Question

You are the pension fund manager for a consulting company. Your workers will begin to retire 10 years from now. You estimate that you will need $1.2 million exactly 10 years from now to fund the first year payments. Due to inflation and growth in the number of retirees, your annual obligations will grow by 5% per year, and will continue forever. Your financial advisors tell you that you can plan on earning 8.0% per year on invested funds. (a) As of now, your company has set aside $12 million to fund its pension obligations. Is this amount sufficient to meet the obligations? What, specifically, is the amount of the shortfall or excess in present value terms? (b) Your company will make annual contributions (assume end-of-year) to its pension fund to ensure that it can meet the obligations. These will grow by 5% per year, and will continue forever. What should be the amount of the first contribution? Suppose that your advisors revise their assessment of the return that you can expect to earn on your invested funds, to 7% per year.    Note that this is a 12.5% (1/8) reduction in the assumed return.   What is the percentage change in your answer to part (a)?  Briefly summarize the real world issue that this calculation highlights.  

Answer 1

All financials below are in $ mn.

Part (a)

PV of all the future obligations at the end of year 9 = PV of a perpetuity with g = P / (R - g) = 1.2 / (8% - 5%) = 40

PV of all obligations today = PV at the end of yer 9 x (1 + R)^-9 = 40 x (1 + 8%)^-9 = 20.01

Hence, shortfall = 20.01 - 12 = 8.01

Part (b)

Let the first contribution be A

Hence, Shortfall = PV of A in perpetuity with growth of 5% = A /(R - g) = A / (8% - 5%) = A / 3% = 8.01

Hence, A = 8.01 x 3% = 0.24

Part (c)

If R = 7% then

PV of all the future obligations at the end of year 9 = PV of a perpetuity with g = P / (R - g) = 1.2 / (7% - 5%) = 60

PV of all obligations today = PV at the end of yer 9 x (1 + R)^-9 = 60 x (1 + 8%)^-9 = 30.01

Hence, shortfall = 30.01 - 12 = 18.01

Hence, %age change = 18.01 / 12.01 - 1 = 24.91% or say 25%

Real life problem:

For every 12.5% change in interest rate, value of the liability changes by 25%. Thus duration of the liability is roughly 2.

The the value of the obligations are sensitive to interest rates. The assets and liabilities must have matching duration to take care of obligations. So, the assets and liabilities need to be hedged on duration.