Variable annuity economic capital: the least-squares Monte Carlo approach

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1 Variable annuity economic capital: the least-squares Monte Carlo approach The complex nature of the guaranteed benefits included in variable annuity products means evaluation by Monte Carlo simulation is computationally challenging. A technique adopted from pricing American options could offer an alternative way to approximate these liabilities and their capital requirements. By MARK CATHCART & STEVEN MORRISON Risk-based capital regulation such as the recently agreed Solvency II directive requires insurers to hold capital against the change in asset and liability values over some time period at a specified confidence level. For products such as variable annuities (VAs) that mix actuarial variables with financial guarantees, this is often calculated using nested Monte Carlo simulation, for which both so-called outer real-world measure scenarios and inner risk-neutral measure valuation scenarios are used. The branching out of the simulations in this process is illustrated in figures 1 and 2. For a single-year projection on a typical insurance product, a full nested simulation approach (using 1, outer scenarios and inner scenarios) on a standard desktop computer will have run-times far too long to make it useful. Most of this time is taken up by the liability valuation calculations performed on the inner scenarios. To make the nested simulation problem less computationally demanding, a number of possible approaches to approximate this valuation component have been suggested. One such approach is to use replicating portfolios, where the liabilities are mapped to a small number of standard financial instruments, giving a faster method for generating market value projections. For more information, see for example Oechslin et al [1]. Though the replicating portfolio technique is conceptually appealing, there are technical challenges in applying it in practice, including choice of appropriate assets and economic scenarios. Also, the long-term nature of insurance business and its exposure to numerous (often non-traded) risk-drivers means it can be challenging to replicate even the simplest liabilities in the capital markets. We now introduce an alternative approach using a form of analytic approximation involving regressing for the liability value on some key economic variables, as shown schematically in figure 3. This method will only take around 15 minutes for a single-year projection on a typical insurance product on a standard desktop computer. Least-squares Monte Carlo simulation We will use the Least-Square Monte Carlo method (LSMC), introduced to value American options by Longstaff and Schwartz in 21 [2]. To see how the method is relevant to calculating variable annuity guarantees consider a put option, expiring in two years, but exercisable after one. The first stage in the simulation task is to generate a number of scenarios for the underlying stock price and calculate the resulting cashflows that arise on exercise. If the option is in-the-money, the holder has the choice between exercising or holding the option until a later time, believing the stock price will fall further. In deciding whether or not to exercise, the holder will compare the amount they will receive from exercising the option early with the (expected) value from continuing to hold the option until expiry. To determine the continuation value of the option, we must generate a set of inner scenarios branching out from each outer scenario (taking us from year one to expiry of the option at year two). We again see this challenge of nested simulation. This approach to pricing American options was originally proposed by Broadie and Glasserman in 1997, who refer to it as a simulated tree [3]. For a large number of exercise opportunities this approach creates problems. Figure 4 shows how the simulated paths of the underlying branch out as we move forward in time through the possible exercise dates of the option. Once these simulated paths have been generated, 44

2 the option is then valued backward through the simulated tree as follows. At maturity, the value of each option at each scenario is just the payoff value. At the penultimate timestep it is the greater of the exercise value and the continuation value, the latter being an average of the option payoffs in the final scenarios reachable from that node. By moving backward in time through the simulated tree and comparing the immediate exercise value with the estimated value from holding the option (until the next exercise opportunity), the time zero value of the option will eventually be found. Under this approach, the simulation of the underlying asset proceeds forward in time, while the option valuation is performed backwards in time. This incompatibility is problematic in pricing American options, where we must estimate a continuation value for each state of the underlying at every possible exercise opportunity in the tree. The amount of simulations and calculations required to proceed in this manner is too computationally demanding, so we must approximate the continuation values from each node with a far smaller number of branches. One method of cutting the number of simulations without losing a lot of accuracy is to use least squares regression, estimating the liability value by averaging over the state variables that govern the inner scenarios. We will see that good results can be obtained this way, even if we restrict to one inner scenario at each node, as shown in figure 5. In this article, we adopt the LSMC algorithm to give a pathwise approximation to the future liabilities associated with some financial product at the projection time. First, let us set t 1, t 2,... as the points in time where the underlying fund level of the product is re-balanced due to the investments made and when a withdrawal is also possible. Let Ω be the probability space of the underlying s paths, with filtration F tk representing the information about the product (for example, the fund level and guaranteed income) and the underlying economy at time t k. Now define the stochastic cashflows from the insurer s reserves to the policyholder at time t as C(ω, t), which will be non-zero only if the guaranteed income level exceeds the underlying fund level at time t. Future liabilities associated with the product at time t k can then be expressed as: t ( ) j = E exp r ω, s L ω,t k K j= k+1 ( ( )ds t ) C ω;t j k ( ) F tk where r(ω, t) is the stochastic risk-free discount rate and the expectation is taken conditional on F tk. The problem is how do we estimate this conditional expectation? The LSMC uses least-squares regression to obtain an approximation for this conditional expectation function at time t k. At this time, the assumption is made that the unknown functional form of L(ω; t k ) can be given as a linear combination of a countable set of F tk -measurable basis functions. Possible choices of basis functions given by Longstaff and Schwartz include (weighted) Laguerre, Hermite, Legendre and Chebyshev polynomials. In fact, Longstaff and Schwartz also found that simply using powers of the state variables as basis functions also gives accurate results. Taking an appropriate set of basis functions, B 1 (X), B 2 (X),..., allows the liability value to be expressed as: ( ) = a m B m ( X ( t k )) L ω;t k m= where the a j are constants and the X are the appropriate explanatory variables (which are found to influence the expected future liabilities of the product). 1 The a j are picked to minimise the squared distance between this approximation and the value gained from the inner scenarios. This LSMC approach to the liability valuation can signifcantly reduce the number of inner scenarios required for each outer scenario projection, perhaps to even just a single inner scenario. This computational task is then easily manageable by a standard computer. Alone, each single inner scenario will provide a poor estimate to the true liability value, but if we regress more than 1, of these single scenario estimates, a refined, more accurate liability valuation for each economic projection (outer scenario) can be achieved. Essentially, we are averaging over all the inner scenarios to obtain a more reliable liability valuation for each one in isolation. Applying the LSMC method to a VA product To investigate how the LSMC method performs in estimating the projected liabilities of complex insurance products, we will apply the technique to a variable annuity (VA) product introduced in [4], which is representative of many of the products currently being offered on the market. VA products have become very popular in Figure 1. Nested Monte Carlo simulation, single projection horizon Time Figure 2. Nested Monte Carlo simulation, multiple projection horizons Inner scenarios Figure 3. Regression method Outer scenarios (real-world) Inner scenarios (risk-neutral) Time step Outer scenarios : 1 2 Liability valuations October 29 45

3 Figure 4. American option simulation tree Simulate underlying Option valuation calculation Figure 5. One inner scenario regression Regress for continuation values, then use to get option value at T-1. Backward iterate through tree Figure 6. Sensitivity of regression estimate to one-year fund value 14, 12, 8, 6, 4, 2, Thousands Fund value at projected time (year 1) Est. from single inner trial (per outer trial) Regression est. the USA and Japan over the past 1 15 years, and despite recent setbacks, many believe this will extend to the UK and Europe in the foreseeable future. A general definition of a VA product is a unitlinked fund offering the choice of optional guaranteed benefits to the customer. Many VA products provide the policyholder with guaranteed periodic payments for a specific period, a feature commonly referred to as a Guaranteed Minimum Withdrawal Benefit (GMWB). Some recent VA products have offered this GMWB feature for the lifetime of the annuitant, and this is a product feature we will consider in the analysis to follow. Further details of the GMWB feature and its sensitivities are given in previous Life & Pensions articles in February 29 [5] and July/August 29 [6]. Variable annuity product for analysis Policyholder: 65-year-old male. GMWB level: 5% of guarantee base. Guarantee Base: Initially funded by policyholder premium. Annually ratchets up alongside an increasing fund value (up to a maximum annual increase of 15%) for first 1 years of policy. Fund Investment: Equity 6%, Bond 4%. 2 Fees: 1.25% of fund value, plus an extra 1% if guarantee active. 3 Results and discussion One-year projection. First, we will consider projecting the VA product liabilities ahead one year into the future. Naturally, the choice of which economic variables will act as state variables in the regression is crucial to success in estimating the future liabilities from year one on. The state variables chosen as being key to estimating the future liabilities were the fund value level (at year one) and the 1-year spot rate on the underlying economy. The Nominal Yield Curve model employed in our implementation was an (annual) Libor market model and the equity (tracker 1 If the LSMC method is used in multi-dimensional problems one may expect the number of basis functions (including cross-terms) to grow exponentially with an increasing number of state variables. Longstaff and Schwartz cite research which shows this number may not increase at such a rate, making the method robust to multi-dimensionality 2 With the bond term re-balanced back to 1 years on an annual basis 3 The policyholder can turn off this guaranteed withdrawal option, or lapse on the policy, canceling this extra guarantee charge. He/she could then buy an open market annuity or remain invested during income drawdown. We will assume a dynamic model for policyholder lapse rate based on fund levels and market conditions, which is a common approach in the industry. Alternative policyholder lapsation models could be used instead fund) asset was modelled to give lognormal equity excess returns with constant volatility. The basis functions we shall consider are simple polynomial powers, up to second-order, including their interaction. For clarity, let us explain in detail how the regression component of the method works in this case of a one-year projection. The value of the GMWB liabilities for the i th real-world scenario is approximated by: L i ) = a + a 1 ) + a 2 ) 2 + a 3 S i ) + a 4 S i ) 2 + a 5 )S i ) where ) and S i ) are the fund value and 1-year spot rate levels, respectively, at the end of year one (along outer scenario i). The a,..., a 5 are the regression coefficients determined by minimising the sum of the squared residuals in fitting against the single scenario liability estimates. Specifically, if y i is the liability calculated from the single inner scenario branching out from outer scenario i, we find the a,..., a 5 which minimise Σ n i =1 (y i L i ))2, where n is the total number of outer scenarios, in this case 1,. The LSMC regression estimates for the future liabilities at the end of year one are then given for each outer scenario i by substituting the values found for a,..., a 5 into the last given formula for L i ). In figure 6, the single inner scenario and corresponding regression estimates for the VA liabilities are plotted for increasing levels of year one fund value. This plot suggests that for lower levels of year-one fund value, the regression estimates for the liability values are slightly larger, as expected for this type of guarantee. If the fund value has fallen significantly by the end of the first year after annuitisation, it is more likely the fund value will be unable to meet the guaranteed income level at some stage early on in the product s lifetime. When this situation occurs, the insurer is then liable to meet this and all subsequent annual incomes to the annuitant out of its own reserves. A plot of the same data for increasing levels of 1-year spot rate is given in figure 7. This shows the future level of interest rate is a crucial factor in determining the regression estimate of the VA liabilities. The regression estimated liabilities increase with decreasing (expected) future interest rate levels, again as expected for this type of guarantee. If future interest rates are large, then any future incomes that must be met by the insurer s own reserves will be discounted by a greater amount. Furthermore, if interest rates are low, policyholder lapse is less likely, resulting in the guarantee rider remaining active on the product. 46

4 A regression surface further examines this dependence of the liabilities on the two explanatory variables. In figure 9, we give a plot of the regression estimated liability values as a surface varying with levels of 1-year spot rate and year one fund value. Now let us consider how accurate the least-squares regression is in approximating the VA product liabilities and their percentiles, which form the basis of capital requirements. We do this by selecting a few different percentiles of the regression distribution and carrying out a full Monte Carlo valuation using scenarios. Of course this process would not be feasible for all 1, outer scenarios, however it allows us to obtain a few highly accurate values for the liability distribution as a benchmark. The comparison of the percentiles of the regression estimates with their corresponding accurate (full Monte Carlo simulation) estimates is given in figure 8 and table 1. Studying this data shows the regression gives a reasonably good approximation across the full range of the liability distribution. The full simulation estimates are given as 99% confidence intervals and the results show that across the liability distribution the LSMC estimates lie reasonably close to these intervals. We must of course remember that the LSMC method is an approximation, thus it would be unexpected to get all the tested percentiles within the 99% full simulation estimate intervals. Naturally, through further investigation of the interest rate explanatory variables this accuracy could be improved upon, however there is enough evidence to conclude that the least-squares regression has been successful in capturing the behaviour of the VA product (projected) liabilities from the end of the first year. Five-year projection. Let us now consider projecting for the VA product liabilities ahead five years into the future. The state variables chosen as being key to estimating the future liabilities in this case were the year-five fund value level, the 1-year spot rate on the underlying economy at year five and the year-five guarantee base to capture the possible ratcheting up effect of the product guarantee base. The choice of which polynomial powers to take in each of the explanatory variables and cross-terms is more important in the success of the five-year projection than the one-year case and is non-trivial. However, by comparing the resultant regression estimates with a selection of target percentiles Table 1: Comparison of regression estimates with full MC simulation estimates. All estimates/variables at year one Percentile Fund value 1-year spot rate Full MC Value (99% C.I.) Reg. Est. 99th 67, % ( 6,33, 6,354) 6,223 95th 69, % ( 5,345, 5,393) 5,181 75th 53, % ( 4,28, 4,247) 4,258 Median 52, % ( 3,859, 3,897) 3,762 5th 61,1 6.2% ( 2,462, 2,496) 2,489 (which are accurately valued by full nested simulation), one can obtain a reasonable choice of basis functions with relative ease. In figure 1, the regression estimates for the future product liabilities are plotted against the levels of year five fund value. This shows the level of fund value at the projection time seems to have a significant effect in the estimated future liability after projection, and this relationship is the same as seen for the one-year projection. The future interest rate explanatory variable seems to be far less significant in the regression than in the one-year projection, but there does appear to be a slight decrease in future liabilities, as the measure of future interest rates increases. The guarantee base influence is less clear, as it is fairly highly correlated to the fund value level. This dependence is made clearer in figure 12, where the regression estimate of the future liabilities is plotted as a surface varying with both fund value and guarantee base levels at projection (for fixed levels of interest rate explanatory variable). This shows that when the guarantee base is large (but the fund value is somewhat smaller) the future liability to the insurer is likely to be large. This is obvious from the product structure, because a large guarantee base means the annual income level to the policyholder will be large, yet the fund value will be relatively small, so is more likely in the subsequent years to not be able to meet these large levels of income. The regression surface plot with varying fund value and interest rate variables (guarantee base fixed) is also informative. In figure 13, we see the estimated liability decreases with an increasing fund value and (less significantly) with increasing expected future rates of interest. Now let us consider how accurate the least-squares regression is in approximating the VA product liabilities. The comparison of the percentiles of the regression estimates with their corresponding accurate (full Monte Carlo simulation) estimates is given in figure 11 and table 2. Here we concentrate more on the approximation in the upper tail of the liability distribution, as is appropriate for capital requirements. Studying Figure 7. Sensitivity of regression estimates to interest rates 12, 8, 6, 4, 2, year spot rate at projection time (year 1) (%) Est. from single inner trial (per outer trial) Regression est. Figure 8. Sensitivity of percentiles to interest rates 12, 11, 9, 8, 7, 6, 5, 4, 3, Percentiles year spot rate at projection time (year 1) (%) Regression est. Full MC est. Percentile reg. est. Figure 9. Regression surface (one-year projection) (projected year 1) ( ) 12, 8, 6, 4, 2, 1-year spot rate (at year 1) (%) Fund value at projection time (year 1) in thousands October 29 47

5 Figure 1. Sensitivity of regression to fifthyear fund value 2, 15, 5, Upper tail of projected liability distribution Fund value at projection (year 5) (Thousands ) Regression estimates Figure 12. Regression surface (five-year projection) interest rate fixed (projected at year 5) ( ) 25, 2, 15, 5, 1-year spot rate fixed at 4.2% Fundvalue at projection (year 5) (Thousands ) 2, -25, 15, -2, -15, 5,- -5, Guarantee base at projection (year 5) 18,5 16,5 14,5 12,5 this data shows the regression gives a reasonably good approximation in the upper region of the liability distribution. The results show that across the upper tail of the liability distribution the LSMC estimates again lie reasonably close to these intervals. Further investigation of the interest rate explanatory variables could improve this accuracy further, but there is evidence to conclude that the least-squares regression shows promise in capturing the behaviour of the VA product s liabilities. Conclusion With the impending introduction of the Solvency II framework for the regulation of insurers in Europe, we are at the dawn of a new age in risk management and economic capital requirements. Over the coming years insurers will become more pro-active in the identification and measurement of possible risks. While the insurance sector prepares for this major change in their governance, it must also plan for the likelihood of a rapidly increasing demand for VA products. In the context of Solvency II, this means insurers must be ready to employ accurate and reliable VA liability Figure 11. Sensitivity of percentiles to fifthyear fund value Percentiles ,5 9 8, Fund value at projection (year 5) (Thousands ) All regression estimates Regression est. (specific percentiles) 14, 12, 8, 6, 4, 2, Full sim. estimate (specific percentiles) Figure 13. Regression surface (five-year projection) guarantee base fixed Guarantee base fixed at 13, Table 2: Comparison of regression estimates with full MC simulation estimates. All estimates/variables at year five Percentile Fund value Guarantee base 1-year S-R Full MC Value (99% C.I.) Reg. Est. 99.5th 33,429 7, % ( 11,241, 11,46) 11,262 99th 34,627 7, % ( 9,498, 9,79) 1, th 28,784 61, % ( 7,966, 8,136) 8,373 95th 31,68 61, % ( 6,924, 7,99) 7,89 9th 42,79 65, % ( 6,14, 6,2) 5,519 Fundvalue at projection (year 5) (Thousands ) 12,-14, -12, 8,- 6,-8, 4,-6, 2,-4, -2, year spot rate 3.5 at projection (year 5) (%) References 1 Oechslin, J et al Replicating Embedded Options, February 27 management and convince regulators they will be able to remain solvent under extreme future market conditions. Generating many different economic scenarios is one way to do this, but the calculation of the liabilities associated with these VA products is a challenge requiring simulation itself, making the natural approach computationally challenging. Thus, as insurers look to project their VA insurance liabilities over the coming years, they will need techniques that are accurate enough, but remain computationally tractable. The LSMC method is one possible approach, providing a simple method while giving a reasonably accurate approximation. With further research, this method could provide even greater levels of accuracy, while remaining computationally attractive. L&P Mark Cathcart is a PhD student at The Department of Actuarial Mathematics and Statistics, Heriot-Watt University. Steven Morrison is head of model research at Barrie & Hibbert. Mark.Cathcart@barrhibb. com and Steven.Morrison@barrhibb.com 2 Longstaff, F and Schwartz, E Valuing American options by Simulation: A Simple Least-Squares Approach Review of Financial Studies 14, pp (21) 3 Broadie, M and Glasserman, P Pricing American-style securities using simulation Journal of Economic Dynamics and Control 21, pp1,323 1,352 (1997) 4 Ledlie, MC et al Variable Annuities British Actuarial Journal (to appear). Available online at actuarial profession web-site, 5 Blamont, D & Sagoo, P Pricing and Hedging of Variable Annuities, February 29 6 Hobbs, C, Krishnaraj, B, Lin, Y and Musselman, J Calculation of variable annuity market sensitivites using a pathwise methodology, July/August 29 welcomes submissions to its peer-reviewed Cutting Edge section. Articles should be sent to laurie.carver@incisivemedia.com. Submission guidelines are available at technical-papers-submission-guidelines 48