Question

This question is about bond duration. You’ll need to use derivatives.
Remember your calculus: The chain rule tells you that the derivative of
ln(P(YTM)), with respect to YTM, is P’(YTM)/P(YTM).
a. Consider a zero-coupon bond which pays off $F in T years. What is
its duration when its yield to maturity (YTM) is zero? Show your
work.
b. Now consider a T year bond which pays a coupon of $C each year. (It
makes no final face payment.) What is its duration when its YTM is
zero? Show your work. (Hint: (1 + 2 + 3 + … + T)/T is (T+1)/2.)
c. Finally, consider a T year bond which pays a coupon of $C each year
and a final face payment of $F in T years. What is its duration when
its YTM is zero? How is its duration related to your answers in parts
(a) and (b)? Show your work

Answer 1

a) Duration of a bond is the weighted average of the cash flows associated with the bond. Zero coupon bond does not pay any coupons, it only pays the face value of the bond upon the maturity of the bond.

Therefore, the duration of a zero coupon bond is equal to its maturity, which is T years in this question.

b) Cash Flows associated with the bond are:

C₁ = C ; C₂ = C ; C₃ = C; ..................... C_T = C (Because all the coupon payments are the same)

Present Value of Coupon Payments will be :

PV₁ = C₁/(1 + YTM)^1 = C/(1 + 0)^1 = C (Because YTM is zero)

PV2 = C2/(1 + YTM)^2 = C/(1 + 0)^2 = C

  PV3 = C3/(1 + YTM)^3 = C/(1 + 0)^3 = C

PV_T = C_T/(1 + YTM)^T = C/(1 + 0)^T = C

Weights of the present values for calculation will be:

W₁ = PV₁ /FV = C/FV (Here, FV = Face Value of the bond)

W₂ = PV₂ /FV = C/FV

W₃ = PV₃ /FV = C/FV

W_T = PV_T /FV = C/FV

So the Duration of the bond will be the weighted average of the no. of years until each cash flow is to be paid, where the weights are the PV of each cash flow as a percentage of bond's full value;

Duration = W₁*1 + W₂*2 + W₃ + ............... + W_T = 1*C/FV + 2*C/FV + 3*C/FV + .......... + T*C/FV

= C/FV (1 + 2 + 3 + .......... + T)

= C/FV * (T*(T+1)/2)

c) Here we have a T year bond which pays coupon C each year and final face payment of F in T years

Here, the duration will be the same as in part b of this question; we just need to put F in place of FV in the final answer of part b

So, Duration = (C/F)*(T(T+1)/2)

Here, C/F is coupon rate