ECN 134 Financial EconomicsAnswer Key for Midterm1. (8) A twent y-year zero coupon with face value $1000 bond sells at $400;what is the five-year spot rate?Answ er : PB= 400 = PVB=1000(1+r)20<=>r=(1000400)1/20− 1=0.0468.Therefore, the 20-year spot rate is roughly 4.7%.2. (10) A rational person will never save at negative real in terest rate. Trueor false? Explain briefly.Answ er : False. If you think saving just as an opportunity for investment,this statement may be true. However, you may really need to save today becausey ou know that you won’t hav e any income and you can’t consume anything inthe future. So, in this case, you would even save at a negative inter est rate. Avery good example is saving for retirement. During the retirement, the futureconsumption is going to be zero unless you saved a part of your income whilew orking. Hence, no matter whether interest rate is positive or negative you haveto save in order to ensure the future consumption for your retirement period.3. (14) i) State Fisher’s separation theorem of the appendix to chapter 4carefully. Why is it called “separation” theorem?Answ er : FST is the fundamental justification for why we compute thepresent values. The theorem states that with a perfect capital market (i.e. con-stant interest rate and lend & borrow freely) one can rank the opportunities ofthe investments independent of people’s preferences. This is the reason why welook ed at present values or future values (or in other words people’s budget sets)and disregarded prefernces. So in this sense, the theorem seperates preferencesfrom the investment decisions. See the appendix to chapte r 4 in the textbookfor a detailed treatment.ii) Describe verbally and graphically a situation in which its assertion fails,and explain brieflywhy.Answ er: There are several cases where F isher’s separation theorem fails.One of them is a situation where one cannot borrow and lend at the samein terest rate (class room example). This implies that the budget line has a kinkattheendowmentpoint. Inthiscasewehavetoknowabouttheconsumer’spreference to decide which investment is better. Another case is the problem 3in PS 1, where we can not borrow freely.14. (16) a) You want to buy a house that costs $400,000. You can mak e adownpayment of $100,000 and take a $300,000 mortgage at 7% interest. Youhate to be in debt, though, an want to pay off the mortgage as soon as possible.Assuming you can come up with the same amount C each year to pay off themortagage, how much do you need to pay annually to be debt-free within 10years.Answ er : Now, your debt is $300,000 with the interest rete 7%. UsingAnn u ity Formula, 300, 000 =C0.07(1 − (11+0.07)10) yields C =42, 713.33 Hence,you need to pay at least$42, 713.33 to be debt-free within 10 years.b) But you can’t really afford that much. On reflection, the maximum youcould come up with each year is $28,000. How long would it take you to pay itback?Answ er :300, 000 =28,0000.07(1 − (11+0.07)t) <=> (1 + 0.07)t=4=221) By applying doubling rule, the "doubling year" td=0.70.07=10. Thus,t =2∗ td=20Hence, after about 20 years you can pay back your debt.2) By taking log on both sides, t =2∗log 2log 1.07=2∗ 10.25 = 20.5c) Compare the total $ payments in a) and b); which one is larger? (Hint:you d on’t need to correctly answer the earlier parts to answer c) correctly)Answ er : 10 ∗ 42, 713.33 = 427, 133.3 < 20 ∗ 28, 000 = 560, 000 Th us, incase of b) your payment is larger. Intuitively, this is because you pay off less ineac h period, you need to pay off the debt over a longer period, which implies bythe "power of the compounding" your debt incresaes over time exponentially,hence you will ultimately pay more.5. (12) Over the 500 months period from 1950-1991, the monthly return onGerman stocks was on average 1%, with a standard deviation of 4.1% . Assumethat monthly returns are (approximately) normally distributed; a table of thenormal distribution is attached at the back.i) In (approximately) how many of these 500 months did the monthly returnexceed 0% but not 6%?Answ er : Here, the monthly return on German stocks are the randomvariable. Let denote X the the return on German stocks. Then X˜N(1, (4.1)2),hence the corresponding z-score has to be z =X−14.1.Pr(0 <X<6) = Pr(0−14.1<z<6−14.1)=Pr(z<6−14.1) − Pr(z<0−14.1)=Pr(z<1.22) − Pr(z<−0.24) = 0.8888 − 0.4052 = 0.48362500 ∗ 0.4836(= 241.8) For about 242 months, the monthly return exceeded0% but not 6%.ii) In the worst 20 months, German stocks lost x % or more of their value;x= __ _ ?Answ er :20500=0.04 = Pr(z<−1.75) = Pr(X−14.1< −1.75) = Pr(X<−6.175)Thus, in the worst 20 months, German stoc ks lost6.175% or more.6. (20) Consider a 20-year bond with face value $1000 and an 5% annualcoupon.i) If the annual interest rate was equal to 6% and constant, what would thebond’s PV be?Answ er :-PVofthefacevalueof1,000;1,000(1+0.06)20= 311.8- PV of the annual coupon stream ; Your constant annual coupon streamwill be 1, 000 ∗ 0.05 = 50500.06(1 − (11+0.06)20) = 573.496- PV of your bond ; 311.8 + 573.496 =885.296ii) If the annual interest rate was equal to 5% and constant, what would itsPV be?Answ er :-PVofthefacevalueof1,000;1,000(1+0.05)20= 376.889- PV of the annual coupon stream ; Your constant annual coupon streamwill be 1, 000 ∗ 0.05 = 50500.05(1 − (11+0.05)20) = 623.11-PV of your bond ; 376.889 + 623.11 = 1, 000Without calculating, since the interest rate(=0.05) is the same ascF=501,000=0.05 we can know that the PV will be 1,000iii) The price of the bond is $ 950. Compute its approximate yield to matu-rity by assuming that the PV is a linear function of the interest rate.Answ er :By linear approximation,1000−8850.05−0.06=1000−9500.05−rr =0.054 Thus, the yieldto maturity is roughly5.4% .7. (20) Exxon is forecast to earn $4/share next year. Each year, Exxon willrein vest 75% of its earnings at a rate of return 6%. Its required rate of returnis 8%.3i) What will Exxon’s dividends be next year?Answ er : The dividend is the remaining fraction of earning for next yearafter reinvestment : 4 ∗ (1 − 0.75) =1ii) What is Exxon’s fair share price now?Answ er : Exxon’s groth rate ; g =0.06 ∗ 0.75 = 0.045P = PV =div1r−g=10.08−0,045= 28.57iii) What will Exxon’s fair share price be in 10 years?Answ er : After 10 years, your dividend next year will be 1 ∗ (1 +0.045)10=1.55P10=