Module 1 - section 4
SHORT CASES SOLUTIONS
Week 3
1. Bob will start to contribute $1,000 per year at the end of this year into his RRSP. Tom decides that would not start now, instead he would prefer to wait 5 years to begin contributing $1,000 at each year-end into his RRSP. How much more would Bob have than Tom in 30 years time? Assume the RRSP funds earn 8% compounded annually.
Solution: Bob invests at the end of this year for the next 30 years @ 8% compounding annually:
CPT
| Mode | P/Y | C/Y | N | I/Y | PMT | PV | FV |
| end | 1 | 1 | 30 | 8 | 1,000.00- | 0 | 113,283.21 |
Alternatively: Tom waits 5 years before investing at the end of each year for the next 25 years @ 8% compounding annually:
CPT
| Mode | P/Y | C/Y | N | I/Y | PMT | PV | FV |
| end | 1 | 1 | 25 | 8 | 1,000.00- | 0 | 73,105.94 |
Therefore, Bob would have invested for an additional 5 years and he would have more than Tom by:
($ 113,283.21 - $ 73,105.94) = $40,177.27
2. Henry is wondering how much money he can save by giving up the smoking habit. He knows that he buys a carton of cigarettes at the end of each month at a cost of $60. His financial advisor tells him that he could have invested these funds into a conservative investment vehicle paying 6% compounded monthly. How much would he have after 5 years if he quit smoking and invested these funds instead?
Solution:
Therefore, Henry would have $4,186.20, as follows:
CPT
| Mode | P/Y | C/Y | N | I/Y | PMT | PV | FV |
| end | 12 | 12 | 60 | 6 | 60- | 0 | 4,186.20 |
Questions 3, 4 and 5 are based on “Justin’s Case” which follows:
Justin just purchased his dream boat. He placed a $2,000 down payment on it and agreed to pay the balance by 36 blended monthly payments of $224.58, which was based on 10% compounded monthly interest.
3. How much was the purchase price of the boat?
Solution:
Step 1: Determine the amount of the loan Justin is assuming:
CPT
| Mode | P/Y | C/Y | N | I/Y | PMT | FV | PV |
| end | 12 | 12 | 36 | 10 | 224.58- | 0 | 6,960.01 |
THEREFORE THE COST OF THE BOAT WILL BE THE PRESENT VALUE OF THE LOAN IN THE AMOUNT OF $ 6,960.01
Step 2: Determine the total cost of the boat:
Cost of the loan (per step 1 above) $ 6,960.01
Add: deposit made 2,000.00
Total cost of the boat $ 8,960.01
4. How much interest did Justin have to pay on his loan?
Solution:
Justin made 36 PAYMENTS OF $ 224.58 = $ 8,084.88
less THE AMOUNT OF THE LOAN 6,960.01
Interest expense $1,124.87
5. Justin decided after 9 payments that he wanted to pay the loan off in its entirety. How much must he pay?
Solution:
The N variable will now change to N = (36 - 9) = 27 payments remain
CPT
| Mode | P/Y | C/Y | N | I/Y | PMT | FV | PV |
| end | 12 | 12 | 27 | 10 | 224.58- | 0 | 5,409.83 |
Therefore, the payout would be $5,409.83
6. The Toronto Firebirds were not happy with their losing streak. As a result, at the end of his second year, they decided to fire their coach as they felt it was his fault. The coach had 5-year contract, which paid $30,000 at the end of each month.
An agreement was made with the coach to buy-out his contract for the remaining term based on 7.5% compounded monthly as the time value of money. How much did the coach receive as a lump-sum payment (ignoring income tax)?
Solution:
After being fired, how much invested capital would be needed today in order to earn 36 payments (3 years remaining in the contract) of $ 30,000 at 7.5%, compounded monthly interest?
Determine the N which represents the time remaining in the contract:N = 3 YEARS LEFT OR 36 PAYMENTS
| Mode | P/Y | C/Y | N | I/Y | PMT | FV | CPT PV |
| end | 12 | 12 | 36 | 7.5 | 30,000- | 0 | 964,437.39 |
THEREFORE, THE COACH WOULD RECEIVE $964,437.39
7. Zina is considering buying a waterfront property. She can purchase a residential building lot for $45,000 cash. Alternatively, she can put $10,000 down and make quarterly payments of $2,500 for the next 4 years. The first payment would be due 3 months after the purchase date and is based on an interest rate of 8% compounded quarterly during the next 4 years. Which option should you choose and by how much?
Solution:
OPTION 1:
PAY $ 45,000 IN CASH OR:OPTION 2:
Make a $10,000 down payment and instalment payments for the balance:
CPT
| Mode | P/Y | C/Y | N | I/Y | PMT | FV | PV |
| end | 4 | 4 | 16 | 8 | 2,500- | 0 | 33,944.27 |
COST OF THE LOAN $33,944.27
Add: DOWNPAYMENT 10,000.00
TOTAL COST $ 43,944.27
THEREFORE OPTION 2 IS LESS EXPENSIVE BY
$ 1,055.73 ($45,000 - $43,944.27)
8. How much interest will Stephanie earn if she contributes $1,500 to an RRSP starting on February 1, 1990 to August 1, 2017, inclusive. She will continue to contribute at the beginning of every six months for this time period. Assume the interest rate would be 8.5% compounded semi-annually?
Solution: Step 1: Determine the future value of the investment:
Note: N = FROM FEB 1/90 TO AUG 1/2017 = 27.5 YRS X 2 P/Y= 55
CPT
| Mode | P/Y | C/Y | N | I/Y | PMT | PV | FV |
| BEG | 2 | 2 | 55 | 8.5 | 1,500- | 0 | 326,252.08 |
Step 2: Determine the portion of the total Future value of the investment that represents interest earned:
THEREFORE, HE WOULD HAVE EARNED IN INTEREST:
VALUE OF INVESTMENT IN 27.5 YRS $326,252.08
LESS OWN CONTRIBUTIONS
(55 PAYMENTS x $ 1,500) = 82,500.00
TOTAL INTEREST EARNED $ 243,752.08
9. Katz Transport is deciding on whether to lease or buy a new transport truck. They are informed that they can either enter into a 5-year lease with GMAC at 11.25% compounded monthly or make a cash purchase of the truck. Whichever alternative they decide, they want to reflect the liability on the long-term liability section on the current Balance Sheet. If they select to lease, they will have to make payments at the beginning of each month, in the amount of $1,700. What would the liability be that they should report on the balance sheet?
Solution: CPT PV
| Mode | P/Y | C/Y | N | I/Y | PMT | FV | PV |
| BEG | 12 | 12 | 60 | 11.25 | 1,700- | 0 | 78,470.45 |
THEREFORE THE INITIAL LIABILITY ON THE BALANCE SHEET WOULD BE $78,470.45
10. Zelda Electronics wants to lease equipment to its customers. A customer has agreed to a lease that would be for five years and would require payments at the beginning of each quarter. Zelda Electronic wants to recover the $20,000 capital cost of the equipment that they would lease over the term of this lease. Assume, Zelda Electronics could earn 16% compounded quarterly on its investment, how much must they charge their customer to accomplish their goal?
Solution:
How much should they charge per lease payment, if they will collect 20 payments (quarterly X 5 years) and need to recover $ 18,000 today? Each payment earns 16% compounded quarterly interest and is paid at the beginning of each quarter.
| Mode | P/Y | C/Y | N | I/Y | PV | FV | CPT PMT |
| BEG | 4 | 4 | 20 | 16 | 20,000- | 0 | 1,415.03 |
Therefore, the customer would have to be charged $1,415.03 in monthly lease payments.
11. How much more will you have in your RRSP 30 years from now if you make fixed contributions of $ 3,000 at the end of each of the next 30 years, instead of waiting 15 years and making annual contributions that are twice as large for half as many years? Assume that interest is 8% compounded quarterly.
Solution:
Alternative 1: contribute $3,000 for 30 years
CPT
| Mode | P/Y | C/Y | N | I/Y | PMT | PV | FV |
| end | 1 | 4 | 30 | 8 | 3,000- | 0 | 355,389.08 |
Alternative 2; contribute $ 6,000 for 15 years
CPT
| Mode | P/Y | C/Y | N | I/Y | PMT | PV | FV |
| end | 1 | 4 | 15 | 8 | 6,000- | 0 | 166,029.67 |
Alternative 1 would be higher by $ 189,359.41 (355,389.08 - 166,029.67)
12. Brad wants to know whether he would have more in his RRSP if he would contribute either contribute $300 at the end of each month or make an annual year-end contribution in the amount of $3,600. Compare the two alternatives at the end of 25 years assuming the investment earns 8.5% compounded semi-annually.
Solution:
Alternative 1: make monthly contributions of $ 300 for 25 years
CPT
-
Mod
P/Y
C/Y
N
I/Y
PMT
PV
FV
end
12
2
300
8.5
300-
0
302,244.75
Alternative 2: Make annual year end contributions of $ 3,600 for 25 years
CPT
-
Mod
P/Y
C/Y
N
I/Y
PMT
PV
FV
end
1
2
25
8.5
3,600-
0
290,846.96
Alternative 1 would be higher by $11,397.79 (302,244.75 - 290,846.96)
13. A couple are saving for a down payment on a home. They will like to save $30,000 over the next 4 years. What amount must they invest from their month-end pay cheques if their savings can earn 5.5% compounded semi-annually?
Solution: CPT PMT
-
Mode
P/Y
C/Y
N
I/Y
FV
PV
PMT
end
12
2
48
5.5
30,000-
0
560.90
Therefore, they must save $ 560.90 each month to reach their goal.
14. What was the selling price of a car that you sold that required your buyer to pay you monthly payments of $160.70 for 42 months after giving you a $2,000 down payment. Assume you charged interest at 12% compounded quarterly.
-
What was the selling price of the car?
-
How much interest was paid?
Solution:a) Step 1: Determine the amount of your customer’s loan: CPT PV
-
Mode
P/Y
C/Y
N
I/Y
PMT
FV
PV
end
12
4
42
12
160.70-
0
5,499.94
Step 2: Determine the total selling price of the car:
Present value of the loan $ 5,499.94
ADD: Deposit 2,000.00
Total selling price $ 7,499.94
b) INTEREST = (42 PAYMENTS X $ 160.70) LESS AMOUNT OF LOAN ($5,499.94) = $ 1,249.46
15. Zina has committed herself to a savings plan. She will contribute $300 at the end of every month to her RRSP for the next 5 years. At that time, her plan is to contribute $2,000 at the end of each calendar quarter for the following 10 years:
a) How much will be in her RRSP at the end of the 15 years if the funds earn 8% compounded semi-annually?
b) How much of this amount is interest?
a) Solutions:
Contributions of $ 300 per month for the first 5 years:
CPT
-
Mode
P/Y
C/Y
N
I/Y
PMT
PV
FV
end
12
2
60
8
300-
0
21,968.43
Contribs of $2,000 per quarter for the following 10 years
CPT
-
Mode
P/Y
C/Y
N
I/Y
PMT
PV
FV
End
4
2
40
8
-2,000
21,968.43-
168,427.29
Therefore, Zina will have $168,427.29 in 15 years
b) The interest will be $ 70,427.29, as follows:
$ 168,427.29 - 98,000 own capital contribution (SEE BELOW)
(60 PAYMENTS X $ 300) + (40 PAYMENTS X $ 2,000)
= $ 98,000
16. Instead of making 12 payments at the beginning of each month in the amount of $1,000, based on 8% compounded semi-annually, how much of a lump-sum payment could you pay today to eliminate the debt?
Solution: a single payment of $11,579.18 is needed. CPT PV
| Mode | P/Y | C/Y | N | I/Y | PMT | FV | PV |
| BEG | 12 | 2 | 12 | 8 | 1,000- | 0 | 11,579.18 |
17. Tommy and his wife wanted to purchase a refrigerator. The retail outlet offered to sell them the refrigerator they selected for six monthly payments of $199, including a payment on the date of purchase. Assume the interest rate built into the financing arrangements is 18% compounded quarterly.
Required:
a) What is the retail or cash price of the refrigerators?
b) If they decide to pay on a monthly basis, how much interest will they pay?
Solution:
a) The cash price or amount of the loan, in this case, of the refrigerator is $1,151.36
Note: N represents P/Y times the number of years (12 X ½ year) = 6 CPT
-
Mode
P/Y
C/Y
N
I/Y
PMT
FV
PV
BEG
12
4
6
18
199-
0
1,151.36
b) Interest that they will pay:
Total payments (6 Payments X $ 199) $ 1,194.00
Less: amount of loan 1,151.36
Interest $ 42.64
18. If you have a $ 100,000 loan at 10.5% compounded monthly and an obligation to make payments of $1,000 per month, how much longer would it take you to repay the entire loan than if you renegotiated to lower the interest rate to 9.75% compounding monthly?
Solution:
At 10.5% compounded monthly, it would take you:
CPT
-
Mode
P/Y
C/Y
FV
I/Y
PMT
PV
N
end
12
12
0
10.5
1,000-
100,000
238.69
At 9.75% compounded monthly, it would take you:
CPT
-
Mode
P/Y
C/Y
FV
I/Y
PMT
PV
N
end
12
12
0
9.75
1,000-
100,000
206.86
Therefore, you would save 31.83 months (238.69 - 206.86) or 2 years and 8 months
19. How long would it take you to pay off a debt in the amount of $11,622.73 @ 10.5% compounded monthly, if you are required to make monthly payments in the amount of $295, at the beginning of each month? Express your answer in months and years (rounded).
Solution:
It would take you approx. 48 months or 4 years .CPT
-
Mode
P/Y
C/Y
FV
I/Y
PMT
PV
N
BEG
12
12
0
10.5
295-
11,622.73
47.999
20. Brian wants to take a leave of absence in the near future. He discusses this goal with his wife Brina and they agree. They want to travel and live in Africa for one year. However, they estimate it will cost $40,000 to do so. If they make month-end contributions of $700 through their employer's salary deferral plan earning 7.5% compounded semi-annually, how long will it take them to meet their goal?
Solution:
It would take you approx. 49 months or 4 years or 4.1 years.
CPT
| Mode | P/Y | C/Y | PV | I/Y | PMT | FV | N |
| end | 12 | 2 | 0 | 7.5 | 700- | 40,000 | 49.12 |
-
Henrico took out a $5,000 bank loan that requires a quarterly payment of $302.07 and is due in 5 ½ years.
-
What is the quarterly compounded nominal rate being charged?
-
What is the effective rate of interest are being charged?
Solution:
a) The quarterly rate being charged is 10.5%. CPT
| Mode | P/Y | C/Y | FV | N | PMT | PV | I/Y |
| end | 4 | 4 | 0 | 22 | 302.07- | 5,000 | 10.5% QUARTERLY |
b) The effective rate is 10.92% annually:
Note: P/Y = 4; C/Y = 1
22. Mr. Loanster purchased appliances from a department store for a total purchase price of $1,934. His conditional sales contract required him to make 12 monthly payments of $175, at the end of each month. What was the effective interest rate charged to Mr. Loanster on the loan?
Solution:
The effective interest rate charged to Mr. Loanster was 16.63%
CPT
| Mode | P/Y | C/Y | FV | N | PMT | PV | I/Y |
| end | 12 | 1 | 0 | 12 | 175- | 1,934 | 16.63% |
-
Mrs. Borrower purchased a stove from a furniture store. The retailer sold the stove to her for $1,195. The conditional sales contract stated that she was required to make monthly payments of $110 for one year, with the first payment on the day of the sale. What was the effective rate of interest that was charged to the customer?
Solution:
The effective interest rate charged was 24.79%.
CPT
| Mode | P/Y | C/Y | FV | N | PMT | PV | I/Y |
| BEG | 12 | 1 | 0 | 12 | 110- | 1,195 | 24.79% |
