Philip H. Dybvig
Washington University in Saint Louis
March, 2005
This is a closed-book examination. Answer all questions as directed. Mark your answers directly on the examination. There are no trick questions on the exam. There are some formulas from the course (including some you will not need) at the end of the exam. All cash flows and interest rates are annual. Good luck!
A. General Concepts Short Anwer: 20 points (Answer each question in no more than two sentences of ordinary length.)
They fall.
The forward rate should typically fall by less than 1%, because current news has less impact on the future than on the present.
The main use of a cap is to limit the risk of rising rates when borrowing at a floating rate.
Rising rates implies later cash flows are relatively less important so that duration falls.
Fallout is lower when rates rise because higher rates make it less attractive to find a new loan elsewhere.
Today we can buy or sell a riskless claim paying $100 a year from now for $75. Or, we can buy a self-amortizing claim paying $100 one year out and $100 two years out for $140.
The one-year discount factor is given by the short bond: 75 D(0,1) = --- = 0.75. 100 The two-year discount factor is given by taking opposite positions in the one-year and two-year bonds. The net cash flow up-front is 140 - 75 = 65, the net cash flow one-year out is 0, and the net cash flow two years out is 100. Therefore, 65 D(0,2) = --- = 0.65. 100
As always, the one-year par coupon yield is the same as the one-year discount yield, which is 100 z(0,1) = --- - 1 = 33 1/3% 75 The two-year coupon yield can be given by the formula 1 - D(0,2) 1 - 0.65 0.35 c(0,2) = --------------- = ----------- = ---- = 25%, D(0,1) + D(0,2) 0.75 + 0.65 1.40 or we note that going long 1.25 2-year bonds and short 1 1-year bond gives net cash flows - $100, $25, and $125, which are the payments from a 2-year par-coupon bond yielding 25%.
Go long $100 worth of the par coupon bond yielding 30%, short 1.25 self-amortizing bonds, and long one short zero-coupon bond. The cash flows are: 0 1 2 -100 30 130 175 -125 -125 -75 100 0 ----- ----- ----- 0 5 5 Other arbs are possible, depending on when you want the cash out.
A pension liability consists of three cash flows: $625 million 5 years out, $800 million 10 years out, and $250 million 15 years out. The liability is funded by a single asset, a pure discount bond maturing in 10 years. The market value of the pension asset equals the market value of the liability, that is, the pension is fully funded in economic terms.
0.8 x 625 + 0.5 x 800 + 0.4 x 250 = 500 + 400 + 100 = $1,000 million = $1 billion By assumption this is also the market value of the asset.
liability: 500 400 100 ---- 5 + ---- 10 + ---- 15 = 2.5 + 4 + 1.5 = 8 years 1000 1000 1000 asset: 1000 ---- 10 = 10 years 1000
The asset has a longer duration so it is more sensitive to interest rate risk. Therefore, the duration measure predicts that a rise in interest rates will cause the market value of the asset to fall more than the market value of the liability.
Assume that the interest rate starts at 4% and in each period and either increases by 2% or decreases by 2% (from 4% up to 6% or down to 2% would be the first move). The risk-neutral probabilities of ups and downs are all 1/2.
This is a one-period discount bond with face of $100 and an interest rate equal to the initial rate of 4%. Therefore, the price is 100 ---- ~ $96.15 1.04
interest tree: 8% / 6% / \ 4% 4% \ / 2% \ 0% discount bond price: 100 / 94.34 / \ 92.49 100 \ / 98.04 \ 100
cash flows: 30 / 10 / \ 0 0 \ / -10 \ -30 price (pre-cash flow) 30 / 24.151 / \ -0.267 0 \ / -24.706 \ -30 calculations: 10 + (30 + 0)/2/1.06 = 24.151 -10 + (0 - 30)/2/1.02 = -24.706 (24.151 - 24.706)/2/1.04 = -.267