Question

Holly bought a 7-year bond, with a 3% coupon paid semi annually. It was priced to yield 3% when she bought it. What is the effective duration assuming a 100-basis point change in interest rates?

Answer 1

Let the par value of the bond be FV = $100

Number of periods to maturity = n = 7*2 = 14 semiannual periods

Coupon Rate = P = 3%/2*100 = $1.5 semiannual

Current Yield = r = 3/2% = 1.5% semiannual

Present Value of the bond = P₀ = P/(1+r) + P/(1+r)² + .... + P/(1+r)ⁿ + FV/(1+r)ⁿ = P[1 - (1+r)^-n]/r + FV/(1+r)ⁿ = 1.5(1 - 1.015^-14)/0.015 + 100/1.015¹⁴ = $100

When yield decreases to r = 2%, (1% semiannual)

Present Value of the bond = P₁ = P/(1+r) + P/(1+r)² + .... + P/(1+r)ⁿ + FV/(1+r)ⁿ = P[1 - (1+r)^-n]/r + FV/(1+r)ⁿ = 1.5(1 - 1.01^-14)/0.01 + 100/1.01¹⁴ = $106.50

When yield decreases to r = 4%, (2% semiannual)

Present Value of the bond = P₂ = P/(1+r) + P/(1+r)² + .... + P/(1+r)ⁿ + FV/(1+r)ⁿ = P[1 - (1+r)^-n]/r + FV/(1+r)ⁿ = 1.5(1 - 1.02^-14)/0.02 + 100/1.02¹⁴ = $93.95

Effective duration = (P₁ - P₂) / (2 x P₀ x Y)

where, Y = 1% or 0.01

Hence, Effective duration = (106.50 - 93.95) / (2 x 100 x 0.01) = 6.275