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?
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.
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.
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.
I just got this problem backwards also. i always thought that when yields go up, the value of whatever (assets or liabilities go down).
I also always thought that when convexity is greater, the move of prices would be greater as the curvature of the yield is bigger (hence prices fall faster than normal per basis point of the yield change).
Using that logic here, when yields go up, I thought that assets would fall at a greater rate then the liabilities since they have greater convexity, which would in turn decrease the surplus.
I obiviously have it backwards, can one of you who clearly understands this, let me know where my logic is faltering please?
By the way, we should all be talking about dollar duration and dollar convexity, not merely duration and convexity.
Without knowing the relative values of the assets and liabilities, we cannot reach a conclusion.
Dear, you’re mixing interest rates and yields. I recommend that you go back and read the topics with a focus on , Interest, vs Yields, vs Yields curve vs Price
in a nutshell. Interest rate is up, Price goes down, yield is up. Yield is up because, the final value is the same (non TIPS), coupons are the same, while Price went down.
The question talks about Yield curve shift.
I think magician means net value decrease if yields rise.
Magician is right. Draw the curves on top of each other with the different convexities to get the idea .
Magician, why you don’t write an article on this in your website ? Seems that the conventional sources are not helping.