1st2nd Seminar and Homework -FAM (1)

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Financial Mathematics
Prof.dr. Paula CURT
1st and 2nd Seminar and Homework
Simple, Compound and Nominal Interest
(1) At what annual rate of simple interest will $500 accumulate to $615 in 2 years?
Answer: i = 0.115
(2) You invest $100 at time 0, at an annual simple interest rate of 9%.
(a) Find the accumulated value at the end of the fifth year.
(b) How much interest do you earn in the fifth year?
Answer: (a)$145; (b)9
(3) In how many years will $500 accumulate to $630 at 7.8 % annual simple interest?
Answer: 313 years; 3years 4months
(4) What simple interest rate is necessary for $10,000 to earn $100 interest in 15 months?
Answer: i = 0.008
(5) A fund is earning 5% annual simple interest. The amount in the fund at the end of the 5th year
is $10,000. Calculate the amount in the fund at the end of 7 years.
Answer: $10, 800
(6) Let i be a simple annual interest rate and suppose that i6 = 0.04. Calculate i.
Answer: i = 24%
(7) Joe deposits 10 today and another 30 in five years into a fund paying simple interest of 11% per
year. Tina will make the same two deposits, but the 10 will be deposited n years from today and
the 30 will be deposited 2n years from today. Tina’s deposits earn an annual effective rate of
9.15% . At the end of 10 years, the accumulated amount of Tina’s deposits equals the accumulated
amount of Joe’s deposits. Calculate n.
Answer: n = 2.33
(8) On January 1, 2000, you invested $1,000. Your investment grows to $1,400 by December 31, 2007.
What was the compound annual interest rate at which you invested?
Answer: i = 4.3%
(9) Jack has deposited $1,000 into a savings account. He wants to withdraw it when it has grown to
$2,000. If the interest rate is 4% annual interest compounded annually, how long will he have to
wait?
Answer: 17.67 years
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(10) You invest some money in an account earning 6% annual compound interest. How long will it
take to quadruple your account balance? (Express your answer in years to two decimal places.)
Answer: 23.79 years
(11) You want to triple your money in 25 years. What is the annual compound interest rate necessary
to achieve this?
Answer: i = 4.49%
(12) At an effective annual interest rate of i, each of the following two sets of payments has present
value K:
(a) A payment of 121 immediately and another payment of 121 at the end of one year.
(b) A payment of 144 at the end of two years and another payment of 144 at the end of three
years.
Calculate K.
Answer: K = 278312
(13) Find the accumulated value of $1,000 after three years at a rate of interest of 24% per year
convertible monthly.
Answer: $2039.89
(14) Find the accumulated value of $3,000 to be paid at the end of 8 years with a rate of compound
interest of 5% per year
(a) convertible quarterly;
(b) convertible monthly.
Answer: (a)$4464.39; (b)$4471.76
(15) Given the interest rate of 12% per year, compounded monthly, find the equivalent effective annual
interest rate.
Answer: ıeff = 12.68%
(16) A bank credits interest on deposits quarterly at rate 2% per quarter. What is the nominal interest
rate (per annum) for interest compounded quarterly? Find the annual effective rate of interest.
Answer: ρ4 = 8%; i = ieff = 8.24%
(17) Investor A deposits X into a savings account at time 0, which pays interest at a annual interest
rate of i, compounded semi-annually. Investor B deposits 2X into a different savings account at
time 0, which pays simple interest at an annual rate of i. Both investors earn the same amount
of interest during the last six months of the 8th year. Calculate i.
Answer: i = 4.729%
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(18) You are given:
(a) Fund X accumulates at an annual interest rate of 8% compounded quarterly;
(b) Fund Y accumulates at an annual interest rate of 6% compounded semiannually;
(c) at the end of 10 years, the total amount in the two funds combined is 1000;
(d) at the end of 5 years, the amount in Fund X is twice that in Fund Y
Calculate the total amount in the two funds at the end of 2 years.
Answer: 560
(19) A borrower receives $1,500 today and must pay back $1,580 in 200 days. What interest rate is
assumed?
Answer: i = 0.096
(20) Alexandra borrows $1,000 on January 8 at 16% per year. She pays $350 on April 12, $20 on
August 10 and $400 on October 3. What is the balance due on December 15?
Answer: $229
(21) A debt of $5,000 is due in 6 month with interest at 5% per year. Partial payments of $3,000 and
$1,000 are made in 2 and 4 months respectively. What is the balance due on the final statement
date.
Answer: $941.67
(22) John invested $500 for 4 years. The interest rate remains 8% each year although in the first
year it is compounded semi-annualy, in the second year compounded quaterly, in the third year
compounded monthly, and in the fourth year compounded daily. How much greater is the final
value in this case than the corresponding final value assuming that the first rate had remained
unchanged for the 4 years.
Answer: $2.48
(23) A sum of money is left invested for three years. In the first year, the interest rate is 4% per year
compounded monthly. In the second year, the interest rate is 8% per year compounded quaterly
and in the third year the rate of interest changes to 5.5% per year compounded daily. Compute
the annual effective rate that would give the same accumulated value at the end of three years.
Answer: ieff = 5.98%
(24) A company wishes to replace the following three debts:
$20,000 due on July 1, 2004
$30,000 due on January 1, 2007 and
$35,000 due on July 1, 2010
with a single debt of Y payable on January 1, 2007. Compute the value of Y , if the interest
rate is 12% per year compounded semi-annually, prior to January 1, 2007 and 10% per year
compounded semi-annually, after to January 1, 2007.
Answer: $81638.36
(25) How much will $2,000 accumulate to in 12 years’ time, if the annual effective interest rate is 4%
for the first 3 years, 6% for the next 4 years and 8% for the last 5 years.
Answer: $4173.23
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