Duration of fixed vs floating for swaps

In chapter 22, it mentions that the duration of a pay fixed position is greater than that of a pay floating position.

Can anyone explain that to me? I figured since duration calculates sensitivity to interest rates, a pay floating position is more sensitive to changes in the IR. Any thoughts?

Your floating rate position will be less sensitive since it moves with interest rates whereas your fixed position has a fixed interest rate that is static and therefore will diverge when interest rise/fall. It is all about their differential movement relative to interest rates.

floating rate position - rates reset every so often. so e.g. if it is linked to the 6 month libor - rates will reset every 6 months. so duration of the floating rate position is on average 3 months (0 at the end, 6 months at the start - average = 3 months).

the fixed rate position per the cfai curriculum has an average duration = 0.75 * term of the position.

A bond stays at par value when its coupon payments = the going interest rate in the market. Since floating payments reset their payments on a periodic basis, they reset their market value to par. Since Fixed Income doesn’t reset its payments, it never resets its market value. Hence, fixed income has greater duration sensitivity since its market value doesn’t reset over its life.

In my understanding, “a pay fixed position has greater duration than that of a pay floating position” is wrong - at least in the context I grasp it.

When you make fixed payments (pay fixed position) you’re duration is negative as you are short bond (eg duration = -3 when bond has 4 year maturity). On the other hand, when you make float payments your duration is less negative ie higher (eg. -0.25 in case of semiannual reset dates).

However, it is true that fixed-payment duration is higher than float-payment durations, although the context of the sentence in question is different.

Also in the book, it says that pay floating position has a positive contract duration whereas pay fixed has a negative contract duration. Is that is true, how can it possibly be that pay fixed has greater duration than pay floating?

its probably in the absolute terms

Use pay fixed to lower duration. If you pay fixed, receive float, it’ll be something like this:

.25 - 3 = -2.75

Float is semiannual (1/2 = .5 then use halflife for duration or .5/2 = .25)

Fixed is usually 75% of maturity so this is a 4 year fixed = 4*.75 or 3 duration

Pay fixed duration is 3 and therefore greater than the receive float (.25)

The coupon payments on a floating-rate bond are sensitive to interest rate changes, but the value (market price) of a floating-rate bond is quite insensitive to interest rate changes. The reverse is true for fixed-rate bonds: insensitive coupon, sensitive price.

thank you! That actually makes sense.

You’re quite welcome.

Occasionally I write things that make sense. They’re rare enough that I should bookmark them.