected return on E ought to be 8%. No need to compute that the arbtrage strategy has a payoff of $. But no credt for a student who gets the drecton of the arbtrage backwards. (a). 4 ponts. pont for each of the four coeffcents. (b) 6 ponts. ponts for sayng that γ s postve, and more ponts for sayng that γ s negatve. Students who correctly dentfy the sgns of γ get full credt, because that s what the queston specfcally asked for. Students who do not correctly dentfy the sgns of γ, but who do make correct statements about γ or γ get one pont for each correct statement.