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University of Connecticut
Math 3615: Financial Mathematics Problems
Fall 2009
Homework Assignment 2 – due Thursday 9/17/ Module 3 Problems
- Using an interest rate of 6% per annum, convertible semi-annually, compute the ratio of the present value of annuity (a) to the present value of annuity (b):
(a) a payment of 25 at the end of each 3-month period for a period of 20 years (payments totaling 2,000) (b) a payment of 400 at the end of each 4-year period for a period of 20 years (payments totaling 2,000)
- Annie, who is now age 60, has an annuity that will pay her 1,000 at the end of each month for the next 30 years.
For the first 10 of these 30 years, Annie intends to deposit these payments into a savings account that pays 6% per annum, convertible monthly. At the end of 10 years (when she retires), she will use the accumulated value of the savings account to purchase an annuity-immediate that will make payments at the end of each month for the following 20 years. (The total payment she will receive each month during those 20 years will equal the sum of 1,000 (from the original annuity) plus the monthly payment from the second annuity.)
If the annuity that Annie purchases at the end of the first 10 years has a monthly payment based on an interest rate of 6.6% per annum, convertible monthly, what is the total amount she will receive each month in years 11 through 30?
- Two 20-year increasing annuities have the same present value, based on an annual effective interest rate of 8%.
One annuity consists of a payment of 100 at t =0, followed by payments that increase by 5% each year thereafter (105 at t =1, 110.25 at t =2, etc.).
The other annuity consists of a payment of X at t =1 (not t =0), followed by payments that increase by 10 each year thereafter (X+10 at t =2, X+20 at t =3, etc.).
Given that these annuities have the same present value at an annual effective rate of 8%, what is the value of X?
Module 4 Problems
- Jerome borrows 1,000 at a 5% effective annual interest rate, and uses the proceeds to purchase a 10-year bond with 8% annual coupons. (Note that Jerome has not used any of his own money.)
Jerome uses each coupon payment from the bond issuer to repay a portion of the loan. When the bond matures after 10 years, he uses a portion of the redemption value to repay the remainder of the loan.
What is the amount of Jerome’s profit at the end of 10 years? (In other words, how much money does he have left from the bond’s maturity value after he pays the remaining amount due on the loan?)
- A 20-year bond issued on September 1, 2008, is callable at par at any time after 5 years. The bond pays semi-annual coupons and its (annual) coupon rate is 7%.
The bond is purchased on the issue date at a price that will provide a yield to maturity of 6% (convertible semi-annually).
What is the earliest coupon date that the bond could be called and still provide the original purchaser a yield of at least 5% (convertible semi-annually)? 9/1/
- A 1,000 face amount bond with semi-annual coupons was originally issued on August 1, 2005, and had an original term was 20 years. It does not have a call provision.
The bond was purchased on September 17, 2009, for a “total price” of 888.60 (including accrued coupon). At that price, its yield to maturity is 7% (compounded semi-annually). What is the bond’s annual coupon rate? (Assume 30-day months.)
- Consider the following two perpetuities-immediate:
(a) A geometric perpetuity-immediate with annual payments; the first payment is 100, and each subsequent payment is 3% larger than the preceding payment.
(b) An arithmetically increasing perpetuity-immediate, consisting of a payment of x at the end of the first year, 2 x at the end of the second year, etc.
If these two annuities are evaluated at an annual effective interest rate of 5%, they have the same present value. What is the value of x?
Module 4 Problems
- Two similar bonds are available for sale. Each has a 1,000 par value, semi-annual coupons, and a 10-year remaining term.
Both bonds are callable after 5 years, with an x % premium
(i.e., if the bond is called, its redemption value is 1, 000 1 100
^ x ⋅ (^) +
Bond A has a 4% annual coupon rate. Bond B has an 8% annual coupon rate.
Based on their current prices, each bond will yield no less than 5.772%.
If the discount on bond A is numerically equal to the premium on bond B, what is the value of x?
