Bond

Coupon Rate

Maturity

Market Price

A

3% paid annually

1 year

998.06

B

4% paid annually

2 years

1011.49

C

7% paid annually

3 years

1094.68

1. Assume the pure expectations hypothesis (PEH) holds, and estimate the term structure for the next three years (i.e. calculate the spot rate for the first year, and the forward rates for the second, and third years).
2. Now assume (setting aside the information in question (1)) that the spot rate for the first year, the forward rate for the second year, and the forward rate for third year are 4.2%, 4.4%, and 4.5%, respectively. What must the price of a three-yearGovernment of Canadanoncallable bond with an annual coupon of 6% be?

3.Assume that the term structure of forward rates is that given in question (2). Suppose that XYZ Inc. issued a new $1,000 face value retractable (puttable) bond with 3 years to maturity, carrying a 5% annual coupon, paid annually. This bond is retractable (puttable) according to the following schedule:

Puttable in 1 year for $975

Puttable in 2 years for $990

Assume that the Pure Expectations Theory of the term structure holds, and that there are no costs involved in issuing new bonds. Also assume that XYZ's bonds are traded with a 2% yield spread over the Government of Canada forward rates. Note that XYZ bondholders can only retract (put) the bond in one year or in two years. Assume that XYZ bondholders choose the retraction date so that their retraction profit is maximized. When do you expect XYZ bondholders to retract (put) the new bond?

Hint:To solve this problem, calculate the expected bond prices in Year 1 and Year 2 using the Pure Expectations theory formulas.

Expected value of the bond one year from now:

Expected value of the bond two years from now:

Then compare the expected bond prices with the exercise price on the "put option" on the bonds for the decision to retract the bonds or not.

Problem 3 (25 marks)

Consider a pension plan that pays beneficiaries in the following manner: at the end of the retirement year of a beneficiary the value of all benefits are transferred to his or her personal account. This means that at the end of every year the pension plan makes lump-sum payments of pension benefits to its beneficiaries.The pension plan'sactuarial team concluded that the pension's obligation stream could not be estimated beyond an 80-year horizon. They further estimate that the plan will have to make annual pension payments of $10 million per year throughout this 80-year horizon. The first payment will take place in exactly one year. Assume the current yield curve is flat at 6%.

Note: In your calculations, use dollar figures rather than millions of dollars.

1)What is the duration of the plan's expected obligation stream?

2)Assume that the pension plan hired a new actuarial team that revised the calculations in (1) and found that the present value of expected pension obligations in the next century is $150 million with a 22-year duration. You decide to utilize an immunization strategy for this obligation that exclusively involves two bondsJandK. BondJis a 7-year bond paying a 5% annual coupon (annually). BondKis a consol (perpetual) bond paying 10% annual coupon rate (annually). Both bonds have a 1,000 face value. How muchmoneyshould the pension plan invest in each of the two bonds? Indicate the position the plan takes in each bond (long or short).

3)A different pension plan calculates a 16-year duration for itsexpected future obligations. It also calculates that itsobligations have a convexity of 29. You were hired by this pension plan and you want to utilize an immunization strategy for its obligations, which exclusively involves three bonds:E, FandG. You calculated the following information for the three bonds:

Bond

Duration

Convexity

E

9.00

29.00

F

21.00

35.00

G

28.00

56.00

Use both duration and convexity in your immunization strategy, to determine theweightsof investment in each of the three bonds.