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# Bond duration and coupons

#### akrushn2

##### Member
Subscriber
Hi guys,

My question is why does duration go down for lower coupon bonds. I have read many many threads on this already on the internet and bionic turtle. The reason I still post this question is b/c I understand the question from the macaulay's duration point of view which is that you get your cashflows faster with higher coupons so less interest rate risk. ok thats fine. but I am stuck on modified duration which literally looks at the math of this. If I compute a simple example where I have a 5% 3 year coupon paying bond and shift yields from 1 to 2% the difference in price I get is much higher than if I do this with the 2% coupon bond example. Please can you actually guide me on only the modified duration explanation for why lower coupons produce more duration than higher coupon bonds b/c the math is not tying this out. Please see my excel file.

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#### ami44

##### Active Member
Subscriber
My question is why does duration go down for lower coupon bonds. ...
Hi akrushn2,

It does not.
Everything else equal a bond with a 2% coupon will have a greater duration than a bond with 5%.

I‘m not sure I get your whole point, but the duration is the weighted average over the cashflows. Lower coupons mean the principal payment in the end gets relative bigger compared to the interest cashflows. In the extreme the interest cashflows are zero (0% coupon) and we only have the principal cashflow at the end. In this case the duration is equal to the Maturity of the bond. This is the maximal possible value (ignoring the possibilitie of negative coupons).

I think what confuses you is the fact, that the duration increases with lower coupon, but the sensitivity (e.g. bpv) decreases, at least in absolute terms. Sensitivity and duration are often almost used synonymously, but in this context they show very different behaviour.

The Sensitivity is the Duration multiplied with the NPV. A lower coupon results in higher Duration but in lower NPV.

$$BPV = - Duration \cdot NPV \cdot \frac{1}{10000}$$

Did that make it clearer?

#### akrushn2

##### Member
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Hi sorry I realize I posted an unhelpful typo. I was actually wondering why duration goes up for lower coupon bonds. You have explained using weighted average time to maturity. I understand this explanation already as I posted in my initial post.

"but the sensitivity (e.g. bpv) decreases, at least in absolute terms. Sensitivity and duration are often almost used synonymously, but in this context they show very different behaviour"

yes thats the whole point. modified duration measures sensitivity- its the other side to duration. why should sensitivity be increasing with lower coupon bonds?

#### akrushn2

##### Member
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Just to add on to this point

"The Sensitivity is the Duration multiplied with the NPV. A lower coupon results in higher Duration but in lower NPV.

BPV=−Duration⋅NPV⋅110000"

i assume by the duration above you're referring to modified duration. my question is why does the modified duration (i.e. sensitivity to changes in yield) meant to increase with lower coupon bonds? if you look at my excel example you will see ive computed a theoretical cashflow for a bond for a 2% versus a 5% and then done a change in yields from 1 to 2%. the difference in bond price is higher for the 5% coupon versus the 2% one. so technically it looks like modified duration (i.e. sensitivity to changes in rates) should be higher for the higher coupon bonds not the lower coupon bonds-- but the theory says the exact opposite and i dont quite know why that is.

like i said i understand the macaulays duration explanation to this which is the weighted time to maturity of cashflows. rather i am more interested in why modified duration (i.e. sensitivity) increases for lower coupon bonds.

does my question now make more sense?

#### ami44

##### Active Member
Subscriber
"The Sensitivity is the Duration multiplied with the NPV. A lower coupon results in higher Duration but in lower NPV.

BPV=−Duration⋅NPV⋅1/10000

i assume by the duration above you're referring to modified duration. my question is why does the modified duration (i.e. sensitivity to changes in yield) meant to increase with lower coupon bonds?
...
Ah, great, now I understand your point better.
You equate sensitivity to changes in interest with duration (mod. duration vs. duration doesn‘t matter here).
The Sensitivity is the BPV in the above formula. As you can see, the Duration is the Sensitivity per NPV.

If you want you can say, that the Duration is the percentage change and the BPV is the absolute change per unit increase in the interest rate.

Lets assume the interest rate moves about +1bp and as a result the NPV decreases from $$NPV_{1}$$ to $$NPV_{2}$$. The the absolute change in NPV per 1bp interest rate change is then
$$\Delta = \frac{NPV_{2}-NPV_{1}}{1bp}$$
and the percentage change is:
$$\Delta\% = \frac{NPV_{2}-NPV_{1}}{NPV_{1} \cdot 1bp} = (\frac{NPV_{2}}{NPV_{1}}-1) \cdot \frac{1}{1bp}$$

Now we lower the coupon rate and as a results both NPVs decrease. They will decrease in a way that the absolute value of $$NPV_{2}-NPV_{1}$$ will decrease, and that $$\frac{NPV_{2}}{NPV_{1}}$$ will decrease as well. That will result in an decrease of the absolute value of $$\Delta$$ but an increase of the absolute value of $$\Delta\%$$. Try the values from your Excel calculation, if you want.

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#### akrushn2

##### Member
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Ah I think I can understand now what you're saying. I was calculating absolute changes in npv after changing interest rate. whereas what you're saying is these absolute change/new npv after interest rate increase is what sensitivity is really measuring. If I do that as per my excel sheet 3.112291/108.6516= 28.645% versus 2.940985/100= 2.940985%. the 2% coupon one is higher therefore it has higher sensitivity.

this is what you mean right?

#### ami44

##### Active Member
Subscriber
Ah I think I can understand now what you're saying. I was calculating absolute changes in npv after changing interest rate. whereas what you're saying is these absolute change/new npv after interest rate increase is what sensitivity is really measuring. If I do that as per my excel sheet 3.112291/108.6516= 28.645% versus 2.940985/100= 2.940985%. the 2% coupon one is higher therefore it has higher sensitivity.

this is what you mean right?
I think, that is what I mean. I would calculate a tiny bit different:
3.112291/111.76394 = 2.7847%
and
2.9409852/102.94099 = 2.8570%

Even though the total sensitivity (absolute value) is higher for the 5% coupon bond (3.11 > 2.94), the percentage sensitivity is actually lower (2.78% < 2.86%).