# Effective duration

On one hand it is said that when interest rates rise, a callable bond exhibits a price profile similar to that of an option free bond.

On the other hand, I hear that the effective duration of an option free bond changes very little in response to interest rate movements.

It’s contradicting. What am I missing!

The first statement is not correct (or at least not complete)

when interest rates rise and the option of callable bond is out of money, a callable bond exhibits a price profile similar to that of an option free bond

That because the delta (or effective duration) of Call option out of money is nearly 0.

If the option of callable bond is in the money. Even interest rare rise, callable bond is non neccessarly similar option free bond

The question remains

As quoted, your statements are inaccurate.

When interest rates are high (whether they’re increasing or decreasing), callable bonds behave like straight bonds.

When interest rates are low (whether they’re increasing or decreasing), callable bonds have negative convexity and very low effective duration.

Practice problems q no. 13

Well, the question asks which Bond’s effective duration will lengthen when interest rates rise.

1. straight bond

2. callable bond

3. putable bond

And the answer is a straight bond. What am I missing? Why then is it said that a straight bond is less affected by changes in interest rates and what about callable being similar to straight in such a scenario…

Where was the question?

EOC q no. 13

First, the answer is that the callable bond’s effective duration will lengthen, not the straight bond’s.

Second, the key bits of information that you omitted were that the coupons on all three bonds are 4% while the YTM is 2.5%.

Under those circumstances, the straight bond’s effective (and modified) duration will shorten as the YTM rises, and the putable bond, with its option out of the money, will behave like the straight bond. The callable bond, on the other hand, has its option in the money, but moving out of the money as interest rates rise. It’s in the region where it has negative convexity, so as the YTM rises, its effective duration lengthens.

The details are important here.

Thank you !

You’re quite welcome.

This is 100% incorrect as interest rate rise , the putable bond is ITM. Don’t forget that a puttable bond gives the right to the investor to sell the bond at par. So when interest rates rise, he will exercise and lend money at the higher rates.

When the coupon is 4% and the YTM is 2.5%, the putable bond’s embedded option is out of the money, which is what I said.

As interest rates rise above 4%, it will come into the money, but that’s not the situation I was describing.

As I wrote, the details are important here.

I see, we’re probably referring to a different question. I’m talking about a question where you have three bond, plain vanilla, puttable and callable. The question is: the bond whose effective duration will lengthen if interest rates rise is?

the answer is callable with “As interest rates rise, a call option moves out of the money, which increases the value of the callable bond and lengthens its effective duration.”

We agree on that.

May be repetition but a few basics worth noting:

1. ED is only valuable to use when valuing bonds with embedded options as ED and MD are same for a straight bond.

2. Since either of the options trigger early retirement hence the ED <= MD at any given time.

3. As interest rate rise the Delta of the Put option starts moving from 0 towards 0.5 and hence it becomes valuable in the hands of investors that would like to retire the bond early. Hence the putable bond exhibits reduced ED

4. As interest rate rise the Delta of the call option starts moving from 1( in this particular case as the call option anyway is deep ITM at 4% cpn and 2.5% YTM) the incentive for the issuer to call reduces (unless it has already called) and thus the ED increases

5. The ED of the straight bond is unaffected with the increase in interest rate on a relative basis.

Hence the callable bond suffers the most in terms of increased ED.

One interesting aspect is if the interest rate were very low and the issuer had not exercised the call option then the chance of the ED reducing further is minimised for a callable bond as the option is again deep ITM.

More accurately, when valuing bonds whose cash flows might change when their YTM changes.

Effective duration is also useful when valuing floating rate bonds, and inverse floaters, for example, neither of which necessarily have embedded options.

Yes. You are right. ED is useful with options and FRNs including the inverse floaters