Pricing and hedging of variable annuities

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1 Cutting Edge Pricing and hedging of variabe annuities Variabe annuity products are unit-inked investments with some form of guarantee, traditionay sod by insurers or banks into the retirement and investment market. Pricing VAs is simiar to pricing ong-dated financia derivatives on a basket of assets. In this paper we expore the pricing, sensitivity and risk management of a specific GMWB contract By Danie Bamont and Pretty Sagoo By far the argest VA market in the word is the United States, where variabe annuities are widey used in retirement panning. In the absence of compusory annuity purchase requirements on retirement such as in the UK, VAs are a common and ong-estabished retirement investment with some tax advantages. Variabe annuities are aso referred to as accumuator products or as GMxBs where, in the atter, the x describes the nature of the guarantee in the product, commony known as the rider. A typica Guaranteed Minimum x Benefit might refer to Accumuation (GMAB), Income (GMIB), Death (GMDB) or Withdrawa (GMWB). In the US, which has the best estabished VA market, the most common VA product is by far the GMWB. Approximatey 78% of VA saes in the US in the first haf of 25 contained a GMWB feature 1. The focus of this paper is therefore a doar-denominated GMWB poicy. More recenty, the GMWB for ife, introduced in 24, has proven extremey popuar. When a GMWB is purchased the initia capita is invested in a sub-account at the hoder s discretion. The hoder can withdraw guaranteed periodic amounts up to the vaue of the initia capita. The GMWB terminates once the initia capita has been withdrawn; any remaining funds in the subaccount are returned to the poicyhoder at maturity. So the sub-account vaue fuctuates with movements in the underying assets and decreases with withdrawas. Therefore, a GMWB effectivey combines: an annuity, in the form of guaranteed periodic withdrawas a ca option on the underying residua sub-account at maturity The poicyhoders are the owners of the sub-account. As such the insurance company is seing protection on an account that is argey managed by the 1 VARDS (Variabe Annuity Research and Data Service) poicyhoder. Typicay, these products offer cients some restricted choice of investment funds for the sub-account with some imited abiity to switch fund aocations during the ifetime of the poicy. An asset aocation spit of 6/4 between equities and fixed income is quite common. The potentia uncertainty arising from fund switching or varying amounts of equity in the underying asset aocation shoud impy high fees. However, finite-ife GMWBs typicay carry a fee of 4-6 basis points per annum, traditionay charged as a percentage of the sub-account vaue. We wi see in Tabe 1: Annua charges GMDB GMWB GMAB GMIB Typica annua charge 15 35bp 4 6bp 3 75bp 5 75bp the next section how fees tend to understate the financia cost of the protection provided. Insurance companies compete against each other not ony on fees but aso on the features of the product. Nevertheess, the typica GMWB carries the foowing features: fee charged to the sub-account some fixed maximum annua withdrawa per $1 invested initiay, typicay $7 withdrawas above the imit are aowed but the investor pays a penaty and may give up some of the guarantee Figures 1 and 2 iustrate the workings of a typica GMWB for two given sub-account scenarios: a bad one and a good one. February 29 39

2 cutting edge Figure 1. GMWB good path Vaue (USD) 4 Guaranteed withdrawas 35 Fina cashfow 3 Stock Vaue (USD) Figure 2. GMWB bad path 14 Periodic withdrawa covered by sub-account Periodic withdrawa borne by the insurer 12 Stock % fat yied curve ognorma (Back-Schoes) underying asset with 2% voatiity $7 yeary withdrawa $1 initia investment/guarantee/stock price no fees and no penaties In other words, the underying asset satisfies the foowing standard stochastic differentia equation under the risk neutra measure: ds S = rdt + σw t, In a good scenario (Fig. 1), when the underying assets perform we, the yeary withdrawa of $7 wi be at east partiay offset by capita appreciation. The sub-account therefore covers the withdrawas over the ife of the GMWB and even has a positive vaue at maturity. The investor benefits from the periodic cashfows, as we as the termina vaue of the sub-account. The insurer receives the annua fees and has no shortfas to cover. In a bad scenario (Fig. 2), there comes a point where the sub-account vaue cannot cover the $7 withdrawa. At that point the sub-account is iquidated and the baance of $7 is covered by the insurer. In addition, unti the GMWB maturity, the insurer wi have to cover the $7 periodic withdrawas. In this instance the investor ony receives a yeary $7 cashfow with no upside. The insurer provides these cashfows when the sub-account reaches $ unti maturity. Pricing The pricing of GMWBs and VA poicies in genera is simiar to pricing ongmaturity exotic financia derivatives. VAs have the foowing features in common with exotic derivatives: a basket of underiers (equity, fixed income and others), ie. hybrid features ratchets (ookbacks), or ro-up features The biggest difference between VA poicies and most financia derivatives is that the former combine financia risks with insurance risks such as surrender, ongevity and mortaity. In this sense, they are compicated products to price, and even more compicated to hedge precisey. In the case of our GMWB, the risk that more, or ess, peope than expected surrender their poicies, and the timings of these surrenders, can have a significant effect on pricing. We wi expore the effects of insurance risks on pricing and risk management in a further paper. In genera, at east for market risks, simiar rues appy as woud to standard derivatives; a onger maturity product woud tend to cost more and wi be sensitive to the eve and voatiity of the underiers. Deterministic rates We initiay assume deterministic rates. We wi reax this assumption ater on and investigate its impact. Throughout this section the foowing wi be assumed as a base case: where: Figure 3. Ca + guarantee vaue on the average path Option (in present vaue terms) Guarantee (in present vaue terms) Stock Guaranteed withdrawas W t is a Weiner process, r is the interest rate, set at 5%, and σ is the voatiity, set at 2%. Two methods of pricing are widey used: Monte Caro (MC) and Numerica Partia Differentia Equations (PDE). The MC method has the advantage of aowing a great number of variabes to be stochastic and any payoff to be priced. In addition it wi ead to a distribution of outcomes, which proves usefu when doing scenario anaysis. The PDE methods can be more accurate and further can be used to consider optima withdrawas, unike MC (Mievsky and Saisbury, 24). Initiay we wi use the MC method for pricing purposes. Foowing put-ca parity arguments, the vaue of the guaranteed periodic withdrawas and the ca option on the sub-account (residua sub-account at maturity) shoud equate to the sub-account pus the cost of insurance (getting the guarantee when the sub-account is zero, ie., a put option): Ca option + Guaranteed withdrawas = Sub account + Insurance cost Figures 3 and 4 show the main variabes aong the average path (average over the simuations). Over the average path the present vaue of the yeary $7 withdrawas is $71.3. The present vaue of the expected residua sub-account vaue is $32.7. This gives a tota $14. vaue to the investor. Figure 4. Cost of insurance on the average path Cost of insurance (in present vaue terms, R.h.s.) Guaranteed withdrawas Life & Pensions

3 Tabe 2: Pricing for various maximum withdrawa eves Withdrawa Option Guarantee Package Insurance Tabe 3: Fair vaue fee (bp per annum) Voatiity Withdrawa 1% 2% 3% hedging the guarantee. In reaity, the fees charged to the poicyhoder need to incude cost of capita for the poicywriter, frictiona costs arising from the basis risk of the theoretica versus actua hedge, a profit margin for the poicywriter and any fees needed to cover insurance risks that have not been accuratey captured in the pricing modes used. On this atter point, the insurance risks in the VA products being offered, such as apse or mortaity, can have a arge impact on pricing. In the case of apse, or poicyhoder behaviour in genera, in the absence of a rigorous pricing methodoogy for the actua risk, VA writers need to price conservativey and thus need to assume that poicyhoders behave competey rationay (ie. they aways exercise in-the-money options). Traditionay, this assumption has not been made and therefore has added to the degree of mispricing that has occurred in this market. Furthermore, many VAs were historicay priced under rea-word assumptions (P-measure), incuding risk premia on assets. For hedging and fair vauation purposes the risk neutra Q-measure is the one that matters, under which a assets earn the risk-free rate. The impication is that indeed, on average, in the rea word, fees might cover for the expected insurance cost. However, in practice there is ony one reaisation of asset returns. In that case, ony if fees are used to impement the appropriate hedging wi they cover for the expected insurance costs. We now investigate the greeks for the typica GMWB. For that purpose numerica PDEs are a more appropriate too. Under the PDE methodoogy, the GMWB is priced recursivey starting at maturity. We can therefore derive a surface of GMWB vaue versus time and sub-account vaue. From this we can derive the insurance vaue, as we as its deta and gamma. We cacuate the basic greeks with respect to the tota account vaue, for demonstration s sake. However, from a risk management practitioner s point of view, risk needs to be anaysed with respect to the eve of component risk factors in the account basket, ie. interest rates and equities (rho and deta respectivey), and their voatiities (vega). The insurance cost (Fig. 5) exhibits a put-ike profie. The main difference with a standard put is that the forward path is one where the underying drops in vaue, because of withdrawas, rather than growing at the risk-free rate. This therefore impies that the cost to the insurer is $4.. In theory, fees on the sub-account shoud cover that cost so that the expected vaue to the investor is ess than $1, impying a negative cost to the insurer. Tabe 2 shows the vaue of the different GMWB components for various aowed withdrawa eves. Since we assume the investor withdraws the maximum aowed, withdrawas are another way of defining the product maturity. A withdrawa of w per year impies a maturity of 1/w years. As such, high withdrawas reduce the option vaue and increase the guarantee vaue. The guarantee vaue is obviousy increased as the time vaue oss is smaer since the $1 nomina vaue is recovered quicker. The option vaue is reduced as there is ess time for the sub-account to grow. More importanty, the higher the withdrawa aowed by the GMWB, the higher the insurance cost. The act of withdrawing a high proportion of the sub-account means it wi converge quicker to zero and have ess time to recover from a bad underying asset performance. As such the insurer is more ikey to have to provide the guarantee. So what shoud be the fair vaue cost of investing in GMWBs? We have seen that it wi depend on the withdrawa eve. It wi aso depend on the voatiity of the underying sub-account assets. The more voatie the assets, the more vauabe the option. Tabe 3 shows some fair vaue fees, assuming the fee is deducted as a portion of the sub-account (effectivey equivaent to a dividend yied on the underying asset). As expected, the fair vaue fee grows with withdrawas as it increases the vaue of the insurance. It aso reduces the maturity of the GMWB, shortening the period over which the fee can be earned. This aspect is ony important at the product design phase. The fair vaue fee grows exponentiay with voatiity and therefore highights how critica the voatiity of the underying asset is. It is therefore important for insurers to consider the ikey voatiity of the assets chosen by the investor. The resuts impy that some insurers may have underpriced GMWBs, shoud investors eect voatie investments. As shown earier, typica fees range from 4 basis points to 6 basis points. This apparent underpricing may have arisen from changes in product design or competitive pressures. The underpricing becomes more apparent when we consider that the fair vaue fees that we have deduced in Tabe 3 are just the theoretica cost of Cost Figure 5. GMWB insurance cost Equa to PV of remaining withdrawas when sub-account = Left to withdraw February 29 41

4 cutting edge Figure 7. GMWB insurance gamma Deta Gamma Figure 6. GMWB insurance deta Left to withdraw Simiar to a vania option, the deta (Fig. 6) exhibits an S-shaped profie and the gamma (Fig. 7) peaks around the at-the-money eve and increases as we get coser to maturity. Vaue (USD) Stochastic rates Unti now we have ony considered deterministic rates and a fat yied curve. Instead, we can consider stochastic rates. We do so here, using a singe factor Vasicek mode, caibrated on the US yied curve (as of August 31, 28). In this mode, the short rate dynamics under the risk-neutra measure Q are given by dr(t) = k[θ r(t)]dt + σdw(t), where k, θ, and the voatiity σ are constants and W(t) is a Brownian motion. As we introduce another stochastic variabe, we ony consider Monte Caro pricing. Under this set-up our resuts are quaitativey simiar to the ones obtained earier. In fact, the most important factor now is the correation between rates and the underying assets. When quoting on hybrid products, traders usuay caibrate their bid-ask spread based on rates-equity correations of -5% and +5%. We cacuate the GMWB insurance cost based on these correations as we as %. As shown in Figure 8, the insurance cost generay increases with the aowed withdrawa amount (or as maturity decreases), just ike under deterministic rates. The insurance cost increases with correation, simiary to a standard put. With positive correation, a put option Figure 8. Insurance sensitivity to wi tend to pay off when rates-equity correation 7 rates are ow. The payoff wi be discounted at a 6 ower rate, giving a 5 higher vaue. The oppo4 site happens with -5% Correation 3 negative correation. % Correation 2 financia risks underying their business. As we move towards Sovency II there is increased focus on market consistent risk measures incuding tai risk, such as vaue-at-risk measures. Hedging can have a dramatic impact on such tai-risk measures. Fees evied on GMWB are set to cover the cost of protection on average. In the rea word there is ony one reaisation of asset performances. It is therefore important to invest fees in assets that wi repicate the behaviour of the guarantee being sod. Hedging versus reserving For reguatory and economic purposes, the rationae for hedging or reserving is to be abe to withstand a significant market shock. The reevant risk eve is typicay the.5% percentie over a one-year horizon. Assuming norma distributions, this requires covering against a 2.58 standard deviation shock. In other words, if assets are kept as a reserve against deterministic shocks, sovency is ony guaranteed at the confidence eve considered. Ony reserves equa to assets can ensure sovency with 1% confidence. On the other hand, one can hedge the tai risk via put options. In this case the tai risk wi be entirey covered with 1% confidence. The burden of providing protection, incuding mode and jump risk wi fa on the option writer. The advantage of hedging is that capita aocated to hedges can be many times smaer than reserving. In Tabe 4 we demonstrate this for the situation where the VA writer is short a put ony, such as for a simpe GMAB on a basket of interest rates and equities. In this case, the hedge cost is the cost of the put whereas the cost of reserving for the risk is equivaent to hoding Tabe 4: Hedging versus reserving Hedge Hedging 5 1 Maximum withdrawa Life & Pensions 15 2 Loca and internationa reguators require insurance companies to appropriatey measure Reserve.4 x σa x T ~2.6 x σa x T Advantage Tota transfer of risk Investment freedom of free assets Disadvantage Negative carry asset (time vaue decay) Voatiity risk premium cost Sti non-zero probabiity of defaut Cost of capita and de-everaging of baance sheet in distress periods Cost 2 5% Correation 1 42 Left to withdraw 2 The first formua is an approximation of the Back-Schoes formua where the option is struck at the money forward (K = SerT). The Back-Schoes formua coapses to Put = Ca = S [N(d1) N(d2)] with d1 = d2 = ½ σ sqrt(t). A Tayor expansion of the function N(.) at d1 and d2 yieds to the foowing approximation: Put = Ca ~.4 S σ sqrt(t). This cost estimate ignores transaction charges.

5 assets sufficient to absorb a shock down to the reserve eve. Assuming a stress equivaent to a Sovency II stress eve of 99.5%, this requires assets of roughy 6 times the cost of hedging. The decision to hedge or not shoud therefore be based on the cost of hedging versus the opportunity cost on reserve capita (Figure 9). Figure 9. NAV profie at maturity Static versus active hedging When it comes to hedging, a wide spectrum of soutions is avaiabe, from static hedging on one side to active hedging on the other. In the case of GMWBs, the perfect hedge is a put option on the sub-account vaue. Given that the sub-account vaue is affected by reative changes (performance of the underying) as we as jumps (withdrawas) it cannot be hedged with standard instruments. Mievsky and Saisbury (24) caim that the vaue of the GMWB is equivaent to a Quanto Asian put on an underying which is the inverse of the sub-account price. This sti keeps the sub-account as underying and introduces an exotic option which woud sti have to be managed using pain vania options. In other words, there wi aways be a basis risk between the hedge and the required hedge. A perfect hedge is ikey to be prohibitivey expensive given the compexity of the underying sub-account process. A static hedge is more ikey to be a portfoio of options chosen to best repicate the insurance cost. A basis risk wi remain. A dynamic hedge shoud reduce the basis risk but wi introduce transaction costs as we as requiring ongoing risk management. To iustrate the pros and cons of both strategies we wi reconsider the GMWB with deterministic rates. For the dynamic hedge we wi assume a monthy rebaancing. Standard deta-hedging is undertaken giving rise to a basis risk given that ony the first derivative with respect to the underying is considered. Other greeks and higher orders are ignored. The reader shoud note that deta hedging aone, abeit on a more frequent basis, has been the hedging approach adopted by many US insurers who have sod VA products. The technique, in the absence of hedging of other and higher order greeks, has proven to be insufficient in stress scenarios, particuary where arge sudden shocks occur in market eves and voatiity. In the static case we can consider a range of potentia soutions: singe put Portfoio of puts with different strikes and maturities Put spreads, designed to ony provide a $7 payoff at option maturity, in accordance with withdrawas The aocation to various instruments is performed by minimising the basis risk at maturity between the insurance cost and the hedge payoff. Effectivey we perform a constrained regression (positive weights, no intercept) to repicate the insurance risk with our universe of hedges. The R 2 of the regression is the metric we use to assess the effectiveness of the hedge. As a resut, the optimisation of 9% a static hedge sti introduces mode risk. 8% Initiay we consider the singe put (Fig. 1). For 7% that we ook at a seection of puts with maturities ranging from 5 to 14 years with strikes between 6% 8% and 11% of spot. We find that at-the-money 5% spot puts work best and the 7-year maturity is optima, giving an R 2 4% of 81.1%. Longer-dated puts are ess effective, given the path dependency of the 3% insurance cost and the greater impact of bad performance of the underying in earier years. Figure 1. Singe puts: goodness of fit Goodness of fit If we consider a portfoio of at-the-money puts (5- to 14-year maturities), we can increase the goodness of fit to 93.7% (Fig. 11). Under this setup most of the weight is attributed to the 5-year option. In genera, onger-dated puts attract smaer weights, in ine with the decreasing R 2 at individua put eve. A portfoio of puts aso has the advantage of not having the hedge concentrated in a singe instrument. The portfoio of put spreads performs marginay better than the singe put hedge (R 2 of 86.4% versus 81.3%), but worse than the portfoio of standard puts (R 2 of 93.7%). The rationa for put spread was ony to get a payoff equa to the required amount, ie. $7 to cover for the withdrawas in any particuar year. However, the underying of the put is different from the sub-account. In fact the binary nature of the payoff of the put spreads introduces further basis risk. But even the best static hedge underperforms the dynamic hedge (R 2 of 96.6%). Figures 12 and 13 show how the dynamic hedge aways eads to a sma basis risk. The static hedge sometimes eads to no basis risk when the underying asset performance is good (no insurance has to be provided) but eads to a basis risk when the underying performance is poor. This is further highighted in Figure 13 showing a much tighter distribution of the basis risk for the deta hedging than for the portfoio of puts. We aso see that the portfoio of puts significanty reduces downside basis risk reative to a singe put. It is important to remember that in this exampe we have ignored insurance risks, in particuar, poicyhoder behaviour. The dynamic approach provides strike 1 strike 9 strike 8 strike Maturity Figure 11. Portfoio of puts: 25% individua weights Put weight 2% 15% 1% 5% % Put maturity February 29 43

6 cutting edge Figure 12. Basis risk and fina equity vaue Figure 13. Basis risk and fina equity vaue 15 1 Deta hedging Put portfoio 35% 3% 25% Singe put: 65.1% Put portfoio: 49.3% Singe put Put portfoio Deta hedging Fina shortfa 5 Density 2% 15% 1% -5 5% Fina equity vaue % Basis at maturity fexibiity in managing insurance risks through the constant re-appraisa of the hedge as insurance risks materiaise. The dynamic approach woud therefore be superior in a rea-word setting. Aternativey, one coud hedge semi-staticay. In other words: dynamicay managing a static hedge because, athough the dynamic hedge appears optima, in practice there may be hurdes to impementing it. First, there is mode risk and in particuar the possibiity of jumps. This is neary impossibe to deta hedge. With a static hedge, the burden of repication in case of jumps (and the options management in genera) woud fa on the option writer, whie risk monitoring woud remain in-house with the VA seer. Furthermore, carrying out a dynamic hedge requires appropriate resources which not a insurers have or coud afford. It is worth mentioning one way in which VA seers can remove most of the risks arising from the VA book. This invoves transacting a reinsurance type contract, with either a bank or reinsurance company. The ast remaining risk then becomes counterparty risk which can further be imited through coateraisation. In current markets, we see the abiity to transact such contracts as being increasingy imited, with many reinsurers reassessing their appetite for such deas. For those who choose to retain insurance risks, in the absence of a (iquid) market for hedging these risks, they normay remain with the VA seer. Nevertheess, they can be mitigated through product design and through reserving expicity for risks that cannot currenty be hedged through the capita markets. Concusion Whie this artice has ooked in some detai at pricing and hedging of one particuar type of VA poicy, it shoud have become cear that pricing and hedging these products correcty is a very product-specific and potentiay compicated exercise. We considered a very simpe product the addition of attractive features such as ratchets or ookbacks further compicates the pricing and hedging process of VA poicies. In the context of an insurer taking such products to market for the first time, as is the case for many European insurers, the ist of practica considerations expands dramaticay: Can current distribution channes fairy market the product? Is product design fuy optimised for risk management? Does inteectua and technica pricing capabiity exist in the firm? What reserves are required? What risk management techniques shoud be adopted? Dynamic or static hedging? are new systems required for risk monitoring and/or hedging? Is reinsurance a possibiity? In the US and Japan, where VA products have been around for a whie, VA seers have aready deat with the above considerations, even if, in our opinion, charges for VA products tended not to cover fuy the impied costs. In these markets, VA seers have entered a new phase of risk management of these products such that a re-appraisa of the risk versus charges is under way. This has ed VA seers to consider outsourcing hedging to third parties more naturay equipped or wiing to dea in dynamic market and insurance risks. L&P Danie Bamont and Pretty Sagoo are vice-presidents of the Strategic Soutions Group and Goba Pensions Strategy Group respectivey, at Deutsche Bank Emai: The opinions or recommendations expressed in this artice are those of the authors and are not representative of Deutsche Bank as a whoe References Mievski, M. and Saisbury, T. Static and Dynamic Vauation of Guaranteed Minimum Withdrawa Benefits August 24 Merton, R. and Perod, A. Theory of Risk Capita in Financia Firms Journa of Appied Corporate Finance, 6(3) (Fa 1993, 16-32) Bauer, D., King, A. and Russ, J. A Universa Pricing Framework for Guaranteed Minimum Benefits in Variabe Annuities Working Paper, Um University 44 Life & Pensions

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