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UNIVERSITY OF TORONTO
Faculty of Arts and Science
JUNE 2007 EXAMINATIONS
ACT240H1F
Duration – 3 Hours
Aids: All calculators allowed
Instructors: Keith Sharp PhD
NOTES:
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Calculators allowed
Scrap paper is to be handed in with this book. It’s OK to write on book.
This is a closed book exam.
Multiple choice: only your letter answer mark sense sheet will be graded.
Each question: 10 points correct, two if blank, zero points if wrong
So expectation if you guess is the same as leaving a blank.
Make sure you’ve indicated your letter answers on the mark sense sheet before time’s up
Please stay in your seats and don’t talk till all question papers and mark sense sheets have
been collected.
Photo ID on desk during exam please.
Name and student ID on this question paper and on mark sense sheet please.
Please code question 31 with the privacy code (A, B, C or D) in the page footer.
Good luck!
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UNIVERSITY OF TORONTO: ACT240H1F SUMMER 2007 FINAL
1.A mutual fund advertises that average annual compound rates of return for various periods
ending December 31, 2005 are as follows:
10 Years
5 Years
2 Years
1 Years
12.6%
15.0%
15.0%
25.0%
Find the 5-year average compound rate of return for the period January 1, 1996 to December 31,
2000.
(A) Less than 9.500%
(B) 9.500% but less than 10.000%
(C) 10.000% but less than 10.500%
(D) 10.500% but less than 11.000%
(E) 11.000% or more
2. Every year a 10 kg rod of Torontonium fuel is added to a multi-rod nuclear machine. The
first fuelling is on January 1, 2010. 15% of the Torontonium is consumed by the machine every
year. For example, by December 31, 2011 the rod deposited 24 months previously contained
only 10* (0.852) kg of fuel.
Calculate the amount of fuel in the machine on December 31, 2025.
(A) Less than 52.400 kg
(B) 52.400 kg but less than 52.500 kg
(C) 52.500 kg but less than 52.600 kg
(D) 52.600 kg but less than 52.700 kg
(E) 52.700 kg or more
3.At an annual effective interest rate of i, i>0, the following are all equal:
(i) the present value of 10000 a the end of 6 years:
(ii) the sum of the present values of 6000 at the end of year t and 56000 at the of year 2t; and
(iii) 5000 immediately.
Calculate the present value of a payment of 8000 at the end of year t+3 using the same annual
effective interest rate.
(A) 1334
(B) 1414
(C) 1604
(D) 1774
(E) 2004
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UNIVERSITY OF TORONTO: ACT240H1F SUMMER 2007 FINAL
4. Jennifer deposits 1000 into a bank account. The bank credits interest at a nominal annual rate
of i convertible semi-annually for the first 7 years and a nominal annual rate of 2i convertible
quarterly for all years thereafter. The accumulated amount in the account at the end of 5 years is
X. The accumulated amount in the account at the end of 9.5 years is 1980. Calculate X to the
nearest dollar.
(A) 1201
(B) 1226
(C) 1251
(D) 1329
(E) The correct answer is not given by (A), (B), (C) or (D)
5. Bruce and Robbie each open up new bank accounts at time 0. Bruce deposits 100 into his
bank account and Robbie deposits 50 into his. Each account earns an effective annual discount
rate of d. The amount of interest earned in Bruce’s account during the 11th year is equal to X.
The amount of interest earned in Robbie’s account during the 16 th year is also equal to X.
Calculate X.
(A) Less than $57.000
(B) $57.000 but less than $58.000
(C) $58.000 but less than $59.000
(D) $59.000 but less than $60.000
(E) $60.000 or more
6. You are given a loan on which interest is charged over a 3-year period , as follows:
(i) an effective rate of discount of 8% for the first year;
(ii) a nominal rate of interest of 5% compounded semiannually for the second year; and
(iii) a force of interest of 4% for the third year.
Calculate the annual effective rate of interest over the 3-year period.
(A) Less than 5.9000%
(B) 5.9000% but less than 6.0000%
(C) 6.0000% but less than 6.1000%
(D) 6.1000% but less than 6.2000%
(E) 6.2000% or more
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UNIVERSITY OF TORONTO: ACT240H1F SUMMER 2007 FINAL
7. An investment of 1000 accumulates to 1320 at the end of 5 years. If the force of interest is δ
during the first year and 1.5δ in each subsequent year, find the equivalent effective annual
interest rate in the first year.
(A)
(B)
(C)
(D)
(E)
Less than 4.25%
4.25% but less than 4.75%
4.75% but less than 5.5%
5.5% but less than 6.25%
6.25% or more
8. Tawny makes a deposit into a bank account which credits interest at a nominal interest rate of
8% per annum, convertible semiannually.
At the same time, Fabio deposits 2000 into a different bank account, which is credited with
simple interest.
At the end of 5 years, the forces of interest on the two accounts are equal, and Fabio’s account
has accumulated to Z.
Determine Z.
(A)
(B)
(C)
(D)
(E)
Less than 3050
3050 but less than 3150
3150 but less than 3250
3250 but less than 3350
3350 or more
9. You run a bank in Tufortiland. The law limits you to offering 5% per annum interest on
savings accounts, but doesn’t specify the compounding (‘convertible’) period. To persuade the
public to come to your bank to deposit money you want to offer them the highest rate possible.
What is the highest effective rate per annum that you can offer without risking a term in the
infamous Tufortese prisons?
(A)
Less than 5.000%
(B)
5.000% but less than 5.050%
(C)
5.050% but less than 5.100%
(D)
5.100% but less than 5.150%
(E)
5.150% or more
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UNIVERSITY OF TORONTO: ACT240H1F SUMMER 2007 FINAL
10. You buy a $400,000 house on January 1, 2008 and arrange with CitiBank a U.S. mortgage
amortized with level payments at the end of each month over a period with last payment
January 1, 2033. The interest rate is 6% per annum compounded monthly. Just after 24
payments have been made, you refinance (pay off Citi) by borrowing the outstanding balance
from Chemical Bank at a rate of 3% per annum compounded monthly. The new mortgage is
amortized over a period with last payment January 1, 2033. Calculate the reduction in your
monthly mortgage payment.
(A) Less than $575.000
(B) $575.000 but less than $600.000
(C) $600.000 but less than $625.000
(D) $625.000 but less than $650.000
(E) $650.000 or more
11. This ad could appear in Varsity:
Only 3% interest per annum (compounded monthly)! Pay for your $10,000 thneed in 48 easy
monthly payments, starting one month from now! Or get a $X discount if you pay in a cash
lump sum!
The merchant is using the current market interest rate of 6% per annum compounded monthly
for her calculations, and wants the same profit on a cash sale as on an ‘easy payments’ sale.
Calculate X. (Note that $560.000 means exactly $560)
(A)
(B)
(C)
(D)
(E)
Less than $560.000
$560.000 but less than $570.000
$570.000 but less than $580.000
$580.000 but less than $590.000
$590.000 or more
12.Victor wants to purchase a perpetuity paying 100 per year with the first payment due at the
end of year 11. He can purchase it by either:
(i) paying 90 per year at the end of each year for 10 years; or
(ii) paying K per year at the end of each year for the first 5 years and nothing for the next 5
years.
Calculate K (nearest 5).
(A) 150
(B) 160
(C) 170
(D) 175
(E) 180
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UNIVERSITY OF TORONTO: ACT240H1F SUMMER 2007 FINAL
13. Jenny’s salary increases by 6% at the beginning of each year of work. She works for 30
years, paying at each year-end a proportion Z of her earnings to a retirement fund which earns
6% per annum tax-free. In her last year she earns $X. She draws at the end of her first year of
retirement a pension of 1.04X (1-Z). The pension continues to increase for inflation at 4%
every year till the money runs out after 20 years of retirement. Calculate Z as a percentage of
earnings.
(A) Less than 35.000%
(B) 35.000% but less than 36.000%
(C) 36.000% but less than 37.000%
(D) 37.000% but less than 38.000%
(E) 38.000% or more
14. Stock in No Thrills Grocery Stores is expected to pay dividends of $5 annually on January 1
of 2009 through 2016 inclusive. The January 1, 2017 dividend is $5.30, and thereafter the 6%
per annum annual increase in dividend continues indefinitely. You buy the stock January 2,
2008, paying a price P which is expected to give you a yield of 8% per annum effective. You
collect the expected dividends and sell on January 2, 2016 at a price $Q which results in you
having earned an actual realized return of 9% per annum effective. Calculate Q.
(A) Less than $385.000
(B) $385.000 but less than $390.000
(C) $390.000 but less than $395.000
(D) $395.000 but less than $400.000
(E) $400.000 or more
15. Amy invests 1000 at an effective annual rate of 14% for 10 years. Interest is payable
annually and is reinvested at an annual rate of i. At the end of 10 years the accumulated interest
is 2341.08. Bob invests 150 at the end of each year for 20 years at an annual effective rate of
15%. Interest is payable annually and is reinvested at an annual effective rate of i. Find Bob's
accumulated interest at the end of 20 years.
(A) 9000
(B) 9010
(C) 9020
(D) 9030
(E) 9040
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UNIVERSITY OF TORONTO: ACT240H1F SUMMER 2007 FINAL
16. A dollar paid at time t has a discounted value (present value) at time 0 given by exp(0.07t). You receive a continuous stream of money, at a rate which varies with time and is given
by $30exp(0.05t) . Calculate the accumulated value of the stream of income at time 10.
(A) Less than $500.000
(B) $500.000 but less than $520.000
(C) $520.000 but less than $540.000
(D) $540.000 but less than $560.000
(E) $560.000 or more
17. All cash flows are just after midnight on New Year’s Eve, at 0001 on January 1. The
time-weighted returns for calendar years are given by:
2010 +13.0%
2011 + 9.2%
The fund begins when you deposit $5,000 into it on January 1, 2010. You make a further
deposit of $10,000 on January 1, 2011. Calculate the money-weighted annual rate of return i on
the fund for the period January 1, 2010 – December 31, 2011.
(A) Less than 9.900%
(B) 9.900% but less than 10.100%
(C) 10.100% but less than 10.300%
(D) 10.300% but less than 10.500%
(E) 10.500% or more
18. You bought your house January 1, 2008. You have a 25-year Canadian mortgage on your
house, with level payments made at the end of each month. The interest rate is fixed for the
period January 1, 2008 – December 31, 2017. During 2015 your mortgage outstanding balance
reduces by $19,543.21. During 2010 it reduces by $14, 232.32. Calculate the amount you
initially borrowed.
(A) Less than $720,000.000
(B) $720,000.000 but less than $730,000.000
(C) $730,000.000 but less than $740,000.000
(D) $740,000.000 but less than $750,000.000
(E) $750,000.000 or more
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UNIVERSITY OF TORONTO: ACT240H1F SUMMER 2007 FINAL
19. ) Jason takes out a 15-year loan, which is repaid with annual payments at the end of each
year. He repays the loan by making payments which are equal to X during years 1-5, 3X during
years 6-10, and 2X during years 11 to 15. Interest is charged on the loan at an annual effective
rate of i, i>0. The amount of interest repaid during year 6 is twice as much as the amount of
interest repaid during year 11. Calculate i.
(A) 14.5%
(B) 14.7%
(C) 14.9%
(D) 15.1%
(E) 15.3%
.
20.A loan is to be repaid by 2n level annual payments, starting one year after the loan is made.
Just after the n-th payment the borrower finds she still owes 3/4 of the original loan amount.
What proportion of the following year's payment will represent interest?
(A) 3/4
(B) 2/3
(C) 1/2
(D) 1/3
(E) 1/4
21. A 2,000 loan is to be repaid with equal payments at the end of each year for 20 years. The
principal portion of the 13th payment is 1.6 times the principal portion of the 5th payment.
Calculate the total amount of interest paid on the loan.
(A)Less than $1,490.000
(B)$1,490.000 but less than $1,510.000
(C)$1,510.000 but less than $1,530.000
(D)$1,530.000 but less than $1,550.000
(E)$1,550.000 or more
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UNIVERSITY OF TORONTO: ACT240H1F SUMMER 2007 FINAL
22. Mr. and Mrs. Chiu need to borrow 80% of the $600,000 price of a house. They use a
Canadian mortgage, amortized over 25 years with payments at the end of each month. While
they are walking to the bank to arrange the mortgage, interest rates fall from 7% per annum
compounded semi-annually to 6% per annum compounded semi-annually. They accumulate the
amount of the monthly mortgage payment reduction in a bank account paying 3% per annum
compounded monthly. Calculate the amount in the bank account just after the 24 th. mortgage
payment.
(A) Less than $7,200.000
(B) $7,200.000 but less than $7,300.000
(C) $7,300.000 but less than $7,400.000
(D) $7,400.000 but less than $7,500.000
(E) $7,500.000 or more
23. BahenBank will lend you mortage money for 25 years, repayable in level payments at the
end of each month. It charges interest at 7% per annum convertible (compounded) semiannually, in the usual Canadian way. Its rule is to limit lending so that your mortgage
payments will not exceed 30% of you and your spouse’s gross combined salary of $108,000 per
annum. You have saved a cash down payment of $60,000. Ignoring payments to realtors,
lawyers, government etc, calculate the maximum that you can agree to pay for a house or condo.
(A)
(B)
(C)
(D)
(E)
Less than $440,000.00
$440,000.00 but less than $450,000.00
$450,000.00 but less than $460,000.00
$460,000.00 but less than $470,000.00
$470,000 or more
24. You borrow $10,000 from Blue Bank, which charges interest of 5% per annum effective,
and agree to pay interest X at the end of each year for 10 years, with repayment of the principal
after 10 years. You fund the repayment by making level payments Y into an account at Red
Bank, which pays interest at 6% per annum effective. For this sinking find arrangement
calculate to the nearest dollar the total annual payment X+Y.
(A)
(B)
(C)
(D)
(E)
$1,259
$1,269
$1,279
$1,289
$1,299
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UNIVERSITY OF TORONTO: ACT240H1F SUMMER 2007 FINAL
25. At January 1, 2008 you borrowed $1,000,000 from Blue Bank to buy a flourishing ‘rent-anerd’ business. You agree to pay to Blue Bank 8% p.a. effective interest annually, with the first
payment on December 31, 2008 and the last payment on December 31, 2027, the day of loan
repayment to Blue Bank. Under a sinking fund arrangement, you make level payments
December 31, 2008 through December 31, 2027 to Red Bank, using an interest rate of 9% per
annum effective.
Just after the 11th payment, Blue Bank announces that to foster good customer relations, it will
now charge only 7% per annum. For your internal business purposes you use an interest rate of
10% per annum effective in calculating present values. You tell Blue Bank, using this basis,
that you would be just as happy to continue to pay at the 8% rate but to receive a lump sum X
from Blue Bank instead of the reduction. The lump sum would be paid just after the 11th.
annual payment. Calculate X.
(A) Less than $55,000.000
(B) $55,000.000 but less than $56,000.000
(C) $57,000.000 but less than $58,000.000
(D) $58,000.000 but less than $59,000.000
(E) $59,000.000 or more
26.Henry has a five-year 1,000,000 bond with coupons at 6% convertible semi-annually. Fiona
buys a 10-year bond with face amount X and coupons at 6% convertible semi-annually. Both
bonds are redeemable at par. Henry and Fiona both buy their bonds to yield 4% compounded
semi-annually and immediately sell them to an investor to yield 2% compounded semiannually. Fiona earns the same amount of profit as Henry. Calculate X (nearest 1000).
(A) 500,000
(B) 502,000
(C) 505,000
(D) 571,000
(E) 574,000
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UNIVERSITY OF TORONTO: ACT240H1F SUMMER 2007 FINAL
27.A bond with coupons equal to 40 sells for P . A second bond with the same maturity value
and term has coupons equal to 20 and sells for Q. A third bond with the same maturity value
and term has coupons equal to 80. All prices are based on the same yield rate, and all coupons
are paid at the same frequency. Determine the price R of the third bond.
(A) 4P – 4Q
(B) 3P-2Q
(C) 4Q-3P
(D) 5P-4Q
(E) The correct answer is not given by (A), (B), (C) or (D)
28.A $100 bond with annual coupons is purchased at a premium of $36 to yield 3.5% per
annum effective. The amount for amortization of premium in the 5th year is $1.00. Find the
term of the bond (in years).
(A) 26
(B) 27
(C) 28
(D) 29
(E) 30
29.A 1000 par value 18-year bond with annual coupons is bought to yield an annual effective
rate of 5%. The amount for amortization of premium in the 10-th year is 20. The book value of
the bond at the end of year 10 is X. Calculate X.
(A) 1180
(B) 1200
(C) 1220
(D) 1240
(E) 1260
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UNIVERSITY OF TORONTO: ACT240H1F SUMMER 2007 FINAL
30.A fund earned investment income of 18,400 during 1999. The beginning and ending
balances of the fund were 200,000 and 258,400 respectively. A deposit was made at time K
during the year. No other deposits or withdrawals were made. The fund earned 8% in 1999
using the dollar-weighted method. Determine K to the nearest week.
(A) Feb 1
(B) March 1
(C) April 1
(D) September 1
(E) The correct answer is not given by (A), (B), (C) or (D)
31.If not already done, please identifying your Privacy ID in the page footer (A, B, C or D) and
code it as the answer for question 31.
Total marks: 300 (30 questions)
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