01+Time+Value+of+Money

Old Exam Questions - Time Value of Money - Solutions Page 1 of 41 Pages Time Value of Money - Solutions 1. Given an equivalent number of payments and interest rates, the future value of an annuity due is typically a little more that the future value of a regular annuity. * A. True B. False 2. The future value of an annuity due will always be greater than the future value of an equivalent ordinary annuity, whereas the present value of an annuity due will always be less than the present value of an equivalent ordinary annuity. A. True * B. False 3. The value of a perpetuity with cash flows starting in Year 1, minus the value of a perpetuity with equivalent cash flows starting in Year N+1, where both perpetuities are valued as of Year 0, is nothing more than the present value of an ordinary annuity with equivalent cash flows in Years 1 through N. * A. True B. False 4. Time value of money is nothing more than mathematical manipulation. In fact, you can find the present value of a cash flow by using a negative interest rate and using a future value calculation. * A. True B. False 5. All other factors held constant, the present value of a given annual annuity decreases as the number of discounting/compounding periods per year increases. * A. True B. False 6. A 15-year mortgage will have larger monthly payments than a 30-year mortgage of the same amount and same interest rate. * A. True B. False 7. Suppose someone offered you the choice of two equally risky annuities, each paying $10,000 per year for five years. One is an ordinary (or deferred) annuity, while the other is an annuity due. Given the mathematics of finance, we know that the present value of Old Exam Questions - Time Value of Money - Solutions Page 2 of 41 Pages the annuity due will exceed the present value of the ordinary annuity, while the future value of the annuity due will be less than the future value of the ordinary annuity. A. True * B. False 8. Suppose someone offered you the choice of two equally risky annuities, each paying $10,000 per year for five years. One is an ordinary (or deferred) annuity, while the other is an annuity due. Given the mathematics of finance, we know that if interest rates increase, the difference between the present value of the ordinary annuity and the present value of the annuity due will remain the same. A. True * B. False 9. As shown in class, it is possible to find the present value of an annuity as the difference of two perpetuities. However, it is impossible to use an equivalent approach to find the future value of an annuity. A. True * B. False 10. Mathematically, discounting a single, lump-sum value over a 10-year period at a fixed interest rate is equivalent to compounding the same lump-sum value, at the same fixed interest rate, over a negative 10-year period. * A. True B. False 1. Determine which of the following investments will have the highest future value at the end of 5 years, assuming that the effective annual rate for all investments is the same. A. The investment pays $100 at the end of every year for the next 5 years (a total of 5 payments). B. The investment pays $50 at the end of every 6-month period for the next 5 years (a total of 10 payments). * C. The investment pays $100 at the beginning of every year for the next 5 years (a total of 5 payments). D. The investment pays $50 at the beginning of every 6-month period for the next 5 years (a total of 10 payments). E. The investment pays $500 at the end of 5 years (a total of one payment). 2. Which of the following statements is not (or least) correct? A. Time value of money calculations allow us to convert values at one point in time to their equivalent values at another point in time. Old Exam Questions - Time Value of Money - Solutions Page 3 of 41 Pages B. For single or lump sum cash flows, present value interest factors and future value interest factors are reciprocal functions of each other. C. The present value, as of Period 0, of an annuity with payments in Periods 1 through N can be found as the difference between a perpetuity with payments in Periods 1 through infinity, and another perpetuity with payments in Periods N+1 through infinity, where both of the perpetuities are evaluated as of Period 0. * D. Except under continuous compounding/discounting, effective annual rates will always be greater than nominal/stated/quoted rates. E. If risk and payments were the same for both, you would not be indifferent between a perpetuity starting in Year 6 and a perpetuity starting in Year 11, even though they might appear to have the same value if you used the equation for the value of a perpetuity (V = D/K). 3. Select the statement that is most correct. A. The effective annual interest rate will always be greater than the nominal interest rate, regardless of the number of compounding periods per year. * B. Given a long enough holding period, the interest-on-interest that an investor earns in a single period will be greater than the interest earned on the original principal (deposit) during that same period. C. For any given annuity, the present value when treating it as an annuity due is always less than the present value when treating it as a regular annuity. D. A loan amortization schedule would show that even though payments on an installment loan are the same in each period, the amount that goes towards interest increases over time while the amount that goes towards principal decreases. E. The present value of a given cash flow is inversely related to the discount rate and the number of discounting periods. It will increase as the interest rate increases and decrease as the number of discounting periods increases. 4. Which of the following investments will have the highest future value at the end of 5 years (at Year 5)? Assume that the effective annual rate for all investments is the same. A. A pays $50 at the end of every 6-month period for the next 5 years (a total of 10 payments). B. B pays $50 at the beginning of every 6-month period for the next 5 years (a total of 10 payments). C. C pays $500 at the end of 5 years (a total of one payment). D. D pays $100 at the end of every year for the next 5 years (a total of 5 payments). * E. E pays $100 at the beginning of every year for the next 5 years (a total of 5 payments). 5. A $10,000 loan is to be amortized over 5 years, with annual end-of-year payments. Given the following facts, which of these statements is most correct? Old Exam Questions - Time Value of Money - Solutions Page 4 of 41 Pages A. The annual payments would be larger if the interest rate were lower. B. If the loan could be amortized over 10 years rather than 5 years, and if the interest rate were the same in either case, the first payment would include less dollars of interest under the 10-year plan than under the 5-year plan. C. The last payment would have a higher proportion of interest than the first payment. D. The proportion of interest versus principal repayment would be the same for each of the 5 payments. * E. The proportion of each payment that represents interest as opposed to repayment of principal would be higher if the interest rate were higher. 6. Which of the following statements is incorrect (least correct)? A. If the discount (or interest) rate is positive, the future value of an expected series of payments will always exceed the present value of the same series. B. To increase present consumption beyond present income normally requires either the payment of interest (because you have borrowed money) or else an opportunity cost of interest forgone (because you have sold off some of your investments). C. Disregarding risk, if money has time value, it is impossible for the present value of a given sum to be greater than its future value. D. Disregarding risk, if the present value of a sum is equal to its future value, either the interest rate = 0, or the number of time periods = 0. * E. If the discount (or interest) rate is positive, the future value of an annuity due will always be less than the future value of an equivalent regular annuity, and the present value of an annuity due will always be less than the present value of an equivalent regular annuity. 7. Which of the following statements is incorrect (least correct)? * A. For an installment loan, the amount of interest paid each period will increase, whereas the amount of principal paid will decrease. Thus, the total payment each period, as shown on an amortization schedule, will remain the same. B. If annual compounding is used, then annual nominal rates will be equal to annual effective rates. C. The present value of an N-period annuity (cash flows in Periods 1 through N), evaluated as of Period 0, may be found as the difference between a perpetuity with its first cash flow in Period 1 and a perpetuity with its first cash flow in Period N+1, as long as both perpetuities are evaluated as of Period 0. D. If compounding is more frequent than once per year, then periodic rates will be less than annual nominal rates. E. The future value of an N-period annuity (cash flows in Periods 1 through N), evaluated as of Period N, may be found as the difference between a perpetuity with its first cash flow in Period 1 and a perpetuity with its first cash flow in Period N+1, as long as both perpetuities are evaluated as of Period N. 8. Which of the following statements is most correct? Old Exam Questions - Time Value of Money - Solutions Page 5 of 41 Pages A. A 5-year $100 annuity due will have a higher present value than a 5- year $100 ordinary annuity. B. A 15-year mortgage will have larger monthly payments than a 30-year mortgage of the same amount and same interest rate. C. If an investment pays 10 percent interest compounded annually, its effective rate will also be 10 percent. D. Statements a and c are correct. * E. All of the statements above are correct. 9. Which of the following statements is most correct? A. The first payment under a 3-year, annual payment, amortized loan for $1,000 will include a smaller percentage (or fraction) of interest if the interest rate is 5 percent than if it is 10 percent. B. If you are lending money, then, based on effective interest rates, you should prefer to lend at a 10 percent nominal, or quoted, rate but with semiannual payments, rather than at a 10.1 percent nominal rate with annual payments. However, as a borrower you should prefer the annual payment loan. C. The present value of a perpetuity (say for $100 per year) will approach infinity as the interest rate used to evaluate the perpetuity approaches zero. D. Statements b and c are correct. * E. All of the statements above are correct. 1. You have just purchased a life insurance policy (whole life) that requires you to make 41 semi-annual payments of $350 each, where the first payment is made immediately (some would define this as an annuity due). The insurance company has guaranteed that these payments will be invested to earn you an “effective” annual rate of 8.16%, although interest is compounded semi-annually. At the end of 20 years (41 payments) you may elect to receive the proceeds of this policy in ten equal annual payments, where the first payment will begin 1 year after the policy terminates (some would define this as a regular annuity). If the “effective” annual interest rate remains at 8.16%, then how much will you receive during each of these ten years? A. $4,967.27 * B. $5,244.62 C. $5,385,72 D. $5,483.19 E. $4,808.48 Given an effective annual rate of 8.16%, the periodic rate can be calculated as: iPER = (1.0816)1/2 - 1.0 = 4% Even though the payments begin immediately, you may wish to treat this as a regular annuity to find the value of the payments as of the last payment, Year 20. The future value of the insurance policy at the end of Year 20 would then be: FV = [$350][FVIFA4%,41] = $34,939.29 Old Exam Questions - Time Value of Money - Solutions Page 6 of 41 Pages Alternatively, N = 41, I/YR = 4%, PMT = -$350, => FV = $34,939.29 This amount is then to be paid out over a 10-year period. Since the first payment begins one year later, you may continue to treat this as a regular annuity. This implies that: $34,939.29 = [PMT][PVIFA8.16%,10] = [PMT][6.6619247] => [PMT] = [$34,939.29]/[6.6619247] = $5,244.62 Alternatively, N = 10, I/YR = 8.16%, PV = -$34,939.29, => PMT = $5,244.62 2. Assume that you take out a 30-year mortgage for $150,000. The nominal annual interest rate quoted for this mortgage is 8.125 percent, although payments are made monthly. If you take the entire 30-years (360 months) to pay off this mortgage, then how much will you pay in total interest and what percentage of the total payments you make over this 30-year period will have gone towards interest? (Hint: if you set your payments per year to 12 for this problem, do not forget to set it back to 1 when you are done.) A. 69.36% B. 59.24% C. 66.83% D. 56.91% * E. 62.59% Periodic Rate = 8.125% / 12 = 0.677083333% per month $150,000 = [PMT][PVIFA0.677083333%,360] PMT = $1,113.75 per month Total Payments = ($1,113.75)(360) = $400,950 Percentage of Total Payments Towards Principal = $150,000 / $400,950 = 37.41% Interest Paid = $400,950 - $150,000 = $250,950 Percentage of Total Towards Interest = $250,950 / $400,950 = 62.59% Alternatively: Set payments to 12 per year, N=360 Old Exam Questions - Time Value of Money - Solutions Page 7 of 41 Pages I/YR = 8.125 PV = -$150,000 FV = $0.00 Solve for PMT = $1,113.74577073 Total Payments = $400,948.48 Use Amortization Function for Periods: 1 (Input) 360  AMORT = Int $250,948.48 = Prin $150,000.00 = bAL = $0.0000 Percentage = $250,948.48 / $400,948.48 = 62.59% 3. If the annual required rate of return associated with the cash flows below is 10 percent, then what is the value of these cash flows evaluated as of Year 6? Year Cash Flow 5 $100 6 $200 7 $300 8 $400 A. $923.16 B. $887.29 C. $861.83 D. $954.87 * E. $913.31 Value6 = ($100)(1.10) + ($200) + ($300)(1/1.10) + ($400)(1/1.10)2 Value6 = $110.00 + $200.00 + $272.73 + $330.58 = $913.31 4. Assume that you are calculating the future value of an annuity that pays $150 in each of Years 1-5. Also assume that the annual required rate of return associated with this annuity is 12 percent. What is the dollar difference between calculating the future value of this annuity as an annuity due versus an ordinary annuity? A. $111.74 B. $107.97 C. $121.82 * D. $114.35 E. $117.28 Set calculator to END for Regular Annuity: Old Exam Questions - Time Value of Money - Solutions Page 8 of 41 Pages N = 5, I/YR = 12, PMT = $150, Solve for FVA = $952.93 FVAD = ($952.93)(1.12) = $1,067.28 Alternatively, set calculator to BEGIN for an Annuity Due: N = 5, I/YR = 12, PMT = $150, Solve for FVAD = $1,067.28 Difference = $1,067.28 - $952.93 = $114.35 5. Assume that you can take out a $30,000 loan to buy a car. This loan will require you to make 60 monthly payments (5-year loan) of $622.75 (you should now be able to calculate the monthly interest rate for this loan). What is the difference between the nominal annual rate and the effective annual rate for this loan? (Note: you may round off your monthly rate to two decimal places.) A. 0.47% * B. 0.38% C. 0.54% D. 0.29% E. 0.31% N = 60, PV = $30,000, PMT = -$622.75, Solve for I/YR = 0.75% Nominal = (0.75%)(12) = 9.00% EAR = (1.0075)12 - 1.0 = 9.38% Difference = 9.38% - 9.00% = .38% 6. A perpetuity has yearly cash flows in Years 20 through infinity, an annual required rate of return of 8 percent, and a value at Year 0 of $645.8974. What is the amount of the perpetuity payment in each of Years 20 through infinity? A. $242.00 B. $229.00 C. $216.00 D. $237.00 * E. $223.00 Value19 = ($645.8974)(1.08)19 = $2,787.50 Payment = ($2,787.50)(.08) = $223.00 7. You have just turned 21 today (October 30th of 2002). You would like to insure that starting on your 65th birthday (44 years from today), that your investments will allow Old Exam Questions - Time Value of Money - Solutions Page 9 of 41 Pages you to withdraw $250,000 each year for 15 years (on October 30th of 2046 through October 30th of 2060), then $100,000 each year for 10 years (on October 30th of 2061 through October 30th of 2070), after which you will live with your children. You plan to make 240 monthly deposits, starting on your 21st birthday (today) and ending one month before your 41st birthday, after which you will simply allow your investment to grow over time. If you can earn a nominal (quoted or stated) annual rate of 12 percent for the next 70 years, but where interest is compounded monthly (giving an effective annual rate of 12.6825 percent), then how much must you invest in each of the next 240 months to meet your investment goals? A. $132.80 * B. $111.37 C. $129.14 D. $117.29 E. $123.51 Periodic Rate = 12% / 12 = 1% per month Effective Annual Rate = (1.01)12 - 1.0 = 12.6825 percent There are a number of different ways in which this problem could be solved. Below is only one of those ways. Set Calculator to END of Period. N = 10, I/YR = 12.6825, PMT = $100,000, Solve for PV at October 30th 2060 = $549,580.24 N = 58, I/YR = 12.6825, FV = $549,580.24, Solve for PV at October 30th 2002 = $539.95 _____ N=15, I/YR = 12.6825, PMT = $250,000 Solve for PV at October 30th 2045 = $1,642,453.33 N=43, I/YR = 12.6825, FV = $1,642,453.33 Solve for PV at October 30th 2002 = $9,675.31 _____ Total equivalent needed as of October 30th 2002 = $539.95 + $9,675.31 = $10,215.26 Set Calculator to Begin of Period. N = 240, I/YR = 1, PV = $10,215.26, Solve for PMT = $111.37 Old Exam Questions - Time Value of Money - Solutions Page 10 of 41 Pages 8. You have taken out a 2-year installment loan that requires a payment every 6 months and has the loan amortization table below associated with it. What is the effective annual interest rate associated with this loan? A. 6.50% B. 6.18% * C. 6.61% D. 6.43% E. 6.32% 6-month interest rate = $162.50 / $5,000 = 3.25% EAR = (1.0325)2 - 1.0 = 6.61% Alternatively, you could have taken a longer approach: Interest Payment = $162.50 Principal Repayment = $5,000.00 - $3,809.31 = $1,190.69 Total Payment each period = $162.50 + $1,190.69 = $1,353.19 N=4, PV = $5,000; PMT = -$1,353.19, Solve for I/YR = 3.25% EAR = (1.0325)2 - 1.0 = 6.61% 9. Assume that you are planning for your child's education. You would like to make deposits in each of the years 0 through 21 (22 deposits) so that your child may make withdrawals in each of the years 18 through 21 for tuition. Tuition is currently $2,000, but is expected to grow at 5% for each of the next 10 years, then at 6% for each of years 11 through 25. If you can earn a stated or nominal annual rate of 8%, but interest is compounded quarterly, then how much must you deposit in each year? * A. $446.65 B. $413.83 C. $492.37 D. $371.96 E. $524.60