TOPIC 5
VALUATION OF FUTURE CASH FLOWS
Solutions to Questions and Problems
Q5.6
Calculating Annuity Values. Your company will generate $45,000 in cash flow each year for the next nine years
from a new information database. The computer system needed to set up the database costs $260,000. If you can
borrow the money to buy the computer system at 8.25 percent annual interest, can you afford the new system?
To find the PVA, we use the equation:
PVA = C({1 – [1/(1 + r)t]} / r )
PVA = $45,000{[1 – (1/1.0825)9 ] / .0825}
PVA = $278,210.88
The present value of the revenue is greater than the cost, so your company can afford the equipment.
Q5.10
Calculating Perpetuity Values. Dawa Financial 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 7 percent, how much will you pay
for the policy?
This cash flow is a perpetuity. To find the PV of a perpetuity, we use the equation:
PV = C / r
PV = $35,000 / .07
PV = $500,000
Q5.14
Calculating EAR. First National Bank charges 10.1 percent compounded monthly on its business loans. First
United Bank charges 10.3 percent compounded semi-annually. As a potential borrower, which bank would you go to
for a new loan?
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 + (.101 / 12)]12 – 1 = .1058 or 10.58%
First United: EAR = [1 + (.103 / 2)]2 – 1 = .1057 or 10.57%
For a borrower, First United would be preferred since the EAR of the loan is lower. 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.
Q5.15
Calculating APR. Magnus Credit Corp. wants to earn an effective annual return on its consumer loans of 16 percent
per year. The bank uses weekly compounding on its loans. What interest rate is the bank required by law to report to
potential borrowers? Explain why this rate is misleading to an uninformed borrower.
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/m – 1]
APR = 52[(1.16)1/52 – 1] = .1486 or 14.86%
This is deceptive because the borrower is actually paying annualized interest of 16% per year, not the 14.86% reported on the
loan contract.
Q5.20
Calculating Loan Payments. You want to buy a new sports coupe for $73,800, and the finance office at the
dealership has quoted you an 6.1percent APR loan for 60 months to buy the car. What will your monthly payments
be? What is the effective annual rate on this loan?
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 )
$73,800 = $C[1 – {1 / [1 + (.061/12)]60} / (.061/12)]
Solving for the payment, we get:
C = $73,800 / 51.6014
C = $1,430.20
To find the EAR, we use the EAR equation:
EAR = [1 + (APR / m)]m – 1
EAR = [1 + (.061 / 12)]12 – 1
EAR = .0627 or 6.27%
Q5.23
Valuing Perpetuities. Mobuto Money Limited is selling a perpetual annuity contract that pays $3,100 monthly. The
contract currently sells for $325,000. What is the monthly return on this investment vehicle? What is the APR? The
effective annual return?
Here we need to find the interest rate that equates the perpetuity cash flows with the PV of the cash flows. Using the PV
of a perpetuity equation:
PV = C / r
$325,000 = $3,100 / r
We can now solve for the interest rate as follows:
r = $3,100 / $325,000
r = .0095 or 0.95% per month
The interest rate is 0.95% per month. To find the APR, we multiply this rate by the number of months in a year, so:
APR = (12)0.95%
APR = 11.45%
EAR = [1 + (APR / m)]m – 1
EAR = [1 + (.1145 / 12)]12 – 1
EAR = .01201 or 12.01%
Q5.28
Discounted Cash Flow Analysis. If the appropriate discount rate for the following cash flows is 9.29 percent per
year, what is the present value of the cash flows?
Year
Cashflow ($)
1
1,800
2
2,300
3
4,500
4
5,100
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 = $1,800 / 1.0929 + $2,300 / 1.0929 2 + $4,500 / 1.09293 + $5,100 / 1.09294
PV = $10,594.61