![Sample our Economics, Finance,Business & Industry journals, sign in here to start your access, latest two full volumes FREE to you for 14 days]({"type":"image" , "src" : "/sda/5931/TOC-Subject-Sample-2021-Economics-Finance-Business-Industry.jpg"})
Abstract
We present a new type of with-profits annuities which offer lifelong, yet hedgeable, guarantees. The rolling annuity gives a minimum lifelong guarantee at the time of contribution complemented with a series of guaranteed increases prior to retirement. Importantly, the initial guarantee and the subsequent increases are all set at prevailing market rates and hence are not known in advance. The structure of the guarantee implies that, prior to the last increase, the liability is equivalent to a zero-coupon bond maturing at the next increase and can therefore easily be hedged in the financial markets. Furthermore, the short duration implies that the financial and regulatory value will (essentially) coincide. We show financial fairness and we derive the reserve and thereby the hedging strategy. We also consider longevity risk, the duration profile, and report on a simulation study of the real value of the final payout.
Notes
No potential conflict of interest was reported by the authors.
1 EIOPA has recently announced a review of the methodology used to derive the UFR. Since the UFR is currently considerably higher than long-dated market rates, a review is likely to result in a downward adjustment.
2 In practice, the age at which you are entitled to receive pension might change over time. We ignore this complication in the present exposition.
3 Note that the increment factor used in is generally increasing in life expectancy since the longer dated returns are then assigned a larger weight relative to the shorter dated returns. This implies that if life expectancy evolves faster than expected, such that
, then the guaranteed pension will generally be larger than it would have been under the original assumptions.
4 In the actual product it is also guaranteed that can never decrease, i.e.
. The members are thus insured against negative rates. In this presentation, we consider only the ‘pure’ product with no option elements.
5 In ATP best estimate mortality forecasts are obtained from the SAINT model, cf. Jarner & Kryger (Citation2011), and longevity risk is calculated by an internal model using the stochasticity of the SAINT model.
6 Data are retrieved from the Human Mortality Database, www.mortality.org. At the time of writing, 2011 is the latest year available.
7 Indeed, interest rate sensitivity is often referred to as duration. We follow this terminology and use the two terms interchangeably.
8 More precisely, we plot the ratio as a function of u, where
with
for
, and zero otherwise. The ‘long-dated’ reserve,
, is the part of the reserve which uses discount factors with maturities of 30 years or longer. Note that the stated expression assumes that the guarantee period is less than 30 years. For illustrative purposes we use an interest rate of
for all maturities, i.e.
. In practice, however, it is not obvious how to value long-dated liabilities and determining proper long term interest rates is a topic of ongoing debate.
9 Formally, the pricing formula (Equation31(31) ) follows from (Equation30
(30) ) by a no-arbitrage argument under the additional assumption that the market price of (interest rate) risk is an affine function of
. The chosen parametrization corresponds to the market price of risk being of the form
. The price of a zero-coupon bond can be obtained by the risk-neutral valuation formula (Equation1
(1) ), where the risk-neutral interest rate dynamics are given by
.
10 In Section 4 we exploited the fact that when the yield curve is flat and the mortality law is of Gompertz-Makeham form, the integral can in fact be evaluated analytically as an ‘life expectancy’ under a modified mortality law, cf. formula (Equation24(24) ). This computational trick is however not available in the more general case considered here.
11 Note that, since higher quantiles are less risky than lower quantiles, the 5%-quantile is a measure of lack of risk rather than of risk. For this reason the ‘risk-reward’ plots of Figures and are plotted with the abscissa inverted.
12 By a limit argument, the special case corresponds to
.