Tutorial 2-Week 2 Solution

1. Huggins Co. has identified an investment project with the following cash flows. If the discount
rate is 10 percent, what is the present value of these cash flows?

Year Cash Flow

Solution
The time line is:

0 1 2 3 4

PV $680 $810 $940 $1,150

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@10% = $680 / 1.10 + $810 / 1.102 + $940 / 1.103 + $1,150 / 1.104 = $2,779.30

2. Investment X offers to pay you $4,700 per year for eight years, whereas Investment Y offers to
pay you $6,700 per year for five years. Which of these cash flow streams has the higher present
value if the discount rate is 5 percent?

Solution
The times lines are:

0 1 2 3 4 5 6 7 8

PV $4,700 $4,700 $4,700 $4,700 $4,700 $4,700 $4,700 $4,700

0 1 2 3 4 5

PV $6,700 $6,700 $6,700 $6,700 $6,700

To find the PVA, we use the equation:

At a 5 percent interest rate:

X@5%: PVA = $4,700{[1 – (1/1.05)8 ] / .05 } = $30,377.10

Y@5%: PVA = $6,700{[1 – (1/1.05)5 ] / .05 } = $29,007.49

3. If you deposit $4,000 at the end of each of the next 20 years into an account paying 9.7 percent
interest, how much money will you have in the account in 20 years?
Solution

Here we need to find the FVA. The equation to find the FVA is:

FVA = C{[(1 + r)t – 1] / r}

0 1 20
…
$4,000 $4,000 $4,000 $4,000 $4,000 $4,000 $4,000 $4,000 $4,000

FVA for 20 years = $4,000[(1.09720 – 1) / .097] = $221,439.14

4. You want to have $50,000 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 6.2
percent interest, what amount must you deposit each year?

Solution
The time line is:

0 1 12

… $50,000
C C C C C C C C C

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}

$50,000 = $C[(1.06212 – 1) / .062]

We can now solve this equation for the annuity payment. Doing so, we get:

C = $50,000 / 17.06825 = $2,929.42

5. The Maybe Pay Life Insurance Co. is trying to sell you an investment policy that will pay you and
your heirs $40,000 per year forever. If the required return on this investment is 5.1 percent,
how much will you pay for the policy?

Solution
The time line is:

0 1 ∞

…
PV $40,00 $40,000 $40,000 $40,000 $40,000 $40,000 $40,00 $40,000 $40,000
0 0

This cash flow is a perpetuity. To find the PV of a perpetuity, we use the equation:

PV = C / r

PV = $40,000 / .051 = $784,313.73

6. First National Bank charges 12.4 percent compounded monthly on its business loans. First
United Bank charges 12.7 percent compounded semiannually. As a potential borrower, which
bank would you go to for a new loan?

Solution
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 + (.1240 / 12)]12 – 1 = .1313, or 13.13%

First United: EAR = [1 + (.1270 / 2)]2 – 1 = .1310, or 13.10%

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.

7. Tai Credit Corp. wants to earn an effective annual return on its consumer loans of 16.5 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? Explain why this rate is misleading to an uninformed
borrower.

Solution
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 = 365[(1.165)1/365 – 1] = .1528, or 15.28%

This is deceptive because the borrower is actually paying annualized interest of 16.5 percent per
year, not the 15.28 percent reported on the loan contract
 8. Fowler Credit Bank is offering 6.7 percent compounded daily on its savings accounts. If you
deposit $7,000 today, how much will you have in the account in 5 years?

Solution
For this problem, we need to find the FV of a lump sum using the equation:

FV = PV(1 + r)t

It is important to note that compounding occurs daily. To account for this, we will divide the
interest rate by 365 (the number of days in a year, ignoring leap year), and multiply the number of
periods by 365. Doing so, we get:

0 1 5(365)
…
$7,000 FV

FV in 5 years = $7,000[1 + (.067 / 365)]5(365) = $9,785.28

9. An investment will pay you $65,000 in 10 years. If the appropriate discount rate is 7 percent
compounded daily, what is the present value?

Solution
The time line is:

0 1 10(365)
…
PV $65,000

For this problem, we need to find the PV of a lump sum using the equation:

PV = FV / (1 + r)t
 It is important to note that compounding occurs daily. To account for this, we will divide the
interest rate by 365 (the number of days in a year, ignoring leap year), and multiply the number of
periods by 365. Doing so, we get:

PV = $65,000 / [(1 + .07 / 365)10(365)] = $32,280.21

10. You want to buy a new sports coupe for $79,500, and the finance office at the dealership has
quoted you an APR of 5.8 percent for a 60-month loan to buy the car. What will your monthly
payments be? What is the effective annual rate on this loan?

Solution
The time line is:

0 1

$79,500 C

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)

$79,500 = $C[1 –
{1 / [1 + (.058 /
12)]60} / (.058 /
12)]

Solving for the

C = $79,500 /
 51.97521 =
$1,529.58

To find the EAR,

EAR = [1 + (APR /
m)]m – 1

EAR = [1 + (.058 /
12)]12 – 1 = .0596,
or 5.96%

11. One of your customers is delinquent on his accounts payable balance. You’ve mutually agreed to
a repayment schedule of $500 per month. You will charge 1.5 percent per month interest on the
overdue balance. If the current balance is $18,000, how long will it take for the account to be
paid off?

Solution
The time line is:

0 1 ?
…
–$18,000 $500 $500 $500 $500 $500 $500 $500 $500 $500

Here we need to find the length of an annuity. We know the interest rate, the PV, and the
payments. Using the PVA equation:

PVA = C({1 – [1 / (1 + r)t] } / r)

$18,000 = $500{[1 – (1 / 1.015)t ] / .015}

Now we solve for t:

1 / 1.015t = 1 – {[($18,000) / ($500)](.015)}

1.015t = 1 / .46 = 2.174

t = ln 2.174 / ln 1.015 = 52.16 months

12. Live Forever Life Insurance Co. is selling a perpetuity contract that pays $1,400 monthly. The
contract currently sells for $215,000. What is the monthly return on this investment vehicle?
What is the APR? The effective annual return?

Solution
The time line is:

0 1 ∞
…
–$215,000 $1,400 $1,400 $1,400 $1,400 $1,400 $1,400 $1,400 $1,400 $1,400

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

$215,000 = $1,400 / r

We can now solve for the interest rate as follows:

r = $1,400 / $215,000 = .0065, or .65% per month

The interest rate is .65% per month. To find the APR, we multiply this rate by the number of
months in a year, so:

APR = (12).65% = 7.81%

And using the equation to find an EAR:

EAR = [1 + .0065]12 – 1 = 8.10%

13. First Simple Bank pays 7.5 percent simple interest on its investment accounts. If First Complex
Bank pays interest on its accounts compounded annually, what rate should the bank set if it
wants to match First Simple Bank over an investment horizon of 10 years?

Solution
The total interest paid by First Simple Bank is the interest rate per period times the number of periods.
In other words, the interest by First Simple Bank paid over 10 years will be:

.075(10) = .75

First Complex Bank pays compound interest, so the interest paid by this bank will be the FV factor
of $1 minus the initial investment of $1, or:

(1 + r)10 – 1

Setting the two equal, we get:

(.075)(10) = (1 + r)10 – 1

r = 1.751/10 – 1 = .0576, or 5.76%

14. You are planning to save for retirement over the next 30 years. To do this, you will invest $850
per month in a stock account and $350 per month in a bond account. The return of the stock
account is expected to be 10 percent, and the bond account will pay 6 percent. When you retire,
you will combine your money into an account with a return of 5 percent. How much can you
withdraw each month from your account assuming a 25-year withdrawal period?

Solution
Although the stock and bond accounts have different interest rates, we can draw one time line, but we
need to remember to apply different interest rates. The time line is:

0 1 360 660
... …
Stock $850 $850 $850 $850 $850
C C C
Bond $350 $350 $350 $350 $350

We need to find the annuity payment in retirement. Our retirement savings ends and the
retirement withdrawals begin, so the PV of the retirement withdrawals will be the FV of the
retirement savings. So, we find the FV of the stock account and the FV of the bond account and add
the two FVs.

Stock account: FVA = $850[{[1 + (.10 / 12) ]360 – 1} / (.10 / 12)] = $1,921,414.74

Bond account: FVA = $350[{[1 + (.06 / 12) ]360 – 1} / (.06 / 12)] = $351,580.26

So, the total amount saved at retirement is:

$1,921,414.74 + 351,580.26 = $2,272,995.00

Solving for the withdrawal amount in retirement using the PVA equation gives us:

PVA = $2,272,995.00 = $C[1 – {1 / [1 + (.05 / 12)]300} / (.05 / 12)]

C = $2,272,995.00 / 171.0600 = $13,287.70 withdrawal per month

15. Suppose an investment offers to triple your money in 12 months (don’t believe it). What rate of
return per quarter are you being offered?

Solution
The time line is:
 0 4

$1 $3

Since we are looking to quadruple our money, the PV and FV are irrelevant as long as the FV is four
times as large as the PV. The number of periods is four, the number of quarters per year. So:

FV = $3 = $1(1 + r)(12/3)

r = .3161, or 31.61%