Duration of portfolio not equal to duration of liability

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johntavv

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This is referring to reading 21, immunization of obligations
So when portfolio duration is > liability duration, portfolio is exposed to price risk.
-If interest rates increase, reinvested coupon and principal increases.
-If interest rates increase, value of bonds decreases.
Overall, losses from the value of outstanding bonds more than offset gains from the additional revenue generated on reinvested principal and coupon. Why is this the case?
Are the bonds associated with portfolio duration while the reinvested coupon and principal are associated with the liability, and so since portfolio duration > liability duration, loss on bonds > gain on reinvested coupon?
 
Try to prove it with your calculator.
Assume a 10 year 5% annual coupon trading at par.
Start with the impact of a yield rise from 5 to 6% on the coupon stream.
First step calculate FV of the coupons at 5%
N=10; I/Y=5; PV=0; PMT=-5; CPT FV=62.9
now at 6%
N=10; I/Y=6; PV=0; PMT=-5; CPT FV=65.9
Additional revenue as result of the increase in yield £3
Now look at how the bond price changes as result of the same rise in yield.
@ 5% yield the bond is priced at par (i.e. 100)
@ 6 you have
N=10; I/Y=6; ; PMT=-5; FV=100 CPT PV=-92.6
Loss due to yield rise 7.4 > 3
In other words when calculating the new bond price you are discounting all cash flows with the new higher yield including the maturity proceeds.
When calculating the future value of the coupon stream the magnitude of your gain is in proportion of the smaller amounts (coupons only vs coupon + maturity proceeds).
Regards, Carlo
 
Carlo, I understand your calculation, loss due to yield dominates gain due to reinvestment.
So the above refers to the case when: portfolio duration > liability duration.
But what happens when: portfolio duration < liability duration?
 
In that case (duration of asset < duration of liabilities) the portfolio is exposed to reinvestment risk.
Assume the manager is trying to meet a 100£ liability which comes due in 10yrs with a bond whose duration is shorther than 10yrs (think for instance a 5 year zero coupon).
The strategy could consists of letting the 5yrs bond mature and, at that point, buy another 5yrs zero coupon bond at the same yield with the maturity proceeds.
i.e. first bond N=10 I/Y=5; PV=-37.7; PMT=0; CPT FV=61.4
second bond N=10 I/Y=5; PV=-61.4; PMT=0; CPT FV=100
If during the first 5yrs yields go up and stay up, the manager will re-invest the proceeds at a higher rate and pocket the difference between
N=10 I/Y=6; PV=-61.4; PMT=0; CPT FV=110
and the 100£ liability.
If yields go down, however, the FV will end up being lower than the amount due (i.e. the manager got the wrong side of the reinvestment risk)
N=10 I/Y=4; PV=-61.4; PMT=0; CPT FV=91
Regards, Carlo
 
If the duration of the assets is greater than the duration of the liabilities, then the assets will be more sensitive to interest rate changes, and vice versa.
 
I get that when duration of assets > duration of liabilities, assets are more senstive to interest rate changes. Hence, the loss in value of the bond > gain in coupon and principal reinvestment.
So does this mean that the bond is the asset and the coupon/principal are the liability?
 
bond is your asset (as a portfolio manager) used to pay off some liability (which is what you are trying to defease using your bond portfolio).
how does the bond help defease the liability - through scheduled payments (of coupon + principal) and a final lumpsum payment of face value.
 
Carlo, s2000 and cpk thank you for the help, i get this concept now.
 
If the duration of the assets is greater than the duration of the liabilities, then the assets will be more sensitive to interest rate changes.
But shorter is still senstivie to interes rate changes?
 
anything that has an interest rate as a factor that affects its price - is affected by interest rate changes.
but what you care about here is the “sensitivity” - how much is the movement and in which direction. That is - do you assets move up when interest rates fall. If so how much? (Same is true for if your assets move down when rates move up/down). How much do you liabilities move and in which direction. So due to this movement what is the net effect on your surplus - did it move up or down?
that is what is meant by sensitivity.
 
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