# Hedging guaranteed liabilities for a pension fund

I'm trying to understand what we mean by linear and non-linear guaranteed liabilities for pension fund ?

Often this exposures are hedged by Interest rates derivatives swaps and swaptions which I can understand just looking from the payoffs of those derivatives but what I can't still understand how liability can be non linear ? Thanks

Consider a very simplistic example.

Imagine a fund with liabilities for this year of 1. These grew by say 3% per annum in perpetuity (call this "G"). The same discount rate applies to all liabilities, say 5% (call this "K").

So the Net Present Value of the future liabilities will be (1+K)/(K-G).

While the derivative of the NPV with respect to either K or G becomes a reciprocal quadratic of the spread between them... which is an explosive function, not a constant!

Please pay no heed to my implicit assumption about the pensioners being immortal... Above was a clearly extreme and stylised scenario; but hopefully reveals the underlying essence of the problem.

Which isn't the rates hedge the fund has on for today's future liabilities; but the mismatch that develops between this hedge and the liabilities once any of the assumptions used to calculate the latter get tweaked a little in any direction.

Maybe this helps?

• Thanks for you reply, but I still can't see it ... – Gogo78 Aug 16 '20 at 15:31
• hopefully the added chart helps? The liabilities might be guaranteed; but that does not mean their present value is fixed in any way. – demully Aug 16 '20 at 15:46
• @gogo78 In what context specifically did you encounter the phrase "non-linear guaranteed liabilities" of a pension fund? Book, article, conversation at work, etc. – noob2 Aug 16 '20 at 19:33
• @noob2 in my internship and still can't understand why – Gogo78 Aug 17 '20 at 5:39
• Monday, your boss says we have 100mio long-term liabilities we need to hedge. You get quotes from your brokers and you hedge by receiving 20 year swaps. All's good until a few months later, LT inflation swaps have moved (so the guaranteed real incomes have changed in value); the actuary has changed his discount rate; and 20 year swap rates have moved. Your liabilities are now 80 or 120 mio, meaning you're now no longer properly hedged. If you still don't get it, google the difference between "delta hedging" and "gamma hedging" – demully Aug 17 '20 at 10:59