Summer 2012Test 1 solution sketches

1(a) If the effective annual discount rate is

12.5%, then what is the effective discount rate for 9 months? The effective rate every three months is the

fourth root of 1.125 – 1, or 2.98836% To get the effective rate every nine months,

we need to compound the quarterly rate three times

(1.0298836)3 = 1.092356 answer is 9.2356%

1(b) Jake invests $5,000 into a bank account

earning 13% annual interest, not knowing whether interests is compounded monthly or yearly. After one year, what is the difference in the amount of interest earned between the two types of compounding? Compounded interest: $5,000(1.0108333)12 =

$5,690.16 Simple interest: $5,000(1.13) = $5,650 The difference between the two is $40.16

1(c) Jeanette is set to receive a perpetuity of

$600 per year forever, starting three years from today. Her effective annual discount rate is 11%. What is the present value of the perpetuity? PV if 1st payment is made 1 yr. from today:

$600/0.11 = $5,454.55 However, the 1st payment will be made 3

yrs. from today, so we need to discount the above value by 2 years

$5,454.55/1.112 = $4,427.03

2: Julius in Las Vegas, NM Julius buys a new house in Las Vegas,

NM. The amount loaned for the mortgage is $200,000. The loan will be paid off in 35 years. The stated annual interest rate is 4.8%, compounded monthly. How much will the monthly payment be if each monthly payment will be the same? Note that the stated monthly rate is 0.4%

2: Julius in Las Vegas, NM

Use the annuity formula, with your time unit in months PV = (C/r)[1 – 1/(1+r)T] 200,000 = (C/0.004)[1 – 1/1.004420] Solve for C to get a monthly payment

of $984.01

3: Joan invests $5,000… What is the IRR?

If Joan invests $5,000 today in a project and receives $4,000 one year from today and $5,000 two years from today in return, what is her annual internal rate of return? You can assume that her annual discount rate is 2%. Note here that the annual discount rate

is IRRELEVANT here

3: Joan invests $5,000… What is the IRR? To find the IRR, we need to find a

discount rate such that the NPV of the investment is zero 0 = -5,000 + 4,000/(1+IRR) +

5,000/(1+IRR)2 This is mathematically equivalent to

0 = -5IRR2 – 6IRR + 4 You can use the quadratic formula to solve

for this: IRR = -1.6770, 0.47703 We cannot have negative #’s for IRR:

Choose 47.703%

4: Find PV of future dividends; eff. ann. discount rate is 23% (a) MacMan’s Computers will pay a

dividend of $2 every 6 months, starting 3 months from today. 6 month discount rate: sqrt(1.23) – 1 =

10.905% PV if the 1st payment is 6 mos. from today

$2/0.10905 = $18.34 PV if the 1st payment is 3 mos. from today

We need 3 mo. discount rate to help us here: Fourth root of 1.23 minus 1: 5.312%

$18.34*1.05312 = $19.31

4: Find PV of future dividends; eff. ann. discount rate is 23% (b) Banana Computers will pay a dividend of

$1.50 today. The dividend will grow by 5% per year for each of the next 2 years. After that, the dividend will remain constant forever.

Year 0 payment has PV of $1.50 Year 1 payment of $1.575 has PV of $1.575/1.23 =

$1.2805 Years 2 and later: PV is (1/1.23)(1.65375/0.23), or

$5.8457 (1/1.23) is due to discounting by one year (1.65375/0.23) is from using the perpetuity formula

Total of the three present value numbers above is $8.626

5:The Va-va-va-voom vacuum The Va-va-va-voom vacuum can be

purchased today for $500. The vacuum lasts for 5 years. A maintenance check is required once, three years from today, which costs $200. The effective annual discount rate is 6%. If you intend on using this vacuum for the next five years, what is the equivalent annual cost of the machine? (Assume that the vacuum has no resale value after five years.) We need to find the PV of costs, followed by

finding the EAC using the annuity formula

5:The Va-va-va-voom vacuum

PV of costs is $500 + $200/1.063 = $667.92

EAC: Use the annuity formula PV = (C/r) [1 – 1/(1+r)T] 667.92 = (C/0.06) [1 – 1/1.065] C = $158.56

6: Assume that Julie will receive a $600 payment made 5 years from today.

(a) If the stated annual interest rate is 6% and compounding occurs continuously, what is the present value of the payment? Standard case of continuous

compounding $600/e0.06*5 = $444.49

6: Assume that Julie will receive a $600 payment made 5 years from today.

(b) If the effective annual interest rate is 6% and compounding occurs continuously, what is the present value of the payment? If the effective rate is 6% annually, we can just

compound a 6% rate every year for 5 years $600/1.065 = $448.35

Alternate method exp(x) = 1.06 x = 0.05827 (this is the annual stated

rate) $600/exp(0.05827*5) = $448.35

7: KG Volleyball… might go out of business in 6½ years KG Volleyball sells volleyballs and knee pads to youth

volleyball programs throughout the country. Stockholders will receive a $3 dividend today, and the annual dividend is expected to grow at 6% every year as long as it stays in business. (Note that dividends are paid annually in this problem.) This company will remain in business forever, except that it has a 30% probability of going out of business 6½ years from today. If KG Volleyball goes out of business, the only money that stockholders will receive after the company goes out of business is a $4 per share payment exactly 7 years from today. What is the present value of all present and future payments that stockholders will receive if the effective annual discount rate for owning this stock is 15%? (Note that your answer is an expected value.)

We need to calculate the PV of both possible outcomes We then take 70% of the PV of the good outcome plus 30% of

the PV of the bad outcome

7: KG Volleyball… might go out of business in 6½ years

Calculating the 70% probability outcome Year 0: PV of $3 Years 1 on: Use the growing

perpetuity formula (note that payment in 1 year is $3.18)

PV = $3.18/(.15-.06) = $35.33 Total of $38.33

7: KG Volleyball… might go out of business in 6½ years Calculating the 30% probability

outcome Year 0: PV of $3 Years 1-6: Use the growing annuity formula

(again note that the Year 1 payment is $3.18)

PV = 3.18 {1/(.15-.06) – [1/(.15-.06)][1.06/1.15]6} = $13.66

Year 7: PV = 4/1.157 = 1.50 Total of $18.16

7: KG Volleyball… might go out of business in 6½ years Final step

70% of $38.33 + 30% of $18.16 $32.28

Approximate breakdown for this problem 2 points for recognizing that you need to

calculate a weighted average 3 points for correctly calculating the 70%

probability PVs 5 points for correctly calculating the 30%

probability PVs