The yield to maturity of a zero-coupon bond that has a stated maturity.
The Forward Rate
The yield to maturity of a zero-coupon bond that an investor agrees to purchase at some future specified date.
Spot- & forward rates for pure discount bonds Spot rate is the yield (return) of a pure discount bond, which is sold att discount, since discount bonds, P < F. If a one-year pure discount bond, just issued, is sold at P = €90.9 and has an F = €100, the spot rate is :
Spot & forward rates for pure discount bonds Forward rate is the interest rate an investor will pay to buy a bond in the future , no matter its true interest rate (or its bond price), that date. If I sign a forward contract to buy next year a two-years bond at a P = SEK892.9 (with F = SEK1000, to be paid in two years from now), the forward rate is : If the interest rate next year becomes 10%, the price of the bond will be SEK909. In that case, I gain since I buy the bond at SEK892.9 .
Spot- & forward rates for pure discount bonds Forward rates are derived from spot rates and provide a good information on the expected interest rates in the future. Maturity (n) Spot (R m ) Forward (f n ) 1 5 - 2 5.8 6.606 3 6.3 7.307 4 6.4 6.701 5 6.45 6.65
Spot- & forward rates for pure discount bonds If we graph the spot yield curve, it is 5 % for 1-year and 6.45 % for 5-years. But why is it f 2 = 6.606 % ? Strategy 1 : Save 1 $ directly in 2-years and get: 1(1 + R 2 ) 2 = 1(1.058) 2 = 1.11936. Strategy 2 : (a) Save first 1 $ for 1-year, and (b) sign a contract to invest your $ and its return in an implied rate , (i.e. f 2 ), in order to get the same as in strategy 1.
Spot- & forward rates for pure discount bonds [1(1.05)]*[1 + r impl ] = 1.11936 , i.e. 1 + r impl = 1.11936 / 1.05, => r impl = f 2 = 0.06606. Alternatively, R 2 = (R 1 + f 2 ) / 2. The Formula to estimate implied forward rates (f n ) from one periods’ spot rates (R n ) is: