Negative Marginal Option Values: The Interaction of. Frictions and Option Exercise in Variable Annuities

Transcription

1 Negative Marginal Option Values: The Interaction of Frictions and Option Exercise in Variable Annuities Thorsten Moenig and Daniel Bauer October 217 Abstract Market frictions can affect option exercise, which in turn affects the value of a marginal option to the writer and may even yield negative marginal option values. We demonstrate the relevance of this mechanism in the context of variable annuities with popular withdrawal guarantees, both theoretically and empirically. More precisely, we show that in the presence of income and capital gains taxation for the policyholder, adding on a common death benefit option allowing to continue the withdrawal guarantee in case of death changes the policyholder s optimal withdrawal behavior. As a consequence, the total value of the contract from the perspective of the insurer may decrease, i.e. the marginal option value is negative. This explains the common practice of including death benefit options without additional charges in these products. Keywords: Risk-Neutral Valuation with Taxation, Negative Option Value, Variable Annuities, Guaranteed Minimum Benefits, Optimal Exercise Behavior. A previous version of the paper was entitled On Negative Option Values in Personal Savings Products. Moenig: moenig@temple.edu; Department of Risk, Insurance, and Healthcare Management; Temple University, 181 Liacouras Walk, Alter Hall 611, Philadelphia, PA 19122; (215) Bauer: dbauer@cba.ua.edu; Department of Economics, Finance, and Legal Studies, University of Alabama, 361 Stadium Drive, Tuscaloosa, AL 35487, (25) We are thankful for helpful comments from participants at the 213 APRIA Annual Meeting and the 48th Actuarial Research Conference. We are also indebted to Glenn Harrison, Ajay Subramanian, Eric Ulm, and Yongsheng Xu for valuable input. Financial support from the Society of Actuaries under a CAE research grant is greatly appreciated. All remaining errors are ours.

2 NEGATIVE OPTION VALUES: FRICTIONS AND OPTION EXERCISE IN VAS 1 1 Introduction Market frictions can have a first-order effect on a security s value, and in contrast to a classical law in financial economics may even yield negative option prices (Longstaff, 1992; Jordan and Kuipers, 1997; Fleckenstein, Longstaff, and Lustig, 214, and references therein). More recently, Jensen and Pedersen (216) show that this logic translates to option exercise. Frictions affect optimal exercise rules, and may even yield an optimal early-exercise for a call option on a nondividend paying stock overturning yet another classical financial economics law. In this paper, we show that the latter mechanism can lead to the former result. That is, financial frictions may affect the exercise rules for a purchaser of a marginal option, which in turn affects the seller s cost of producing the payoff and may even yield negative marginal option values. We demonstrate the relevance of this mechanism in the context of variable annuities (VAs). More precisely, we show that frictions VA policyholders face (preferential taxation rules, incomplete markets, transaction costs, etc.) affect their exercise of withdrawal options, and adding on a death benefit option is associated with a negligible or even a significant negative cost i.e., a profit for the issuer. This may explain why VA issuers incorporate free death benefit options in these contracts. VAs are investment vehicles offered by U.S. life insurers since the 197s. They are equipped with certain option features that can be exercised upon survival to a certain age or contract year so-called Guaranteed Living Benefits (GLBs) or upon death so-called Guaranteed Minimum Death Benefits (GMDBs). The VA market is large. With nearly USD 2 trillion in net assets, VAs account for almost a quarter of the US insurance industry s total assets (Dept. of the Treasury, 216; Insurance Information Institute, 217). And despite a recent downtrend in sales and GLB election rates, annual new premiums still well exceed $1bn, and over 75% of all new VAs include GLBs. In particular, various versions of withdrawal guarantees so-called Guaranteed Minimum Withdrawal Benefits (GMWBs) and Guaranteed Lifetime Withdrawal Benefits (GLWBs) have been the most popular living benefits since the mid-2s. Despite the size of the market, VAs have received limited coverage in the mainstream finance literature, with the exception of a few recent papers (Moenig and Bauer, 216; Koijen and Yogo,

3 NEGATIVE OPTION VALUES: FRICTIONS AND OPTION EXERCISE IN VAS 2 217). A possible explanation is that these products and associated problems are generally perceived as applications of familiar option pricing techniques (Black and Scholes, 1973; Merton, Brennan, and Schwartz, 1977), the specifics of which are covered in the more specialized financial mathematics and actuarial literature (Milevsky and Posner, 21; Milevsky and Salisbury, 26; Bauer, Kling, and Russ, 28; Dai, Kwok, and Zong, 28, among many others). However, Moenig and Bauer (216) point out that the resulting exercise rules for optimal withdrawals do not align with empirical observations and show that the misalignment can be explained by frictions the buyer faces, specifically by the preferential tax treatment that VAs enjoy in comparison to conventional investments (Brown and Poterba, 26). The current paper makes the related but distinct point that these frictions will thus also affect the value of the option to the producer/writer. Indeed, using a simple two-period model, we show that the impact of the tax friction on exercise behavior can have the perverse consequence that including an additional option feature (a GMDB) will decrease the total option value from the perspective of the option writer (the insurance company). We go on to analyze ten VA plus GMWB products offered in the marketplace, accounting for product differences and details in the contracts, in a Black-Scholes economy. We find that while adding on a GMDB to a VA+GWMB policy with a conventional proportional fee structure increases the policyholder s valuation by an average 1.5% of the investment amount, the insurer itself profits by a little more than 1.3% thus the marginal value of producing the option is negative. Remarkably, we obtain negative option values for all considered VA+GMWB products with a conventional fee structure, although a less common modification to the fee structure that is present in some VA+GMWB products appears to invert the result. Formally, the VA investment is placed in a sub-account and invested in a combination of mutual funds. GMWB riders are an optional add-on feature available in many VA policies, which grant the policyholder the right to withdraw small guaranteed installments over the course of the policy. More precisely, each year, the policyholder may withdraw up to the guaranteed annual amount g, regardless of the performance of his VA investment. If sufficient funds are available, the withdrawal

4 NEGATIVE OPTION VALUES: FRICTIONS AND OPTION EXERCISE IN VAS 3 is taken out of the VA sub-account; otherwise the insurer has to finance the shortfall out of its own funds. Withdrawals in excess of g (including policy surrenders) are possible but are not subject to the guarantee and may result in a penalty. Of course, the policyholder may also choose to forego a withdrawal in order to keep the money in the VA sub-account. Typical GMDB riders, on the other hand, ensure that the VA pays at least a guaranteed amount and allow the beneficiary to continue the GMWB, in the case of the policyholder s premature death. Nearly all VA+GMWB policies offered in the U.S. market include such GMDBs, without an explicit additional charge for the policyholder. So-called dynamic functions for exercise behavior in industry models that are based on empirical observations stipulate that policyholders withdraw funds when the total amount they are able to withdrawal exceeds the current account value (American Academy of Actuaries, 25). This aligns with optimal behavior when considering taxation since withdrawals are tax-free in these situations but not with optimal exercise rules when ignoring taxes (Moenig and Bauer, 216). Generally, withdrawals are costly for the insurer, since they reduce future fee income (charged as a fixed fraction of the account value for most products) and increase the likelihood that the company will have to finance a shortfall. Including a GMDB makes policyholders less likely to exercise, since withdrawing also decreases the guaranteed death benefit amount. In addition, withdrawing may result in frictional costs (e.g. due to foregone tax benefits) to the policyholder, but not to the insurer. Therefore, the present value of the change in exercise behavior can exceed the marginal value of adding on the death benefit option. We first show this mechanism more formally in a two-period Binomial model in Section 2, and then carry out our analyses of empirical products in a Black-Scholes pricing model in Section 3. Finally, Section 4 concludes. Proofs and technical material is collected in the Appendix.

5 NEGATIVE OPTION VALUES: FRICTIONS AND OPTION EXERCISE IN VAS 4 2 Negative Marginal Option Values in a Two-Period Model 2.1 Model Setup We model the VA as a special two-period investment. At time, the agent places 1 into an account managed by an insurance company. The money is invested in a risky asset, which evolves according to a standard binomial tree. That is, a dollar invested in the asset either increases to u dollars or decreases to d dollars by the end of the period. In addition, agents have access to a risk-free asset that earns an effective rate of return r per period. The VA policy includes a withdrawal guarantee that grants the right but not the obligation for the investor to withdraw 5 each at times 1 and 2, provided that she is alive at the time, and regardless of her account value. If the VA account value is not sufficient to cover the withdrawal(s), the insurer will have to pay the shortfall. In exchange, the insurer collects fees that are charged at rate φ against the VA account value at the beginning of each period (that is, at times and 1). A withdrawal reduces the VA account value accordingly. At time 2, the investor receives either the guaranteed amount of 5 or the account value, net of taxes. We reflect the preferential tax treatment of U.S. VAs by imposing a personal income tax at rate τ [, 1] on all earnings from the VA upon their withdrawal and a capital gains tax at rate κ [, 1] on earnings from any other investment at the end of each period. We assume that the initial investment (principal) of 1 is after-tax income and can be withdrawn tax-free. However, earnings are withdrawn before the principal. Moreover, we model the investor s mortality risk in the form of a mortality rate q in the first period and q 1 in the second period, both conditional on the investor being alive at the beginning of the period. For valuation purposes, we consider the product measure using the risk-neutral measure for financial risk and the physical measure for the investor s biometric risk. 1 We now illustrate how the valuation perspectives of insurer and investor differ, and we demon- 1 This is the minimal martingale measure, which provides us with a canonical benchmark for pricing by valuing unspanned risk in a risk-neutral way (Björk and Slinko, 26). See also Basak and Chabakauri (21); Møller (21); and Bauer, Phillips, and Zanjani (213) for more details and motivation for this choice.

6 NEGATIVE OPTION VALUES: FRICTIONS AND OPTION EXERCISE IN VAS 5 Table 1: Parameter choices for the two-period model. Parameter Value Parameter Value Parameter Value u 1.5 d.5 r.25 q q 1.25 fee 4.44% Note: The financial parameters u, d, and r describe the binomial tree and the risk-free investment. q and q 1 are the first- and second-period mortality rates of the individual, respectively. The fee is calibrated so as to make the insurer break even in the absence of a death benefit guarantee and under optimal withdrawal behavior. strate that this can lead to a change in optimal withdrawal behavior that makes the addition of a free death benefit guarantee strictly beneficial to both parties. Purely for the purpose of simplifying our exposition, we make a few parametric assumptions, as depicted in Table 1. Note that under these specifications the risk-neutral probability that the asset increases in a period is given by p := r d u d = Valuation without Death Benefit Guarantee Valuation of Withdrawal Guarantee The investor s withdrawal decision at maturity is straightforward, but whether to withdraw the guaranteed amount of 5 at time 1 depends on the VA account value at the time. In particular, depending on the movement of the underlying asset, the VA account can be in either the up -state or the down -state at time 1. Figure 1 depicts the evolution of the VA account in the case where the investor chooses to withdraw either in both states, or in the down -state only; the investment tree for other withdrawal strategies can be constructed analogously. Starting with an initial investment of 1, the insurer first deducts the guarantee fee at rate φ for the first period. The remainder is invested in the risky asset for one period, and following the assumptions made in Table 1 moves to either V u := 1(1 φ)u = or V d :=

7 NEGATIVE OPTION VALUES: FRICTIONS AND OPTION EXERCISE IN VAS 6 1(1 φ)d = If the investor chooses to withdraw, her VA account value is reduced by 5 (down to a minimum of ). The insurer then deducts the guarantee fee for the second period again at rate φ and invests the remainder of the VA account in the risky asset for the second period. We denote the time-2 VA account value by V ij, where i, j {u, d} correspond to the first-period and second-period performance of the risky asset, respectively, and w denotes the time-1 withdrawal amount. Then, V ij (w) = max{v i w, }(1 φ)j. If the investor is alive at maturity, and if her VA account has fallen below 5, the issuer makes up the difference. If the investor withdraws in both states see Figure 1(a) the expected present value of fees paid equals the expected present value of the guarantee payouts: P V Fees = 1φ + p max{v u 5, } + (1 p) max{v d 5, } φ r and P V Guar = p max{5 V u, } + (1 p) max{5 V d, } 1 + r P r(ij) max{5 V ij (5), } +(1 q 1 ) i,j {u,d} (1 + r) That is, the assumed guarantee fee of 4.44% makes the insurer break even under this withdrawal strategy. We now determine under what conditions (specifically, under what tax rates τ and κ) this is indeed the investor s optimal strategy. Optimal Withdrawal Strategy The optimal time-1 decision is the one that maximizes the investor s subjective expected present value of future after-tax payouts. We illustrate the respective calculations leading to the optimal decision in the up state, that is when the time-1 account value is V u = First, consider the case where the investor withdraws the guaranteed amount of 5 in the up - state, as depicted in Figure 1(a). Note that the amount withdrawn is composed of max{v u

8 NEGATIVE OPTION VALUES: FRICTIONS AND OPTION EXERCISE IN VAS 7 Figure 1: Evolution of investment account in basic model. w = Fees Issuer: (133.79). Fees Guar. Issuer: 5 (44.6) 5.4 w = Fees Guar. 5 Issuer: 5. Guar. 5 Issuer: 5. (a) If investor withdraws 5 in both states. w = Fees Issuer: (25.46). Fees Issuer: (68.49). w = Fees Guar. 5 Issuer: 5. Guar. 5 Issuer: 5. (b) If investor withdraws only in the down state. Note: Payouts in case of death in parentheses.

9 NEGATIVE OPTION VALUES: FRICTIONS AND OPTION EXERCISE IN VAS 8 1, } = in (taxable) earnings since these are withdrawn first and 6.66 in (tax-free) principal. Since the investor pays income taxes (at rate τ) on these earnings, her after-tax payout at time 1 is thus (1 τ) = τ, with certainty. Her second-year payout, on the other hand, will depend on both the movement of the asset and her death or survival. In particular, following her earlier withdrawal, only = can be taken out tax-free at time 2. Hence, if the VA account value increases to V uu (5) = , she receives (1 τ) ( ) = τ, net of taxes. Since τ 1, this quantity exceeds the guaranteed withdrawal amount of 5 so that the investor s time-2 payout in the up-up -state is given by τ, regardless of her survival or death. Conversely, if her investment account decreases in the second period, her terminal payout will be tax-free: Either V ud (5) = 44.6 if she dies, or 5 due to the guarantee in case of survival. The investor s time-2 post-tax cash flow which we denote by the random variable Y is therefore either τ, 5., or 44.6, with respective probabilities (beginning in the time-1 up -state) of p, (1 p)(1 q 1 ), and (1 p)q 1. Her subjective expected present value of Y at time 1 is denoted by V 1 and defined as the amount of money she needs at time 1 to set up a portfolio that exactly replicates these after-tax payouts. Since earnings in this replicating portfolio are taxed at rate κ at the end of the period, we apply the corresponding valuation framework developed by Moenig and Bauer (216). In particular, V 1 is given implicitly as the unique solution to the following equation: (1 + r) V 1 = E Q [Y ] + κ 1 κ EQ [max { Y V 1, }]. (1) Under our parameterization, we thus have 2 V 1 = τ κ κ. 2 Sketch of proof: Since V 1 is unique, we simply need to verify that V 1 (as specified here) indeed solves Equation (1). Since 5 < V τ, we have E Q [max { Y V 1, }] = p ( τ V 1 ). It can then be shown that V 1 verifies Equation (1).

10 NEGATIVE OPTION VALUES: FRICTIONS AND OPTION EXERCISE IN VAS 9 Hence, combined with her after-tax payout from the time-1 withdrawal, the investor holds the monetary equivalent of V 5 u := ( τ) + V 1 = τ κ τ κ κ (2) in the up -state at time 1, if she withdraws 5. Conversely, if she chooses not to withdraw in the up state, we find following Figure 1(b) a time-1 value of V u := τ κ κ. (3) Combining Equation (2) and Equation (3) leads to the following result: Lemma 1. In the two-period model without a death benefit guarantee, withdrawing is optimal in the up -state if and only if V 5 u > V u, that is if and only if τ κ 4 τ κ <.7. This inequality cannot be satisfied when κ 7/37. For κ < 7/37, the left hand side is increasing in the withdrawal tax rate τ: A larger value of τ increases the cost of withdrawing, which makes withdrawing less likely. On the other hand, for fixed τ, a larger value of κ makes outside investments less attractive, and therefore also reduces the likelihood of withdrawing. In similar fashion, it can be shown that withdrawing is optimal in the down -state for all κ, τ [, 1]. 2.3 Adding a Death Benefit Guarantee We now add a death benefit guarantee to the VA investment, at no extra charge. This additional option promises to return the original investment of 1 minus proportional withdrawals in case the investor dies before the policy matures. We find that at time 1, withdrawing the guaranteed amount of 5 is still optimal in the down -state for all κ, τ [, 1]. However, the presence of the

11 NEGATIVE OPTION VALUES: FRICTIONS AND OPTION EXERCISE IN VAS 1 death benefit guarantee may impact the optimal withdrawal strategy in the up -state. In particular, we obtain the following result its derivation can be found in Appendix A. Lemma 2. In the presence of the death benefit guarantee, the investor prefers not to withdraw in the up -state if and only if τ κ 4 τκ >.57. Interestingly, if both inequalities of Lemma 1 and Lemma 2 hold, we observe a change in the investor s optimal behavior: While she prefers to withdraw the guaranteed amount if the GMDB is absent, she is better off forgoing this withdrawal if the GMDB is included in the VA. Moreover, such a change in withdrawal behavior has some benefits for the insurer as well: Delaying the withdrawal reduces the insurer s expected obligations from the withdrawal guarantee and also increases the amount of fees that he will collect in the second period. In fact, these benefits outweigh the insurer s additional costs of providing the death benefit guarantee. This leads to the following result, which we prove in Appendix A. Lemma 3. Under the conditions of both Lemma 1 and Lemma 2, that is if.7 > τ κ 4 τ κ > κ, (4) the death benefit guarantee has a negative marginal value to the insurer. Figure 2 demonstrates that the constraint in Lemma 3 is feasible. For values of (marginal income tax rate) τ and (marginal capital gains tax rate) κ above and to the right of the upper boundary, withdrawing in the up -state is never optimal, with or without the GMDB rider. And for (κ, τ) below and to the left of the lower boundary, withdrawing is optimal, again irrespective of the presence of the death benefit guarantee. However, the shaded area between the boundaries corresponds to the tax parameters for which the GMDB alters the investor s withdrawal behavior: withdrawing is optimal in the case without the rider, but not when it is included in the VA. This change in the investor s behavior turns out to be highly beneficial for the insurer. At the initial break-even fee rate φ = 4.44%, the insurer makes an expected net profit of.87 (see

12 NEGATIVE OPTION VALUES: FRICTIONS AND OPTION EXERCISE IN VAS 11 Figure 2: Feasible region of Inequality (4) τ Note: The shaded region represents the combinations of τ and κ for which the investor optimally withdraws the guaranteed amount in the up -state at time 1 if and only if the VA does not include the death benefit guarantee. κ Appendix A) in the form of a risk-neutral present value. In fact, when the GMDB rider is included in the VA, the break-even fee drops to 3.93%. That is, the insurer would be willing to offer the death benefit guarantee at a negative price, even though the guarantee is also beneficial to the investor. To summarize, the insurer only values the payouts from the withdrawal guarantee and the death benefit guarantee, while the investor also values her tax obligations. Deferring the withdrawal in the up -state reduces the value of the withdrawal guarantee but increases the value of the death benefit rider (since in this state a withdrawal reduces the guaranteed death benefit amount by more than the amount withdrawn). Under our parameter assumptions, this has a positive net effect on the insurer. Therefore, these two factors alone would not suffice for the investor to actually defer withdrawing. However, deferring the withdrawal also has a tax benefit to her, since she can delay or if the investment loses value in the second period avoid tax payments on her earnings from the VA. Under the conditions of Lemma 3, the combination of tax benefit and increased death benefit value is sufficient to overcome the reduced value of the withdrawal guarantee, and therefore the investor chooses not to withdraw the guaranteed amount in the up -state when the GMDB

13 NEGATIVE OPTION VALUES: FRICTIONS AND OPTION EXERCISE IN VAS 12 rider is present. Without the death benefit rider, on the other hand, the investor s tax benefit does not on its own outweigh the reduction in the value of the withdrawal guarantee, so that deferring the withdrawal in not in her best interest in this situation. This is possible because of the financial involvement of a third party that is inactive in the decision making, namely the government that collects fewer taxes from the investor. However, since the government established the preferential tax treatment of VAs in order to encourage and support individuals and households to save for their own retirement, the welfare implications of this tax transfer are not clear. Indeed, the inclusion of the death benefit guarantee encourages the investor to keep her VA intact longer, which is in the spirit of the government s VA tax law. Thus, in this example, the addition of a free death benefit guarantee is financially beneficial to both investor and insurer for certain combinations of income and capital gains tax rate. The practical relevance of this mechanism depends on the empirical question of whether we obtain negative marginal values for real-world VA contracts under more realistic model and parameter choices. We present corresponding analyses in the next section. 3 Death Benefit Valuation in VAs with Withdrawal Guarantees To demonstrate that negative marginal option values are relevant in the U.S. VA market, we analyze ten empirical contracts offered between 27 and 214. More precisely, we determine the valuation of the contracts from the perspective of the buyer (policyholder) and the writer (insurance company) in a calibrated option pricing model. We compare the values including and excluding the death benefit guarantee, and calculate the marginal value from both perspectives. We find that for all eight considered contracts with a conventional proportional fee structure, including the GMDB option increases the policyholder s valuation but decreases the replication value for the insurer. This can explain the practice of not charging for plain these embedded benefit guarantees. However, we find that the result inverts for fee structures that were introduced more recently in the marketplace, i.e. here the death benefit guarantees carry a positive marginal production value.

14 NEGATIVE OPTION VALUES: FRICTIONS AND OPTION EXERCISE IN VAS Description of Empirical VA Policies We implement ten VA+GMWB contracts offered in the U.S. market between 27 and 214. Our selection of contracts was based on a web search of product details, where we include them in the analysis if we found all the necessary supporting information (detailed product prospectus) and if an initial perusal of contract features seemed feasible in view of implementation. Importantly, we show all analyses for VA+GMWB that we implemented, that is we did not select which contracts to present based on the results. 3 These products are offered by different providers and are distinct in a variety of features, such that the ten contracts present a reasonable cross section of product features available in the marketplace. In particular, the contracts differ with regards to when the product was available for purchase; the guaranteed annual withdrawal amount; the overall guaranteed amount; the VA base fee (formally known as the Mortality and Expense Risk Charge plus the Administration Charge) and the GMWB rider fee, and whether the latter can potentially be waived; the surrender fee schedule and whether a percentage of the VA account value can always be withdrawn free of charge; the presence and frequency of a step-up (or reset ) feature on the guaranteed amount; and the way in which the guaranteed amount is adjusted following an excess withdrawal. Table 2 shows the key features of each of the ten policies. All ten contracts contain a return-of-premium death benefit guarantee: similar to our amended two-period model, the insurer pays the maximum of the VA account value and the initial investment (minus prior withdrawals) upon the policyholder s premature death. In addition, all considered VA policies offer a continuation option for the GMWB rider; that is, beneficiaries can alternatively choose to receive the guaranteed annual withdrawal amount each year, until the total remaining guaranteed amount is reduced to. In the standard case with the GMDB included we allow the beneficiary to select either the VA account value, the lump-sum death benefit payout, or the GMWB continuation option, based on which of the three cash flow streams offers the largest subjective risk-neutral present value. To determine the marginal value of the death benefit guarantee, we also 3 We implemented one additional VA+GMWB contract but the results were not finalized by the time of writing. This contract will be included in a future version of the paper.

15 NEGATIVE OPTION VALUES: FRICTIONS AND OPTION EXERCISE IN VAS 14 Table 2: Description of empirical VA+GMWB contracts. Issuer VA name Rider name VA fee Rider fee Guarantee Step-Up Surr. fees Year AG5 Jackson National Perspective II AutoGuar 5 13 bps 85 bps 1 $5, annual, 7 years 214 Life Insurance Comp. elective AG6 AutoGuar 6 1 bps 1 $6, ASL2 Prudential Annuities Advanced Series GMWB 165 bps 35 bps 2,3 $7, 5 years n/a 27 Life Assurance Corp. Lifevest II HF Hartford Life Director M The Hartford s 115 bps 5 bps 2 $7, n/a 7 years 5 27 Insurance Company Principal First HFP Principal First 2 bps 2 $5, n/a 7 years 5 Preferred RAVA RiverSource Annuities Retirement Advisor Withdrawal 95 bps 15 bps 2 $7, annual, 1 years Advantage Plus 6 Benefit Rider autom. RAVA< 6 bps 2,7 SB5 Security Benefit AdvisorDesigns GMWB 12 bps 55 bps 2 $5, 4 5 years 7 years 27 Life Insurance Comp. SB6 $6, 4 SB7 $7, Notes: Dashes are short for identical entries as in the cell above. 1 The rider fee is charged in proportion to the remaining guaranteed withdrawal amount. 2 The rider fee is charged in proportion to the VA account value. 3 The rider fee is waived after 7 years if no withdrawal is made in the first 7 years, and until a step-up occurs. 4 For SB5, total guaranteed amount is 13% of premium (G = $13,); for SB6, guaranteed amount is 11% of premium. 5 The investor can withdraw 1% of the account value each year without incurring the surrender fee. 6 The investor receives a 2% purchase payment credit at inception of the policy (that is, the initial VA account value is $12,). 7 RAVA< is identical to RAVA GMWB, but when investing in a low-equity fund.

16 NEGATIVE OPTION VALUES: FRICTIONS AND OPTION EXERCISE IN VAS 15 Table 3: Parameter choices for Empirical VAs. Description Parameter Value Policyholder & contract specifications Age at inception x 55 VA principal 1, Years to maturity T 4 Annual guaranteed amount see Table 2 Mortality rates 212 IAM Basic Male Mortality table Financial market parameters Interest rate r t based on yield curve 1 Volatility σ 16% Tax rates Income tax rate τ 3% Capital gains tax rate κ 23% Early withdrawal penalty 1% 2 Note: 1 The annual interest rate r t is implied by the yield curve from the first day of trading in the contract year (that is, year of inception) for each VA policy, as stated in Table 2. 2 The early withdrawal penalty of 1% is applied by the U.S. government on the amount withdrawn from the VA if the policyholder is below age consider the case without a GMDB rider whereby the beneficiary receives simply the VA account value upon the policyholder s death. 3.2 Calibration and Implementation We evaluate the contracts in a Black-Scholes option pricing model. More specifically, we assume that the underlying fund evolves according to a geometric Brownian motion with constant volatility. The risk-free rate of return is determinist but time-varying, based on the yield curve from 1/1/27 or 1/1/214 corresponding to the inception date of the VA policy in view (as indicated in Table 2). All considered VA providers impose investment restrictions on accounts with an embedded withdrawal guarantee. The providers objective is to reduce their risk exposure from the guarantee. We reflect this by assuming that the VA investment is placed in a combination of stocks and

17 NEGATIVE OPTION VALUES: FRICTIONS AND OPTION EXERCISE IN VAS 16 bond and its evolution can be represented in a Black-Scholes framework with an annual volatility of 16%, which corresponds to an investment mix of roughly 8% in the S&P 5 index the historical volatility between 1989 and 28 is around 19% and 2% in fixed-income derivatives. However, we also consider alternative volatility assumptions. 4 All VAs enjoy the preferential tax treatment as described in our two-period model. Effective tax rates are difficult to calibrate since there is considerable heterogeneity and since tax rules vary for different financial instruments. Here we rely on the tax parameters from Moenig and Bauer (216), which are based on representative values for the U.S. tax code. In particular, we assume that the investor faces a marginal income tax rate of τ = 3% and a marginal tax rate on her capital gains of κ = 23%. In addition, we now also account for the 1% early withdrawal penalty that the U.S. government imposes on withdrawals the policyholder makes prior to age 59.5 (IRS, 23). Finally, the investor s mortality rate is based on the 212 IAM Basic Male mortality table for U.S. individual annuitants. Due to path-dependence and (optimal) exercise rules embedded in the valuation problem, closed-form solutions are not feasible and we have to rely on numerical methods. Similarly to Bauer, Kling, and Russ (28) and Moenig and Bauer (216), we rely on recursive dynamic programming. More specifically, we use a grid for the underlying state space consisting of account value, tax base, and benefit bases, and for each point on the grid determine the value of the payoff at maturity, adjusted for taxes if the valuation is from the policyholder s perspective. We then work backward in time, and determine the corresponding value for each point in the grid at times T 1, T 2,..., by solving one-period valuation problems. Here we rely on the risk-neutral expected value or the equivalent of Equation (1) for the insurer s and for the policyholder s valuation problems, respectively. Since the end-of-period value function is only available on the state-space grid, we derive approximations by interpolating between grid points and by discretizing the return space similar to Bauer, Kling, and Russ (28). We determine the (optimal) exercise rules that maximize 4 An additional justification for our parameter choice is that 16% produces a valuation from the policyholder s perspective that is closest among all integer-valued volatility parameters for the majority of the contracts. We provide valuation results for the integer-valued volatility parameter that produces a valuation closest to the one-time premium for all considered contracts.

18 NEGATIVE OPTION VALUES: FRICTIONS AND OPTION EXERCISE IN VAS 17 the value function from the policyholder s perspective. For the presentation of our results in the next section, we consider the case of a 55-year old investor who purchases a single-premium VA policy with an up-front investment of $1, (these assumptions are innocuous in view of our results). The policy matures when the investor turns age 95. All relevant assumptions and parameter choices are listed in Table Results for Empirical Contracts Table 4 and Table 5 display valuation statistics for the ten empirical contracts, both with and without the death benefit guarantee. The two tables differ based on the chosen value of the volatility parameter σ, 16% and 15%, reflecting different investment restrictions. Here, V denotes the risk-neutral value for the contract from the policyholder s perspective, that is accounting for the preferential tax treatment. In particular, even though the contract value is close to the initial investment of $1, for all contracts, there is still a substantial surplus for the insurance company, evaluated as the difference in the risk-neutral values of fee income and costs for producing the contingent option benefits. For the contracts as they are offered (including a GMDB, rows 4 through 8), this surplus ranges from a little under $5, to almost $25,, and it is decreasing in volatility, which is not surprising given option sensitivities. We note that while the company may face additional costs including expenses, costs for holding risk capital, etc., the observation that the values to the policyholders are close to their initial investment which trivially corresponds to the reservation price for a risk-neutral investor suggests that the companies reap a significant portion of the tax benefit. For five of the ten considered contracts, σ = 16% is the integer-valued volatility parameter that produces a subjective policy value closest to the the initial investment, and we mark the contracts with an asterisk in Table 4. For three more contracts, 15% produces a value close to $1,, and similarly we mark them with an asterisk in Table 5. For two remaining contracts, RAVA and RAVA<, the volatility parameters leading the value to align with the investments are 14% and 12%, respectively, and we use those figures for producing the results in Table 5. Here, it is worth

19 NEGATIVE OPTION VALUES: FRICTIONS AND OPTION EXERCISE IN VAS 18 Table 4: Valuation results for empirical VA+GMWB contracts, with σ = 16%. Contract AG5* AG6* ASL2* HF HFP RAVA RAVA< SB5 SB6* SB7* Fee proportional to Guar. Guar. Account Account Account Account Account Account Account Account σ 16% 16% 16% 16% 16% 16% 13% 16% 16% 16% With GMDB V 98,96 99,53 99,933 13,24 13,835 16,564 12,524 11,935 11,37 11,346 Costs GMWB 1,619 16,1 1,271 9,746 3,33 19,859 13,513 12,36 1,46 9,921 GMDB 1,47 1,865 2,16 2, ,91 2,12 2,144 2,225 2,236 Fees 22,17 24,54 31,865 28,955 28,686 27,463 22,394 26,816 28,988 29,6 Surplus 1,144 6,575 19,434 16,898 24,815 4,73 6,779 12,636 16,717 17,443 Without GMDB V 97,975 98,29 98,8 1,922 13,355 13,65 1,698 1,119 99,33 99,19 Costs GMWB 9,19 12,5 8,19 9,652 2,967 19,433 12,413 11,141 8,986 9,154 Fees 2,835 2,196 24,564 27,716 27,46 24,386 18,22 22,326 24,241 26,164 Surplus 11,726 8,146 16,374 18,64 24,493 4,953 5,87 11,185 15,255 17,1 Marginal Values To Insurer (1,582) (1,571) 3,6 (1,166) 322 (25) 972 1,451 1, To Policyholder 931 1,24 1,925 2, ,959 1,826 1,816 2,4 2,156 Note: The table displays the subjective risk-neutral present value to the policyholder ( V ) as well as the risk-neutral present values of the collected fees and of the guarantee costs. Surplus is the difference between fees and costs, to the insurer. The Marginal Value of the GMDB rider to the insurer is the difference between the respective surplus with and without the GMDB. The Marginal Value of the GMDB rider to the policyholder is the difference between the subjective policy values V. Contracts marked with an asterisk (*) indicate that the policy value V is closest to the initial investment of 1, for the chosen (integer) value of σ. Descriptions of the contracts can be found in Table 2, and the underlying parameter specifications are displayed in Table 3.

20 NEGATIVE OPTION VALUES: FRICTIONS AND OPTION EXERCISE IN VAS 19 Table 5: Valuation results for empirical VA+GMWB contracts, with σ = 15%. Contract AG5 AG6 ASL2 HF* HFP* RAVA* RAVA<* SB5* SB6 SB7 Fee proportional to Guar. Guar. Account Account Account Account Account Account Account Account σ 15% 15% 15% 15% 15% 14% 12% 15% 15% 15% With GMDB V 97,344 97,813 97,216 99,956 11,144 1,536 99,799 99,423 98,222 98,316 Costs GMWB 8,774 12,443 6,713 8,48 2,339 15,81 1,824 9,919 7,68 7,833 GMDB 1,99 1,22 1,257 1, ,669 1,298 1,432 1,465 1,617 Fees 18,733 19,181 23,37 27,855 25,529 21,716 17,762 2,496 23,67 25,557 Surplus 8,86 5,536 15,67 17,52 22,633 4,966 5,64 9,145 13,922 16,17 Without GMDB V 96,487 96,919 96,16 98,226 1,773 99,133 98,726 98,259 96,978 96,97 Costs GMWB 8,368 11,198 5,176 7,233 2,237 13,769 1,162 9,154 6,712 6,541 Fees 18,414 17,939 16,885 23, 24,486 17,692 15,23 16,642 18,261 2,296 Surplus 1,46 6,741 11,79 15,767 22,249 3,923 4,861 7,488 11,549 13,755 Marginal Values To Insurer (1,186) (1,25) 3,358 1, , ,657 2,373 2,352 To Policyholder ,56 1, ,43 1,73 1,164 1,244 1,49 Note: The table displays the subjective risk-neutral present value to the policyholder ( V ) as well as the risk-neutral present values of the collected fees and of the guarantee costs. Surplus is the difference between fees and costs, to the insurer. The Marginal Value of the GMDB rider to the insurer is the difference between the respective surplus with and without the GMDB. The Marginal Value of the GMDB rider to the policyholder is the difference between the subjective policy values V. Contracts marked with an asterisk (*) indicate that the policy value V is closest to the initial investment of 1, for the chosen (integer) value of σ. Descriptions of the contracts can be found in Table 2, and the underlying parameter specifications are displayed in Table 3.

21 NEGATIVE OPTION VALUES: FRICTIONS AND OPTION EXERCISE IN VAS 2 noting that indeed RAVA< is a contract with substantial investment restrictions (see Footnote 7 in Table 2). In what follows, if not mentioned otherwise, we will be referring to the value marked with an asterisk for each contract. Clearly, the policyholder benefits from the addition of the GMDB rider, since it is offered at no additional charge. As a consequence, the value of each contract to the policyholder, V, is larger when the GMDB rider is included, as can be seen when comparing row 4 and row 9 in Tables 4 and 5. The difference, i.e. the marginal value to the policyholder, is provided in the bottom of the tables. It ranges from a little under $4 to more than $2,15, with an average benefit of roughly $1,4. However, and this is our key empirical finding, for all eight VA policies that charge the fee proportional to the account value, the marginal value to the insurance company is also positive. That is, the company stands to benefit from the addition of the free option. The intuition is consistent with our two-period model from Section 2: the addition of the death benefit guarantee incentivizes the policyholder to defer withdrawals, which is reflected in the larger present value of fees under all ten contracts and under both values of σ. In particular, while the cost for producing the GMWB payoff decreases, and while of course there are no longer costs incurred for producing the contingent GMDB payment, the reduction in fee income dominates both of these effects. The financial benefit of including the free GMDB rider ranges from slightly under $4 to more than $3,, an average financial benefit of approximately $1,3 across companies. It is worth noting that while we find negative marginal option values for proportional fees, this observation does not carry over if the fee is charged proportionally to the benefit base, see contracts AG5 and AG6. The former fee structure has been the conventional fee structure within VA contracts and is still the one dominating in practice, although alternatives such as this one have been emerging sporadically over the last decade. 5 Our results suggest that product management decisions have to carefully distinguish observations between these generations of contracts. Empirical 5 More recently, a few insurers have even started to offer VA guarantees with varying fee rates, possibly tied to the VIX-index in order to reflect the increased guarantee value in times of high market volatility (CBOE, 213; Bernard and Tang, 216; Cui, Fen, and MacKay, 217), to the moneyness of the guarantee (Bernard, Hardy, and MacKay, 214), or to time (Bernard and Moenig, 217).

22 NEGATIVE OPTION VALUES: FRICTIONS AND OPTION EXERCISE IN VAS 21 observations based on an insurer s experience may not carry over when modifying the baseline. 4 Conclusions This paper explores the impact of exercise behavior on the valuation of marginal dynamic options in the context of financial frictions. In such environments, the valuations of the option writer and the option holder can deviate. Due to this gap, a value-maximizing option holder does not necessarily behave as to maximize the production costs for the option writer. Indeed, this can lead to the perverse situation that adding on a marginal option free of charge which of course benefits the option holder distorts exercise behavior in such a way that the production costs decrease. Thus, the marginal option will have a negative value. We show that this mechanism is not only a theoretical curiosity, but indeed is relevant in the context of VA investments. Their preferential tax treatment relative to other savings products affects the valuation from the perspective of the holder, even under the assumption of risk neutrality and complete markets. We demonstrate that for eight considered standard VAs with withdrawal guarantees, including a death benefit continuation option free of charge will increase the policy value to both the policyholder and the insurance company offering the VA contract. This explains why this type of death benefit is automatically included in most withdrawal guarantees. However, our results also show that this mutual benefit may not prevail under alternative fee structures that insurers recently started to integrate into their VA products. We limit our considerations to tax frictions, which have a first-order effect on VAs (Brown and Poterba, 26), although it is conceivable that other aspects such as incompleteness in the market of personal savings products or financial frictions that the company faces (e.g. various expenses or costs for holding regulatory risk capital) further drive apart the valuation. However, it is remarkable that our calculated policy values all are very close to the initial investment under reasonable model calibrations. This suggests a pricing strategy that contemplates the reservation price of a risk-neutral investor, which may be reasonable given the heterogeneity and elusiveness

23 NEGATIVE OPTION VALUES: FRICTIONS AND OPTION EXERCISE IN VAS 22 of preference parameters. We consider the further exploration of the pricing behavior as well as the question of what drives product innovation in the VA space as interesting avenues for future research. References American Academy of Actuaries (25). Variable Annuity Reserve Working Group (VARWG) Analysis Report, Attachment 5: Modeling Specifications. Basak, Suleyman, and Georgy Chabakauri (21). Dynamic mean-variance asset allocation. Review of Financial Studies, 23: Bauer, Daniel, Alexander Kling, and Jochen Russ (28). A Universal Pricing Framework for Guaranteed Minimum Benefits in Variable Annuities. ASTIN Bulletin the Journal of the International Actuarial Association, 38: Bauer, Daniel, Richard D. Phillips, and George H. Zanjani (213). Financial pricing of insurance. In Handbook of Insurance (G. Dionne Ed.), Springer, New York, NY. Bernard, Carole, Mary Hardy, and Anne MacKay (214). State-dependent fees for variable annuity guarantees. ASTIN Bulletin, 44: Bernard, Carole, and Junsen Tang (216). Variable Annuities with fees tied to VIX. Working paper, Grenoble Ecole de Management and University of Waterloo. Bernard, Carole, and Thorsten Moenig (217). Where Less Is More: Reducing Variable Annuity Fees to Benefit Policyholder and Insurer. Journal of Risk and Insurance, forthcoming. Björk, Tomas, and Irina Slinko (26). Towards a general theory of good-deal bounds. Review of Finance, 1: Black, Fischer, and Myron Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81:

24 NEGATIVE OPTION VALUES: FRICTIONS AND OPTION EXERCISE IN VAS 23 Brown, Jeffrey R., and James M. Poterba (26). Household ownership of variable annuities. In Tax Policy and the Economy (J. Poterba Ed.), Vol. 2: MIT Press, Cambridge, MA. CBOE (213). VIX for Variable Annuities Part II. Chicago Board of Options Exchange Working Paper. Cui, Zhenyu, Runhuan Feng, and Anne MacKay (217). Variable Annuities with VIX-Linked Fee Structure under a Heston-Type Stochastic Volatility Model. North American Actuarial Journal, 21: Dai, Min, Yue Kuen Kwok, and Jianping Zong (28). Guaranteed minimum withdrawal benefit in variable annuities. Mathematical Finance, 18: Fleckenstein, Matthias, Francis A. Longstaff, and Hanno Lustig (214). The TIPS-Treasury Bond Puzzle. Journal of Finance, 69: Insurance Information Institute (217). Facts + Statistics: Industry overview. iii.org/fact-statistic/industry-overview. Internal Revenue Service (IRS), U.S. Department of the Treasury (23) General Rule for Pensions and Annuities (Publication 939). Jensen, Mads V., and Lasse H. Pedersen (216). Early option exercise: Never say never. Journal of Financial Economics, 121: Jordan, Bradford D., and David R. Kuipers (1997). Negative option values are possible: The impact of Treasury bond futures on the cash US Treasury market. Journal of Financial Economics, 46: Koijen, Ralph S.J., and Motohiro Yogo (217). The Fragility of Market Risk Insurance. Working Paper, New York University and Princeton University. Longstaff, Francis A. (1992). Are negative option prices possible? The callable US Treasury- Bond puzzle. Journal of Business, 65:

25 NEGATIVE OPTION VALUES: FRICTIONS AND OPTION EXERCISE IN VAS 24 Moenig, Thorsten, and Daniel Bauer (216). Revisiting the Risk-Neutral Approach to Optimal Policyholder Behavior: A Study of Withdrawal Guarantees in Variable Annuities. Review of Finance, 2: Merton, Robert C., Michael J. Brennan, and Eduardo S. Schwartz (1977) The valuation of American put options. Journal of Finance, 32: Milevsky, Moshe A., and Steven E. Posner (21) The titanic option: valuation of the guaranteed minimum death benefit in variable annuities and mutual funds. Journal of Risk and Insurance, 68: Milevsky, Moshe A., and Thomas S. Salisbury (26) Financial valuation of guaranteed minimum withdrawal benefits. Insurance: Mathematics and Economics, 38: Møller, Thomas (21). Risk-minimizing hedging strategies for insurance payment processes. Finance and Stochastics, 5: US Department of the Treasury (216). Annual Report on the Insurance Industry (September 216). reports-and-notices/documents/216_annual_report.pdf.

26 NEGATIVE OPTION VALUES: FRICTIONS AND OPTION EXERCISE IN VAS 25 Appendices A Technical Appendix Proof of Lemma 2. The death benefit option guarantees a payout of 1 in case the investor dies, minus proportional withdrawals. For instance, if the investor withdraws 5 in the up -state, the time-2 death benefit payout is reduced to 1 1(1 φ)u 5 1(1 φ)u 65.12, and, similarly, to upon withdrawal in the down -state. Naturally, payouts in the survival state remain unaffected. This yields the VA account trees with and without time-1 withdrawal in the up -state as displayed in Figures 3(a) and 3(b), respectively. To determine the investor s optimal withdrawal strategy in the up -state, we again compute her subjective time-1 value of the resulting after-tax cash flows from the VA investment, in analogy to the process described in Section 2.2. We begin with the case where the investor withdraws the guaranteed amount of 5, depicted in Figure 3(a). From our earlier discussion we know that the investor receives τ after taxes at time 1. In addition, her (random) time-2 payout Y can be either τ (with probability p), 5. due to the withdrawal guarantee (with probability p(1 q 1 )), or due to the death benefit guarantee (with probability pq). The investor s subjective time-1 value V 1 of this random cash flow Y is then given implicitly and uniquely by Equation (1). In particular, under our parameter specifications (see Table 1), it can be easily verified that τ κ V 1 = κ Therefore, the investor s time-1 subjective risk-neutral value from withdrawing 5 in the up -state is Vu τ κ τ κ := ( τ) + V 1 = κ In similar fashion, and with the help of Figure 3(b), we can determine the investor s time- 1 subjective risk-neutral value from not making a withdrawal in the up -state. We denote this quantity by Vu and find that it is identical to V 1, the time-1 subjective value of the investor s time-2 cash flow from the VA. It turns out that here this value depends on the relation between τ and κ. In particular, if τ κ, then V 1 1 and therefore V u := τ 19.1 κ κ In this case, the investor prefers to withdraw the guaranteed amount of 5 in the up -state if and only if τ κ 4 τκ <.57. (5).