# Surplus immunization and convexity

From AM 2013 , Q8, I take that :

Given a small parallel change in yields, the value of a surplus can be affected if the convexities of assets and liabilities are not matched, even if the Durations are matched.

If I am not mistaken, this is not the case when the surplus is zero (PV of assets = PV of liabilities) : in this case convexities is not a concern, as far as I can tell.

Would anyone care to explain that ?

it might affect even if the surplus is zero.

Change in value = -D*Change in Rate + Convexity * Change in Rate^2

so A=100, L=100 Da=DL=3, Ca=4, CL = 5, Change in Rate = 1%

Change in Asset => -3(0.01) + 4(0.01^2) = -0.0296

Asset = 97.04

Change in Liab => -3(0.01) + 5 (0.01^2) = -.0295

Liab = 97.05

Surplus = -0.01

Dude thanks but you are using a formula that isn’t in the curriculum as far as I can tell. Also, I am 95% positive that according to the books, convexity is not an issue if A=L and Durations are equal in a classical immunization context.

Formula may not be there in the LEvel III curriculum directly - but it is definitely there from past levels, and it applies, no matter what you say.

Classical Immunization is based on a big approximation - that interest rate changes will be SMALL - and PARALLEL - and we definitely know that is not the case.

I am looking at the Exhibit 15 and 16 on my 2014 book - Section 4.1.2.1 - “Duration and Convexity of Assets and Liabilities” - and in the analysis there after

and there is no mention anywhere of this being true only for when PV(A) = PV(L). The example does take A > L and then do the analysis, but it is equally relevant and applicable for when A = L as well.

CPK, is it true to say the disbursion of the asset cashflow around the liability horizon helps alleviate some of the potential impacts from convexity differences between the asset and the liability? Realizing alot of variable come into play but thinking convexity differences - assuming durations are matched would cause durations to extend or contract for a given move in interest rates.

BTW, your contributions to the Level-III forum are greatly appreciated (S2000 as well, of course)

yes, disbursion of the cashflows around liability horizon would alleviate

What you are saying is probably right. I’ll think about convexity from now on in this topic.