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PV Problem Set 1. Assume a .05 (5%) time value of money. We have a debt to pay and are given a choice of paying $1,000 now or some amount X five years from now. What is the maximum amount that X can be for us to be willing to defer payment for 5 years? Loan = $1,000, time value of money 5%. We have two options: pay it now or defer the payment for 5 years. (i) if we pay it now, we should pay a $1,000. (ii) If we defer the payment 5 years, we must calculate the value of this $1,000 in 5 years from now. Using your calculator: PV=1000 I/YR=5 N=5 Press FV to compute FV PV (1 r ) n , Future value of a single value 1,000 0 1 2 3 4 5 Same ThAng!!! FV $1,000 (1.05) 5 1,276.28 Answer: The maximum amount for us to be willing to defer the payment for 5 years would be $1,276.28. 1 2. How much are you willing to pay today for the following cash-flows (CF’s): 100$ in year one, $200 in years two and three, and 400 in years four and five if the current interest rate is 7%? How about in year 6? How about in year 3? First, draw a time line and enter the cash-flows: 0 1 2 3 4 5 100 200 200 400 400 6 (Now, after all that work, take a break and have a beer—like most MBA’s, your probably exhausted at this point. Once you have had a couple MGD’s, and maybe a shot of mad dog, you will be ready do the hard part.) To find the price today, you compute the present value of the CF’s: PV 100 200 200 400 400 1021.76 1.07 1.07 2 1.07 3 1.07 4 1.07 5 Another way of writing this equation is: PV 5 100 3 200 400 t t 1.07 t 2 1.07 t 4 1.07 It is the same thing—study how the sum sign works. Payments 2 and 3 are 200 dollars, and payments 3 and 4 are 400 dollars. So this is the range of the indices on the respective sum signs. To compute the price in year 6—that is the future value in year 6—we simply future value the PV at the rate at which interest can be earned: FV6 1021.76 1.07 1533.38 6 Alternatively, we could compute it directly: 4 2 t 3 t 1 PV 100 1.07 5 200 1.07 t 400 1.07 t 1533.38 Which is the same as: 4 2 t 3 t 1 PV 100 1.07 5 200 1.07 t 400 1.07 t 1533.38 2 or: PV 100 1.07 5 200 1.07 4 200 1.07 3 400 1.07 2 400 1.07 (This is probably the way that makes the most sense to you. However, it is the same thing.)—Remember, you multiply and divide before adding and subtracting. And finally, to compute the value in year 3, we either FV the value from time zero, discount the value from time 6, or we do it the hard way, and start from scratch. Either way, done correctly, we still get the same answer. You should do it all three ways so you convince yourself. i. FV 3 1021.76 1.07 3 1251.70 ii. PV 3 1533.38 1.07 3 1251.70 iii. PV 100 2 1.07 200 1.07 200 400 400 1251.70 1.07 1.07 2 Thus, the answers are 1021.76, 1533.38, and 1251.70, respectively. Note: I have gone to a lot of trouble in this problem to show you how the sum sign functions. Study how it works so that you are fluent in it. Additionally, you should carry through the multiplication and compute answers two and three using the alternative methods, not just the quickest method. This will reinforce the concept of time value of money and how we are able to move money to different points in time. Finally, after this very rough evening, you deserve another beer! Join your brethren down at the local tavern, Gentleman’s (or Lady’s) club, or whatever haunt piques your interest. Remember, you’re an MBA, or at least you getting there. So go behave like one, and take the evening off—after all, this was a lot of work and you deserve a treat! 3 3. Suppose that you are considering an investment with the following cash flows that are being guaranteed by your kid brother: $100 in years 1-3, $300 in years 5-8. If the return that you want on the investment is 9%, what price should you pay today? Suppose that you are asked to make a single payment in year 4 for the entire set of cash flows, how much would you pay? 0 1 2 3 100 100 100 3 4 5 6 7 8 300 300 300 300 8 PV 1100 1300 941.66 .09t .09t t 1 t 5 Using your calculator – use the cash-flow function: CF0 = 0 CF1 = 100, (Number of times) N = 3 CF2 = 0, Nj = 1 CF3 = 300, N j= 4 I% or I%/YR = 9 Press NPV to compute the PV. To solve for the amount in period 4: 3 4 FV4 100 1.09 t t 1 t 1 300 1.09t , or, FV4 941.66 1.09 4 1,329.23 Using your calculator – use the TVM function: PV=941.66 I%/YR = 09 N=4 PMT = 0 Press FV. 4 4. Compute the present value for a coupon bond that promises to pay a coupon (interest payment) of $50 a year for 30 years and has a face value of $1,000. The first interest payment is one year from now. Use a rate of discount of .05. 30 years bond with coupon of $50 (5%), face value of $1,000 and 5% of discount rate. 30 PV i 1 50 1000 (1.05) i (1.05) 30 Using your calculator: PMT= 50 FV=1,000 I/YR=5 N=30 Press PV to compute Answer: The present value (price) for this bond is $1,000 5. A 20-year $1,000 coupon bond promises to pay a 4.5% coupon rate annually. The current interest rate is 5%. How much is the bond worth now? How much would the bond be worth if the current interest rate were 4%? 20 years bond with coupon of 4.5%, face value of $1,000 and 5% of discount rate. 20 PV i 1 45 1000 i (1.05) (1.05) n Using your calculator: PMT= 45 FV=1,000 I/YR=5 N=20 Press PV to compute Answer: The present value for this bond is $937.689 if the inters rate fall to 4% 20 PV i 1 45 1000 (1.04) i (1.04) n Using your calculator: PMT= 45 FV=1,000 I/YR=4 N=20 Press PV to compute Answer: The present value for this bond is $1,067.95 5 6. The newspaper headline states “Baseball Player Signs for $14 Millions.” A reading of the article revealed that the player will receive $1 Million per year for the next 6 years. He will then receive $400,000 per year for 20 years (i.e.: 6 * $1,000,000 + 20 * $400,000 = $14,000,000). Assuming that the player can borrow funds at 10% per year, what is the present value of this contract? Contract: 1 million for 6 years and then 0.4 millions for the next 20 years. This is the same as 0.6 million for 6 years plus 0.4 million for 26 years. The rate of return is 10%. 0 1 1 1 1 1 1 .4 .4 .4 … 1 2 3 4 5 6 7 8 9 … .4 .4 .4 .4 .4 .4 .4 .4 .4 .4 6 1,000,000 26 400,000 PV i (1.10) i i 1 i 7 (1.10) or, rewriting the equation to make it more calculator friendly: 6 600,000 26 400,000 PV i i i 1 (1.10) i 1 (1.10) .4 .4 .4 .4 .4 .4 25 26 Using your calculator: PMT= 600,000 I/YR=10 N=6 Press PV to compute first sum sign PMT= 400,000 I/YR=10 N=26 Press PV to compute and add the results together Answer: The present value of this contract is $6,277,534.61 6 7. It is your lucky day: you have been fired from your job. But when your stupid boss kicked you out of the office, he accidentally handed you two $100,000 dollar bearer bonds (non-registered bonds – you only need to present the coupons for payment) with the stack of papers that he cleared from your desk. The bonds have a 10% annual coupon and maturity of five years. If the current market rate on the bonds is 8.7%, what is this severance package worth? 0 1 2 10,000 10,000 3 10,000 4 5 10,000 110,000 P/YR = 1 I/yr =8. 7 N=5 PMT = 10,000 FV = 100,000 Press PV 5 PV 1.087t 10, 000 t 1 100, 000 1.0875 Answer: Package 2 PV $210,192.31 8. Suppose that you decide to hold the bonds in the previous question for a year, and then sell them. The next year, the interest rate on comparable bonds increases to 9.5%. How much will you be able to sell them for at that time? If you received a 5% rate of return on your money and you stick the coupons in the bank, in retrospect, would you have been better off selling the bonds when you first got them, or, holding them for a year, and selling them at the higher yield (9.5%)? In one year, the future CF’s from the bonds will be: 0 1 2 10,000 10,000 3 4 10,000 100,000 4 , 000 , 000 PV 10 100 101,602.24 1.095t 1.0954 t 1 Package 2 * PV 2 * Coupons 2 * 101,602.24 2 * 10,000 223,204.48 If you sold the bonds when they were received, you would have gotten $210,192.31 (See #8). Then re-investing the proceeds at 5% for 1 yr: 210,192.31(1.05) = 220,701.93 Answer: You are better off retaining them – getting 10% interest on your money. 7 9. Assume that you have just purchased a $75,000 house. One bank will give you a 9% mortgage with repayment in equal annual installments over 20 years with $15,000 down payment. Another bank wants a 10% rate of interest but will give you a 25-year equal-annual-installment mortgage with a $15,000 down payment. Assuming that you have that $15,000, which of the two deals will minimize the annual payments? Purchase a house for $75,000, bank (a) 9% on equal payment over 20 years with $15,000 down payment; bank (b) 10% on equal payment over 25 years with $15,000 down payment. (a) The $15,000 of down payment will decrease the amount borrowed to $60,000 (amount borrowed=$75,000-15,000) 20 PMT 60,000 PMT 6,572.79 t 1 . 09 t 1 (b) 25 PMT 1.10 t 1 t 60,000 PMT 6,610.08 PV=60,000 FV=0 I/YR=9 N=20 Press PMT PV=60,000 FV=0 I/YR=10 N=25 Press PMT Answer: Bank (a) at 9% interest minimizes the annual payment 8 10. Assume that a bank charges .01 interest per month. You borrow $50,000, to be paid by equal payment over a 35-month period, first payment to be due one month from now. How much will you have to pay each month? What is the annual effective interest cost? Loan $50,000, 1% month, over 35 month 35 PMT 1.01 t t 1 50,000 PMT 1,700.18 PV=50,000 FV=0 I/YR=1 N=35 Press PMT to compute EAR 1 rmonth 1 (1.01)12 1 12.68% 12 Answer: The annual payment will be $1,700.18 and the effective annual rate EAR is 12.68%. 11. Suppose the tooth fairy wants to buy a car, the cost of the car is $14,000. If you agree to finance it at an interest rate of 9% per annum (that is an EAR of 9%), and she agrees to make monthly payments at the end of each month for the next 48 months, how much will her payments be? 0 1 2 3 -14,000 PMT PMT PMT r 1.09 monthly 48 PMT 1.0072 t 1 t 1 12 1 .0072 14,000 … … 48 PMT Using your calculator – use the TVM function: P/YR =1 PV=14,000 I%/YR = .72% N = 48 FV = 0 Press PMT Answer: 346.00 per month. 9 12. If the time value of the money is .10 how much do you have to save per year for 20 years to have $50,000 per year for perpetuity? Assume that the first deposit is immediate and that the first payment will be at the beginning of the 21st year. Time value of money is 10%, the period is 20 years, $50,000 per year as a perpetuity, deposits are made at the beginning of the period, the first payment of the perpetuity will be made at the beginning of the 21st year or at the end of 20th year (same thing). 0 -X 1 -X 2 -X 3 -X 4 -X 5 … … 19 -X -X 20 0 21 50 22 50 … … Step 1: Because the perpetuity begins in year 21, it’s PV using the perpetuity formula gives a PV in YEAR 20. Therefore, the PV of the perpetuity in YEAR 20 is: 50,000 $500,000 0.10 PV In other words, we will need $500,000 in YEAR 20 to support a 50K annual perpetuity thereafter. . Step 2: Now we PV the $500,000 to year zero: PV 500,000 74,321.814 (1.10) 20 PV= 0 PMT= 0 FV=500,000 I/YR=10 N=20 Press PV to compute Step 3: Now we compute the corresponding annuity due for the first 20 years: 19 20 PMT PMT 74,321.814 1.10 t t t 0 1.10 t 1 1.10 PMT 7,936.19 Set at BEG PV=74,321.814 FV= 0 I/YR=10 N=20 Press PMT to compute Answer: The savings per year will be $7,936.19 made at the beginning of each year. 10 13. If an investor can earn 8% per year, how much will the investor have at the end of 10 years, if he contributes $100 each year? What is the present value if cost of money is 8%? 0 1 2 3 4 5 6 7 8 9 10 100 100 100 100 100 100 100 100 100 100 Cash flow $100 per year, rate of return r=8%, period of time 10 years. th (i) The future value at the end of the 10 year the investor will be: 10 FV 100 (1.08) i 1,448.66 PV= 0 PMT= 100 I/YR=8 N=10 Press FV to compute i 1 Answer: The investor will have $1,448.66 at the end of the 10th year. (ii) The present value of this investment is: 10 PV i 1 100 671.01 (1.08) i PMT= 100 FV= 0 I/YR=8 N=10 Press PV to compute Answer: The present value of this investment is $671.01 11 14. Determine the annual payments and amortization schedule for a $1000, 8% per year installment loan, to be repaid over 3 years. 3 $1000 t 1 PMT 1.08t PV=1,000 FV= 0 I/YR=8 N=3 Press PMT PMT 388.034 Amortization Schedule Year1 Year2 Principal 1,000.00 Interest (8%) 80.00 Amortization 308.03 Payment 388.03 Year3 Total 691.97 359.29 0.00 55.36 28.74 164.10 332.68 359.29 1,000.00 388.03 388.03 1,164.10 15. Now suppose that payments are to be made semi-annually, what are the payments, and is the total amount in interest and principle less or more than the annual loan? Explain. Payment made semiannually as a follow, 8% annually—APR (4% semiannually): 6 1,000 t 1 PMT 1.04t PMT 190.762 PV=1,000 FV= 0 I/YR=4 N=6 Press PMT The total of the payments for the annual loan is $1,164.10, for the semi-annual loan, $1,144.56. The total of the payments is less than for the semi-annual payment loan because we amortize the principal sooner, so the interest expense will be lower. 12 16. A has borrowed $100,000. Repayment is over four years, and the interest rate is 20% per year (first payment at the end of the year). Determine an amortization schedule. Then compute payments for a five year loan with the first payment to be made at the beginning of the year. 4 $100,000 t 1 PV=100,000 FV= 0 I/YR=20 N=4 Press PMT to compute PMT 1.20t Answer: The annual payment will be $38,628.91 Amortization Schedule Year1 100,000.000 Principal Interest (20%) 20,000.000 Amortization 18,628.912 38,628.912 Payment Year2 81,371.088 16,274.218 22,354.694 38,628.912 Year3 59,016.394 11,803.279 26,825.633 38,628.912 Year3 32,190.760 6,438.152 32,190.760 38,628.912 Total 32,190.760 54,515.648 100,000.000 154,515.648 The first payment of a five payment loan with the first payment at the beginning is essentially an annuity due: Using your calculator: 4 100,000 t 0 PMT 1.20t Set at BEG PV=100,000 I/YR=20 N=5 Press PMT to compute Answer: The annual payment will be $27,864.975 13 17. Assume a 10% discount rate (assume you can invest and borrow at 10%). Compute: (a) $10,000 in cash or $1,000 per year for perpetuity (first payment at the end of the first period). (b) $10,000 in cash or $1,100 per year for perpetuity (first payment at the end of the first period). (c) $10,000 in cash or $900 per year for perpetuity (first payment at the beginning of the first period). a) $1,000 per years as a perpetuity: PV 1,000 10,000 0.10 b) $1,100 per years as a perpetuity: PV 1,100 11,000 0.10 c) $900 per years as a perpetuity, but first payment received at the first year. PV 900 900 9,900 0.10 14 18. Convert 6% APR to the corresponding EAR, assuming a) Annual compounding, b) quarterly compounding, c) monthly compounding, and d) daily compounding: a. Because compounding is Annual, the EAR and APR are 6%. b. The corresponding monthly rate is: 6% 1.5% 4 4 4 EAR 1 rQuarterly 1 1.015 1 6.13636% .5% 12 1 1.00512 1 6.16778% c. The corresponding monthly rate is: 6% EAR 1 rMonthly 12 .016438% 365 365 EAR 1 rDaily 1 1.0001643812 1 6.18313% d. The corresponding daily rate is: 6% 19. Convert 6% EAR to the corresponding EAR, assuming a) Annual compounding, b) quarterly compounding, c) monthly compounding, and d) daily compounding: a. Again, for annual compounding, the period rate is the annual rate, so the APR equals the EAR. b. The corresponding PERIOD RATE is: rQuarterly 1 EAR 4 1 1.06 4 1 1.467385% APR 4 1.467385% 5.86954% 1 1 c. The corresponding period rate is: 1 rMonthly 1.06 12 1 .486755% APR 12 .486755% 5.84106% d. The corresponding period rate is: 1 rDaily 1.06 365 1 .015965% 1 APR 1.06 365 1 5.82736% 15 20. Suppose that you just turned 25, and you decide to contribute $100 a month at the end of each month until age 60. Compute how much you will have, also compute how much you will have if you leave the money in the account until age 67. Assume that at age 67, you plan to withdraw equal payments for the next 30 years—how much can you withdraw at the end of each month? Finally, realizing that you may live past the ripe old age of 97 (your hoping that won’t be the case—at that point what do you have to live for except your morning constitutional and the one day every four years that you put on a suit, and vote with a bunch of other senile codgers), so you also compute what can be withdrawn on a monthly basis in perpetuity. Assume that you can get 12% APR with monthly compounding. This seems like a complex problem, but it is not, its verbose—deal with it. If you read carefully, the problem has already been broken into smaller chunks to illustrate the steps of converting one annuity to another (or, in the second part a perpetuity). The initial problem looks like this: 25-0 25-1 25-2 … -100 -100 60-0 60-1 … -100 FV60 67-0 67-1 … FV67 97-12 Where, FV60 is the amount in the bank at the beginning of age 60 (60-0), and FV67 is the amount at age 67 (67-0). (Note that 60-0 is 59-12, the end of one year is the beginning of the next.) Step 1: Figure out what your annuity is worth in present value terms at age 25. PV 420 100 100 100 100 ... $9846.88 1 2 3512 t (1.01) (1.01) (1.01) t 1 (1.01) Step 2: Compute the future value for age 60 and 67. This can be done two different ways. Either compute the future value based on compounding for 420 monthly periods using the monthly rate of interest of 1%, FV 60 $9846.88 (1.01) 420 $643,095.95 FV 67 $9846.88 (1.01) 504 $1,483,444.05 Or, compute the EAR—the corresponding annual interest rate—and use annual compounding: EAR 1.0112 1 12.683% FV 60 $9,846.88 (1.12683) 35 $643,095.95 FV 67 $9,846.88 (1.12683) 42 $1,483,444.05 16 Now you are ready to solve the more complex problem regarding the size of the annuity payments you can withdraw. The revised problem looks like this: 25-0 25-1 25-2 … -100 -100 60-0 60-1 … -100 67-0 67-1 … PMT… 97-12 PMT Step 3: This is just another straight-forward annuity problem that my dead mother with dementia could solve. Because we have already computed the value of the account at time 67-0, we can re-write the problem as follows: 67-0 67-1 -1,483,444 … PMT 97-12 … PMT So now you just have to compute what the payments will be: 1,483,444 3012 PMT (1.01) t 1 t PMT $15,258.89 If you wish to compute the PMT corresponding to a perpetuity: PMT .01 PMT $14,834.44 1,483,444 17 21. Let’s try something a little simpler. Suppose that you are again a viral 25 years old, and wish to make annual contributions of $2000 at the end of each year until age 60—you will be making annual contributions at the end of each year age 25 through 59—a total of 35 contributions. Suppose again that you want to know what you’ll have in the bank when you turn 60 and 67, and plan to make annual withdrawals at the beginning of each year starting a the beginning of age 67 (an annuity due). You want to compute the size of the withdrawal so that your account will carry you for 35 years. Assume that your money compounds at an APR rate of 9% compounded monthly. This is ultimately what you want to do (remember, the end of age 59 is the beginning of age 60): 25 26 27 … -2000 60 61 -2000 … 67 68 -2000 … 96 PMTPMT… PMT Step 1: compute the present value of the CF’s: There is a little trick here. You have annual CF’s and a monthly compounding rate of .75% (9% divided by 12 months). So, you must compute the EAR or annual rate of interest to conduct your analysis. EAR (1.0075)12 1 9.381% 35 PV t 1 2000 20,395.38 (1.09381) t Step 2: Compute the FV at ages 60 and 67: FV 60 20,395.38 (1.09381) 35 $470,433.28 FV 67 20.395.38 (1.09381) 42 $881,234.05 Step 3: Using FV67 as the PV at age 67, compute the value of an annuity due: 35 881,234 1.09381 t 1 PMT (1.09381) t PMT $79,003.71 If you wanted to know what the payments for an ordinary annuity would be: 35 PMT t t 1 (1.09381) PMT $86,415.04 881,234 18 22. Let’s work it the other way. Suppose that you would like to receive $10,000 per month at the end of each month for the rest of your life (a perpetuity), beginning at age 67. How much would you have to contribute in terms of an ordinary annuity today until age 65, if you just turned 30? Assume that you can get an EAR of 11.5% on your money. The problem looks like this: 30-0 30-1 30-2 … PMTPMT 65-0 65-1 … PMT 67-0 67-1 … -10,000 forever … -10,000 Step 1: Since your are getting an effective annual rate of 11.5%, the real annual rate, you must compute a monthly rate to deal with the monthly cash-flows: 1 rmonthly (1.115) 12 1 .911% Step 2: Compute the value of your annuity at 67-0: PV 67 10,000 $1,097,397.49 .00911 Step 3: Now you just PV the amount back to 30-0: PV 1,097,397.49 1.00911 3712 19,573.37 Step 4: Compute the corresponding monthly annuity: 19,573.37 3512 PMT (1.00911) t 1 t PMT $182.36 19 23. Lets redo problem 3, but this time assume that you want a 30-year monthly annuity starting at age 67 as opposed to a perpetuity. Because we have already done some of the work, we can start at Step 2. Step 2: Compute the value of your annuity at 67-0: PV 67 3012 10,000 1.00911 t t 1 1,055,755.84 Notice, when you get out to 30 years, there is not a great deal of difference between having an annuity and a perpetuity. The reason is that the more distant payments add very little to the present value. Step 3: Now you just PV the amount back to 30-0: PV 1,055,755.84 1.00911 3712 18,830.64 Step 4: Compute the corresponding monthly annuity: 18,830.64 3512 PMT (1.00911) t 1 t PMT $175.44 20 Appendix: For those of you interested in old formulas in the pre-calculator days, the following identity holds: 1 (11r ) N PMT PVA PMT t t 1 (1 r ) r N It is known as the annuity formula. And what it says is that for a series of equal payments, also known as an “annuity”, there is a reduced form expression that does away with the sum sign to compute the present value. In fact, to compute the expression, all that you need do is carry through the algebra. This is the one formula that I ask you to remember, and it is handy. But it is only for an annuity. Specifically, the present value of the annuity (PVA) that falls in period zero, is the PV of the subsequent equal payments in periods 1 through N. So by definition, the PVA is in the period immediately prior to the first payment. In other words, if you have payments in periods 3 though 7, this formula computes the value of those payments in period 2, not period zero. To find the true present value of those payments using this formula, you must first compute the value in period 2, then, discount the amount you computed back two periods, to time zero. The formula is handy because most annuities start in time one. But, you must understand exactly what the formula is giving you. You may also use it as an alternative to the sum sign, where appropriate. Moreover, unlike the sum sign, the formula can actually be used to compute the value of an annuity by simply plugging in the payment, number of periods (n), and the discount rate, r. 21