Independent solution

How to solve this Macaulay and Modified Duration question

Setup

Setup

Let i be the common annual yield. For either bond, Macaulay duration equals modified duration multiplied by one plus i.

DMac=(1+i)DmodD_{\mathrm{Mac}}=(1+i)D_{\mathrm{mod}}

Model

Model

Since i is positive, Macaulay duration exceeds its own modified duration. The value d lies between a and b, so it must pair with modified duration a.

d=(1+i)ad=(1+i)a

Compute

Compute

Use the common multiplier on the bond whose modified duration is b.

1+i=da1+i=\frac da
DMac,other=bda=bdaD_{\mathrm{Mac,other}}=b\frac da=\frac{bd}{a}

Answer

Answer

The required expression is bd divided by a, which is choice A.

bda(A)\boxed{\frac{bd}{a}\quad\text{(A)}}