Act 2120 University Of Manitoba Actuarial Club – 2016/2017

Interest Theory Midterm 2 – Time: 70 min 1) The following table shows the annual effective interest rates being credited by an investment account, by calendar year of investment. The investment year method is applicable for the first 3 years, after which a portfolio rate is used: Calendar Year of Investment

Investment Yr. Rates Calendar Year of Portfolio Rate

Portfolio Rate i1 i2 i3

1990 10% 10% t% 1993 8% 1991 12% 5% 10% 1994 (t - 1)% 1992 8% (t - 2)% 12% 1995 6% 1993 9% 11% 6% 1996 9% 1994 7% 7% 10% 1997 10%

Sophie makes an investment of 100 at the beginning of years 1990, 1991, and 1992. The total amount of interest credited by her fund during the year 1993 is equal to $28.40. Calculate t. 2) You are given the following information about the activity in two different investment accounts:

Account K Fund Value Before

Activity Activity

Date Deposit Withdrawal January 1, 2015 100.0

July 1, 2015 125.0 X October 1, 2015 110.0 2X

December 31, 2015 125.0

Account L Fund Value Before

Activity Activity

Date Deposit Withdrawal January 1, 2015 100.0

July 1, 2015 125.0 X December 31, 2015 105.8

During 2015, the dollar-weighted return for investment account K equals the time-weighted return for investment account L, which equals i. Calculate i. 3) Patrick borrows $10,000 for 10 years at an annual effective interest rate of i. He accumulates the amount necessary to repay the loan by using a sinking fund. He makes 10 payments of X at the end of each year, which includes interest on the loan and the payment into the sinking fund, which earns an annual effective rate of 8%. If the annual effective rate of the loan had been 2i, his total annual payment would have been 1.5X. Calculate i.

Act 2120 University Of Manitoba Actuarial Club – 2016/2017

4) A continuously increasing annuity with a term of n years has payments payable at an annual rate t at time t. The force of interest is equal to 1/n. Calculate the present value of this annuity in terms of n. 5) A fund is built with annual payments increasing by $1 from $1 to $10 and then decreasing by $1 to $0. The first payment of $1 is made today. If the fund is used to purchase a 10 year level annuity with the first payment at 20 years from today, what is the amount of the level payment? Assume an annual effective rate of interest of 4%. 6) Calvin wishes to purchase a top-market stereo system so he and his neighbours could both listen to his favorite album, Beyoncé’s ”Lemonade.” He is offered the following payment options: Option 1: $0 down $432 in 1 year $300 in 2 years Option 2: $86.56 down $250 in 1 year $400 in 2 years Determine the range of interest rates for which the present value of Option 1 is less than the present value of Option 2. 7) A loan, at a nominal annual interest rate of 24% convertible monthly, is to be repaid with equal payments at the end of each month for 2n months. The nth payment consists of equal payments of interest and principal. Calculate n. 8) You are given a perpetual annuity-immediate with annual payments increasing in geometric progression, with a common ratio of 1.07. The annual effective interest rate is 12%. The first payment is $1. Calculate the PV of this annuity. 9) Martin borrows $10,000 for 25 years, at an effective annual interest rate of 5%. A sinking fund is used to accumulate the principal by means of 25 annual deposits earning an effective annual interest rate of 4%. Calculate the sum of the net amount of interest paid in the 13th installment and the increment in the sinking fund for the ninth year. Disclaimer: this exam is not one that has actually been previously tested in Interest Theory. This was an exam created by UMAC in order to provide students with a more recent realistic representation of what one may expect on both Interest Theory Exams, and Exam FM. There is a strong possibility that this exam is of easier difficulty than what will be tested in Interest Theory.

- Sergiu Buda