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 plans actuarial team concluded that the pensions 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 plans expected obligation stream?

FV = P*[(1+r)^n-1)/r]

n = 66.08 years

FV = P*[(1+r)^n-1)/r]

FV = 80*10,000,000 = 800,000,000

Annuity = 10,000,000

800,000,000 = 10,000,000*[(1+6%)^n-1)/6%]

800*6%-1 = 106%^n

n log 1.06 = log 47

n = log 47/log 1.06

n = 66.08 years

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 bonds J and K. Bond J is a 7-year bond paying a 5% annual coupon (annually). Bond K is a consol (perpetual) bond paying 10% annual coupon rate (annually). Both bonds have a 1,000 face value. How much money should 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 its expected future obligations. It also calculates that its obligations 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, F and G. 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 the weights of investment in each of the three bonds.