Question

Consider the following investment strategies: a. Buying and holding an n-year zero-coupon bond b. Buying an (n-1)-year zero-coupon bond and rolling over the proceeds into a 1. year bond Under certainty, the yield curve suggests that neither entails any risk and so both strategies must provide equal returns. Mathematically, this can be expressed as (1 + Yn)" = (1 + Yn-1)-1 X (1+ (1) where n denotes the period of maturity, Yn is the yield to maturity of a zero-coupon bond with an n-period maturity, and is the short-rate in the last period. b. Let In denote the forward rate for period n. Construct the formula for (1 + n) based on the result in Equation (1). [2.5 marks] Suppose rı = 5% and E(r) = 6%. In addition, suppose that the liquidity premiums are zero. Calculate y2. (5 marks) In Question 1b. above, suppose that we observe yz = 6.49%. Calculate f. [5 marks]

Answer 1

a)The short rate of the nth year r_n should be equal to the forward rate of the nth year f_n

So, the forward rate for the nth year f_n is given by

(1+y_n)ⁿ = (1+y_n-1)^n-1​*(1+f_n)

So, ​(1+f_n) = (1+y_n)ⁿ / (1+y_n-1)^n-1

which is the required formula

b) y₂ the yield rate for two years is dependent upon r₁ and r₂

where r₁ is the rate for one year and r₂ is the rate from 1st year to 2nd year

So, (1+y₂)² = (1+r₁) * (1+E(r₂))

=> (1+y₂)² = (1+0.05)*(1+0.06) =1.113

=> (1+y₂) = sqrt (1.113) = 1.054988

=> y₂ = 0.054988 = 5.499%

c) If   y₂ = 6.49%, r₁=5%

So, we have (1+y₂)² = (1+r₁) * (1+f₂)

=> (1+0.0649)² = (1+0.05)* (1+f₂)

=> (1+f₂) = (1+0.0649)² /(1+0.05) =1.080011

=> f₂ = 0.080011 =8.00%