Hedging Segregated Fund Guarantees

Transcription

1 Hedging Segregated Fund Guarantees Heat A. Windcliff Dept. of Computer Science University of Waterloo, Waterloo ON, Canada N2L 3G1. Peter A. Forsyt Dept. of Computer Science University of Waterloo, Waterloo ON, Canada N2L 3G1. and Kennet R. Vetzal Centre for Advanced Studies in Finance University of Waterloo, Waterloo ON, Canada N2L 3G1. February 18, Abstract Segregated funds ave become a very popular investment instrument in Canada. Segregated funds are essentially mutual funds tat ave been augmented wit additional insurance features, wic provide a guarantee on te initial principal invested after a specified time orizon. Tey are similar in many respects to variable annuities in te United States. However, segregated funds often ave complex embedded optionality. For instance, many contracts provide a reset provision. Tis allows investors to increase teir guarantee level as te value of te underlying mutual fund goes up. Tese contracts typically also offer features suc as mortality benefits, were te guarantee is paid off immediately upon deat of te investor. Tese contracts are furter complicated by te possibility of investor lapsing since te payment for te guarantee is amortized over te life of te contract. In tis work we describe edging strategies wic allow underwriting companies to reduce teir risk exposure to tese contracts. Te edging tecniques incorporate te strengts of bot actuarial and financial approaces. In particular, we look at some of te difficulties tat arise due to te fact tat in many cases te underwriting company is not able to take sort positions in te underlying mutual fund. An alternative is to edge using oter actively traded securities, suc as index participation units, stock index options, or stock index futures contracts. However, due to te mismatc between te edging instrument and te segregated fund contract being edged, tere is additional basis risk. We investigate te performance of tese types of edging strategies using stocastic simulation tecniques and investigate te return on te initial capital outlay mandated by new regulatory requirements. H. Windcliff, P.A. Forsyt and K.R. Vetzal, February 18,

2 Segregated funds a ve become a very popular investment instrument in Canada. Segregated funds are essentially mutual funds tat ave been augmented wit additional insurance features tat provide a guarantee on te initial principal invested after a specified time orizon. Tey are similar in many respects to variable annuities in te United States. However, segregated funds often ave complex embedded optionality. For instance, many contracts provide a reset provision. Tis allows investors to increase teir guarantee level as te value of te underlying mutual fund goes up. Tese contracts typically also offer features suc as mortality benefits, were te guarantee is paid off immediately upon te deat of te investor. Furtermore, because te payment for te guarantee is usually amortized over te life of te contract, tere are additional complications due to investor lapsing. In Canada, te Office of te Superintendent of Financial Institutions (OSFI) as recently imposed stricter new regulations for tese contracts, requiring tat insurers set aside a substantial amount of capital to back up tese guarantees. Tese reserve requirements can be reduced if appropriate edging strategies ave been put in place. In tis work we describe edging strategies wic allow underwriting companies to reduce teir risk exposure to tese contracts. Te edging tecniques incorporate te strengts of bot actuarial and financial approaces. We investigate te performance of tese types of edging strategies using stocastic simulation tecniques. It is well known tat a contract wit a maturity guarantee attaced to it can be tougt of as an investment in te underlying asset combined wit a put option, wic may ave very complex features suc as mortality benefits or reset provisions. Hedging strategies tat reduce downside risk for put options typically involve sorting te underlying mutual fund. For obvious regulatory reasons, it may not be possible for te underwriting company to take sort positions in H. Windcliff, P.A. Forsyt and K.R. Vetzal, February 18,

3 te underlying mutual fund. One alternative is to set aside a bond wit face value equal to te guarantee level and dynamically edge a (possibly complex) call option position. Te advantages of tis formulation are tat te insurer can easily take long positions in te underlying mutual fund and downside risk as been completely edged. However, now te insurer is exposed to a considerable amount of upside risk if te edging of te call option does not work as planned. Anoter alternative is to edge using oter actively traded securities, suc as index participation units, stock index options, or stock index futures contracts. However, due to te mismatc between te edging instrument and te segregated fund contract being edged, tere is additional basis risk. Using stocastic simulation we can quantify te risk associated wit selling tese contracts in a more realistic setting wic includes non-optimal investor beaviour and basis risk. We would like to empasize tat te decision of weter or not to edge tese contracts in many situations is a management issue. In some cases, edging may be necessary to reduce te capital requirements due to te new regulations. Te edging strategies described ere are capable of reducing te downside risk associated wit writing tese contracts. Of course tis comes at an expense, as te expected profit of te edged position is lower tan te expected profit of te unedged position. Te purpose of tis paper is to develop te tools required to make an informed decision as to weter or not an institution would benefit from implementing a edging strategy for tese contracts. Description of te Segregated Fund Contract Te term, segregated fund, often refers to a mutual fund combined wit a long-term maturity guarantee (typically 10 years) wit additional complex features. One popular provision H. Windcliff, P.A. Forsyt and K.R. Vetzal, February 18,

4 tat is included wit many of tese contracts is te reset feature. Wen te investor resets, tey excange teir existing guarantee for a new 10-year maturity guarantee set at te current value of te mutual fund. Hence, te reset feature allows te investor to lock in market gains as te value of te underlying mutual fund increases. Typically, te investor is able to reset te contract up to a maximum of two or four times per calendar year. Tis introduces an optimization component to tese contracts were te investor must decide wen tey sould reset and lock in at te iger guarantee level. In addition to te reset feature, many oter exotic features are included in segregated fund guarantees. For example, many segregated funds include a deat benefit so tat te guarantee is paid out immediately, even if te investor expires before te maturity date of te guarantee. As te investor ages, te mortality benefits may become more valuable and typically resets are not permitted after, say, te investor's 70 t birtday. Alternatively, more complex variations of te reset feature can also be introduced as te investor becomes older. For example, after te investor's 70 t birtday te guarantee level upon reset may be some fraction of te value of te underlying mutual fund at te time of te reset. In practice te investor is not carged an initial fee for te segregated fund guarantee. Instead te investor pays a iger management expense ratio (MER) over te life of te contract to cover te cost of providing te guarantee. Te total MER can be considered to be te sum of te proportional fee, r m, wic is allocated to te management of te underlying mutual fund, togeter wit te proportional fee, r e, wic is allocated to fund te guarantee portion of tese contracts. It may be optimal for te investor to lapse and avoid paying te iger MER if te guarantee is unlikely to be in-te-money at maturity. For tese and oter reasons, suc as need for liquid assets, an investor may witdraw teir investment from te segregated fund contract. H. Windcliff, P.A. Forsyt and K.R. Vetzal, February 18,

5 Te reset feature described above will elp to reduce te amount of investor lapsing since te guarantee can be reset to a new at-te-money guarantee. Furter, a proportional deferred sales carge (DSC) is often applied if te investor witdraws te investment during te first several years. Based on communications wit vendors of tese contracts, it appears as toug tis fee is paid to te underlying mutual fund and tat in practice none of tis fee is allocated to fund te guarantee portion of te contract. It sould be reiterated tat investor lapsing is not always beneficial to te insurance company writing te guarantee portion of tese contracts. Specifically, since te payment of te guarantee is deferred over te life of te contract, any edging costs incurred by te insurance company may not be recovered if te investor lapses prematurely. Te numerical experiments presented in tis paper will be based on two contracts tat are described in Table 1. Te first contract is a simple 10-year maturity guarantee wit no reset provisions. Te second contract is a prototypical segregated fund guarantee tat incorporates te reset feature. Bot contracts provide mortality benefits so tat te guarantee is paid out immediately upon te deat of te investor. No initial fee is carged to enter into tese contracts and te investor pays for tese guarantees by te increased MER as described in te table wit te sliding scale DSC to mitigate investor lapsing. Table 1 also describes te key market parameters used in te simulations suc as te risk-free interest rate and volatility of te underlying mutual fund. Table 1 approximately ere. Due to te complexity of tese contracts it is difficult to draw general conclusions from individual numerical experiments. Te numerical results in tis paper are intended to provide a study of te beaviour of a realistic contract but small canges in te contractual details or H. Windcliff, P.A. Forsyt and K.R. Vetzal, February 18,

6 variations in te market settings suc as te volatility and risk-free interest rate will affect te pricing and edging of tese contracts. Interested readers are referred to (Windcliff et al. 2002) for te effects of tese and oter parameters, suc as te level of investor optimality, on te valuation of segregated fund guarantees. Te Distribution of Returns for Unedged Positions In order to quantify te risk involved wit writing te segregated fund guarantees described in te previous section, we can investigate te distribution of returns for an unedged position. At tis point, we will not assume optimal investor beaviour, but will presume tat investors use euristic rules for te reset feature and optimal lapsing. We will find tat tere can be a substantial amount of downside risk to te insurer wen writing segregated fund guarantees wit a reset provision, even wen investors act non-optimally. Let S represent te value of te underlying mutual fund and let K be te current guarantee level. In tis section we will use te following euristic rules for applying te reset feature and lapsing. Heuristic reset rule: Investors will reset te guarantee level if tere are reset opportunities remaining and S > 1. 15K ; i.e. if te value of te underlying mutual fund as risen so tat te current guarantee level is 15% out-of-te-money. Heuristic lapsing rule: Investors will lapse out of te contract at time t *, and tereby avoid paying te remaining proportional fees, if tere are no remaining reset opportunities at time t * and S > 1. 4K ; I.e. if te value of te underlying mutual fund as risen so tat te guarantee level is 40% out-of-te-money. Investigation of te optimal reset region sows tat resetting te guarantee wen te H. Windcliff, P.A. Forsyt and K.R. Vetzal, February 18,

7 underlying asset as risen by 15% can be a reasonable approximation to te optimal exercise boundary during te first few years of te contact or during te first few years after a reset as taken place (see (Windcliff et al. 2002)). In fact, tis euristic rule as been adopted by a Canadian Institute of Actuaries task force on segregated funds during teir assessment of risk management strategies. Te euristic lapsation rule is based on a plot of te optimal lapsation boundary given in (Windcliff et al. 2001). Wen te investor as no remaining reset opportunities tey may be better off lapsing out of te contract to avoid paying te proportional fees since te guarantee is unlikely to be in-te-money at maturity. In fact, as discussed in (Windcliff et al. 2001), even if reset features are not explicitly offered, investors can syntetically create tem by lapsing and re-entering te contract, tereby obtaining a new at-temoney guarantee. In te Monte Carlo simulations provided in tis paper, it is assumed tat te investor makes decisions regarding te reset feature and optimal lapsing 100 times per year, or approximately twice every week. Numerical experiments indicate tat more frequent exercise decisions by te investor do not appreciably affect te results. To quantify te risk associated wit writing an unedged segregated fund guarantee, we consider te 95% conditional tail expectation (CTE) and te annualized rate of return on an initial capital requirement. Recent regulatory canges from OSFI ave introduced stricter capital requirements for companies offering tese contracts to ensure tat sufficient resources are available to back up tese guarantees. Specifically, if no edging strategy is put in place OSFI requires tat te insurer set aside te 95% CTE in liquid, risk-free instruments. Te 95% CTE is te expected value of te outcomes tat lie in te worst case 5% tail. In oter words, te 95% H. Windcliff, P.A. Forsyt and K.R. Vetzal, February 18,

8 CTE is te mean value of te worst case outcomes tat are ignored by a 95% value at risk (VaR) measurement. In comparison wit VaR measures, te CTE is muc more conservative wen setting aside capital for contracts tat exibit a long tail of values, wic occur wit relatively low probability, suc as segregated fund guarantees. If te insurer implements a edging strategy, te OSFI capital requirement can be reduced by up to a maximum of 50% of te reduction in te 95% CTE indicated by te proposed edging strategy. Te capital must be invested in safe, liquid instruments, and for tis paper we will assume tat te capital investment grows at te risk-free rate. It sould be pointed out tat we ave cosen te proportional fee r e so tat te cost of edging, net of future incoming fees for tese contracts is initially zero; in oter words, te reserve amount is zero. As a result te total balance seet requirement and capital requirements are identical. We estimate te required capital by generating simulations of te mutual fund pat, tereby generating a profit and loss (P&L) distribution for te writer of te guarantee as sown in Figure 1. Te P&L for a simulation is given by P & L = (edge value payoff value ) * e rt t* were r is te discounting rate used and * t is te time tat te contract is terminated. If we use te risk-free rate as te discounting rate, r = r, ten te P&L distribution can be used to estimate te 95% CTE. Tis is te amount tat must be set aside in risk-free instruments so tat te insurer as sufficient resources to back up te guarantee in te average of te worst case situations. Figure 1 approximately ere. Table 2 gives statistics for te P&L distribution. It is difficult to draw useful comparisons between P&L distributions since tere are many risk/reward tradeoffs to consider H. Windcliff, P.A. Forsyt and K.R. Vetzal, February 18,

9 and te duration of te contract is uncertain due to investor lapsing, mortality and te investor's use of te reset feature. In tis work we use te distribution of te annualized return on te capital set aside for tese guarantees as a measure of te profitability of offering tese products. Te outlay of capital to satisfy te OSFI requirements can be tougt of as introducing an associated cost wit selling tese guarantees, and we are interested in te rate of return on tis investment. We define te annualized return on capital (ARC) as 1 (capital value + edge value - payoff value) = t* ARC 1 * t initial capital requiremen t Tis can be regarded as te return on te initial capital per year for te writer of te guarantee. Te ARC is similar to te risk adjusted return on capital (RAROC) described in (Jameson 2001), but as been converted to an annualized rate of return to facilitate comparisons between contracts of different durations. It sould be noted tat te ARC cannot be tougt of as a compounded rate of return. We ave cosen tis specification of te return on te initial investment since te final value of our position at maturity can be negative, and tis cannot be quantified as a compounded rate of return on te (positive) initial investment. In order to facilitate comparisons wit compounded rates of return we can define an effective continuously compounded rate were 1 * r eff = ln(1 + ARC t * t ) * t is te average duration of te contract during te simulation and we use te mean ARC in tis calculation. Tis definition of te effective rate r eff incorporates te fact tat, upon selling tese contracts te insurer is locked into tis position for a duration of time wic depends upon te investor's actions. We find tat * t is approximately 6.3 years for te contract wit no resets and is approximately 21.2 years for te contract wit two resets per year wen te investor uses H. Windcliff, P.A. Forsyt and K.R. Vetzal, February 18,

10 euristic rules described in tis section to determine teir use of te reset feature and antiselective lapsing beaviour. Wen te investors act optimally, te average duration of te contracts become 6.4 years and 17.9 years respectively. Table 2 approximately ere. Te results in Table 2 indicate tat, wen no edging strategy is in place and investors act non-optimally, te expected effective return on a 95% CTE capital requirement is is approximately 9.6% for te guarantee tat does not offer any resets, and is about 8.5% for te guarantee wic offers two resets per annum. Many segregated funds are currently offered wit substantially lower proportional fees. Of course wit a lower proportional fee carged to cover te cost of providing te guarantee te return on capital will be reduced. It is interesting to note in Table 2 tat, wen no edging strategy is implemented, investor non-optimality does not significantly increase te rate of return on te initial capital investment required by te insurer to satisfy te OSFI guidelines. In a later section, we will find tat wen a dynamic edging strategy is in place, investor non-optimality can result in a significant increase in te effective rate of return on capital. Altoug te expected return indicates tat writing tese contracts and leaving tem unedged can be profitable, te capital requirements of $8.65 per undred dollars of underlying mutual fund for te contract wit no resets and $13.46 per undred dollars for te contract wit two resets per annum may be proibitive. As mentioned above, te OSFI capital requirements for tese products can be reduced if appropriate edging strategies ave been put in place. Furtermore, in Figure 1 we see tat tere is a substantial amount of variability in te ARC, particularly for te contract tat offers two resets per annum, wit many outcomes generating losses. In te following sections we will investigate te statistical performance of various H. Windcliff, P.A. Forsyt and K.R. Vetzal, February 18,

11 edging strategies for segregated fund guarantees. Hedging Risk Exposure for Segregated Fund Guarantees Te edging strategies tat we will investigate in tis paper will incorporate te strengts of bot actuarial and modern financial teory approaces. An insurer offering a segregated fund guarantee is exposed to several sources of risk. For example, due to te mortality benefits offered by many of tese contracts, te value of te contract will depend upon te demograpic profile of te investor wo is purcasing te contract; i.e. female, aged 50 years, non-smoker. If a large number of tese contracts ave been sold to investors from a similar demograpic profile, ten we can assume tat mortality risk is diversifiable. We can consider edging an aggregate contract from wic a fraction of te investors die during eac year wit a rate specified by a standard mortality table. Anoter source of uncertainty, wic may be considered to be diversifiable, is deterministic investor lapsing. Here we may be able to treat te fraction of te investors tat witdraw teir accounts (for non-optimal reasons) eac year as a deterministic function. We would like to empasize tat pricing tese contracts under te assumption tat investors will act non-optimally may be dangerous and may result in mis-pricings by te insurer. Altoug te majority of individual investors may not ave te expertise to utilize tese complex features efficiently, we ave eard of incidents were financial planners ave assisted teir customers in doing so as an additional service. Te insurer is also exposed to risk due to te uncertain movements of te underlying mutual fund since te guarantee will only ave positive value to te investor if te mutual fund is below te guarantee level at maturity. It is well known tat market risk exposure is not readily H. Windcliff, P.A. Forsyt and K.R. Vetzal, February 18,

12 diversifiable. In tis case tecniques from modern financial teory described in Hull (1997) or Wilmott (1998) can be applied. We denote V ( S, K, U, T, t) to be te value of te segregated fund guarantee wic depends upon te value of te underlying mutual fund S, te current guarantee level K, te number of reset opportunities used tis year U, te current maturity date T, and time t. In tis work we consider simple dynamic edging strategies wic create delta-neutral ( =VS ) positions for te insurer over brief time intervals. As a result, over sort time intervals, te value of te edged portfolio is immune to canges in te value of te mutual fund. If it is not optimal to utilize te reset feature or lapse ten te value V satisfies te partial differential equation (PDE) V t ( r r r ) SV + 1 σ S V rv R( t) r S + M ( t) max( K S,0) = 0 m e S 2 SS e were R (t) denotes te number of investors remaining in te contract (wo ave not perised or lapsed) at time t and M (t) is te mortality rate at time t. Tis equation is very similar to te classical Black-Scoles equation from option pricing teory, but contains two additional terms. Te final two terms tis equation represent te rate of incoming proportional fees collected from investors remaining in te contract at time t and te rate of payments made to deceased investors at time t respectively. Also, te convection coefficient (in front of te V S term) is sligtly different from tat in te classical Black-Scoles model. Tis is because we ave assumed tat edging is performed wit a perfectly correlated asset tat does not ave management fees deducted, and ence te drift rate is affected. As an example, we can consider edging a guarantee on an index tracking mutual fund by trading index participation units. In a later section we will generalize tese tecniques to allow for edging wit an imperfectly correlated asset. H. Windcliff, P.A. Forsyt and K.R. Vetzal, February 18,

13 If we let segregated fund contract, ten if satisfy te constraint U max denote te maximum number of resets permitted per annum by te V ( S, K, U, T, t) V ( S, S, U + 1, t + T, t) U < U max tere are reset opportunities remaining and V must ext were T ext is te amount tat maturity is extended by upon resetting te guarantee level. Effectively, tis models te fact tat te investors can receive a new guarantee wit guarantee level K = S and maturity T = t + Text, and tat one more reset opportunity as been used. It will be optimal for investors to lapse if te value of te guarantee, net of te proportional fees required to maintain te contract, is more negative tan any deferred sales carges tat must be paid upon terminating te contract. As a result, te writer cannot allow te value of te edging position to become negative in anticipation of future incoming fees. We can model optimal investor lapsing by imposing te additional constraint V ( S, K, U, T, t) 0. Finally, at maturity, te value of te contract is V ( S, K, U, T, T ) = R( T )max( K S,0) wic states tat only investors remaining in te contract at maturity receive te final payoff. Typically te investor is not carged an initial premium to enter into tese contracts. As a result, te fair value for tese contracts is determined by te expense ratio, r e, tat makes te value of te contract initially zero. In (Windcliff et al. 2002) we determine te fair proportional fee rate for various contracts and various models for te underlying security. In tis work we will assume tat te expense ratios are fixed at te levels given in Table 1 and our focus will be to investigate our ability to edge te risk exposure due to price movements by te underlying security. Oter sources of risk, suc as interest rate risk and implied volatility risk, basis risk, H. Windcliff, P.A. Forsyt and K.R. Vetzal, February 18,

14 liquidity risk, etc., will affect te value and edging of tese contracts. Te new actuarial reserving guidelines and OSFI's new capital standards require insurers to explicitly provide for tese risks if tey intend to take credit for edging strategies. Interested readers are referred to (Windcliff et al. 2001) for a complete and detailed description of te matematical model and computational tecniques used to obtain te edging strategies for te numerical experiments in tis paper. Statistical Results for Dynamically Hedged Positions Te insurer may wis to implement a edging strategy for several reasons. First, by implementing a edging strategy, te downside risk associated wit writing tese contracts may be reduced. Second, if te insurer implements a edging strategy, te OSFI capital requirement can be reduced by up to a maximum of 50% of te reduction in te 95% CTE indicated by te proposed edging strategy. For te numerical results provided in tis paper te required capital for a dynamically edged position will be given by OSFI Capital Requiremen t = CTE edge + (CTE noedge - CTE were te conditional tail expectations are taken wit a 95% confidence level. 1 2 In Table 3 we provide numerical results wen a delta-neutral edging strategy wic is re-balanced 50 times per year (i.e. approximately on a weekly basis) is implemented. Comparing tese results wit tose for unedged positions contained in Table 2, we see tat te 95% CTE is reduced dramatically due to te edging strategy, resulting in muc smaller capital requirements. Of course tis reduction in downside comes also comes wit lower expected profits from te contract. However, since less capital is required wen a edging strategy is implemented, te rate of return on te initial capital outlay is only moderately affected. For te contract wit no edge ) H. Windcliff, P.A. Forsyt and K.R. Vetzal, February 18,

15 resets te rate of return on te initial reserve is approximately 7.6%, wereas for te contract wit two resets per annum te return is between 6.3% and 6.9%, depending upon te degree of optimality displayed by te investor in teir use of te reset feature. Table 3 approximately ere. It sould be noted tat wen tis dynamic edging strategy is implemented, non-optimal investor beaviour could now lead to additional profits by te insurer, wic was not te case wen no edging strategy was implemented. Tis indicates tat te insurer is not penalized for edging te worst case situation wic assumes tat te investor acts optimally as profits will be accrued as non-optimal decisions occur. Altoug it may presently be safe to assume tat te majority of investors will act sub-optimally, it would be dangerous to build tis assumption into te pricing and edging of tese products. Due to te fact tat te capital requirement is only reduced by a maximum of 50% of tat indicated by te proposed edging strategy, te effective rate of return on te initial capital outlay by te insurer is quite low. Te guidance note issued by OSFI tat specifies tis maximum capital offset due to edging (OSFI 2001) states tat as te industry and OSFI gain confidence in implementing suc strategies, tis limitation will be reviewed. In Table 3 we also provide numerical results in te cases wen te capital requirement can be reduced by up to a maximum of 75% and 100% of te reduction in te 95% CTE indicated by te proposed edging strategy. As expected, in tis case te return on capital can improve quite dramatically; particularly wen investors act sub-optimally. In Figure 2(a) we can compare te relative strengts and weaknesses of te edged position. Tis figure depicts te ARC for a standard OSFI capital requirement, wic allows te initial capital outlay to be reduced by up to a maximum reduction of 50% as a result of te H. Windcliff, P.A. Forsyt and K.R. Vetzal, February 18,

16 proposed edging strategy. For te edged position te initial capital is $7.22 per undred dollars of underlying mutual fund, wile for te unedged position te initial capital is $13.40 per undred dollars of underlying mutual fund. In tis case we see tat for te edged position, te number of outcomes wic generate a loss on te initial capital is negligible; wic is not te case for te unedged position. On te oter and, tis reduction in downside risk as come at te expense of upside potential and we see tat te edged position also as relatively fewer outcomes tat generate large profits. Figure 2 approximately ere. Figure 2(b) compares te effects of optimal and euristic investor beaviour for a edged position. We see tat te profit for te writer increases wen te investor does not act optimally. Notice tat non-optimal investor beaviour introduces a positive skew in te distribution and te downside risk is not dramatically affected by te euristic investor beaviour. Hedging wit a Correlated Asset Classical edging strategies mitigate te insurer's downside risk by taking a sort position in te underlying mutual fund. For obvious reasons, due to regulatory policy tis is not possible. If te mutual fund is tracking an actively traded index, ten one can use index participation units to accurately edge risk exposure. Te numerical results provided tus far in tis paper ave assumed tis situation. However, it is often te case tat te mutual fund is not constructed to closely track an index and edging must be performed using a basket of securities tat closely replicate te performance of te mutual fund. In general, te price movements of tis basket of edging securities will not be perfectly correlated wit te underlying mutual fund. Oter possible motivations for studying edging wit an imperfectly correlated asset include H. Windcliff, P.A. Forsyt and K.R. Vetzal, February 18,

17 illiquidity in te underlying asset (Sircar and Papanicolaou 1998) and transaction costs (Wilmott 1998). In tis situation, te positions taken by te edging strategy can affect te price of te underlying asset. Sufficient illiquidity may make edging wit a correlated liquid asset te preferred coice. In tis section we extend te Black-Scoles framework to allow for te pricing and edging of option contracts wen it is not possible to establis a edging strategy wic trades directly in te underlying asset. Wit te exception of a brief note (Seppi 1999) on minimum variance cross edging strategies, very little work as been done in te matematical modelling of edging strategies in tis setting. Given an asset wic is correlated wit te underlying, we provide a partial differential equation (PDE) wic determines a cross-edging strategy wic can be used by te insurer to minimize te variance of te risk exposure to price movements by te underlying asset. Te resulting option pricing model includes te Black-Scoles and present expected value models as special cases. Anoter metod tat can be applied wen edging wit an imperfectly correlated asset formulates te option pricing problem in an incomplete market setting and uses a utility maximization approac (Davis 2000). Matematical Model. For expositional simplicity we will develop te model for a edging wit an imperfectly correlated asset in te context of a simple vanilla put option. In particular we ignore exotic features suc as mortality benefits, te deferred payment of tese contracts troug proportional fees, te reset feature and lapsing. Te numerical results provided in tis paper will be based on a generalized model tat incorporates tese effects. Consider an option written on te underlying asset wit price given by S u, wic satisfies te stocastic differential equation (SDE) ds u = µ S dt + σ S u u u u dz u. H. Windcliff, P.A. Forsyt and K.R. Vetzal, February 18,

18 Here µ u is te drift rate and σ u is te volatility of tis asset and Wiener process. dzu is an increment from a If it is not possible to trade directly in te underlying asset edge for tis option by trading in anoter asset wit price process S u, we can try to establis a S, wic satisfies ds = µ S dt + σ S dz, were µ is te drift rate and σ is te volatility of te asset S. Te Wiener increment dz is correlated wit te increment for S u wit corr( dz, dz ) = ρ. u Following standard tecniques as described in (Wilmott 1998) we establis a portfolio tat contains te option, wose value is given by V ( S u, t), and a sort position of sares of te second asset S, Π = V ( Su, t) S. Using Ito's Lemma we can estimate te cange in value of tis portfolio over small increments of time. Te coice of wic minimizes te variance of te returns on tis portfolio is given by S = ρ S u σ σ u V S u. We can loosely tink of tis model as edging as muc of te risk exposure to underlying price movements in ligt of te basis risk introduced by edging wit a non-perfectly correlated asset. If tis partially edged portfolio earns te rate of return r (wic we discuss below) ten we find tat te value of te option satisfies te PDE V t ( µ r ) S V + σ S V r = 0. σu + µ u ρ u S 2 V u SuSu σ In order to determine an appropriate specification for te discounting rate r we consider H. Windcliff, P.A. Forsyt and K.R. Vetzal, February 18,

19 tis equation under several special circumstances. ρ = ± 1: In tis case we are in a standard Black-Scoles setting and olding sares of te edging asset eliminates all risk from te portfolio Π to leading order. In tis case we sould discount at te risk-free rate, r. ρ = 0 : If ρ = 0 ten = 0 and te portfolio Π contains only a long position in te option. In tis case we sould discount te portfolio Π by te expected rate of return on te option, * r, wic can be empirically estimated using market option prices and te real-world (P-measure) drift rate. A simple way to model te discounting rate, r = r( ρ) described above is to specify r * ( ρ ) = ( 1 ρ ) r + ρr, were r is te risk-free rate and, wic is consistent wit te cases * r is te expected rate of return on te option. We remark tat wen ρ = 1 we recover te Black-Scoles model and wen ρ = 0 we recover present expected valuation metods using te real drift rate of te underlying security and a risk-adjusted discounting factor. Numerical Experiments. In Table 4 we provide estimates for te risk adjusted discounting rate * r for tese contracts. In tis work, we cannot estimate * r using market prices since tese exotic long-term options are not traded on excanges. Instead, we estimate * r by determining te discounting rate so tat te present expected value of tese contracts (using te real drift rate for te underlying security) is initially zero. In oter words, we determine te discounting rate tat te customer must implicitly be using to warrant entering into tese contracts. Table 4 approximately ere. H. Windcliff, P.A. Forsyt and K.R. Vetzal, February 18,

20 Since te drift rate of te underlying asset is greater tan te risk-free rate, taking into account te fees required to maintain te guarantee, te older of a long position in te guarantee will encounter a loss on average. Te results sown in Table 4 are consistent wit te findings in (Coval and Sumway 2001) were, using market prices for excange traded options, te autors find tat put options ave returns tat are bot statistically and economically negative. In our case, tis refers to te rate of return on te proportional fees paid by te customer to maintain te segregated fund guarantee. As expected te deferred sales carge as a dramatic impact on te expected rate of return for a contract wit no reset features but as very little impact for te contract wit two resets per annum. If tere are no reset opportunities, te investor will anti-selectively lapse out of te contract if te guarantee becomes out-of-te-money; tereby avoiding te remaining fees required to maintain te guarantee. On te oter and, if te contract offers te customer te ability to reset te guarantee level ten anti-selective lapsing does not play as large a role. In Table 5 we provide results for a partially edged position wen te correlation between te underlying mutual fund and te edging assets varies between ρ = 0 to ρ = 1. Wen ρ = 0 we are unable to edge and te outcome is identical to te unedged position described in Table 2. Wen ρ = 1 tere is no basis risk and te results are identical to te edged positions described in Table 3. We see tat wen te assets are partially correlated te performance of te edging strategy degrades and te reduction in te required reserve capital is minimal wen ρ =. 9 and is in fact worse tan te unedged position wen ρ =. 75. However, te rate of return on te initial capital outlay is not any worse tan te case wen te edging asset is perfectly correlated. Tis is because te portion of te contract tat remains unedged still earns a larger expected profit. Unfortunately, it is probably e case tat tese strategies will H. Windcliff, P.A. Forsyt and K.R. Vetzal, February 18,

21 not be useful in practice since we assume tat te main advantage of edging is to reduce te regulatory capital requirements. We do point out tat tis edging strategy is in some sense an optimal edging strategy wen basis risk exists in tat te position in te edging asset,, was cosen to minimize te variability. As a result, we contend tat te management of basis risk is of extreme importance wen edging wit a partially correlated asset and sould be approaced wit care. Table 5 approximately ere. Furter, we notice tat te re-balancing interval does not significantly affect te performance of te dynamic edging strategy wen using a non-perfectly correlated asset. In Table 5 we see tat wit a correlation of ρ =. 9, adjusting te edging position 50 times per year only marginally reduces te 95% CTE wen compared wit edging 10 times per year, from $9.60 to $9.27 per undred dollars of underlying mutual fund. Tis is because te majority of te risk is due to te basis risk between te underlying and edging instruments. In te absence of basis risk it is possible to dramatically improve te performance of te classical Black-Scoles delta-neutral edging strategy by matcing te option s curvature using a gamma edge. Gamma-neutral edging strategies reduce te risk exposure to large asset price movements by trading in oter option contracts written on te same underlying asset. As noted above, te majority of te risk is due to basis risk and consequently we expect tat gamma-neutral strategies would do little to improve te performance wen edging wit an imperfectly correlated asset. Figure 3 plots te profit and loss distributions and distributions of return on capital wen edging wit assets wic ave various degrees of correlation wit te underlying mutual fund. We see tat even for quite a ig correlation of ρ =. 9 te distributions are very broad and muc of te downside risk is not effectively reduced. In particular, we note tat wen using a edging H. Windcliff, P.A. Forsyt and K.R. Vetzal, February 18,

22 asset wit ρ =. 75, te lower tail of te profit and loss distribution sown in Figure 3(a) is actually ticker tan te unedged case (corresponding to ρ = 0 ). We sould point out tat many segregated fund guarantees are offered on mutual funds tat are actively managed. In tis case it may be very difficult to determine a basket of securities tat as a very ig degree of correlation wit te underlying mutual fund. Figure 3 approximately ere. Conclusion Te decision of weter or not to actively edge tese contracts is a management issue. In essence, edging can be tougt of as constructing insurance in te market for te writer. If one ignores te cost of capital, ten edging reduces te profitability associated wit providing tese contracts to te customer. However, due to regulatory requirements tere is an associated cost wit providing tese guarantees. In tis paper we describe metods for quantifying te return on tis initial capital outlay wic can be considered wen making decision as to weter an institution wises to continue sale of tese contracts and/or weter or not edging is appropriate. Te annualized return on capital (ARC) described in tis paper is a form of a riskadjusted return on capital. In some cases, due to available resources, management may be forced to establis edging positions in order to reduce tis capital requirement. Te results given in tis paper indicate tat some of te risks involved wit offering tese contracts can be reduced dramatically using simple dynamic edging strategies. In fact, as a result of te reduced capital requirements wen a edging strategy is implemented, it may be possible to actually increase te return on te initial capital investment. Currently owever, OSFI as taken a conservative position and allows H. Windcliff, P.A. Forsyt and K.R. Vetzal, February 18,

23 only a reduction of 50% of te indicated reduction in te 95% CTE due to te proposed edging plan. In tis case, te return on capital decreases wen edging is implemented. In some cases it is not possible to trade directly in te underlying asset. For tis situation, we describe a PDE wic determines a cross-edging strategy wic trades in a correlated asset and can be used to minimize te variance of te risk exposure of te insurer. Te resulting option-pricing model includes te Black-Scoles and present expected valuation models as special cases and te necessary parameters can be estimated using market data. As te correlation between te underlying asset and edging instrument increases, te variability of te relative edging error using tis edging model decreases. Wen edging wit an asset wic is not perfectly correlated wit te underlying, te majority of te residual risk is due to basis risk between te edging and underlying instruments. As a result, very frequent re-balancing and more complex gamma-neutral strategies are not very elpful in reducing te variability of te partially edged position. To conclude, we point out tat te numerical experiments provided in tis paper represent a rater conservative contract. In practice many guarantees are offered on more volatile underlying mutual funds and frequently te proportional fees carged to maintain te guarantee are muc lower tan tose used ere. In tese cases te profitability associated wit offering tese products can be muc lower or can even result in an expected loss. Interested readers are referred to (Windcliff 2002) for additional insigts as to wy tese contracts can be suc a liability and ow variations in te product design and market parameters suc as te volatility and risk-free interest rate affect te cost of providing tese guarantees. H. Windcliff, P.A. Forsyt and K.R. Vetzal, February 18,

24 Acknowledgment Tis work was supported by te Natural Sciences and Engineering Researc Council of Canada, te Social Sciences and Humanities Researc Council of Canada, and te Royal Bank of Canada. References Coval, Josua D. and Tyler. Sumway (2001). Expected option returns. Journal of Finance 106(3), Davis, Mark (2000). Optimal Hedging wit Basis Risk. Working paper, Financial and Actuarial Matematics Group, Tecnisce Universitat Wien, Vienna, Austria. Hull, Jon (2000). Options, Futures and oter Derivative Securities, Fourt Edition. Prentice Hall, New Jersey, United States. Jameson, Rob (2001, February). Between a RAROC and a ard place. ERisk, 1-5. OSFI (2001, August). Capital Offset for Segregated Fund Hedging Programs (MCCSR). Office of te Superintendent of Financial Institutions of Canada. Seppi, Duane J. (1999). Matematical Finance: Class notes. Graduate Scool of Industrial Administration, Carnegie Mellon University. Sircar, K. Ronnie and George Papanicolaou (1998). General Black-Scoles Models Accounting for Increased Market Volatility from Hedging Strategies. Applied Mat Finance 5(1), Wilmott, Paul (1998). Derivatives. Jon Wiley and Sons, West Sussex, England. Windcliff, Heat A., Peter A. Forsyt, and Ken R. Vetzal (2001). Segregated funds: Sout options wit maturity extensions. Insurance: Matematics & Economics 29(1), Windcliff, Heat A., Martin K. Le Roux, Peter A. Forsyt, and Ken. R. Vetzal (2002). Understanding te Beaviour and Hedging of Segregated Funds Offering te Reset Feature. To appear in te Nort American Actuarial Journal. H. Windcliff, P.A. Forsyt and K.R. Vetzal, February 18,

25 Feature Description Investor profile 50 year old Canadian female (from Deterministic lapse rate 5% per annum Optimal Lapsing Yes, investors will lapse out of te contract if te value of te guarantee becomes less tan te value of te remaining fees tat will be deducted to maintain te guarantee. Initial investment $100 Maturity term 10 years, wit a maximum expiry on te investor s 80 t birtday. Resets Contract 1: No resets. Contract 2: Two resets per year permitted until te investors 70 t birtday. Upon reset, te guarantee level is set to te value of te underlying mutual fund and te maturity is extended by 10 years from te reset date. Mortality benefits Guarantee is paid out immediately upon te deat of te investor. MER For bot contracts, a proportional fee of r m =1% is allocated to te manager of te underlying mutual fund. In addition, to fund te guarantee portion of tese contracts, additional fees are deducted at te following rates. Contract 1: (no resets) r e = 50 b.p. is allocated to fund te guarantee for a total MER of 1.5%. Contract 2: (two resets p.a.) r e = 90 b.p. is allocated to fund te guarantee for a total MER of 1.9%. DSC A deferred sales carge is levied upon early redemption using a sliding scale from 5% in te first year to 0% after five years. Tis fee is paid to te management of te underlying mutual fund and none of it is allocated to te guarantee portion of te contract. Volatility σ = 17.5% Risk-free interest rate r = 6% Drift rate (before fees) µ = 10% Table 1: Specification of te guarantee contracts and market information used in te numerical experiments provided in tis paper.

26 Profit and Loss Return on Initial Capital Mean 95% CTE Capital Mean reff Investor Contract ($) ($) ($) ARC Heuristic No resets % 9.6% Two resets p.a % 8.5% Optimal No resets % 9.5% Two resets p.a % 8.5% Table 2: Statistics for te profit and loss distribution and te return on investment for a 95% CTE capital requirement for an unedged segregated fund guarantee. Te contract wit no resets carges a proportional fee of r = 50 b.p. to fund te guarantee wile te contract wit two reset e opportunities per annum carges a proportional fee of r e = 90 b.p. Te euristic rules used by te investor to determine teir use of te reset feature and anti-selective lapsing are described in te accompanying text.

27 Profit and Loss Return on Initial Capital Mean 95% CTE Capital Mean reff Investor Contract ($) ($) ($) ARC Heuristic No resets % 7.6% % 8.6% % 12.6% Two resets p.a % 6.9% % 7.5% % 12.5% Optimal No resets % 7.6% % 8.6% % 12.6% Two resets p.a % 6.3% % 6.5% % 7.6% Table 3: Statistics for te profit and loss distribution and te return on investment for a segregated fund guarantee tat is edged 50 times per year. Te tree capital amounts sown for eac scenario represent a maximum 50% reduction, a 75% reduction and a 100% reduction from te unedged 95% CTE. Te contract wit no resets carges a proportional fee of r e = 50 b.p. to fund te guarantee wile te contract wit two reset opportunities per annum carges a proportional fee of r = 90 b.p. e

28 Contract re Lapse penalty * r No resets 50 b.p. DSC -14.5% No DSC 0.0% Two resets per annum 90 b.p. DSC -7.8% No DSC -7.0% * Table 4: Risk adjusted discounting rates, r, for te segregated fund guarantees studied in tis * paper. We obtained r by determining te discounting rate tat makes te present value of tese contracts zero initially using te real world (P-measure) drift rate for te underlying mutual fund. Te deferred sales carge (DSC) used to mitigate investor lapsing utilizes a sliding scale from 5% to 0% during te first five years of te contract.

29 Profit and Loss Return on Initial Capital Mean 95% CTE Capital Mean reff Reedge Frequency ρ ($) ($) ($) ARC 50 p.a % 6.9% % 6.9% % 6.9% % 8.5% 50 p.a p.a Table 5: Performance of edging strategies wic use edging assets wit varying levels of correlation, ρ, wit te underlying mutual fund. Te guarantee offers two resets per annum and it is assumed tat investors use te euristic rules as described in te accompanying text for teir use of te reset feature and anti-selective lapsing.

30 0.8 No resets 2 resets p.a. 0.7 Relative Frequency Profit and Loss Figure 1(a): Te profit and loss distribution for unedged segregated fund guarantees tat offer no resets and two resets per annum. We assume tat investors utilize te euristic rules described in te accompanying text to determine teir use of te reset feature and anti-selective lapsing. 25 No resets Two resets p.a. Relative Frequency Annualized Return on Capital Figure 2(b): Te return on investment for a 95% CTE capital requirement for unedged segregated fund guarantees wic offer no resets and two resets per annum. We assume tat investors utilize te euristic rules described in te accompanying text to determine teir use of te reset feature and anti-selective lapsing.