Financial Markets and Valuation  Tutorial 1: SOLUTIONS. Present and Future Values, Annuities and Perpetuities


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1 Financial Markets and Valuation  Tutorial 1: SOLUTIONS Present and Future Values, Annuities and Perpetuities (*) denotes those problems to be covered in detail during the tutorial session (*) Problem 1. (Ross, Westerfield & Jaffe) You have won the Florida state lottery. Lottery officials offer you the choice of the following alternative payouts: Alternative 1: $10,000 one year from now. Alternative 2: $20,000 five years from now. Which should you choose if the discount rate is a. 0%? b. 10%? c. 20%? d. What rate makes the options equally attractive to you? (a) PV(Alt. 1) = 10,000 < PV(Alt. 2) = 20,000 => Choose Alt.2 which has higher NPV (b) PV(Alt. 1) = 10,000/(1+0.10) = 9, < Alt. 2 = 20,000/( )^5 = 12, => Choose Alt.2 which has higher NPV (c) PV(Alt. 1) = 10,000/(1+0.20) = 8, > PV(Alt. 2) = 20,000/(1+0.20)^5 = => Choose Alt. 1 which has higher NPV (d) Equate 10,000/(1+r)=20000/(1+r)^5, to find r => (1+r)^4 = 2, r = 18.9% Problem 2. (Ross, Westerfield & Jaffe) Suppose you place $1000 in an account at the end of each of the next 4 years. If the account earns 12%, how much will be in the account at the end of 7 years? Solution : The $1,000 that you place in the account at the end of the first year will earn interest for six years. The $1,000 that you place in the account at the end of the second year will earn interest for five years, etc. Thus, the account will have a balance of $1,000 (1.12) 6 + $1,000 (1.12) 5 + $1,000 (1.12) 4 + $1,000 (1.12) 3 = $6, (*) Problem 3. (Ross, Westerfield & Jaffe) Assuming an interest rate of 10%, calculate the present value of the following streams of yearly payments: a. $1000 per year forever, with the first payment one year from today. b. $500 a year forever, with the first payment 2 years from today. c. $2,420 a year forever with the first payment 3 years from today. FMV/Tutorial 1 Solutions/Sept.Oct
2 a. PV (today) = 1000/0.1 = $10,000 b. PV (today) = [500/0.1]/1.1=5,000/1.1 = $4, c. PV (today) = [2420/0.1]/(1.1)^2= $20,000 (*) Problem 4. (Ross, Westerfield & Jaffe) You are saving for your retirement. You have decided that one year from today you will place 2% of your annual salary in an account which will earn 8% p.a. Your salary is 50,000 today, but it will grow at 4% p.a. throughout your career. How much money will you have for your retirement which will begin in 40 years? You are going to put aside 2% of your annual salary (which grows at 4% per year). The first payment one year from today is 2% * 50,000 * 1.04 = 1,040. To find the Future Value in 40 years time, it is easier to first determine the present value of this growing annuity, using the Growing Annuity formula. 40 Present Value of these savings is PV = 1, = 20, And them to compound the PV for 40 periods, to determine the Future Value in 40 years time when you retire: FV = 20, * [(1.08)^40] = $440,011 (*) Problem 5. (Ross, Westerfield & Jaffe) What is the future value three years hence of $1,000 invested in an account with a stated annual interest of 8 percent, a. compounded annually? b. compounded semiannually? c. compounded monthly? d. Why does the future values increase as the compounding period shortens? a. $1.000 (1.08) 3 = $1, b. $1,000 [1 + (0.08 / 2)] 2 3 = $1,000 (1.04) 6 = $1, c. $1,000 [1 + (0.08 / 12)] 12 3 = $1,000 ( ) 36 = $1, d. The future value increases because of the compounding. The account is earning interest on interest. Essentially, the interest is added to the account balance at the end of every compounding period. During the next period, the account earns interest on the new balance. When the compounding period shortens, the balance that earns interest is rising faster. (*) Problem 6. Societe Generale offers a 4.1% SAIR compounded quarterly, while BNP Paribas offers a 4.05% SAIR compounded monthly. Which bank offers the higher rate for your deposit? FMV/Tutorial 1 Solutions/Sept.Oct
3 At SG the effective annual rate is (1 + r/4)^41 = ( /4)^41 = At BNP Paribas the effective annual rate is (1 + r/12)^121 = ( /12)^121 = => Societe Generale is better Problem 7. (Grindblatt & Titman) You are considering a new business venture and want to determine the present value of seasonal cash flows. Historical data suggests that quarterly flows will be $3,000 in quarter 1, $4,000 in quarter 2, $5,000 in quarter 3, $6,000 in quarter 4. The annualized rate is 10 percent, compounded annually. a. What is the PV if this quarterly pattern will continue into the future (that is, forever)? b. How would your answer change if same quarter growth is 1 percent per year in perpetuity? a. Treat each quarter s cash flow stream separately and them add up to have full PV. PV(CF 1 st quarter) = PV(3,000 in 1 st quarter) + PV(3,000 perpetuity in every 1 st quarter) = 3,000/[(1.10)^(1/4)] + [3,000/0.10]/[(1.10)^(1/4)] = $32,223 PV(CF 2 nd quarter) = 4,000/[(1.10)^(2/4)] + [4,000/0.10]/[(1.10)^(2/4)] = $41,952 PV(CF 3 rd quarter) = 5,000/[(1.10)^(3/4)] + [5,000/0.10]/[(1.10)^(3/4)] = $51,206 PV(CF 4 th quarter) = 6,000/[(1.10)] + [6,000/0.10]/[(1.10)] = $60,000 TOTAL PV = $185,381 b. Same as in a. but now each quarterly cash flow stream is a growing perpetuity. And need to take into account that first cash flow of the perpetuity is already after the 1% annual growth. PV(CF 1 st quarter) = PV(3,000) + PV(3,030 GROWING perpetuity in every 1 st quarter) = 3,000/[(1.10)^(1/4)] + [3,030/( )]/[(1.10)^(1/4)] = $35,803 PV(CF 2 nd quarter) = 4,000/[(1.10)^(2/4)] + [4,040/( )]/[(1.10)^(2/4)] = $46,614 PV(CF 3 rd quarter) = 5,000/[(1.10)^(3/4)] + [5,050/( )]/[(1.10)^(3/4)] = = $56,895 PV(CF 4 th quarter) = 6,000/[(1.10)] + [6,060/( )]/[(1.10)] = = $66,667 TOTAL PV = $205,979 FMV/Tutorial 1 Solutions/Sept.Oct
4 Bonds (*) Problem 8. (Ross, Westerfield & Jaffe) A bond with the following characteristics is available. Principal: $1,000 Term to maturity: 20 years Coupon rate: 8 percent, annual payments Calculate the price of the bond if the stated annual interest rate is: a. 8 percent b. 10 percent c. 6 percent. a. Since the coupon rate coincides with the discount rate, the present value of the bond should be equal to the face value, i.e. the bond is traded at par. PV (8%) = (0.08)*1,000 * A (20 periods, 8%) + 1,000/(1+0.08)^20 = = 80* ,000* = = 1,000 = Face value b. Since the coupon rate is less than the discount rate, the present value of the bond is less than the face value, i.e. the bond is traded at a discount. PV (10%) = (0.08)*1,000 * A (20 periods, 10%) + 1,000/(1+0.10)^20 = = 80* ,000* = = < 1,000 c. Since the coupon rate exceeds the discount rate, the present value of the bond exceeds the face value, i.e. the bond is traded at a premium. PV (6%) = (0.08)*1,000 * A (20 periods, 6%) + 1,000/(1+0.06)^20 = = 80* ,000 * = = 1, > 1,000 (*) Problem 9. (Ross, Westerfield & Jaffe) You have just purchased a newly issued $1,000 5year Vanguard Company bond at par. This 5year bond pays $60 in interest semiannually. You are also considering the purchase of another Vanguard Company bond that returns $30 in semiannual interest payments and has six years remaining before it matures. This bond has a face value of $1,000. a. What is the effective annual return on the fiveyear bond? b. Assume that the rate you calculated in part (a) is the correct rate for the bond with six years remaining before it matures. What should you be willing to pay for that bond? c. How will your answer to part (b) change if the fiveyear bond pays $40 in semiannual interest? a. Since issued at par, the semiannual coupon rate equals the semiannual discount rate = 6% per semester or, equivalently, the annual coupon rate is 12% per year. To determine the effective annual discount rate (EAIR), we have to take care of the effect of first annual coupon being reinvested and earning interest. => 1 + EAIR = [( 1 + annual coupon rate / m ) ]^(m) FMV/Tutorial 1 Solutions/Sept.Oct
5 where m=2 semesters per year => EAIR = [(1+0.12/2 )]^2 1 = 12.36% b. To determine the price of bond it s better to work with the effective semiannual rate of 6% (although the annual rate is 12.36% as shown in a). => PV (6year bond) = 30 * A(12 periods, 6%) + 1,000/(1+0.06)^12 = = 30 * *0.497 = = = < 1,000 c. The new effective semiannual discount rate is now 4% => PV (6year bond) = 30 * A(12 periods, 4%) + 1,000/(1+0.04)^12 = = 30 * * = = = < 1,000 FMV/Tutorial 1 Solutions/Sept.Oct
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