Insurance Liabilities and Option Prices

1. Insurance Liabilities and Option Prices A case study in market-consistent model calibration Andrew Smith8 September 2004andrewdsmith8@deloitte.co.uk

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

3. 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.

4. Closest Match: Swaptions • A receiverswaption 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

5. Market DataSwaption Volatilities at 30/06/2004 most reliable data Source: Royal Bank of Scotland

6. 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

7. 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.

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

9. 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.

10. 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.

11. 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.

12. Insurance Liabilities and Option Prices A case study in market-consistent model calibration Andrew Smith8 September 2004andrewdsmith8@deloitte.co.uk