convexity

[derswap07]

Suppose the convexity of the liabilities is less than the convexity of the assets. The yield curve shifts 75 BPS upwards. What will be the effect on the surplus?

 

[cpk123]

Convexity alone is not enough. Convexity is a 2nd level movement. And since you are using Change in R^2 for that effect - it will usually be a much smaller # effect.
You need to consider both Duration and Convexity.
Net Change in Price = -Duration * Change in R + Convexity * Change in R^2
So if D=5, C=7, and r = 1%
Change in price = -5*0.01 + 7 * 0.0001 = -0.0493 = -4.93%
and then consider the movement in value of each of the Asset and the Liability
which will be the net effect on the surplus.

 

[patso]

Hmmmmmmm not sure i get this.
Take a look at 2013 AM paper -see here part D which says “where the convexity of the liabilities is less than the convexity of the assets, the decline in value of the liabilities as a result of the yield curve shift will be greater than the decline in value of the assets, thus increasing economic surplus”.
If im to use the formula CPK refers above, i would expect the bond to move in propotion to the convexity since P = ….+ convexity x change^2. I would then expect the liabilities to decrease less than the decline in value in assets, decreasing the economic suplus. Hope this makes sense.
Come back CPK , where thou art CPK or Magician?????????????????????

 

[S2000magician]

The durations are matched; i.e., equal. That’s the missing piece about which CPK was talking.
The convexity of the assets is greater than the convexity of the liabilities; you’re net zero duration and positive convexity. If yields decline, the net value increases; if yields rise, the net value increases.

 

[cpk123]

Put in numbers and you will get the concept.
Durations are the same.
A=100, L=100 Net economic Surplus = 0 (A-L)
Now Ca = 5, Cl = 4, DA=DL=5 say.
say rate changes by 1%
Effect on Asset = -5 * 0.01 +5*0.01^2 * 100 = 0
A = 100.0
Effect on Liab = -5 * 0.01 + 4 * 0.01^2 * 100 = -0.01
Liab = 99.99
Surplus = 100 - 99.99 = 0.01
increase in surplus.