Forward rates and pricing a three-year coupon bond
A life insurer must pay £50 million at the end of three years. The market has liquid zero-coupon government bonds with annual effective spot yields of 3.00% for one year, 3.60% for two years and 4.00% for three years. A bank offers the insurer a three-year annual-coupon bond with a 4.00% coupon and a face value of £100, quoted at £99.80 per £100 nominal. The insurer's investment committee wants to compare the bond with a dedicated three-year zero-coupon purchase. The committee also asks what one-year forward rates are implied between years one and two, and between years two and three, and whether the offered bond is cheap or expensive on an arbitrage-free basis. No taxes, transaction costs or default risk apply.
1. Calculate the implied one-year forward rates between years one and two, and between years two and three. Show the formulas.
Read the worked answer
Let s1=3.00%, s2=3.60% and s3=4.00%. Discount factors are v1=1/1.03=0.970874, v2=1/1.036^2=0.931710 and v3=1/1.04^3=0.888996. The one-year forward rate from time 1 to 2 satisfies (1+s1)(1+f12)=(1+s2)^2. Thus f12=(1.036^2/1.03)-1=1.073296/1.03-1=0.042036, or 4.2036%. Similarly f23=(1.04^3/1.036^2)-1=1.124864/1.073296-1=0.04804, or 4.804%. The forward curve is upward sloping: the implied one-year rate rises from 3.00% now to about 4.20% and then 4.80% for later periods.
2. Calculate the arbitrage-free price per £100 nominal of the 4% three-year bond and state whether the quoted £99.80 is cheap or expensive.
Read the worked answer
Price=4 x v1 + 4 x v2 + 104 x v3 =4 x 0.970874 + 4 x 0.931710 + 104 x 0.888996 =3.8835 + 3.7268 + 92.4556=100.066 per £100 nominal. The quoted price of £99.80 is therefore about £0.27 per £100 cheap, or 0.27% of nominal. On an arbitrage-free basis using the given spot curve, the fair value is £100.07, so a buyer is paying less than the present value of the bond's cashflows. The cheapness is small and could reflect liquidity, tax, credit or quotation timing; it is not a risk-free arbitrage unless those factors are absent.
3. Explain why the zero-coupon bond is the exact match for the liability and identify the risks that remain with the coupon bond even if it is cheap.
Read the worked answer
The liability is a single £50 million cashflow at time 3. A three-year zero-coupon bond paying at exactly time 3 matches the timing and amount, subject to rounding, and, if held to maturity, removes reinvestment risk. The 4% coupon bond pays coupons at times 1 and 2, which must be reinvested at unknown future rates; its realised return therefore depends on those reinvestment rates even if it is held to maturity. It also has different credit and liquidity characteristics from a government zero. Buying it because it is £0.27 cheap does not remove that reinvestment risk. A dedicated zero, or a matching asset swap, is the cleaner liability hedge; the coupon bond may be used only if the cheapness adequately compensates for the residual risk.