Fixed Income: Duration

[Hanuman]

I am unable to understand the logic in the below statements :

Holding other factors constant:

• Duration increases when maturity increases.
• Duration decreases when the coupon rate increases.
• Duration decreases when YTM increases.

Is duration = Bond Price/ Yield to Maturity?

If so, Bnd price should increase with increase in coupon payment. So, duration should increase. What am I missing?

Thanks

[cfageist]

Duration tells us how sensitive bond prices are to changes in interest rates. Duration is not Bond Price/YTM. Duration is really the weighted average time it takes to receive your cash flows (coupons and principal).

Duration increase when maturity increases because it takes longer on average to receive cash flows. The long the maturity, the longer your duration.

Duration decreases when coupon increases because the bond price relies less on the principal value which come at maturity. So you have larger cash flows occurring earlier in the life of the bond. These cash flow reduce your duration.

Duration decreases when YTM increases because of the convex nature of price/YTM relationship. The convex relationship means that the bond price decrease at a decreasing rate.

[S2000magician]

With all due respect, you need to be a bit more careful with your replies here.

@cfageist

Duration tells us how sensitive bond prices are to changes in interest rates.

Modified duration and effective duration tell us how sensitive a bond’s price is to a change in its yield to maturity (but not to changes in interest rates in general). Macaulay duration does not.

@cfageist

Duration is not Bond Price/YTM.

True.

@cfageist

Duration is really the weighted average time it takes to receive your cash flows (coupons and principal).

This is true for Macaulay duration, but not for modified duration, nor for effective duration. And the weights are specifically the present values of the cash flows to be received at each time.

@cfageist

Duration increase when maturity increases because it takes longer on average to receive cash flows. The long the maturity, the longer your duration.

True for Macaulay duration and modified duration. Not necessarily true for effective duration.

@cfageist

Duration decreases when coupon increases increases because the bond price relies less on the principal value which come at maturity. So you have larger cash flows occurring earlier in the life of the bond. These cash flow reduce your duration.

True for Macaulay duration and modified duration. Not necessarily true for effective duration.

@cfageist

Duration decreases when YTM increases because of the convex nature of price/YTM relationship. The convex relationship means that the bond price decrease at a decreasing rate.

True for modified duration and true-ish for Macaulay duration (the convex nature of the price/YTM relationship and the change in the Macaulay duration both result from the nature of the present value calculation; the convexity doesn’t cause the change in Macaulay duration). Not necessarily true for effective duration.

[cfageist]

Thank you for pointing this out! I was focusing on Macaulay duration but should have been more careful about generalizing.

[Leo_XCVI]

Duration is a measure of sensitivity of the price of a fixed-income security to a change in interest rates. So, technically, duration is the %(Change in Price) over %(Change in YTM).

Macauly Duration (MacDur), specifically, is the weighted average term to maturity of cash flows.

1. If maturity increases, the weighted average term to maturity of cash flows will increase. Result: Increase in MacDur.
2. If coupon rate increases, cash flows of the fixed-income increase and less weight is placed on the par value. Result: Decrease in MacDur.
3. If YTM increases, the present value of cash flows decreases. Result: Decrease in MacDur.

[S2000magician]

Please reread what I wrote above.

@Leo_XCVI

Duration is a measure of sensitivity of the price of a fixed-income security to a change in interest rates. So, technically, duration is the %(Change in Price) over %(Change in YTM).

This is true for modified duration and effective duration, but not Macaulay duration. Macaulay duration _ is not _ a measure of the price sensitivity of a bond.

@Leo_XCVI

Macauly Duration (MacDur), specifically, is the weighted average term to maturity of cash flows.

1. If maturity increases, the weighted average term to maturity of cash flows will increase. Result: Increase in MacDur.
2. If coupon rate increases, cash flows of the fixed-income increase and less weight is placed on the par value. Result: Decrease in MacDur.
3. If YTM increases, the present value of cash flows decreases. Result: Decrease in MacDur.

Concerning your point 3: your first sentence is correct, but your conclusion doesn’t follow from it; all that follows from the first sentence is that the price of the bond decreases. The reason that Macaulay duration decreases with an increase in YTM is more complex than that.

[qcw512]

Very great summary, help people to fresh their mind.