Insurance Liabilities and Option Prices A case study in market-consistent model calibration Andrew Smith 8 September 2004 andrewdsmith8@deloitte.co.uk

Agenda Example product – guaranteed annuity options Swaps and swaptions Assumptions needed to bridge the gap between swaps and annuity guarantees Importance of different assumptions

What is a Guaranteed Annuity Option? An add-on to an existing pension savings product When the savings product matures (with uncertain annuity value), the policyholder can choose to: Take the money and buy a life annuity in the open market Purchase an annuity at a guaranteed rate - for example £1 per annum for every £9 maturity value. This is the guaranteed annuity option Guaranteed annuity options are valuable if interest rates are low (because then market annuities are expensive) and expires worthless if interest rates rise.

Closest Match: Swaptions A receiver swaption entitles the bearer to: Receive a series of fixed cash flows (six monthly, between the strike date and swap maturity date) While paying a floating rate But the bearer can walk away with no obligation on the strike date (eg if floating rates are expected to remain higher than the fixed cash flows) As for GAO’s, a swaption is more valuable as interest rates fall, and can expire worthless as interest rates rise

Market Data Swaption Volatilities at 30/06/2004 most reliable data Source: Royal Bank of Scotland

Zero Coupon Yield Volatilities: A Calibration Input A fitted model usually produces a smoother surface, described by a small number of parameters, and so does not capture all swaption prices. The reason for building this model is to price GAOs. Source: Bootstrap of swaption data

Market Consistent Prices (Step 1) 10 Year Annuity Certain, £1 Maturity Its more costly to guarantee a higher rate – but the shape of the curve (“smile”) is a chosen assumption, as only observed vols are at the money.

Swaps have a credit (+other) risk premium of 30-40 bp over strips Spot yield @ 30/06/2004 Source: intercapital / datastream and DMO

Effect of using Risk-Free rates (mix swaption vols with gilt strip curve) Largest increase in GAO cost (as % swap based value) is for short dated out-of-the -money options.

Further Adjustments Required Stochastic mortality, expense and capital loads increase variability of annuity yields relative to swap comparison – likely that firms will have to reflect (at least) stochastic mortality in RBS in future, further increasing stated GAO costs. Quanto effects – bad news if the GAO is in the money at the same time as large fund maturity values Take-up rates likely to be a guess – but possible that firms will have to disclose a worst case.

Conclusions Robust and well-established techniques exist for market-consistent valuation of short-dated market guarantees Typical insurance guarantees require some interpolation or extrapolation to allow for a range of strikes and maturities Converting from inter-bank credit to risk-free is subtle and requires extra assumptions – stated liabilities increase by varying amounts. Allowance may be required for stochastic mortality, expense loads, capital cost, quanto effects and other “basis risks” between swap rates and annuity rates. Uncertainty over take-up rates has a huge effect, dominating everything else, and is very difficult to quantify.

Insurance Liabilities and Option Prices A case study in market-consistent model calibration Andrew Smith 8 September 2004 andrewdsmith8@deloitte.co.uk