EDPA 5521 – 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.
100 a month for 20 years (100*12*20) = 24000 without interest or 1200 a year. I would rather use a 5% interest rate for calculating the lump-sum settlement, as if we do so, the sum give to me as a lump-sum settlement will be higher. In addition, if then there is a higher than 5% interest rate in the market, I would stand to gain an even greater sum over time! Here are the calculations:
PV = Ct / (1+r)t-1
PV = Present Value
C = Cost
R = Discount Rate
T = Year (T = 1 for start year)
PV = Ct / (1+r)t-1
PV = 24000 / (1+0.05)19
PV = 24000 / 2.5269501953756382228051721572876
PV = 9497.6149886041085756423239984798
PV = Ct / (1+r)t-1
PV = 24000 / (1+0.10)19
PV = 24000 / 6.1159090448414546291
PV = 3924.1918086093171105063858863825
If I chose the 10% interest rate lump-sum settlement I would receive close to 4000 dollars or 3924.19 dollars while if I choose the 5% interest rate lump-sum I would receive close to 10000 dollars or 9497.61. Because I would want the most money I would choose the lowest interest rate possible when calculating the payment of a lump-sum in my favor.
- 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.
PV = Ct / (1+r)t-1
PV = Present Value
C = Cost
R = Discount Rate
T = Year (T = 1 for start year)
PV = Ct / (1+r)t-1
PV = 1200 / (1+0.11)8
PV = 1200 / 2.30454
PV = 520.71129 (Part 2) – Present value of $1,200 in 9 years.
or
X * (1.11)^8 = 1200
X * 2.30454 = 1200
X = 520.71129162435887422218750813611
PV = Ct / (1+r)t-1
PV * (1+r)t-1 = Ct
PV = Present Value
C = Cost
R = Discount Rate
T = Year (T = 1 for start year)
PV * (1+r)t-1 = Ct
FV = PV * (1 + r)t
FV = 800 * (1.06)^11
FV = 1518.64 (Part 1) – Future value of $800 in 12 years.
Checking
FV = 800 * (1.06)^11
FV = 1518.63884666833987125248
- 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? A small positive change in yearly interest rate can mean a large financial gain over time.
PV * (1+r)t-1 = Ct
FV = PV * (1 + r)t
At 5% interest rate
FV = 15,000 * (1.05)^17
FV = 15000 * 2.29202
FV = 34380.27477
At 8% Interest Rate
FV = 15,000 * (1.08)^17
FV = 15000 * 3.70002
FV = 55500.27082
- 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?
With an interest rate of 6% she would need a minimum investment of $15,686 to reach $25,000 in 9 years. With an interest rate of 7% she would need a minimum investment of $12,546 to reach $25,000 in 9 years.
PV = Ct / (1+r)t-1
PV = Present Value
C = Cost
R = Discount Rate
T = Year (T = 1 for start year)
At 9% interest rate
PV = 25,000 / ((1.06)^8)
PV = 25,000 / 1.5938
PV = 15685.78241
At 6% interest rate
PV = 25,000 / ((1.09)^8)
PV = 25,000 / 1.9926
PV = 12546.42176
Checking:
15685 * 1.06^8 = 24999.5
12546 * 1.09^8 = 24998.7
- 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%? The higher the interest rate, the less money that will need to be deposited for the fund to last all four years.
Annualization Formula
A(r,n) = (r (1+r)n)/((1+r)n-1)
R = interest rate
N = lifetime of asset for depreciation
Interest Rate – 6% over 4 years
A = (0.06(1+0.06)^4 / (1+0.06)^4-1 or 0.06*1.06^4 / 1.06^4-1
A = 0.07574 / 0.26248 = 0.28856 (multiply by this number for an annual cost)
X * 0.28856 = 5,000
X = 17327.4189
Interest Rate – 10% over 4 years
A = (0.10 (1+0.10)^4 / (1+0.10)^4-1 or 0.10*1.10^4 / 1.06^4-1
A = 0.14641 / 0.4641 = 0.315471 (multiply by this number for an annual cost)
X * 0.315471 = 5,000
X = 15849.3174
- 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?
9000 * years [Symbol] 9000 * 10 = 90000 (Cost)
PV = Ct / (1+r)t-1
PV = Present Value
C = Cost
R = Discount Rate
T = Year (T = 1 for start year)
PV = 90000 / (1+0.07)9
PV = 90000 / 1.8384592
PV = 48954.03716 or 48954.04 for a one-time payment (lump sum)
Checking [Symbol] 48954.03716 * (1.07)9 = 90000.0006
- 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%? This technology would be quite expensive.
Annualization Formula
A(r,n) = (r (1+r)n)/((1+r)n-1)
R = interest rate
N = lifetime of asset for depreciation
Cost Per Year – Equipment at 6%
A = (0.06(1+0.06)^5 / (1+0.06)^5-1
A= 0.06*1.06^5 / 1.06^5-1
A = 0.08029 / 0.338
A = 0.23755 (multiply by this number for an annual cost)
170000 * 0.23755 = 40383.5 per year
40383.5 * 5 = 201917.5 total
- 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?
PV = Present Value
C = Cost
R = Discount Rate
T = Year (T = 1 for start year)
Summing all the values we obtain: $92,500 (simple summation)
(Simple present value) PV = 92,500 / 1.05^4 [Symbol] 92500 / 1.2155 [Symbol] $76100.37
PV = Ct / (1+r)t-1
PV = 12,000 / 1.05^4 [Symbol] 12,000 / 1.2155 [Symbol] 9872.480
PV = 12,500 / 1.05^3 [Symbol] 12,500 / 1.1576 [Symbol] 10798.203
PV = 21,000 / 1.05^2 [Symbol] 21,000 / 1.1025 [Symbol] 19047.619
PV = 16,000 / 1.05^1 [Symbol] 16,000 / 1.0500 [Symbol] 15238.095
PV = 31,000 / 1.05^0 [Symbol] 31,000 / 1.0000 [Symbol] 31000.000
Adding all PVs at 5% [Symbol] 85956.40
PV = Ct / (1+r)t-1
PV = 12,000 / 1.10^4 [Symbol] 12,000 / 1.4641 [Symbol] 8196.1615
PV = 12,500 / 1.10^3 [Symbol] 12,500 / 1.3310 [Symbol] 9391.4350
PV = 21,000 / 1.10^2 [Symbol] 21,000 / 1.2100 [Symbol] 17355.3719
PV = 16,000 / 1.10^1 [Symbol] 16,000 / 1.1000 [Symbol] 14545.4545
PV = 31,000 / 1.10^0 [Symbol] 31,000 / 1.0000 [Symbol] 31000.000
Adding all PVs at 10% interest rate [Symbol] 80488.42
When the PV are calculated for each year we notice that the amount of money that would be needed is higher than if we calculated the present value for all $92,5000 for all 5 years. The present values of each of new these investments are different because they will be implemented at different times during the five years. The more we can wait to make a payment (without a higher negative interest rates from having obtained a loan) the better. In this case, it was better (it seems cheaper) for the school to calculate costs with a higher discount rate. Also, the 5% discount rate was still better (or gave a lower $ amount) than if the school payed all $92500 dollars up front.
- 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.
Yearly cost disposable
Price per Case – 12.75
Number of Cases – 24.00
Total per Year – 306.00 (or 12.75 * 24)
Interest she could gain if she deposited the money she would otherwise deposit.
Total per Year – 306.00
Total per Year – 324.00
324.00 – 306.00 = 18
Bank Investment (9%) – 1.09
18 * 1.09 = 19.62
Yearly cost refundable bottles (no deposit)
Price per Case – 12.00
Number of Cases – 24.00
Total per Year – 288.00 (or 12 * 24)
Yearly cost refundable bottles (deposit only)
Price per Deposit – 1.50
Number of Deposits – 24.00
Total Deposit – 36.00
Yearly cost refundable bottles (total cost)
Price with Deposit – 13.50
Number of Cases – 24.00
Total per Year – 324.00
Interest over deposit (if she had deposited the deposit in a bank instead)
36.00
1.09
39.24
Bank Account Excel Sheet
|
324 |
308 |
18 |
1.09 |
19.62 |
0.00 |
19.62 |
|
|
324 |
0 |
0 |
0 |
-36.00 |
36.00 |
|
|
288 |
36 |
1.09 |
39.24 |
0.00 |
39.24 |
36.00 – 19.62 = 16.38 dollars
Gain or Loss
If she buys the disposable bottles, she gains interest to an 18 dollar bank account which will result in 19.62 dollars. However, when subtracting the 36 dollars she obtains back from the deposit one finds that by buying returnable bottles she saved money. She ends up with 16.38 more dollars in her bank account as a result.
- 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?
I would ask to receive the money over time, since the money that I would earn from putting the money in the 10 million dollars in the bank would be less after 12 years than if I ask for the money over time in 1.2 million dollar segments. A basic, but unfair calculation would argue that 12 years at 1.2 million is the same as 14.4 million dollars which is greater than 10 million dollars. Yet, by receiving the money quicker, I have the opportunity of investing it in a bank from year ONE and gain 11 years of interest rate.
Nevertheless, the interest gained is not likely to match what I could gain if I receive the money over time. For example, if the interest rate is 3.5%, I could have up to 145.9 million dollars 12 years later from the lump sum in comparison to 175 million if I invest the 1.2 million every year at a 3.5% interest rate. The difference is reduced the greater the interest rate and eventually flips to provide a greater return to the person that received the lump sum, but even at 7%, I would stand to gain 210 million over 12 years if I received the lump sum in comparison to 214 million if I received 1.2 million a year and reinvested it.
However, if the average interest rate on saving further increases, the 10 million dollar lump sum will generate more income over 12 years. This was true of an 8% interest rate and any interest higher than 8%. Yet, 7% is to some, including myself (particularly in today’s economy) a high interest rate, and since I would actually be expecting for the interest rate to be lower, I would prefer receiving the 1.2 million payments a year. Since it is only 12 years, it is not as if I would not (likely) see the money (I hope to be alive by then). Also if my grandfather was the one who won the Powerball, I would advise him the same. Apart from the fact that it’s hard to get 7% today, it is probably easier (mentally) to keep some perspective (and sanity) and be more judicious with the prize money if you receive it over time. More than one lottery winner has gone back to poverty a few years after asking for a lump sum payment and making a series of poor investments.
