Question

Suppose a 2-year 7% coupon-paying bond is priced at par, and a 1-year zero (maturing at $100) is priced at $95.238095238. What is the implied 2-year zero rate? What is the implied one-year zero rate, one year from now? These bonds have annual coupon periodicities. Use discrete discounting (1+r)^(-t).

Answer 1

Price of 1 year zero coupon bond = Par value/(1+r₁)^t

Where, r₁ is one year zero rate

t is time to maturity

Therefore, $95.238095238 = $100/(1+r₁)¹

1+r₁ = $100/$95.238095238

1+r₁ = 1.05

r₁ = 1.05 - 1

r₁ = 0.05 or 5%

Price of a two year coupon paying bond = Coupon payment/(1+r₁)¹ + (Coupon payment + par value)/(1+r₂)²

Coupon payment = 7%*$100

= $7

Since the bond is priced at par,

$100 = $7/(1+0.05)¹ + ($7+$100)/(1+r₂)²

$100 = $6.666667 + $107/(1+r₂)²

$100 - $6.666667 = $107/(1+r₂)²

$93.33333 = $107/(1+r₂)²

(1+r₂)² = $107/$93.33333

(1+r₂)² = 1.146429

1+r₂ = 1.146429^(1/2)

1+r₂ = 1.0707

r₂ = 1.0707-1

r₂ = 0.0707 or 7.07%

Therefore, implied two year zero rate is 7.07%

According to forward rate model,

[1+f(T*,T)]^T = [1+r(T*+T)]^T*+T / [1+r(T*)]^T*

Where, f(T*,T) denotes forward rate T* years from today with maturity of T years

r(T*+T) indicates the spot rate with T*+T years of maturity

r(T*) is spot rate with T* years of maturity

The one year zero rate one year from now is calculated as

[1+f(1,1)]¹ = [1+r(1+1)]⁽¹⁺¹⁾ / [1+r(1)]¹

[1+f(1,1)]¹ = [1+r(2)]² / [1+r(1)]¹

[1+f(1,1)]¹ = (1+0.0707)² / (1+0.05)¹

[1+f(1,1)]¹ = 1.0918

f(1,1) = 1.0918 - 1

f(1,1) = 0.0918 or 9.18%

Therefore, implied one year zero rate one year from now is 9.18%