Duration

Recall from Level I that:

Modified duration measures a bond’s price sensitivity to interest rate changes, assuming that the bond’s cash flows do not change as interest rates change. The standard measure of convexity can be used to improve price changes estimated from modified duration.

Modified duration and convexity are not useful for bonds with embedded options, however, because the cash flows from these bonds will change if the option is exercised. To overcome this problem, effective duration and effective convexity should be used, because these measures take into account how changes in interest rates may alter cash flows.

The following expressions can be used to compute effective duration and effective convexity for any bond:

Calculating effective duration and effective convexity for bonds with embedded options is a complicated undertaking because we must calculate values of $B{V}_{+\Delta y}$ and $B{V}_{-\Delta y}$. Here’s how it is done:

Step 1: Given assumptions about benchmark interest rates, interest rate volatility, and any calls and/or

puts, calculate the OAS for the issue using the current market price and the binomial model.

Step 2: Impose a small parallel shift in the benchmark yield curve by an amount equal to +Δy.

Step 3: Build a new binomial interest rate tree using the new yield curve.

Step 4: Add the OAS from step 1 to each of the one year rates in the interest rate tree to get a “modified”

tree.

Step 5: Compute BV+Δy using this modified interest rate tree.

Step 6: Repeat steps 2 through 5 using a parallel rate shift of −Δy to obtain a value of BV−Δy .

Both call and put options have the potential to reduce the life of a bond, so the duration of callable and putable bonds will be less than or equal to the duration of their straight counterparts.

Effective duration (callable) ≤ effective duration (straight).

Effective duration (putable) ≤ effective duration (straight). Effective duration (zero coupon) ≈ maturity of the bond. Effective duration of fixed rate coupon bond < maturity of the bond.

Effective duration of floater ≈ time (in years) to next reset.

While effective duration of straight bonds is relatively unaffected by changes in interest rates, an increase in rates would increase the effective duration of a callable bond, and decrease the effective duration of a putable bond.