Interest Rates and Inflation: Present Value and Future Value Calculations - Prof. Don Neil, Assignments of Finance

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Problem Sets FINA 3770-004 Spring 2005

Problem Set 1: Compound Interest Problems

Prior to attempting problems 1-37, please review examples 1, 2, 3, 4, and 5 of Lecture Topic 4.

1. Suppose you invest $1,000 on January 31, 200 5 , earning 10% compounded annually. How much will you have on January 31, 201 5?
2. Suppose you invest $1,000 on January 31, 200 5 , earning 10% compounded semi-annually. How much will you have on January 31, 201 5?
3. At 10% compounded quarterly, what will $1,000 grow to in 10 years?
4. At 10% compounded continuously, what will $1,000 grow to in 10 years?
5. At 12% compounded annually, what will $1,000 grow to in 10 years?
6. At 12% compounded semi-annually, what will $1,000 grow to in 10 years?
7. At 12% compounded quarterly, what will $1,000 grow to in 10 years?
8. At 12% compounded monthly, what will $1,000 grow to in 10 years?
9. At 12% compounded continuously, what will $1,000 grow to in 10 years?
10. At 9% compounded annually, what will $1,000 grow to in 10 years?
11. At 9% compounded semi-annually, what will $1,000 grow to in 10 years?
12. At 9% compounded monthly, what will $1,000 grow to in 10 years?
13. At 9% compounded continuously, what will $1,000 grow to in 10 years?
14. What is the effective rate that is equivalent to 10% compounded semi-annually?
15. What is the effective rate that is equivalent to 10% compounded quarterly?
16. What is the effective rate that is equivalent to 10% compounded continuously?
17. What is the effective rate that is equivalent to 12% compounded semi-annually?
18. What is the effective rate that is equivalent to 12% compounded quarterly?
19. What is the effective rate that is equivalent to 12% compounded monthly?
20. What is the effective rate that is equivalent to 12% compounded continuously?
21. What is the effective rate that is equivalent to 9% compounded quarterly?
22. What is the effective rate that is equivalent to 9% compounded monthly?
23. What is the effective rate that is equivalent to 9% compounded continuously?

Notice that as the interest rate increases, the future value (not surprisingly) also increases. At a given annualized interest rate, the future value increases as the number of compounding periods per year increases.

24. Suppose you can earn 10% interest compounded annually. How much would you need to invest today (assume it’s January 31, 200 5 ) in order to have $1,000 on January 31, 2013?
25. Suppose you can earn 10% interest compounded semi-annually. How much would you need to invest today January 31, 200 5 ) in order to have $1,000 on January 31, 201 3?
26. At 10% interest compounded quarterly, what is the present value of $1,000 to be received 8 years from now?

Problem Sets FINA 3770-004 Spring 2005

27. At 10% interest compounded continuously, what is the present value of $1,000 to be received 8 years from now?
28. At 12% interest compounded annually, what is the present value of $1,000 to be received 8 years from now?
29. At 12% interest compounded semi-annually, what is the present value of $1,000 to be received 8 years from now?
30. At 12% interest compounded quarterly, what is the present value of $1,000 to be received 8 years from now?
31. At 12% interest compounded monthly, what is the present value of $1,000 to be received 8 years from now?
32. At 12% interest compounded continuously, what is the present value of $1,000 to be received 8 years from now?
33. At 9% interest compounded annually, what is the present value of $1,000 to be received 8 years from now?
34. At 9% interest compounded monthly, what is the present value of $1,000 to be received 8 years from now?
35. At 9% interest compounded continuously, what is the present value of $1,000 to be received 8 years from now?
36. Now, duplicate some of these using the effective rates you computed earlier, in order to prove to yourself that you get the same answer either way.
37. How much must you deposit now at 6% compounded quarterly in order to have $1,000 in 5 years?

Notice that as the interest rate goes up, the present value goes down, and vice versa. Also, at a given annualized interest rate, the present value decreases as the number of compounding periods per year increases. That is, the more frequent the compounding, the higher the effective rate.

Prior to attempting problems 38-40 please review example 6 of Lecture Topic 4.

38. An opportunity offers to double your investment in 12 years. What rate of annually compounded interest is implied?
39. A house purchased for $40,000 in 1975 sold for $85,000 in 1985. What was the average compound annual rate of appreciation?
40. If you can earn 10% interest compounded monthly, how long will it take for an investment of $1,000 to grow into $2,000?

Prior to attempting problems 41-46 please review example 9 of Lecture Topic 4.

41. If inflation averages 3% per year, what will a dollar stuffed into the mattress be worth in 25 years?
42. If inflation averages 3% per year, what will a dollar stuffed into the mattress be worth in 50 years?
43. A dollar at the end of 1976 could purchase about as much as 58 cents could at the end of 1967. What was the average annually compounded inflation rate over that period?
44. If it seems to you that everything costs four times as much now as it did twenty years ago, are you surprised? What average annually compounded rate of inflation is implied?
45. One hundred years ago an unskilled laborer earned $1 a day. Now such a laborer earns $32 (at about $4 an hour for 8 hours). What is the average compound annual rate of wage inflation?

Problem Sets FINA 3770-004 Spring 2005

46. If this rate of inflation continues, what will the daily wage be after another hundred years?

Prior to attempting problems 47-51 please review example 10 of Lecture Topic 4.

47. Suppose there is deflation of 10% annually (that is, prices drop at an annually compounded rate of 10%). What would a dollar stuffed into the mattress be worth in 1 year?
48. With deflation of 10% compounded annually, what would a dollar stuffed into the mattress be worth in 5 years?
49. With deflation of 10% compounded annually, what would a dollar stuffed into the mattress be worth in 10 years?
50. With deflation of 5% compounded annually, what would a dollar stuffed into the mattress be worth in 5 years?
51. With deflation of 5% compounded annually, what would a dollar stuffed into the mattress be worth in 10 years?

Prior to attempting problems 52-59 please review examples 11 & 12 of Lecture Topic 4.

52. What is the present value of $1,000 to be received 10 years from now if the required real rate of return is 3% compounded annually and the expected rate of inflation is 5% compounded annually?
53. What is the present value of $1,000 to be received 10 years from now if the required real rate of return is 3% compounded annually and the expected rate of inflation is 7% compounded annually?
54. What is the present value of $1,000 to be received 10 years from now if the required real rate of return is 4% compounded annually and the expected rate of inflation is 6% compounded annually?
55. What is the present value of $1,000 to be received 10 years from now if the required real rate of return is 4% compounded annually and the expected rate of inflation is 8% compounded annually?
56. What is the present value of $1,000 to be received 10 years from now if the required real rate of return is 5% compounded annually and the expected rate of inflation is 7% compounded annually?
57. What is the present value of $1,000 to be received 10 years from now if the required real rate of return is 5% compounded annually and the expected rate of inflation is 9% compounded annually?
58. What is the present value of $1,000 to be received 10 years from now if the required real rate of return is 5% compounded continuously and the expected rate of inflation is 7% compounded continuously?
59. What is the nominal rate in each of the seven preceding problems?

Notice that for a given real rate, the spread between the real rate and the nominal rate exceeds the rate of inflation (except in the case of continuous compounding). This excess increases as inflation increases.

Prior to attempting problems 60-72, please review examples 13 & 14 of Lecture Topic 4.

60. In terms of today's purchasing power, how much would you expect to have in 10 years as the result of investing $1,000 today, if the nominal return is 10% compounded annually and the inflation rate is expected to be 8% compounded annually?

Problem Sets FINA 3770-004 Spring 2005

61. In terms of today's purchasing power, how much would you expect to have in 10 years as the result of investing $1,000 today, if the nominal return is 5% compounded annually and the inflation rate is expected to be 3% compounded annually?
62. In terms of today's purchasing power, how much would you expect to have in 10 years as the result of investing $1,000 today, if the nominal return is 8% compounded annually and the inflation rate is expected to be 5% compounded annually?
63. In terms of today's purchasing power, how much would you expect to have in 10 years as the result of investing $1,000 today, if the nominal return is 12% compounded annually and the inflation rate is expected to be 9% compounded annually?
64. In terms of today's purchasing power, how much would you expect to have in 10 years as the result of investing $1,000 today, if the nominal return is 20% compounded annually and the inflation rate is expected to be 16% compounded annually?
65. In terms of today's purchasing power, how much would you expect to have in 10 years as the result of investing $1,000 today, if the nominal return is 6% compounded annually and the inflation rate is expected to be 2% compounded annually?
66. In terms of today's purchasing power, how much would you expect to have in 10 years as the result of investing $1,000 today, if the nominal return is 6% compounded continuously and the inflation rate is expected to be 2% compounded continously?
67. In terms of today's purchasing power, how much would you expect to have in 10 years as the result of investing $1,000 today, if the nominal return is 8% compounded annually and the inflation rate is expected to be 10% compounded annually?
68. Calculate the real rate for each of the previous eight problems.

Notice that even with the same spread between the nominal rate and the rate of inflation, the real rate is higher when inflation is lower.

69. If you borrow money to buy durable goods at an interest rate of 12% and inflation is 9%, what is the real rate of interest you are paying?
70. If you borrow money to buy durable goods at an interest rate of 9% and inflation is 5%, what is the real rate of interest you are paying?
71. If you borrow money to buy durable goods at an interest rate of 12% and deflation is 9%, what is the real rate of interest you are paying?
72. If you borrow money to buy durable goods at an interest rate of 10% and deflation is 5%, what is the real rate of interest you are paying?

Prior to attempting problem 73, please review example 15 of Lecture Topic 4.

73. You must decide whether to borrow money to buy a car now, or save your money and pay cash later. You expect car prices to be rising at about 4% per year. The best savings opportunity you can think of is a money market CD that you expect will earn a return of 3% per year. You can get a car loan at 5%. If you borrowed the money, what would be the real interest rate, after tax, if you are in the 25% marginal tax bracket and can deduct the interest as a business expense? Should you buy or wait? Assume annual compounding throughout, including the inflation.
74. A parcel of land purchased for $500,000 in 1974 sold for $2,500,000 in 2004. Assume that over this period inflation averaged 6% compounded annually. What was the average compound annual real rate of change in the value of this land?
75. What sort of people (or institutions) would prefer higher-than-anticipated inflation, those with more debts than assets or those with more assets that debts?
76. During times when inflation is higher than anticipated, which would be more valuable, money in the bank or productive skills?

Fina 3770-004 SOLUTIONS: PROBLEM SET 1 Spring 2005

1. Data input: PV is –1000, I/YR is 10, P/YR is 1, N is 10, PMT is 0, compute FV. Result is $2,593.
2. Data input is the same as the previous problem, except that P/YR is 2 and N is
   20. FV is $2,653.
3. Data input is the same as the previous problem, except that P/YR is 4 and N is
   40. FV is $2,685.
4. Type 0.1 and multiply times 10 (annual interest multiplied by amount of time). Then press the ex^ button, and multiply times 1000. The result is $2,718.
5. Data input: PV is –1000, I/YR is 12, P/YR is 1, N is 10, PMT is 0, compute FV. Result is $3,105.
6. Data input is the same as the previous problem, except that P/YR is 2 and N is
   20. FV is $3,207.
7. Data input is the same as the previous problem, except that P/YR is 4 and N is
   40. FV is $3,262.
8. Data input is the same as the previous problem, except that P/YR is 12 and N is
   120. FV is $3,300.
9. Type 0.12 and multiply times 10 (annual interest multiplied by amount of time). Then press the ex^ button, and multiply times 1000. The result is $3,320.
10. Data input: PV is –1000, I/YR is 9, P/YR is 1, N is 10, PMT is 0, compute FV. Result is $2,367.
11. Data input is the same as the previous problem, except that P/YR is 2 and N is
    20. FV is $2,411.
12. Data input is the same as the previous problem, except that P/YR is 12 and N is
    120. FV is $2,451.
13. Type 0.09 and multiply times 10 (annual interest multiplied by amount of time). Then press the ex^ button, and multiply times 1000. The result is $2,459.
14. Reff = (1.05)^2 – 1 = 10.25%
15. Reff = (1.025)^4 – 1 = 10.38%
16. Reff = e 0.1^ – 1 = 10.52%
    17. Reff = (1.06)^2 – 1 = 12.36%
    18. Reff = (1.03)^4 – 1 = 12.55%
    19. Reff = (1.01)1 2^ – 1 = 12.68%
    20. Reff = e 0.12^ – 1 = 12.75%
    21. Reff = (1 + 0.09/4)^4 – 1 = 9.31%
    22. Reff = (1 + 0.09/12)1 2^ – 1 = 9.38%
    23. Reff = e 0.09^ – 1 = 9.42%
    24. Data input: FV is 1000, I/YR is 10, P/YR is 1, N is 8, PMT is 0, compute PV. Result is $466.51. The negative sign in the display indicates that you must make an investment (outflow of funds) in time zero, in order to receive an inflow in the future.
    25. Data input is the same as the previous problem, except that P/YR is 2 and N is
    26. PV is $458.
    27. Data input is the same as the previous problem, except that P/YR is 4 and N is
    28. PV is $453.
    29. Type –0.1 and multiply times 8 (annual interest multiplied by amount of time, with negative sign). Then press the ex^ button, and multiply times 1000. The result is $449.
    30. Data input: FV is 1000, I/YR is 12, P/YR is 1, N is 8, PMT is 0, compute PV. Result is $403.88. The negative sign in the display indicates that you must make an investment (outflow of funds) in time zero, in order to receive an inflow in the future.
    31. Data input is the same as the previous problem, except that P/YR is 2 and N is
    32. PV is $393.
    33. Data input is the same as the previous problem, except that P/YR is 4 and N is
    34. PV is $388.
    35. Data input is the same as the previous problem, except that P/YR is 12 and N is
    36. PV is $384.
    37. Type –0.12 and multiply times 8 (annual interest multiplied by amount of time, with negative sign). Then press the ex^ button, and multiply times 1000. The result is $382.

Fina 3770-004 SOLUTIONS: PROBLEM SET 1 Spring 2005

33. Data input: FV is 1000, I/YR is 9, P/YR is 1, N is 8, PMT is 0, compute PV. Result is $501.87. The negative sign in the display indicates that you must make an investment (outflow of funds) in time zero, in order to receive an inflow in the future.
34. Data input is the same as the previous problem, except that P/YR is 12 and N is
    96. PV is $488.
35. Type –0.09 and multiply times 8 (annual interest multiplied by amount of time, with negative sign). Then press the ex^ button, and multiply times 1000. The result is $486.
36. Answers should agree exactly. Errors result from rounding the effective rate. Even a small difference between the exact effective rate and the rounded version can throw off the calculation by a surprising extent.
37. Data input: FV is 1000, I/YR is 6, P/YR is 4, N is 20, PMT is 0, compute PV. Result is $742.47. The negative sign in the display indicates that you must make an investment (outflow of funds) in time zero, in order to receive an inflow in the future.
38. PV is –1 and FV is 2. P/YR is 1 and n is
    12. Calculate I/YR. Result is 5.95%
39. PV is –40,000 and FV is 85,000. P/YR is 1 and n is 10. Calculate I/YR. Result is 7.83%
40. P/YR is 12, PV is –1000, FV is 2000, I/YR is 10. You will find n is almost 84 months (7 years), so it would take 7 years to pass the goal. After 83 months you would have $1991.33, and after 84 months you would be across the goal line with $2007.
41. Data input: Fat Value (FV) is 1, I/YR is 3, P/YR is 1, N is 25, PMT is 0, compute Puny Value (PV). Result is 48¢. (The negative sign in the display is due to the sign convention).
42. Data input is the same as the previous problem, except that and N is 50. PV is 23¢.
    43. Data input: Fat Value (FV) is 1, Puny Value (PV) is –0.58, P/YR is 1, N is 9, PMT is 0, compute I/YR. Result is 6.24%. (The negative sign for PV recognizes the sign convention).
    44. Data input: Fat Value (FV) is 4, Puny Value (PV) is –1, P/YR is 1, N is 20, PMT is 0, compute I/YR. Result is 7.18%. (The negative sign for PV recognizes the sign convention).
    45. Data input: Fat Value (FV) is 32, Puny Value (PV) is –1, P/YR is 1, N is 100, PMT is 0, compute I/YR. Result is 3.53%. (The negative sign for PV recognizes the sign convention).
    46. Leave data from the previous problem, except that PV is –32. Then compute FV. Result is $1,
    47. Data input: FV is 1, P/YR is 1, N is 1, PMT is 0, I/YR is –10, compute PV. The result tells us that $1 to be received a year from now would purchase as much as $1.11 does today. (The negative sign for PV recognizes the sign convention).
    48. Data input is the same as the previous problem, except that N is 5. The result tells us that $1 five years from now would purchase as much as $1.69 does today. (The negative sign for PV recognizes the sign convention).
    49. Data input is the same as the previous problem, except that N is 10. The result tells us that $1 ten years from now would purchase as much as $2.87 does today. (The negative sign for PV recognizes the sign convention).
    50. Data input is the same as the previous problem, except that I/YR is –5 and N is
    51. The result tells us that $1 five years from now would purchase as much as $1.29 does today. (The negative sign for PV recognizes the sign convention).
    52. Data input is the same as the previous problem, except that N is 10. The result tells us that $1 ten years from now would purchase as much as $1.67 does today. (The negative sign for PV recognizes the sign convention).

Fina 3770-004 SOLUTIONS: PROBLEM SET 1 Spring 2005

52. This can be done in one step using the nominal rate. Data input follows: FV is 1000, P/YR is 1, I/YR is 8.15, N is 10, PMT is 0, compute PV. Result is $456.81. (The negative sign for PV recognizes the sign convention). Alternatively, the calculation can be done in two steps. First deflate the future amount, as follows: Fat Value (FV) is 1000, P/YR is 1, I/YR is 5, N is 10, PMT is 0, compute Puny Value (PV). Intermediate result is –631.91. Second step: change sign to positive, input as FV, change I/YR to 3, and compute PV. Final result is $456.81. (The negative sign for PV recognizes the sign convention).
53. Data input is the same as the previous problem, except that the nominal rate is 10.21 (if you do the one-step method). Result is $378.26. (The negative sign for PV recognizes the sign convention). For the two-step method I/YR is 7 in the first step, and 3 in the second step.
54. Data input is the same as the previous problem, except that the nominal rate is 10.24 (if you do the one-step method). Result is $377.23. (The negative sign for PV recognizes the sign convention). For the two-step method I/YR is 6 in the first step, and 4 in the second step.
55. Data input is the same as the previous problem, except that the nominal rate is 12.32 (if you do the one-step method). Result is $312.92. (The negative sign for PV recognizes the sign convention). For the two-step method I/YR is 8 in the first step, and 4 in the second step.
56. Data input is the same as the previous problem, except that the nominal rate is 12.35 (if you do the one-step method). Result is $312.08. (The negative sign for PV recognizes the sign convention). For the two-step method I/YR is 7 in the first step, and 5 in the second step.
    57. Data input is the same as the previous problem, except that the nominal rate is 14.45 (if you do the one-step method). Result is $259.32. (The negative sign for PV recognizes the sign convention). For the two-step method I/YR is 9 in the first step, and 5 in the second step.
    58. This can be done in one step using the nominal rate, 7% + 5% = 12% (remember, the Fisher Effect simplifies with continuous compounding). Type –0.12 and multiply times 10 (annual inflation multiplied by amount of time). Then press the ex^ button, and multiply times 1000. Result is $301. Alternativey, the calculation can be done in two steps. First deflate the future amount, as follows: Type –0.07 and multiply times 10 (annual inflation multiplied by amount of time). Then press the ex^ button, and multiply times
    59. Intermediate result is $469.59. Store this in one of the memory registers. Second step: Type –0.05 and multiply times 10 (annual real rate multiplied by amount of time). Then press the ex button, and multiply times the amount stored in memory. Final result is $301.
    60. This problem is included in the set in order to make sure everyone can find the nominal rates, even those who choose to do the calculations using the two-step approach. To find the nominal rate, first add the real rate to the rate of inflation, then add the product of the two. The nominal rates are as follows: 8.15%, 10.21%, 10.24%, 12.32%, 12.35%, 14.45%, 12.00%

Fina 3770-004 SOLUTIONS: PROBLEM SET 1 Spring 2005

60. This calculation can be done in two steps. First find the future amount, as follows: PV is –1000, P/YR is 1, I/YR is 10, N is 10, PMT is 0, compute FV. Intermediate result is $2,593.74. Second step: change I/YR to 8, and compute PV. Final result is $1,201. Now you can find the real return as follows: change sign in the display to positive and input as FV. Then PV is –1000; compute I/YR. Result is approximately 1.85%
61. Data input is the same as the previous problem, except that I/YR is 5 in the first step, and 3 in the second step. Final result is $1,212. You can find the real return the same way as in the previous problem. The result is approximately 1.94%
62. Data input is the same as the previous problem, except that I/YR is 8 in the first step, and 5 in the second step. Final result is $1,325. You can find the real return the same way as in the previous problem. The result is approximately 2.86%
63. Data input is the same as the previous problem, except that I/YR is 12 in the first step, and 9 in the second step. Final result is $1,311. You can find the real return the same way as in the previous problem. The result is approximately 2.75%
64. Data input is the same as the previous problem, except that I/YR is 20 in the first step, and 16 in the second step. Final result is $1,403. You can find the real return the same way as in the previous problem. The result is approximately 3.45%
65. Data input is the same as the previous problem, except that I/YR is 6 in the first step, and 2 in the second step. Final result is $1,469. You can find the real return the same way as in the previous problem. The result is approximately 3.92%
    66. This calculation can be done in two steps. First find the future amount, as follows: Type 0.06 and multiply times 10 (annual interest multiplied by amount of time). Then press the ex^ button, and multiply times 1000. Intermediate result is $1,822.12. Store this in one of the memory registers. Second step: Type –0.02 and multiply times 10 (annual interest multiplied by amount of time). Final result is $1,491. Altenatively, the calculation can be done in one step, using the real rate, 6% – 2% = 4%. Then type 0.04 and multiply times 10 (annual interest multiplied by amount of time). Then press the ex^ button, and multiply times 1000. Result is $1,491.
    67. This calculation can be done in two steps. First find the future amount, as follows: PV is –1000, P/YR is 1, I/YR is 8, N is 10, PMT is 0, compute FV. Intermediate result is $2,158.92. Second step: change I/YR to 10, and compute PV. Final result is $832.36. Now you can find the real return as follows: change sign in the display to positive and input as FV. Then PV is –1000; compute I/YR. The result is negative, approximately –1.82%
    68. Approximate real rates are given with the solutions above.
    69. (^) R = (0.12 – 0.09)/1.09 ≈ 2.75%
    70. (^) R = (0.09 – 0.05)/1.05 ≈ 3.81%
    71. (^) R = (0.12 – (– 0.09))/(1–.09) ≈ 23.08%
    72. (^) R = (0.10 – (– 0.05))/(1–.05) ≈ 15.79%

Fina 3770-004 SOLUTIONS: PROBLEM SET 1 Spring 2005

73. Nominal rate after tax is 5% times 0.75 = 3.75%. Then subtract inflation, leaving a negative result, –0.25%. Finally, divide this by 1.04 (one plus the rate of inflation), for a result of approximately –0.24% The real rate of interest for borrowing is negative after tax. Furthermore, the after- tax real return from investing in the CD would be negative, so borrow the money and buy the car now.
74. Easiest approach is to deflate the selling price, then calculate rate of return. So in the, FV is 2,500,000, N is 30, I/YR is 6, P/YR is 1, PMT is 0, compute PV and obtain $435,275.33. Second step: change the sign in the display to positive and enter this result as FV. PV is –500000, compute I/YR. The result is –0.46%, indicating that in real terms you lost money on this investment. Or, you could inflate the purchase price and then calculate rate of return. Then in the first step, PV is 500,000, N is 30, I is 6%, P/YR is 1, PMT is 0, compute FV and obtain $2,871,745.59. For the second step change the sign to negative and enter this result as PV, input the sales price of 2,500,000 as FV, and compute interest (don’t forget the sign convention—either PV or FV must be negative). The answer again is –0.46%, indicating that in real terms you lost money on this investment. Another alternative is to calculate the nominal rate and plug it into the Fisher effect calculation. Then in the first step, FV is 2,500,000, PV is –500,000, N is 30, P/YR is 1, PMT is 0, compute interest. The result is approximately 5.51%. Next, subtract 6% and divide by 1.06. The answer again is –0.46%.
75. More debt than assets
76. (^) Skills