EDPA 5521
October 12, 2011
PROBLEM SET #2: Compound interest, discounting, annualization and present value
- Rather than pay you $100 a month for the next 20 years, the person who injured you in an automobile accident is willing to pay a single amount now to settle your claim for injuries. Would you rather an interest rate of 5% or 10% be used in computing the present value of the lump-sum settlement? Comment or explain. To calculate the present value of this stream of payments, use the formula in the textbook on page 93. Each payment of $1200 per year must be discounted by the appropriate factor. See #6, below, for an example. The most common error is to discount the entire set of $24,000 payments by the discount factor that is appropriate for the $1200 payments in year 20. This is clearly not appropriate, since the $24,000 payments are spread out over 20 years and each year’s payments must be discounted separately, by the appropriate discount factor. You would prefer a 5% discount rate because it results in a larger present value lump-sum payment.
- Compute the future value of $800 invested for 12 years at 6% compounded annually. Compute the present value of $1,200 due in 9 years at 11% annually. FV = 800 (1+.06)12 = $1609.76
PV = 1200/(1+.11)9 = $469.11 [Recall that the present value of $1200 due in one year is $1200/(1+.11). In other words, t=2. The present value of $1200 due in 9 years means that t=10]. The most common error is to assume that t=9.
- Mr. Jones has $15,000 to invest for the future college education of his new granddaughter. He wishes to know how much it will amount to if he invests it at 5% per year for 18 years. What if he were to invest it at 8% for 18 years?
FV = 15000 (1 + .05)18 = $36,099.29 (t=19)
FV = 15000 (1+ .08)18 = $59,940.29 (t=19)
- Ms. Olson wishes to be sure that she has at least $25,000 in 9 years when her daughter will begin college. How much must she invest today to accomplish this purpose if the interest rate is 6% per year? How much would she need if the interest rate is 9 percent per year?
PV = 25000/ (1+ .06)9 = $14,797.46 (t=10)
PV = 25000/(1+ .09)9= $11,510.69 (t=10)
- In order to establish a fund that will provide a scholarship of $5,000 a year over four years, with the first award to occur now, how much must be deposited if the fund earns 6%? How much is required if the fund earns 10%?
PV = 5,000 + 5000/(1+.06) + 5000/(1+.06)2 + 5000/(1+.06)3 = $18,365.06
PV = 5,000 + 5000/(1+.1) + 5000/(1+.1)2 + 5000/(1+.1)3 = $17,434.26
- A local foundation helps the University to establish a new Center for the Study of Educational Efficiency and has agreed to pay $9,000 now and every year for another 10 years. Because of a potential change in the leadership at the University, the Foundation wishes to discharge its obligation by paying a single lump sum to the University now in lieu of the payment due and all future payments. How much should the Foundation pay the University if the discount rate is 7 percent per year?
PV= [9000/(1+.07)1-1]+ [9000/(1+.07)2-1]+ [9000/(1+.07)3-1]+ [9000/(1+.07)4-1]+ [9000/(1+.07)5-1]+ [9000/(1+.07)6-1]+ [9000/(1+.07)7-1]+ [9000/(1+.07)8-1]+ [9000/(1+.07)9-1]+ [9000/(1+.07)10-1]+ [9000/(1+.07)11-1]
PV= 9000 + 8411.215 + 7860.949 + 7346.681 + 6866.057 + 6416.876 + 5997.08 + 5604.748 + 5238.082 + 4895.404 + 4575.144= $72,212.24
The foundation should pay the institution $72, 212.24 if the discount rate is 7% per year. This discount rate was applied to the initial payment and the subsequent 10 additional years using the PV formula. The most common error is to assume that there are only 10 payments of $9,000. Another error is to add up the payments and divide $99,000 by the discount factor that is only appropriate for the last $9,000 payment received in year 11.
- Your school district, bowing to parent pressure, just purchased a new set of “smartboards” to install in classrooms, at a total cost of $170,000. The system is expected to wear out in 5 years. How much in costs for this equipment should be charged against calculating instructional costs each year when the district is borrowing funds at 6%?
From Table 4.1 annualization factors: 5 years at 5% = .2310; 5 years at 7% = .2439. Average these factors to obtain approximate rate for 6 years = .23745.
.23745 * 170,000 = $40,366.50 (see the textbook, pp. 64-70, for an explanation of the annualization factor)
- Assume that you are asked to review the costs of a 5-year technology upgrade project for the Cranberry Schools. After doing a careful identification and specification of ingredients and their costs, you obtain the following costs:
Year 1: $12,000
Year 2: $12,500
Year 3: $21,000
Year 4: $16,000
Year 5: $31,000
What is the present value of this stream of costs for both a 5% and 10% discount rate. Compare the present values obtained with these calculations with a simple summation. Why do they differ?
Five year cost with 5% and 10% discount rates:
|
|
No Discount |
5% Disc Rate |
Present Value @5% |
10% Disc Rate |
Present Value @10% |
|
Y1 |
$12,000.00 |
$12,000.00 |
$12,000.00 |
||
|
Y2 |
$12,500.00 |
12,500/1.05 = |
$11,904.76 |
12,500/1.1 = |
$11,363.64 |
|
Y3 |
$21,000.00 |
21,000/1.052 = |
$19,047.62 |
21,000/1.12 = |
$17,355.37 |
|
Y4 |
$16,000.00 |
16,000/1.053 = |
$13,821.40 |
16,000/1.13 = |
$12,021.04 |
|
Y5 |
$31,000.00 |
31,000/1.054 = |
$25,503.78 |
31,000/1.14 = |
$21,173.42 |
|
Total |
$92,500.00 |
$82,277.56 |
$73,913.46 |
Without discounting, the sum of costs totals $92,500 and is different from the discounted totals because it does not include any discounting, meaning the time delay in spending for the second through fifth years is not considered. For those years the money could be invested and earn interest thus lessening the loss for those years.
- Drinking beer! Donna likes beer! She consumes 24 cases of beer over the course of a year. Her local store tells her that she can buy beer in disposable bottles for $12.75 per case or for $12.00 a case of returnable bottles if a $1.50 refundable deposit is paid per case at time of purchase. She must buy all 24 cases at the same time and then return any returnable bottles at the end of the year. If she is currently getting 9% per year on her savings, how much does she save by buying the returnable and thereby losing the use of her deposit money for one year.
Option 1: 24*$12.75 = $306
Option 2: 24*$12 = $288
Initial savings = 306-288 = $18
Deposit = 24*$1.50 = $36
Have $36 – $18 = $18 less to invest
Lost interest from deposit = $18 *(.09) = $1.62
Net savings = $18 – $1.62 = $16.38
- You have just won the powerball! They are offering you either $1,200,000 a year for the next 12 years, or $10,000,000 today. Which option will you select and why? Would the decision likely be different if your rich grandfather won the powerball?
There are two methods of solving this problem. The first method is to compare the future value of the yearly payments with the future value of the lump sum payment.
METHOD #1. Select the $1,200,000 a year for the next 12 years. Assuming a 4% discount rate, the first $1,200,000 may be invest for 12 years at an interest rate of 4%. The next payment may be invested for 11 years at 4%. Etc. At the end of 12 years I will have a total of $18,752,205.22 (see below for calculations).
However, if I took the $10,000,000 today and invested it for the next 12 years at a 4% interest rate I would have a total of $16,010,322.19 which is much less money in the end.
Since my grandfather may not live to receive 12 years of payouts, he should elect to receive the immediate payment of $10,000,000.
|
1,200,000*1.0412 = |
$1,921,238.66 |
|
1,200,000*1.0411 = |
$1,847,344.87 |
|
1,200,000*1.0410 = |
$1,776,293.14 |
|
1,200,000*1.049 = |
$1,707,974.17 |
|
1,200,000*1.048 = |
$1,642,282.86 |
|
1,200,000*1.047 = |
$1,579,118.14 |
|
1,200,000*1.046 = |
$1,518,382.82 |
|
1,200,000*1.045 = |
$1,459,983.48 |
|
1,200,000*1.044 = |
$1,403,830.27 |
|
1,200,000*1.043 = |
$1,349,836.80 |
|
1,200,000*1.042= |
$1,297,920.00 |
|
1,200,000*1.04 = |
$1,248,000.00 |
|
Total: |
$18,752,205.22 |
Compare this with the future value of $10 million in 12 years:
10,000,000*1.0412 = $16,010,322.19
This result holds for discount rates of up to 7.45%, far above typical discount rates of 3-5%. Beyond 7.45%, it is better to select the $10 million payment now.
METHOD #2. The second method of solving this problem is to compare the present value of the yearly payments with the present value of the lump sum payment.
PV= ∑ [1200000/(1+.03)1-1]+ [1200000/(1+.03)2-1]+ [1200000/(1+.03)3-1]+ [1200000/(1+.03)4-1]+ [1200000/(1+.03)5-1]+ [1200000/(1+.03)6-1]+ [1200000/(1+.03)7-1]+ [1200000/(1+.03)8-1]+ [1200000/(1+.03)9-1]+ [1200000/(1+.03)10-1] + [1200000/(1+.03)11-1] + [1200000/(1+.03)12-1]
PV= 1200000+ 1165049+ 1131115+ 1098170+ 1066184+ 1035131+ 1004981+ 975709.8+ 947291+ 919700+ 892912.7+ 866905.5= $12,303,149.00
Now compare this amount with the lump sum payment of $10,000,000 (which is, by definition, in present value terms). Clearly, it would be preferable to select the yearly payments. Again, the result holds for discount rates up to 7.45%.
The most common error is to compare the present value of the yearly payments with the future value of the lump sum payment. This is clearly not appropriate. You must either compare the present value of both options, or compare the future value of both options. Do not mix present and future values.
