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11/4/13 keyspring 03 www2.econ.iastate.edu http://www2.econ.iastate.edu/classes/econ353/kilkenny/key2f03.htm keyspring 03 KEY #2 Econ 353 Fall 2003 Prof. Kilkenny last updated: November 19, 2003 1. Ch4 Q6 What is the yield to maturity on a $1000 face-value discount bond maturing in one year that sells for $800? The yield to maturity ("i*")of this security with F=$1000 and P = $800 is found as follows. n Rearrange the DPV formula (which is F = P(1+i) ) to solve for i*: i* = (F/P) - 1 for n=1 => 1000/800 - 1 = 5/4 - 1 = 1/4 = 25% 2. Ch4 Q2 You have just won $20 million in the state lottery, which promises to pay you $1 million (tax free) every year for the next 20 years. Have you really won $20 milion? Getting annual installments of $1 mil. over 20 years is not worth $20 million today for at least three reasons. One, you would have to forgo the interest you could have earned if you got all the $20mil right away. This is shown by rearranging the discounted present value formula (Equation (1), page 63, to solve for F given a P=$20 million today. After 20 years, you’d have a lot more than $20 million if they gave $20 million to you now (at i=4%, you’d have $31 million after 20 years). Two, money later is just not as desirable as money now. Using Equation 3 page 66 (where all the Cs and the F are $1 million), the discounted present value formula shows that the sum of payments over 20 years is worth less than $20 million dollars today (at i=4%, its only worth $13.6 million today). Three, inflation between now and then also reduces the real value of the later payments (or later earnings). 3. Ch4 Q7 What is the yield to maturity on a simple loan for $1 mil. that requires a repayment of $2 million in five years time? The yield to maturity of this security with F= $2 mil, P = $1 mil., and n=5 is found by rearranging the DPV formula, and solving for i*: i* = = 0.149 => 14.9% 4. “The recent Hot Lotto jackpot was $2.7 million. The winner was allowed to choose between receiving $1.64 mil. cash immediately, or $2.7 mil. over 25 years at the rate of $108 thousand each year.” Note: using equation (2) or (3), the present discounted value of the stream of $108K for 25 years, at an i=4% is = $1,754,672 (or $1.75 million). Prof K used both EXCEL formulas and summation (like equation 2), and the financial calculation [=PV(prime rate,n,C,,1)] to calculate it. a) P = F/(1+i) discounts a future value F to a present value P for a simple loan. (Mishkin, pg 63). b) The Hot Lotto immediate cash payout, $1.64 mil.= P. www2.econ.iastate.edu/classes/econ353/kilkenny/key2f03.htm 1/5 11/4/13 keyspring 03 The future (“face”) value, F = $2.7mil. c) The time to maturity, n = 25 years. d) The yield to maturity that discounts the lump sum future value to the present value is: i*= = 0.0201 => 2.01% e) The prime rate, the interest rate charged by large commercial banks on loans to their best corporate customers, is currently 4%(9/11/03; data source: http://federalreserve.gov/releases/h15/update/). f) If we invested $1.64 mil. at the prime rate for 25 years, we’d have: n 25 F = P(1+i) = $1.64(1.04) = $4,371,972 g) If we invested all of each annual payment, C = $108,000; each and every year at the same prime rate, after 25 years we’d have: F = ΣnC(1+i) = $4,667,668 Note: Prof K uses EXCEL formulas and [=FV(prime rate,n,C,,1)] to verify this solution. *)Which would you rather have, the cash now or the stream of payments? Prof K. would rather have the cash now, as do most Lottery winners. But that is just a personal preference (“Life is short”). Because this last question (only) is about opinions, there are no wrong answers. 5. Ch4 Q9 Which $1000 bond has the higher yield to maturity, a 20-year bond selling for $800 with a current yield of 15%, or a one-year bond selling for the same price ($800) with a current yield of 5%? On page 70, Mishkin says, "When a coupon bond has a long term to maturity (say, 20 years or more), …you would expect the current yield to be a rather close approximation of the yield to maturity.” On pages73- 75, Mishkin “shows how misleading the current yield can be for a bond with short maturity…" 20 Year Bond One Year Bond F = $1,000 F = $1,000 n = 20 = n = 1 = short term, so the current yield is “misleading.” long term, so the current yield is a “rather If the bond is trading at a premium (Pis higher than F) as in the close approximation of the yield to example on pages 73-75, current yield will overestimate the maturity.” Prof K: this is because the DPV yield to maturity because it ignores the capital loss at maturity. of the capital gain ($1000-$800= $200) If it is trading at a discount, current yield will underestimate it by after 20 years is negligible. ignoring the capital gain. www2.econ.iastate.edu/classes/econ353/kilkenny/key2f03.htm 2/5 11/4/13 keyspring 03 P = $800 P = $800 i = C/P = 15% i = C/P = 5% CY CY => C = .15*800 = $120 => C = .05*800 = $40 note: CR = C/F = 12% current yield estimate of i* = (F+C)/P - 1 = (1040/800) - 1 = .3 i* is approx 15% i* = 30% In sum, we can safely say that this one year bond has a much higher yield to maturity (30%) than the 20 year bond (about 15%). **PLEASE NOTE how unsatisfactory is the answer in the back of the book. 6. “The best time to take out a mortgage is when the real cost of borrowing is relatively low.” a) The Fisher equation is i = R + Π, so i =Ri - Π , and eΠ = i -Ri . b) The nominal interest rate, i, on 30-year T-bonds (on 9/11/03) was 5.22% c) The real interest rate on 30-year T-bonds (9/11/03): 2.7% sources: http://www.bloomberg.com/markets/rates/index.html NOTE: interest rates vary hour-to-hour. So don’t worry if your quotes aren’t exactly the same as these in the key. It is a GOOD IDEAto date your rate quotes (like I do here). d) Given (a),(b) and (c), the current expected rate of inflation Π = i - R = 5.22 - 2.7 = 2.52% e) The nominal interest rate on a 30 year fixed-rate mortgage is 6.26% f) i(30) mortgage rate last September was 6.09% sources: http://www.freddiemac.com/pmms/pmms30.htm NOTE: mortgage interest rates vary regionally (as well as day-to-day). For comparability, PLEASE use data only from the assigned data site(s). Note that this question asked for past month and year data to avoid the problem of date/time variability. g) Given (d), the expected real interest rate on a 30yr mortgage isRi = i - Π = 6.26 – 2.52 = 3.74% h) Given last year’s inflation rate of 1.5%, the real interest rate on a 30 year mortgage last year was www2.econ.iastate.edu/classes/econ353/kilkenny/key2f03.htm 3/5 11/4/13 keyspring 03 R = i - Π = 6.09 – 1.5 = 4.59% i) In sum, this year’s real cost of a mortgage is even lower than last year’s, so now is still a good time to buy a house. Table (not required) This year Last year N 30yr T-Bonds 5.22% R 30yr TIIS 2.7% eπ, actual π 2.52% 1.5% (calculate) N 30yr mortgage 6.26% 6.09% R 30yr mortgage 3.74% 4.59% (calculate) 7. Ch4 Q12: “If there is a decline in interest rates, which would you rather be holding, long-term bonds or short-term bonds? Why? Which type of bond has the greater interest rate risk?” Interest-rate risk is “the riskiness of an asset’s return resulting from interest-rate changes,” (Miskin, 2004, pg 78.) On the same page he says that “Prices and returns for long-term bonds are more volatile than those for shorter-term bonds,” so the answer to the third part of this question is that long-term bonds carry greater interest-rate risk. Prof K: Remember that the ONLY case in which the rate of return (RoR = i CY + g; equation 10) equals the yield to maturity is when the security is held for the whole time to maturity. Remember also the facts (shown in Ch 1, figure 1) that short-term interest rates (the yield to maturity on 3-month T-Bills) are more volatile than long-term interest rates (yield to maturity on bonds n>20.) There is no contradiction here because returns and yields are two different things. Prof. K prefers to define interest rate risk as the possibility of a capital loss. That is, what is scary about volatility in security markets is not the possibility of gaining, but the possibility of losing. Because bond prices and interest rates are inversely related, bond-holding savers have reason to fear interest rates rising (bond prices falling). www2.econ.iastate.edu/classes/econ353/kilkenny/key2f03.htm 4/5 11/4/13 keyspring 03 Bond-holding savers are happy when interest rates fall, because bond prices rise, and could even sell at a premium (P>F). In this case, by selling those bonds they could get an even larger capital gain, and the rate of return will be higher than the yield to maturity. Remember also that all interest rates tend to move together (rise (or fall) at the same times). If all interest rates decline, all bond prices rise. Given that long-term bond prices will rise more than short-term prices, we can answer that we’d rather on▯termbond, because long-term bonds will offer a larger capital gain rate. If we sell them then, we may obtain a rate of return that is higher than the yield to maturity. Prof K: note also that the possibility of capturing a higher return would be needed by most savers to compensate them for the larger interest-rate risk on long-term bonds. www2.econ.iastate.edu/classes/econ353/kilkenny/key2f03.htm 5/5
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