Huggins Co. has identified as investment project with the following cash flows.
Yr 1 = 800
Yr 2 = 1090
Yr 3 = 1350
Yr 4 = 1475
If the discount rate is 7%, what is the present value of these cash flows?
What is the present value at 17%?
What is the present value at 25%?
answer
To solve this problem, we must find the PV of each cash flow and add them. To find the PV of a lump sum, we use:
PV = FV / (1 + r)^t
PV@7% = $800 / 1.07 + $1,090 / 1.07^2 + $1,350 / 1.07^3 + $1,475 / 1.07^4 = $3,926.98
PV@17% = $800 / 1.17 + $1,090 / 1.17^2 + $1,350 / 1.17^3 + $1,475 / 1.17^4 = $3,110.05
PV@25% = $800 / 1.25 + $1,090 / 1.25^2 + $1,350 / 1.25^3 + $1,475 / 1.25^4 = $2,632.96
Calculator Solution:
CFo = 0
C01 = 800
F01 = 1
C02 = 1090
F02 = 1
C03 = 1350
F03 = 1
C04 = 1475
F04 = 1
I=7
NPV CPT
3926.98
continue
question
Cannonier, Inc., has identified an investment project with the following cash flows.
Yr 1 = 1030
Yr 2 = 1260
Yr 3 = 1480
Yr 4 = 2220
If the discount rate is 8%, what is the future value of these cash flows in Yr 2?
What is the future value at a discount rate of 11%?
What is the future value at a discount rate of 24%?
answer
To solve this problem, we must find the FV of each cash flow and sum. To find the FV of a lump sum, we use:
FV = PV(1 + r)^t
FV@8% = $1,030(1.08)3^ + $1,260(1.08)^2 + $1,480(1.08) + $2,220 = $6,585.57
FV@11% = $1,030(1.11)^3 + $1,260(1.11)^2 + $1,480(1.11) + $2,220 = $6,823.91
FV@24% = $1,030(1.24)^3 + $1,260(1.24)^2 + $1,480(1.24) + $2,220 = $7,956.40
Notice, since we are finding the value at Year 4, the cash flow at Year 4 is added to the FV of the other cash flows. In other words, we do not need to compound this cash flow.
Calculator Solution:
CFo = 0
C01 = 1030
F01 = 1
C02 = 1260
F02 = 1
C03 = 1480
F03 = 1
C04 = 2220
F04 = 1
I=8
NPV CPT
6585.57
question
An investment offers $6,800 per year for 20 years, with the first payment occurring one year from now.
If the required return is 7 percent, what is the value of the investment?
What would the value be if the payments occurred for 45 years?
What would the value be if the payments occurred for 70 years?
What would the value be if the payments occurred forever?
answer
To find the PVA, we use the equation:
PVA = C({1 − [1 / (1 + r)^t]} / r)
PVA@20 yrs: PVA = $6,800{[1 − (1 / 1.07^20)] / .07} = $72,039.30
PVA@45 yrs: PVA = $6,800{[1 − (1 / 1.07^45)] / .07} = $92,517.55
PVA@70 yrs: PVA = $6,800{[1 − (1 / 1.07^70)] / .07} = $96,290.65
To find the PV of a perpetuity, we use the equation:
PV = C / r
PV = $6,800 / .07 = $97,142.86
Notice that as the length of the annuity payments increases, the present value of the annuity approaches the present value of the perpetuity. The present value of the 70-year annuity and the present value of the perpetuity imply that the value today of all perpetuity payments beyond 70 years is only $852.21.
Calculator Solution:
Enter
N=20
I/Y=7%
PMT=6800
Solve for PV: 72039.30
question
Your company will generate $62,000 in annual revenue each year for the next seven years from a new information database. If the appropriate interest rate is 7.75 percent, what is the present value of the savings?
answer
To find the PVA, we use the equation:
PVA = C({1 − [1 / (1 + r)^t]} / r)
PVA = $62,000{[1 − (1 / 1.0775^7)] / .0775}
PVA = $325,573.40
Calculator Solution:
Enter
N=7
I/Y=7.75
PMT=62000
Solve for PV: 325573.40
question
If you deposit $5,200 at the end of each of the next 15 years into an account paying 11.3 percent interest, how much money will you have in the account in 15 years?
How much will you have if you make deposits for 30 years?
answer
Here we need to find the FVA. The equation to find the FVA is:
FVA = C{[(1 + r)^t − 1] / r}
FVA for 15 years = $5,200[(1.1130^15 − 1) / .1130]
FVA for 15 years = $183,255.01
FVA for 30 years = $5,200[(1.113030 − 1) / .1130]
FVA for 30 years = $1,096,281.39
Notice that because of exponential growth, doubling the number of periods does not merely double the FVA.
Calculator Solution:
Enter
N=15
I/Y=11.30%
PMT=5200
Solve for FV: 183,255.01
continue
question
You want to have $78,000 in your savings account 12 years from now, and you're prepared to make equal annual deposits into the account at the end of each year. If the account pays 6.80 percent interest, what amount must you deposit each year?
answer
Here we have the FVA, the length of the annuity, and the interest rate. We want to calculate the annuity payment. Using the FVA equation:
FVA = C{[(1 + r)^t − 1] / r}
$78,000 = $C[(1.0680^12 − 1) / .0680]
We can now solve this equation for the annuity payment. Doing so, we get:
C = $78,000 / 17.67928
C = $4,411.94
Calculator Solution:
Enter
N=12
I/Y=6.80%
FV=78000
Solve for PMT: 4411.94
question
The Maybe Pay Life Insurance Co. is trying to sell you an investment policy that will pay you and your heirs $35,000 per year forever. If the required return on this investment is 5.3 percent, how much will you pay for the policy?
answer
PV = C / r
PV = $35,000 / .0530
PV = $660,377.36
question
First National Bank charges 14.5 percent compounded monthly on its business loans. First United Bank charges 14.8 percent compounded semiannually.
Calculate the EAR for First National Bank and First United Bank.
As a potential borrower, which bank would you go to for a new loan?
answer
For discrete compounding, to find the EAR, we use the equation:
EAR = [1 + (APR / m)]^m − 1
So, for each bank, the EAR is:
First National: EAR = [1 + (.145 / 12)]^12 − 1 = .1550, or 15.50%
First United: EAR = [1 + (.148 / 2)]^2 − 1 = .1535, or 15.35%
Notice that the higher APR does not necessarily mean the higher EAR. The number of compounding periods within a year will also affect the EAR.
Calculator Solutions
Enter
NOM=14.5%
C/Y=12
Solve for EFF=15.35
question
Tai Credit Corp. wants to earn an effective annual return on its consumer loans of 16 percent per year. The bank uses daily compounding on its loans. What interest rate is the bank required by law to report to potential borrowers?
answer
The reported rate is the APR, so we need to convert the EAR to an APR as follows:
EAR = [1 + (APR / m)]^m − 1
APR = m[(1 + EAR)1^1/m − 1]
APR = 365[(1.160)^1/365 − 1]
APR = .1485, or 14.85%
This is deceptive because the borrower is actually paying annualized interest of 16 percent per year, not the 14.85 percent reported on the loan contract.
Calculator Solution:
Enter
EFF=16.0%
C/Y=365
Solve for NOM=14.85
question
You want to buy a new sports coupe for $93,500, and the finance office at the dealership has quoted you an APR of 7.5 percent for a 48 month loan to buy the car.
What will your monthly payments be?
What is the effective annual rate on this loan?
answer
We first need to find the annuity payment. We have the PVA, the length of the annuity, and the interest rate. Using the PVA equation:
PVA = C({1 − [1 / (1 + r)^t]} / r)
$93,500 = $C[1 − {1 / [1 + (.075 / 12)]^48} / (.075 / 12)]
Solving for the payment, we get:
C = $93,500 / 41.35837
C = $2,260.73
To find the EAR, we use the EAR equation:
EAR = [1 + (APR / m)]^m − 1
EAR = [1 + (.075 / 12)]^12 − 1
EAR = .0776, or 7.76%
Calculator Solution:
Enter
N=48
I/Y=7.50%/12
PV=93500
Solve for PMT=2260.73
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