Valuation of large variable annuity portfolios: Monte Carlo simulation and synthetic datasets

Abstract Metamodeling techniques have recently been proposed to address the computational issues related to the valuation of large portfolios of variable annuity contracts. However, it is extremely diffcult, if not impossible, for researchers to obtain real datasets frominsurance companies in order to test their metamodeling techniques on such real datasets and publish the results in academic journals. To facilitate the development and dissemination of research related to the effcient valuation of large variable annuity portfolios, this paper creates a large synthetic portfolio of variable annuity contracts based on the properties of real portfolios of variable annuities and implements a simple Monte Carlo simulation engine for valuing the synthetic portfolio. In addition, this paper presents fair market values and Greeks for the synthetic portfolio of variable annuity contracts that are important quantities for managing the financial risks associated with variable annuities. The resulting datasets can be used by researchers to test and compare the performance of various metamodeling techniques.


Introduction
A variable annuity is an insurance product created and sold by insurance companies as a tax-deferred retirement vehicle to address many people's concerns about outliving their assets [29,32].Essentially, a variable annuity is a deferred annuity with two phases: the accumulation phase and the payout phase.During the accumulation phase, the policyholder makes purchase payments to the insurance company.During the payout phase, the policyholder received bene t payments from the insurance company.The policyholder's money is invested in a set of investment funds provided by the insurance company.The policyholder has the option of allocating the money among this set of investment funds.A major feature of a variable annuity is that it includes guarantees or riders.Due to this attractive feature, lots of variable annuity contracts were sold.According to [32], the annual sales of variable annuities in the U.S. were more than $100 billion for every year from 1999 to 2011.
The guarantees embedded in variable annuities are nancial guarantees that cannot be adequately addressed by traditional actuarial approaches [6,26].Dynamic hedging is adopted by many insurance companies to mitigate the nancial risks associated with these guarantees.Dynamic hedging requires calculating the fair market values and Greeks (i.e., sensitivities) of the guarantees.Since the guarantees embedded in variable annuities are relatively complex, their fair market values cannot be calculated in closed form except for special cases [14,24].In practice, insurance companies rely on Monte Carlo simulation to calculate the fair market values and the Greeks of the guarantees.However, using Monte Carlo simulation to value a large portfolio of variable annuity contracts is extremely time-consuming because every contract needs to be projected over many economic scenarios for a long time horizon [11].
In the past few years, metamodeling techniques have been proposed to address the computational issues associated with the valuation of large variable annuity portfolios.See, for example, [15], [20], [16], [21], [22], [27], [19], [28], and [23].The main idea of metamodeling techniques is to construct a surrogate model on a set of representative variable annuity contracts in order to reduce the number of contracts that are valued by Monte Carlo simulation.This is achieved by selecting a small number of representative contracts, using Monte Carlo simulation to calculate the fair market values (or other quantities of interest) of the representative contracts, building a regression model (i.e., the metamodel) based on the representative contracts and their fair market values, and nally using the regression model to value the whole portfolio of variable annuity contracts.
However, it is di cult for researchers to obtain real datasets from insurance companies to assess the performance of those metamodeling techniques.As a result, the aforementioned papers on variable annuity portfolio valuation used synthetic datasets to test the performance of the proposed metamodeling techniques.Di erent groups of researchers created di erent synthetic datasets to test various proposed methods.For example, the synthetic datasets used by [15] and [27] are di erent in portfolio composition.
In this paper, we create synthetic datasets to facilitate the development and dissemination of research related to the e cient valuation of large variable annuity portfolios.In particular, we create a large synthetic portfolio of variable annuity contracts based on the properties of real portfolios of variable annuities and implement a simple Monte Carlo valuation engine that is used to calculate the fair market values and the Greeks of the guarantees embedded in those synthetic variable annuity contracts.The purposes of this work are to relieve researchers from spending time on creating such datasets and to provide common datasets that can be used to evaluate the performance of metamodeling approaches.
The remaining part of this paper is organized as follows.Section 2 describes how the synthetic portfolio of variable annuity contracts is created.Section 3 presents a Monte Carlo simulation engine for valuing the guarantees embedded in variable annuities.In Section 4, we present synthetic datasets that can be used to test the performance of metamodeling techniques.Section 5 concludes the paper with some remarks.The software that is used to generate the synthetic portfolio and implement the Monte Carlo simulation engine is described in Appendix B.

Synthetic Portfolio of Variable Annuity Contracts
In this section, we describe how to create a synthetic portfolio of variable annuity contracts to mimic a real portfolio of variable annuity contracts.In particular, we create a synthetic portfolio of variable annuity contracts based on the following major properties typically observed on real portfolios of variable annuity contracts: • Di erent contracts may contain di erent types of guarantees.
• The contract holder has the option to allocate the money among multiple investment funds.
• Real variable annuity contracts are issued at di erent dates and have di erent times to maturity.

. Guarantee Types
Guarantees embedded in variable annuities can be divided into two broad categories: the guaranteed minimum death bene t (GMDB) and guaranteed minimum living bene ts (GMLB).The GMDB rider guarantees the policyholder a speci c amount upon death during the term of the contract [26].The death bene t is paid to the designated bene ciary of the policyholder upon the death of the policyholder.The death bene t comes in several forms [5]: return of premium death bene t, annual roll-up death bene t, and annual ratchet death bene t.The return of premium death bene t is the most basic form of the death bene t.Under this form, the death bene t paid is equal to the maximum of the account value at time of death and the premium.This option is usually o ered without additional charges.Under the annual roll-up death bene t option, the death bene t increases at a speci ed interest rate.Under the annual ratchet death bene t option, the death bene t is reset to the account value if it is higher than the current death bene t.
There are several types of GMLBs: guaranteed minimum accumulation bene ts (GMAB), guaranteed minimum income bene ts (GMIB), guaranteed minimum maturity bene ts (GMMB), and guaranteed minimum withdrawal bene ts (GMWB).The GMAB rider guarantees that the policyholder has the option to renew the contract during a speci ed window after a speci ed waiting period, which is usually 10 years.The speci ed widows typically begins on an anniversary date and remains open for 30 days [7].The GMIB rider guarantees that the policyholder can convert the lump sum accumulated during the term of the contract to an annuity at a guaranteed rate [26].The GMMB rider guarantees the policyholder a speci c amount at the maturity of the contract [26].The GMWB rider gives a policyholder the right to withdraw a speci ed amount during the life of the contract until the initial investment is recovered.Similar to the death bene t, the living bene t can be the original premium or subject to regular or equity-dependent increases.
The riders can be purchased individually or in combination for additional fees.For example, the GMDB and the GMWB riders can be purchased simultaneously.To create a synthetic portfolio of variable annuity contracts, we consider 19 products shown in Table 1.For the synthetic variable annuity policies, we set the rider fees of individual riders in the range of 0.25% to 0.75% according to the ranges given in [5].The rider fee of the combined guarantees is set equal to the sum of the fees of the individual guarantees minus 0.20%.

. Investment Funds
In practice, the policyholder's money is invested in one or more investment funds provided by the insurance company.The policyholder is allowed to select the investment funds.In dynamic hedging, a fund mapping is used to map an investment fund to a combination of tradable and liquid indices such as the S&P500 index.A fund mapping is used for the following reasons.First, di erent policyholders may invest their money in di erent combinations of investment funds.Second, most of the investment funds are not tradable and guarantees need to be hedged by derivatives on tradable indices such as S&P500.Third, using tradable and liquid indices in the asset model is also convenient in terms of calibrating the asset model parameters from the market.A fund mapping that maps an investment fund to k indices is denoted by a vector of k weights (w , w , . . ., w k ) such that The rate of return r f of the investment fund at a period is calculated as where r I j is the rate of return of index I j at the same period for j = , , . . ., k.The weights of an investment fund can be estimated by the method of least squares from the historical returns of the investment fund and the indices.
Table 2 shows the fund mappings of ten investment funds.Funds 1 to 5 are the index funds that replicate US large-cap equity, US small-cap equity, international equity, xed income, and money market fund, respectively.Fund 6 is a balanced mix of US large-cap equity and US small-cap equity.Other funds are di erent combinations of the indices.In the synthetic portfolio, we generate the account values of the investment funds of a policy as follows.First, we generate randomly the total account value AV from a speci ed range.Second, we generate a random integer l between 1 and 10, inclusive.Third, we select randomly l investment funds from the ten investment funds.Finally, we set the account values of those l selected investment funds to be AV /l, that is, the total account values are allocated to the l investment funds equally.

. Aging
Aging refers to the process of adjusting a variable annuity contract from an old date to a new date to re ect the changes of the account values and other relevant items (e.g., withdrawals, bene t base).In practice, variable annuity policies in a portfolio are issued at di erent dates.To value the policies at the valuation date, the policies are aged from the issue dates to the valuation date.
To create the synthetic portfolio of variable annuity policies, we make some assumptions for the sake of simplicity.In particular, we assume that all policies are issued on the rst day of a month and the policyholders' birth dates are also on the rst day of a month.The birth dates of policyholders are randomly generated from an interval of dates and the issue dates of the policies are randomly generated from another interval of dates.Table 3 shows some parameters used to create synthetic policies.Once we generate a variable annuity policy, we age it to the speci ed valuation date.In practice, the aging process re ect what happens actually to the policies.To generate the synthetic portfolio, aging a policy is just projecting the policy from the issue date to the valuation date based on one economic scenario of the investment funds.Details of the liability cash ow projection are discussed in Section 3.2.

Monte Carlo Valuation
During the past decade, some studies have attempted to value variable annuity contracts in a unifying way.For example, [5] developed a framework to value various guarantees embedded in variable annuity contracts.[3] proposed a unifying framework to value variable annuities under general model assumptions.[4] developed a dynamic programming algorithm for pricing the GMWB under a general Lévy processes framework.[31] presented an optimal stochastic control framework to price variable annuity guarantees.However, these existing studies focused on contracts that are written on a single asset.
In this section, we present a simple Monte Carlo valuation model for valuing guarantees of the synthetic portfolio of variable annuities.In particular, we present a multivariate risk-neutral scenario generator, liability cash ow modeling, and fair market value and Greek calculation.An early version of this Monte Carlo valuation model was presented at a conference by one of the authors [17].The purpose of this Monte Carlo valuation model is to calculate the fair market values and related Greeks of the synthetic variable annuities so that metamodeling techniques can be tested.As a result, we made many simplifying assumptions in the Monte Carlo valuation model.For example, we consider only single-premium contracts and do not model dynamical policyholder behaviors.Monte Carlo valuation models used in practice are much more complicate than the one presented in this paper.Although the Monte Carlo valuation model presented in this paper is simple, the datasets it creates are useful to validate metamodeling techniques.If a metamodeling technique does not work well for the datasets created in this paper, it is unlikely that it will work well for real datasets in practice.

. Risk-Neutral Scenario Generator
Economic scenario generators are used to simulate movement scenarios of the indices according to an asset model.There are two types of scenarios: risk-neutral and real-world.Risk-neutral scenarios are simulated under the risk-neutral measure; while real-world scenarios are simulated under the real-world measure.Riskneutral scenarios are used to calculate the fair market values of nancial derivatives such as the guarantees embedded in variable annuities.Real-world scenarios are used to calculate solvency capitals or evaluate hedging strategies.
Most economic scenario generators remain proprietary, but two economic scenario generators are in the public domain: the one developed by the CAS (Casualty Actuarial Society) and the SOA (Society of Actuaries) and the one developed by the AAA (American Academy of Actuaries) and the SOA [2].The CAS-SOA scenario generator is used to generate economic scenarios for asset-liability analysis for property-liability insurers [1,10].
The AAA and the SOA have created an economic scenario generator, named Academy's Interest Rate Generator (AIRG), for regulatory reserve and capital calculations.The latest version of the economic scenario generator can be obtained from https://www.soa.org/tables-calcs-tools/research-scenario/.It is a real-world economic scenario generator and can be used to generate both interest rate and equity scenarios.
Both the CAS-SOA generator and the AAA-SOA generator can generate interest rate and equity scenarios.However, the resulting scenarios generated by these two generators di er signi cantly.In particular, the interest rates generated by the CAS-SOA generator have a wider distributions than those generated by the AAA-SOA generator.For a detailed comparison of the two economic scenario generators, readers are referred to [2].
Although using economic scenario generators is the only practical way to value many life insurance contracts, it has received little attention in the academic literature.The paper by [33] is among the few papers devoted to this subject.[33] gives a brief background of the Solvency II and discusses the use of economic scenario generators in the context of Solvency II.
In this paper, we present a simple economic scenario generator to generate risk-neutral scenarios.In this simple generator, we model xed income indices directly rather than use an interest rate model.The inputs to the generator consists of the yield curve, the correlation matrix, and the volatilities.Let ∆ be the time step and m be the number of time steps.For example, ∆ = and m = if we use a monthly time step and a horizon of 30 years.The yield curve can be bootstrapped from swap rates [18,25].For example, Table 4 gives 8 swap rates of di erent tenors from the US market.We can bootstrap the 8 swap rates to get 8 discount factors at the maturity dates of corresponding swaps.Then we can interpolate the discount factors to get the discount factors at all months.Figure 1 shows the monthly forward rates interpolated by the loglinear method [25].

Forward Rate
Figure 1: The monthly forward rates bootstrapped from the swap rates given in Table 4.
Now let us introduce a multivariate Black-Scholes model.Suppose that there are k indices S ( ) , S ( ) , . .., S (k)  in the nancial market and their risk-neutral dynamics are given by [8]: where B ( ) t , B ( ) t , . .., B (k) t are independent standard Brownian motions, r t is the short rate of interest, and the matrix (σ hl ) is used to capture the correlation among the indices.The stochastic di erential equations given in Equation ( 2) have the following solutions [8]: Let t = , t = ∆, . . ., tm = m∆ be time steps with equal space ∆.For j = , , . . ., m, let A (h) j be the accumulation factor of the hth index for the period (t j− , t j ), that is, Suppose that the continuous forward rate is constant within each period.Then we have exp where f j is the annualized continuous forward rate for period (t j− , t j ).The above equation leads to Combining Equations ( 3) and ( 4), we get where By the property of Brownian motion, we know that Z (l)  , Z (l) , . .., Z (l) m are independent random variables with a standard normal distribution.
From Equation ( 5), we can calculate the continuous return for the period (t j− , t j ) as The mean and covariance matrix of the returns are given by Let Σ be the covariance matrix of the annualized continuous returns of the k indices and let where σ is the transpose of σ.From Equation (9), we see that σ is the Cholesky decomposition of the covariance matrix Σ.The simple scenario generator described above requires two inputs: the forward curve and the covariance matrix.In this generator, the bond index and the equity index are simulated in the same way by considering their covariance structure.
Once we have index scenarios simulated from Equation ( 5), we can obtain the fund scenarios by blending these index scenarios.Let n be the number of risk-neutral paths.For i = , , . . ., n, j = , , . . ., m, and h = , , . . ., k, let A (h)  ij be the accumulation factor of the hth index at time t j along the ith path.Suppose that there are g investment funds in the pool and the fund mappings are given by Then the simple returns of the hth investment fund can be blended as where F (h) ij is the accumulation factor of the hth fund for the period (t j− , t j ) along the ith path.Since the sum of weights is equal to 1, we have

. Liability Cash Flow Projection
Once we have the risk-neutral scenarios for all the investment funds F (l) ij , i = , , . . ., n, j = , , . . ., m, l = , , . . ., g, we can project the cash ows of the contract according to contract speci cations and the purpose of valuation.If we are interested in the value of the whole contract, we can project the cash ows of the whole contract.For example, the valuation method proposed by [5] is based on the whole contract.In this paper, we are interested in the market-consistent value (or fair market value) of the guarantees embedded in variable annuity contracts.To do so, we only project the cash ows arising from the guarantees.
Without loss of generality, we assume that there are three types of cash ows: death bene t, guaranteed bene ts, and guarantee risk charges for providing such guaranteed bene ts.For a general variable annuity contract, we use the following notation to denote these cash ows that occur within the period (t j− , t j ] along the ith risk-neutral path: GB ij denotes the guaranteed death or living bene t.DA ij denotes payo of the guaranteed death bene t.LA ij denotes payo of the guaranteed living bene t.RC ij denotes the risk charge for providing the guarantees; PA (h)  ij denotes the partial account value of the hth investment fund, for h = , , . . ., g. TA ij denotes the total account value.In general, we have We use the following notation to denote various fees: ϕ ME denotes the annualized M&E fee of the contract; ϕ G denotes the annualized guarantee fee for the riders selected by the policyholder; ϕ (h) F denotes the annualized fund management fee of the hth investment fund.Usually this fee goes to the fund managers rather than the insurance company.Then we can project the cash ows in a way that is similar to the way used by [5].For the sake of simplicity, we assume that events occur in the following order during the term of the contract: • fund management fees are rst deducted; • then M&E and rider fees are deducted; • then death bene t is paid if the policyholder dies; • then living bene t is paid if the policyholder is alive.We also assume that the fees are charged from the account values at the end of every month and the policyholder takes withdrawal at anniversaries of the contracts.
Once we have all the cash ows, we can calculate the fair market values of the riders as follows: where x is the age of the policyholder, p is the survival probability, q is the probability of death, and d j is the discount factor de ned as The risk charge value can be calculated as For the sake of simplicity, we did not use dynamical lapse models or stochastic mortality models [12,13] in our Monte Carlo valuation.How the cash ows of various guarantees are projected is described in Appendix A.

. Fair Market Value and Greek Calculation
We use the bump approach [9] to calculate the Greeks.Speci cally, we calculate the partial dollar deltas of the guarantees as follows: ( )  , . . ., PA (l− ) , ( + s)PA (l)  , PA (l+ )  , . . ., PA (k) s − V PA ( )  , . . ., PA (l− ) , ( − s)PA (l)  , PA (l+ )  , . . ., for l = , , . . ., k, where s is the shock amount applied to the partial account value and V (• • • ) is the fair market value written as a function of partial account values.Usually, we use s = .to calculate the dollar deltas.The partial dollar delta measures the sensitivity of the guarantee value to an index and can be used to determine the hedge position with respect to the index.We calculate the partial dollar rhos in a similar way.In particular, we calculate the lth partial dollar rho as follows: where V (r l + s) is the fair market value calculated based on the yield curve bootstrapped with the lth input rate r l being shocked up s bps (basis points) and V (r l − s) is de ned similarly.A common choice for s is 10 bps.

Synthetic Datasets
In this section, we present the synthetic portfolio and the corresponding fair market values and greeks calculated by the Monte Carlo simulation method described in the previous section.The datasets can be downloaded from http://www.math.uconn.edu/~gan/software.html.

. Synthetic Portfolio
We generated 10,000 synthetic variable annuity policies for each of the guarantee types given in Table 1.
The synthetic portfolio contains 190,000 policies.The elds of the synthetic variable annuity policies are described in Table 5.There are 45 elds in total, including 10 fund values, 10 fund numbers, and 10 fund fees.The synthetic portfolio contains about 40% policies with female policyholders.The distribution of gender by product type is shown in Table 6.Table 7 shows the summary statistics of the age, the time to maturity, and the dollar elds.The age is the years between the birth date and the current date.The time to maturity is calculated from the current date and the maturity date.The fund fees and the M&E fee are given in Table 3.The rider fees of di erent guarantee types are presented in Table 1.• The synthetic portfolio does not contains variable annuity contracts that have di erent tax treatments.
In real portfolios, some contracts are quali ed investments under the Income Tax Act.• The Monte Carlo simulation engine does not use any lapse models.In practice, some lapse model is used in the Monte Carlo valuation.In spite of the limitations, the synthetic datasets can be used to test the performance of various metamodeling techniques in terms of speed and accuracy.If a metamodeling technique does not work well for the synthetic datasets, it is not likely to work well for the real portfolio of variable annuity contracts.The full datasets or subsets can be used to test di erent models.Interested researchers and practitioners can download the source code of the software from http://www.math.uconn.edu/~gan/software.htmland possibly extend it to consider other guarantee types or other Monte Carlo valuation methods.
• The payo of the living bene t is zero, i.e., LA i,j+ = .
• After the maturity of the contract, all the state variables are set to zero.

A. GMAB and DBAB Projection
Di erent speci cations for the GMAB rider exist.See [26] and [31] for examples.We follow the speci cation given in [26] and consider GMAB riders that give policyholders to renew the policy at the maturity date.As a result, a policy with the GMAB rider may have multiple maturity dates.
At the maturity dates, if the guaranteed bene t is higher than the fund value, then the insurance company has to pay out the di erence and the policy is renewed by reseting the fund value to the guaranteed bene t.If the guaranteed bene t is lower than the fund values, then the policy is renewed by reseting the guaranteed bene t to the fund value.Let T = T be the rst renewal date.Let T , T , . .., T J be the subsequent renewal dates.Under such a GMAB rider, the guaranteed bene t evolves as follows: where T = {T , T , . . ., T J } is the set of renewal dates.We assume that the policyholder renews the policy only when the account value at a maturity date is higher than the guaranteed bene t.The payo of the living bene t is calculated as follows: The payo of the death bene t is zero if the policy contains only the GMAB rider.For the DBAB policy, the death bene t is calculated according to Equation (17).
If the payo is larger than zero, then the fund value is reseted to the guaranteed bene t.In other words, the payo is deposited to the investment funds.We assume that the payo is deposited to the investment funds proportionally.Speci cally, the partial account values are reseted as follows: for h = , , . . ., g, where LA (h) i,j is the amount calculated as,

A. GMIB and DBIB Projection
A variable annuity policy with a GMIB rider gives the policyholder three options at the maturity date [5,30] where äT and äg are the market price and the guaranteed price of an annuity with payments of $1 per annum beginning at time T, respectively.In this paper, we determine äT by using the current yield curve.We specify äg by using a particular interest rate, i.e., äg = ∞ n= n px e −nr , where r is an interest rate set to 5%, which is about 1% higher than the 30 year forward rate shown in Figure 1.
The guaranteed bene ts and guarantee risk charges are projected according to Equations ( 16) and ( 15), respectively.The payo of the death bene t is zero if the policy contains only the GMIB.For the DBIB policy, the death bene t is projected according to Equation (17).

A. GMMB and DBMB Projection
For the GMMB and DBMB guarantees, account values, guarantee risk charges, and guaranteed bene ts are projected according to the GMDB case speci ed in Equation ( 14), Equation (15), and Equation ( 16), respectively.The payo of the living bene t is projected as LA i,j+ = , if t j+ < T, max{ , GB i,j+ − TA i,j+ }, if t j+ = T. (22) For the GMMB guarantee, the payo of the guaranteed death bene t is zero.For the DBMB guarantee, the payo of the guaranteed death bene t is projected according to Equation (17).

A. GMWB and DBWB Projection
To describe the cash ow project for the GMWB, we need the following additional notation: WA G ij denotes the guaranteed withdrawal amount per year.In general, WA G ij is a speci ed percentage of the guaranteed withdrawal base.WB G ij denotes the guaranteed withdrawal balance, which is the remaining amount that the policyholder can withdrawal.WA ij denotes the actual withdrawal amount per year.
For j = , , . . ., m − , the cash ows of the GMWB from t j to t j+ are projected as follows: • Suppose that the policyholder takes maximum withdrawals allowed by a GMWB rider at anniversaries.
Then we have • The partial account values evolve as follows: i,j+ (24) for h = , , . . ., g, where ∆ is the time step and WA (h) i,j is the amount withdrawn from the hth investment fund, i.e., WA (h)  i,j+ = WA i,j+ If the account values from the investment funds cannot cover the withdrawal, the account values are set to zero.• The guarantee risk charges are projected according to Equation (15).

Figure 2 :Figure 3 :
Figure 2: Histogram of the fair market values and the deltas of individual policies.

Table 1 :
Variable annuity contracts in the synthetic portfolio.

Table 2 :
Ten investment funds.Each row is a mapping from an investment fund to a combination of ve indices.

Table 3 :
Parameter values used to generate the synthetic portfolio.

Table 4 :
The US swap rates at various tenors as ofJune 11, 2014.

Table 5 :
Description of the policy elds.

Table 6 :
Distribution of gender by product type.

Table 7 :
Summary statistics of some elds.Note that age and ttm are calculated from the birth date, valuation date, and maturity date.
:• get back the accumulated account values, • annuitize the accumulated account values at the market annuitization rate, or • annuitize the guaranteed bene t at a payment rate rg per annum.As a result, the payo of the GMIB rider is given by − TA i,j+ , if t j+ = T, (