ECN 134 Financial Economics Answer Key for Midterm 1 8 A twenty year zero coupon with face value 1000 bond sells at 400 what is the five year spot rate 1000 1000 1 20 Answer PB 400 P VB 1 r 1 0 0468 20 r 400 Therefore the 20 year spot rate is roughly 4 7 2 10 A rational person will never save at negative real interest rate True or false Explain briefly Answer False If you think saving just as an opportunity for investment this statement may be true However you may really need to save today because you know that you won t have any income and you can t consume anything in the future So in this case you would even save at a negative interest rate A very good example is saving for retirement During the retirement the future consumption is going to be zero unless you saved a part of your income while working Hence no matter whether interest rate is positive or negative you have to 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 4 carefully Why is it called separation theorem Answer FST is the fundamental justification for why we compute the present values The theorem states that with a perfect capital market i e constant interest rate and lend borrow freely one can rank the opportunities of the investments independent of people s preferences This is the reason why we looked at present values or future values or in other words people s budget sets and disregarded prefernces So in this sense the theorem seperates preferences from the investment decisions See the appendix to chapter 4 in the textbook for a detailed treatment ii Describe verbally and graphically a situation in which its assertion fails and explain briefly why Answer There are several cases where Fisher s separation theorem fails One of them is a situation where one cannot borrow and lend at the same interest rate class room example This implies that the budget line has a kink at the endowment point In this case we have to know about the consumer s preference to decide which investment is better Another case is the problem 3 in PS 1 where we can not borrow freely 1 4 16 a You want to buy a house that costs 400 000 You can make a downpayment of 100 000 and take a 300 000 mortgage at 7 interest You hate to be in debt though an want to pay o the mortgage as soon as possible Assuming you can come up with the same amount C each year to pay o the mortagage how much do you need to pay annually to be debt free within 10 years Answer Now your debt is 300 000 with the interest rete 7 Using C 1 Annuity Formula 300 000 0 07 1 1 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 a ord that much On reflection the maximum you could come up with each year is 28 000 How long would it take you to pay it back Answer 300 000 28 000 0 07 1 1 1 0 07 t 1 0 07 t 4 22 0 7 1 By applying doubling rule the doubling year td 0 07 10 Thus t 2 td 20 Hence after about 20 years you can pay back your debt 2 By taking log on both sides t 2 log 2 log 1 07 2 10 25 20 5 c Compare the total payments in a and b which one is larger Hint you don t need to correctly answer the earlier parts to answer c correctly Answer 10 42 713 33 427 133 3 20 28 000 560 000 Thus in case of b your payment is larger Intuitively this is because you pay o less in each period you need to pay o the debt over a longer period which implies by the 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 on German stocks was on average 1 with a standard deviation of 4 1 Assume that monthly returns are approximately normally distributed a table of the normal distribution is attached at the back i In approximately how many of these 500 months did the monthly return exceed 0 but not 6 Answer Here the monthly return on German stocks are the random variable 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 1 4 1 6 1 0 1 Pr 0 X 6 Pr 0 1 z Pr z 6 1 4 1 4 1 4 1 Pr z 4 1 Pr z 1 22 Pr z 0 24 0 8888 0 4052 0 4836 2 500 0 4836 241 8 For about 242 months the monthly return exceeded 0 but not 6 ii In the worst 20 months German stocks lost x or more of their value x 20 Answer 500 0 04 Pr z 1 75 Pr X 1 4 1 1 75 Pr X 6 175 Thus in the worst 20 months German stocks lost 6 175 or more 6 20 Consider a 20 year bond with face value 1000 and an 5 annual coupon i If the annual interest rate was equal to 6 and constant what would the bond s PV be Answer 1 000 PV of the face value of 1 000 1 0 06 20 311 8 PV of the annual coupon stream Your constant annual coupon stream will be 1 000 0 05 50 50 1 20 0 06 1 1 0 06 573 496 PV of your bond 311 8 573 496 885 296 ii If the annual interest rate was equal to 5 and constant what would its PV be Answer 1 000 PV of the face value of 1 000 1 0 05 20 376 889 PV of the annual coupon stream Your constant annual coupon stream will be 1 000 0 05 50 50 1 20 0 05 1 1 0 05 623 11 PV of your bond 376 889 623 11 1 000 Without calculating since the interest rate 0 05 is the same as Fc 50 1 000 0 05 we can know that the PV will be 1 000 iii The price of the bond is 950 Compute its approximate yield to maturity by assuming that the PV is a linear function of the interest rate Answer By linear approximation 1000 885 0 05 0 06 1000 950 0 05 r r 0 054 Thus the yield to maturity is roughly 5 4 7 20 Exxon is forecast to earn 4 share next year Each year Exxon will reinvest 75 of its earnings at a rate of return 6 …