MTH 360

Theory of Mathematical Interest

Question 1

Homework 6

(Mon. Nov. 20)

Suppose a 2 year bond with F = C = 1000, and a nominal annual coupon rate convertible semian- nually of 10% is purchased at a price to yield 8% convertible semiannually.

(a) Calculate the price of the bond using the basic formula, and the premium/discount formula. (b) Calculate the BV after 1 year (2 terms).

(c) Calculate the amount of interest in the third coupon.

(d) Calculate the amount of amortization of premium in the third coupon.

Question 2

A 10-year bond with semi-annual coupons is bought at a discount to yield 9% convertible semian- nually. If the amount for accumulation of discount in the next-to-last coupon is 8, find the total amount for accumulation of discount during the first four years in the bond amortization schedule.

Question 3

A 40 year bond with a par value of 5000 is redeemable at par and pays semi-annual coupons at a rate of 7% convertible semi-annually. The bond is purchased to yield annual effective rate of 6%. Calculate the amortization of the premium in the 61st coupon.

Question 4

A 10 year bond is redeemable at par of 100,000. The bond has semi-annual coupons of 4000. The bond is bought to yield 6% convertible semi-annually. Four months after purchase, calculate the book value based on the semi-theoretical method (which we learned in class).

Question 5

A 100 par value 4% bond with semi-annual coupons is callable at the following times:

• 109.00, 5 to 9 years after issue

• 104.50, 10 to 14 years after issue

• 100.00, 15 years after issue (maturity date).

What price should an investor pay for the callable bond if they wish to realize a yield rate of

(1) 5% payable semi-annually and (2) 3% payable semi-annually?

References

S.A. Broverman (2017), Mathematics of Investment and Credit, 7th Ed. (ACTEX) M.B. Finan (2016), A Basic Course in the Theory of Interest and Derivatives Markets: A Preparation for the Actuarial Exam FM/2 (http://faculty.atu.edu/mfinan/actuarieshall/mainf.pdf)

MTH 360 Theory of Mathematical Interest

Question 6

You are given the following information about an investment account:

Date January 1 July 1 December 31

Value Immediately Before Deposit 10

Deposit

12 X X

Over the year, the time-weighted return is 0%, and the dollar-weighted return is Y . Calculate Y .

Question 7

A large pension fund has a value of 500,000,000 at the start of the year. During the year, the fund receives contributions of 100,000,000, pays out benefits of 40,000,000 and has interest income of 60,000,000. Estimate the yield rate on the fund if the contributions, benefits and interest are uniformly spread throughout the year.

Question 8

You are given the following term structure:

s0(1) = 0.15, s0(2) = 0.10, s0(3) = 0.05

These are effective annual rates of interest for zero coupon bonds of 1, 2, and 3 years maturity, respectively. A newly issued 3-year bond with face amount 100 has annual coupon rate 10%, with coupons paid once per year starting one year from now. Find the price and effective annual yield to maturity of the bond.

Question 9

According to the current term structure of interest rates, the effective annual interest rates for 1, 2, and 3 year maturity zero coupon bonds are

Year 1 2 3

Interest Rate 0.08 0.10 0.11

Find the one-year forward effective annual rate of interest and find the two-year forward effective annual rate of interest.

References

S.A. Broverman (2017), Mathematics of Investment and Credit, 7th Ed. (ACTEX) M.B. Finan (2016), A Basic Course in the Theory of Interest and Derivatives Markets: A Preparation for the Actuarial Exam FM/2 (http://faculty.atu.edu/mfinan/actuarieshall/mainf.pdf)