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University of Toronto Scarborough

Finance

MGFB10H3

Derek Chau

Summer

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1
Chapter 5
25. Calculate the effective annual rates for the following:
A. 24 percent, compounded daily
B. 24 percent, compounded quarterly
C. 24 percent, compounded every four months
D. 24 percent, compounded semi-annually
E. 24 percent, compounded continuously
F. Calculate the effective monthly rate forAto D.
Level of difficulty: Medium.
Solution:
.24
k = (1 + )365 − 1 = 27 .11 %.
365
A. m = 365:
.24 4
k = (1 + ) − 1 = 26 .25 %.
4
B. m = 4:
k = (1 + .24 )3 − 1 = 25 .97 %.
3
C. m = 3:
k = (1 + .24 )2 − 1 = 25 .44 %.
2
D. m = 2:
k = e.2−1 = 27.12%.
E. Continuous compounding:
F. The effectively monthly rate is:
m 365
k = (1+ QR ) −1 (1+ .24 )12 −1
m 365
A. m=365, f=12 = =2.02%
m
QR f .24 4
k = (1+ ) −1 (1+ )12−1
m 4
B. m=4, f=12. = =1.96%
m 3
k = (1+QR ) −1 (1+ .24) −1
m 3
C. m=3, f=12. = =1.94%
m
QR f .24 2
k = (1+ ) −1 (1+ ) −1
m 2
D. m=2, f=12. = =1.91% 2
28. When Jon graduates in three years, he wants to throw a big party, which will cost $800. To
have this amount available, how much does he have to invest today if he can earn a
compound return of 5 percent per year?
Topic: Discounting
Level of difficulty: Medium
Solution:
Jon needs $800 in three years; that is the future value amount. The present value equivalent
is:
1 1
PV 0 FV ×3 3 = $800× 3= $691.07
(1+ k) (1+.05)
Or using a financial calculator (TI BAII Plus),
N=3, I/Y=5, PMT=0, FV= -800, CPT PV=691.07
38. Jimmie’s new car (see Problem 20) will cost $29,000. How much will his monthly car
payments be if he obtains a loan that is amortized over 60 months, and the nominal interest
rate is 8.5 percent per year with monthly compounding?
Topic: Effective Interest Rates and LoanArrangements
Level of difficulty: Medium
Solution:
First, find the effective interest corresponding to the frequency of Jimmie’s car payments (f
=12); with monthly compounding, set m=12,
mf 1212
k =1+ QR −1= 1 8.5% −1= 0.7083%
monthly m 12
The 60 car payments form an “annuity” whose present value is the amount of the loan (the
price of the car): 3
1
1− 60
$29,000 = PMT (1+ 0.007083) ⇒ PMT = $594.98
0.007083
47. Josephine needs to borrow $180,000 to purchase her new house in Yarmouth, Nova Scotia.
She would like to pay off the mortgage in 20 years, making monthly payments. For the initial
three-year term, Providence Bank has offered her a quoted annual rate of 6.40 percent.
A. What is the effective annual interest rate?
B. What is the effective monthly interest rate?
C. How much will Josephine’s monthly mortgage payments be?
Topic: Mortgage Loans and Effective Interest Rates
Level of difficulty: Medium
Solution:
A. In Canada, fixed-rate mortgages use semi-annual compounding of interest, so m=2. The
effective annual ratemis therefore: 2
QR 0.064
k = + −1= 1 −1= 6.5024%
m 2
B. With monthly payments, f=12. We can find the effective monthly interest rate from the
effective annual rate, k:
1 f 112
kmonthly 1+ k ) −1= 1+6.5024% ) −1= 0.5264%
C. The amortization period is 20 years, or 20 x1 2 = 240 months. Josephine’s monthly
payments can be computed as:
1
1− 240
(1+ 0.005264)
$180,000 = PMT 0.005264 ⇒ PMT = $1,322.69
4
50. A lakefront house in Kingston, Ontario is for sale with an asking price of $499,000. The real-
estate market has been quite active, so the house will almost certainly attract several offers,
and may sell for more than the asking price. Charlie is very eager to purchase this house, but
is concerned that he may not be able to afford it. He has $130,000 available for a down
payment, and can pay up to $1,950.00 per month on a mortgage loan. As a long-time
customer, Charlie’s bank has offered him a great mortgage rate of 3.90 percent on a one-year
term. If the loan will be amortized over 25 years, what is the most that Charlie can afford to
pay for the house?
Topic: Mortgage Loans
Level of difficulty: Medium
Solution:
With semi-annual compounding (the norm in Canada) and monthly payments, m=2 and f=12.
The effective monthly rate is:
m 2
QR f 0.039 12
kmonthly + m −1= 1 2 −1= 0.3224%
The present value of the mortgage payments over the amortization period (25 years x 12 =
300 months) is:
1
1− 300
PV = $1950.00× (1+ 0.003224) = $374,553.72
0 0.003224
In addition, Charlie has $130,000 available as a down payment; the most he can pay for the
house is, therefore, $374,553.72 + $130,000 = $504,553.72. 5
51. Timmy sets himself a goal of amassing $1 million in his retirement fund by the time he turns
61. He begins saving $3,000 each year, starting on his 21 birthday (40 years of saving).
A. If his savings earn 10 percent per year, will Timmy achieve his goal?
B. Will Timmy be able to retire before he turns 60? That is, at what age will the value of his
savings plan be worth $1 million?
Topic: Investing Early
Level of difficulty: Medium
Solution:
A. Timmy’s savings extend right to age 61 (end of each year), so this

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