cutting edge Guaranteed links to the life market This article addresses a frequent problem The approach requires the determination of the sensitivity of the relevant for dynamically hedging unit- value to small changes in a range of risk factors. These sensitivities are called linked life insurance contracts with ‘’ and have specific names for each risk factor: Delta, for the first-order sensitivity towards the assets that are part of guarantees: the determination of the any underlying funds; hedge sensitivities in a numerically Rho, for first-order-interest rate sensitivity; Vega, for first-order- sensitivity; efficient manner. For this purpose we Theta, for first-order-time sensitivity; use a replication portfolio approach Gamma, for the second-order sensitivity towards the assets, which are part of the underlying funds – that is, the sensitivity to delta. by tigran kalberer Additionally there is a requirement that instruments that THE EUROPEAN life insurance market is currently being flooded by unit- show the same dependency on the risk factors as the value of the guarantee linked products with investment guarantees (ULG). While these products are available. It is assumed that these instruments can be sold. are relatively new to the European market, they are very well known in the If an amount of these instruments is short sold that exactly offsets the US and Canada, where they are called ‘variable annuities’. The design, pric- sensitivity of the value of the guarantee, at the end of a small period the ing and risk-management processes of these products present new challenges value of the hedge portfolio is still the value of the guarantee and the hedg- to the industry. Unfortunately the technology currently available does not ing can continue. This paper addresses the task of determining the provide solutions to all of the problems that these products can generate. sensitivities of the value of the guarantee over a whole portfolio of contracts, A popular risk management approach for these products is dynamic hedg- in an efficient way at each future point in time for a large range of possible ing. This approach is based on a stochastic simulation approach for valuing situations. liabilities, which typically is very time-consuming. Valuation of ULG products Unit-linked products with investment guarantees The current best practice approach for valuing ULG products is to use sto- The wide variety of ‘typical’ unit-linked life insurance contracts are well chastic simulation – that is, to produce a sufficiently large set of known throughout the industry. The policyholder pays a single premium or market-consistent scenarios , , describing all relevant market Xi i = 1 … n regular premiums. The insurance company deducts expense charges and risk parameters. If the number of scenarios is sufficiently large, the law of large premiums, and invests the remaining part of the premium in fund units – numbers applies and either mutual funds or funds created internally within the insurance  Z  1 n ZX() Value= EQ ℑ ≈ i company for this purpose.  0  ∑ N n i=1 NX()i The insurance company charges fees regularly to the funds account of the Here Z is the contingent cashflow at time T;N is a reference asset, called policyholder. Upon occurrence of a defined event – such as maturity, annu- Numeraire; ℑ is the information available at time 0; and Q is the so-called itisation or death – the policyholder receives the value of the funds account. 0 risk-neutral measure (see Kalberer (2006)). Alternatively, the funds account can be converted into a guaranteed tradi- tional annuity or another guaranteed benefit for the policyholder. Numerical problems in a dynamic hedging approach When determining the above-mentioned sensitivities (the greeks), two Dynamic hedging problems arise: Dynamic hedging is an approach where a hedge portfolio is purchased and actively managed in such a way that all shortfalls arising from an investment Problem 1: The calculation of the greeks usually requires repeated stochas- guarantee can be financed by this portfolio under all possible financial tic simulation for a large number of scenarios over a huge portfolio of market situations. contracts, potentially condensed into a considerable number of model points

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Figure 1. Value of the maturity guarantee at maturity Figure 3. Payoff of liability portfolio compared to payoff of the replication portfolio 1,000 1,000

800 all)

500 600 y (shortf 400 urit

alue of the guarantee 200

0 V at mat 0 0.5 1.0 1.5 2.0 2.5 3.0 Price of a funds unit 0 –100 0 100 200 300 400 500 600 700 800 900 Payoff of replication portfolio

Figure 2. Value of the maturity guarantee at maturity Figure 4. Hedge slippage and replication-portfolio hedge error in comparison 40 1,000

y 20 Value of maturity guarantee 500 Value of replication portfolio 0

0 e probabilit –20

ice of a funds unit Hedge slippage

Pr –40 Deviation replication portfolio to actual portfolio payoff

–500 Cumulativ Worst case: slippage and error added 0 0.5 1.0 1.5 2.0 2.5 3.0 –60 Price of a funds unit –80 0 200 400 600 800 1,000 Scenarios

in order to determine their market-consistent value and the sensitivities of The properties these replicating assets should possess are as follows: this value. This procedure is usually very time consuming. Their value should be easy to determine, preferably using closed-form solutions. This allows the determination of the value of the guarantees Problem 2: The scenarios used for this purpose have to be calibrated such with the minimum of numerical error but preserves the potential cali- that they are market-consistent – that is, they reflect the market prices of bration error. financial instruments, which are liquidly traded and where prices are availa- If possible, there should be market prices (in contrast to just model- ble. The asset models used for generating such scenarios typically do not prices). If this is possible, the calibration error can also be minimised. adequately allow for the reflection of all features of the market prices of financial instruments. Features such as the dependency of volatility on time The idea to replicate complex and possibly large portfolios of liabilities using and (‘smile’) are difficult to reflect in the calibration of the sce- replication portfolios is an old one, but only recently has there been widespread narios used for valuation. This effect can lead to significant errors in valuing application of this process within the insurance industry. ULG products, sometimes referred to as ‘model error’. An introduction to this approach, which focuses on traditional life busi- ness, is given by Oechslin et al (2007). A potential solution: The replication portfolio approach The approach is based on a list of candidate assets, which have the proper- Both problems could be solved by using a replication portfolio approach to the ties described above, and on finding the linear combination of these assets, valuation of ULG products. The underlying idea is very simple and the accu- which replicates the given liability cashflows in an optimal way – for exam- racy of this approach can be quantified. In most cases, the replication approach ple, by minimising the average deviation between the cashflows of the gives very accurate approximations for the required calculation results. liabilities and the replication-portfolio measured using a convenient metric, such as least squares. The replication portfolio In most cases the weights of the optimal portfolio can be determined using The replication portfolio of a set of liabilities is defined as a portfolio consist- linear algebra, as also described in Oechslin et al (2007). ing of potentially fictitious assets (candidate assets) with certain properties that generate very similar cashflows at all points of time under all possible Replication portfolios for ULG-products investment scenarios. For some purposes it is only necessary to replicate the For illustrative purposes, I focus on path-independent guarantees in the market value and its dynamics at a certain point in time. following example.

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technical_december.indd 40 5/12/07 10:41:13 Table 1: Optimal linear combination of candidate assets points of time and the cashflows. This would be slightly harder to visualise but the same principles would apply. In these cases, path-dependent candi- Strike Notional date assets would have to be chosen. 0.2 0 0.4 39.20392 The choice of candidate assets 0.6 133.0466 Determining the candidate assets is the real challenge of applying the repli- 0.8 218.4072 cation portfolio approach and requires high levels of skill and experience. The example used here is a very simple one and in reality the features of the 1 282.6702 guarantees would be far more complex – for example, they would include 1.2 204.7578 path dependency. This does not mean that the approach no longer works. It 1.4 82.24168 just implies that the candidate assets used in determining the replication 1.6 34.89139 portfolio as linear combination of these candidate assets, may be more com- 1.8 10.96966 plex also – that is, these also include path-dependency. There are several approaches that can be taken in order to determine these 2 –0.60765 candidate assets, including: Analysing the relative importance of each potential candidate asset from Figure 1 examines ULG contracts that mature in a certain point of time t, a large list of potential assets in the first step, and then only focusing on where the maturity payment depends on the same underlying funds’ value the most important ones in the second step, as a means of of avoiding at t only. This implies that there is no dependency on either the path or any over fitting; other financial instruments (aside from the underlying ones). The overall Using a priori knowledge of the liabilities to choose appropriate candi- pay-off generated by the guarantees of the contracts will be a decreasing date assets – for example, floating strike look-back options to reflect function of the funds value. As figure 1 illustrates, this looks surprisingly certain advanced features of ULG products. smooth. For this and the following examples I have used a sample portfolio of Typically these candidate assets are based on total-return indices in order to 1,000 single premium ULG-contracts with varying guarantee-levels and avoid risk. Gp volumes , a term of 10 years and make the initial unit price = 1. Vp The pay-off function is The importance of asymptotic behaviour When estimating sensitivities, it is very important that the replication port- folio has the same asymptotic behaviour concerning extreme capital market VG⋅max − FV, 0 ∑ p ()p situations as the pay-off of the liability portfolio of guarantees. policies p For the ULG products under consideration, it is easy to determine the Where FV is the funds value in year 10. value of the guarantees – as the unit price goes to infinity, it is zero. The The graph contains all the information necessary to describe the pay-off of value of the guarantees as the unit price goes to zero is as easy to determine; the guarantee over the entire liability portfolio. If this graph is successfully it is simply the sum of the guarantees. This asymptotic behaviour can be approximated by a linear combination of simple functions (dependent on enforced by adding additional constraints to the optimisation process. the same underlying), it is possible to represent a potentially large portfolio Using put options as candidate assets automatically ensures that the value of of contracts (1,000 in this case) by a small set of functions. replication portfolio approaches zero as the unit price increases. An obvious candidate to approximate the graph is a set of plain vanilla put options on the underlying with a 10-year term and a range of different strikes. Advantages of the replication portfolio approach This example chooses strike prices 0.2, 0.4 and so on, up to 2. The strike To illustrate this approach, it is assumed that the dynamics of the funds’ prices are chosen such that they cover the area where the pay-off function prices follow a simple Geometric Brownian motion with drift µ = 3% and needs to be approximated, here in the range between 0 and 2. The optimal volatility σ = 20% per annum. linear combination of these candidate assets is shown in table 1.  S  1 t+1 =µ − σ2 dt + σ dW The approximation given by these instruments can be visualised as shown ln   () t  St  2 in figure 2. The optimal replication portfolio for our purposes is defined as the linear combination of the candidate assets minimising the squared devi- The closeness of fit of the replication portfolio can be illustrated by plot- ations between the pay-off function and the approximating replication ting the value of the replication portfolio against the pay-off generated by portfolio for a sufficiently large number of scenarios. Depending on the the guarantee for a number of stochastic scenarios based on this process purpose, it might be necessary to choose another metric for this optimisa- (1,000 in this case), as shown in figure 3. tion. It is important to note that this approach uses the results produced for The standard deviation of the difference of the pay-offs (replication portfolio the purpose of pricing – that is, the scenarios used for pricing, building on minus liabilities) is about 1.3 – that is, 0.13% of the aggregated premiums. these results in a natural way. The value of the cashflows is: If the liability cashflows were path dependent then there would be multi- Exact value, calculated contract-by-contract using a closed-form dimensional dependencies between the different risk factors at different solution: 124.62

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Table 2: Comparison of sensitivities derived from a range of approaches

Sensitivity Exact Bumping the model Replication portfolio approach Base case 124.62 125.46 124.66 Delta, Initial funds price +.01 –214.10 –219.06 –214.08 Delta, Initial funds price +.1 –196.49 –201.40 –196.47 Delta, Initial funds price +.5 –139.55 –142.90 –139.55 Rho –3,370.26 –3,427.90 –3,370.36

Value determined using above mentioned market-consistent scenarios: Determining sensitivities (greeks) 128.33 The replication portfolio approach can also be used to determine the sensi- Value of the replication portfolio, determined using closed-form solutions: tivities of the value of the liabilities to changes in the underlying economic 124.66 variables, the so-called ‘greeks’. The most popular method of determining these sensitivities is known as An introduction into stochastic simulation in an insurance environment is the ‘bumping the model’ approach where the value of the liabilities is deter- given in Kalberer (2006). mined for the current value of the variable and a ‘shocked’ value. However, this approach has severe disadvantages: The control-variate approach The sensitivities estimated this way usually have a large estimation error; The difference between the liability portfolio and the replication portfolio Determination of the sensitivities is time consuming, as the stochastic can be considered as a new asset, the ‘difference asset’. The pay-offs of this simulation has to be performed repeatedly for each sensitivity, and for asset have a considerably lower volatility than the pay-off of the liability each model point. portfolio itself (in our example: the above mentioned 1.3 for the ‘difference asset’ versus 210 for the liability cashflow). Using the replication portfolio approach to produce Thus, the estimation of the value of this difference asset using stochastic sce- sensitivities narios has a considerably higher degree of accuracy than the direct estimation of It is relatively straightforward to use a replication portfolio approach to the value of the pay-offs. In this example, the estimated value of the new differ- determine the sensitivities. After the replication portfolio has been deter- ence asset is –0.02 at time 0 (in other words, the average of the discounted mined, its sensitivity can easily be determined, as the replication portfolio is difference of the cashflows of the replication portfolio and the liabilities). based on instruments that have closed-form solutions for their values and Therefore, the value of the liability portfolio can be determined with very usually also for their sensitivities. low stochastic error as: In the example, the sensitivities derived by the different approaches can be Value of replication portfolio (closed form solution available, thus no sto- compared, the results of which are shown in table 2 (where 1,000 scenarios chastic error) + value of ‘difference-asset’ (low stochastic error due to low were used). variance of pay-offs) in this example: 124.66 – 0.02 = 124.64, much closer In this instance, we did not use the control-variate approach to produce to the exact value. the ‘bumping the model’ sensitivities. The replication method shows a This effect can be explained by the central limit theorem, which shows that superior degree of accuracy compared to all other approximation lower variance leads to higher accuracy. approaches. It involves considerably less computation time than the bump- This approach is called the control-variate approach and is widely used in ing the model approach. determining the value of contingent payments using stochastic simulation. If the replication portfolio approach were not used for anything else but as Error bounds for the replication portfolio approach control-variate, this alone would justify its determination. In order to be able to use this method in a reliable manner, the error bounds Using the replication portfolio as the control variate requires considerably need to be established for the sensitivities that are determined using the fewer scenarios for estimating the value of the guarantees, while preserving the replication portfolio approach. To estimate the error potentially introduced level of accuracy. Alternatively, the level of accuracy can be considerably increased by this approach, two methods are discussed below: while using the same number of scenarios as under the naive approach. If it is possible to determine direct market values for the assets of the replica- Error bounds for estimating sensitivities using the tion portfolio then the calibration problem can also be addressed. In fact, the likelihood-ratio approach potential model error introduced by inadequate calibration impacts the stochas- The likelihood ratio approach uses re-weighting of the scenarios in order to tic valuation of the difference asset only, which, by construction, is small. The determine the value of a cashflow under changed assumptions. valuation of the replication portfolio is usually exposed to far lower model error, Let Q denote the probability measure based on the current calibration and as the assets in the replication portfolio can be valued directly in the market. let Q´ denote the probability measure for a calibration where the economic So the impact of both the above mentioned problems is substantially variable has been changed by a small amount. Let dQ and dQ´ denote the reduced. associated probability densities and assume all necessary requirements on

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technical_december.indd 42 5/12/07 10:41:16 dQ, dQ´ are fulfilled, such that: of the approximation of the expected value by stochastic simulation to the real expected value is sufficient such that we can assume that the error is really normally distributed with the empirical volatility as standard deviation. Q′  Z  Q  Z dQ′ E   =E  ⋅  .  N   N dQ  Large movements in the risk factors, such as the funds price in our exam- ple, generate too few scenarios to guarantee that the results of the central This result can be used to estimate the error produced, by determining limit theorem and law of large numbers apply. the sensitivities using the replication portfolio approach compared to the One indication for this is that exact sensitivity. Exact sensitivity – estimated sensitivity (via replication portfolio) = n 1  dQ′()Xi  ∑ ⋅   n i=1  dQ ()Xi   R ′ R   Q  Z  Q′  Z  Q  Z  Q  Z   E   − E   −  E   − E   is considerably smaller than 1 in such situations. The approach described   N   N    N   N  above for estimating error bounds is not applicable in these situations. The  R  Q  Z  dQ′  Q Z  dQ′  effect of this is important for one of the potential applications mentioned =E  ⋅ −1 − E  ⋅ −1  N  dQ   N  dQ  below. However, the approach is perfect for just estimating sensitivities from  − R  a given set of scenarios, as very small shocks are required, which is exactly the Q ()ZZ  dQ′  = E  ⋅ −1 case when the approach to estimate error bounds works well.  N  dQ  where ZR is the cashflow as generated by the replication portfolio. Do we need error bounds? This error term is now estimated by using the empirical estimator for this A more robust approach is to estimate the impact the potential error has on error term, using the already generated results for the stochastic scenarios: the overall hedging process. In the end, this is what matters and this approach can be used easily for path-  − R  dependent options and options dependent on several economic variables. Q ()ZZ  dQ′  E  ⋅  −1 A so-called ‘hedge assessment’ is performed for this approach. This is a task  N  dQ    that would have to be prepared in order to assess the quality of the dynamic N − R hedge process anyway. 1 (ZT(), XZi ()TX, i )  dQ′()Xi  ≈ ∑ ⋅  −1 For this purpose, a sufficiently large set of scenarios is generated under the N = NT(), X  dQ()X  i 1 i i physical measure (that is, using the best estimate ‘real world’ probabilities, including risk-premiums) with sufficient granularity, which in most cases The variance of this estimator can also be approximated. This allows error means on a daily basis. The aim is to simulate the effect of the hedge opera- bounds for any required level of security to be estimated using the law of tions along these scenarios, measuring the shortfall at maturity. The shortfall large numbers. is defined as the difference between the amount necessary to fill up the If the replication portfolio is not determined using the scenarios to deter- fund’s value to a potentially higher guarantee level and the value of the mine the error bound, the estimator for the error is unbiased. hedge portfolio at maturity. The likelihood ratios dQ’/dQ should be determined using the analytical Due to the fact that the hedging instruments used are typically linear instru- formulas for the distributions, if available. The economic scenario generator ments and the change in value of the liabilities is non-linear, a hedge slippage producing the asset scenarios for the stochastic valuation is typically based can be observed. The accrued sum of all these hedge slippages at maturity is on stochastic differential equations, which can be used to determine the the total hedge slippage for each scenario. The hedge slippages for all scenarios likelihood ratios. can be used to derive an empirical distribution of the hedge slippages. However, caution must be exercised, as the random variables could be mul- If an initially fitted replication portfolio is used to determine the sensitivi- tivariate distributed, meaning that the pay-off could depend on more than ties and greeks at each future point in time for each scenario, instead of one economic variable. ‘properly’ calculated sensitivities, then the error produced by this approach For our example, the error bounds for a 99% level of security (for estimat- can be estimated very easily. In fact, it can be assumed that the exposure of ing the error as above) are astonishingly small: the replication portfolio is hedged and the difference between the pay-off of Error estimate (exact sensitivity – estimated sensitivity (via replication- the hedge portfolio and the pay-off implied by the guarantee is exactly the portfolio)): –0.023; error introduced by using the replication portfolio. This error can be deter- 1%/99% quantile of the error estimate: ±0.022 around the mean, assum- mined for the scenarios considered, which in turn can be regarded as an ing the law of large numbers is applicable. empirical estimate of the error. In addition to this error, the hedging process will not work perfectly and Considering that the exact delta is 214.1, this is a remarkably small error. produces various losses and gains, mainly because of hedge slippages, and It could be argued that this article could finish here, as this approach seems basis risk. Typically the underlying funds are actively managed and cannot to work so nicely. be short sold, so for hedging purposes they have to be replicated using The answer is that, so far, the law of large numbers has been used to estimate market indices. This approach gives rise to basis risk. error bounds. At this point, it is not known beforehand when the convergence In my experience, the error introduced by hedging the replication portfo-

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lio will be negligibly small compared to the hedge slippage itself. If a hedge case can lead to unreliable estimations of the error bonds and convergence assessment is performed for the sample portfolio, assuming a weekly delta problems; The stochastic simulation and the determination of the repli- hedge, the result seen in figure 4 is produced. cation portfolio has to be repeated if the risk-factors move too far away The graph shows the P&L from the hedging operations and the error from the base-case or if the liability portfolio has changed, at least implied by using the replication portfolio per scenario. The biggest losses are monthly to the left. Perform hedge assessment calculations. Performing a hedge assessment The fact that the replication portfolio was hedged instead of the actual based on a full stochastic approach is not always feasible, as discussed portfolio does not add a significant amount of risk. The hedge slippage itself above. The main problem is that this is a nested stochastic approach and has a far bigger magnitude than the error introduced by the replication port- needs considerable run-times. Performing the hedge assessment based on folio approach. Owing to tracking error, in most cases, the basis risk alone is the replication portfolio is a very efficient alternative and sufficient to far bigger than the deviation caused by using the replication portfolio. This discuss the main issues. justifies the use of the replication portfolio approach. Of course, in this example, the hedge strategy is simplified, in that we Whether any of these approaches are applicable depends on the nature of assume a weekly delta hedge only. In reality, a more frequent hedge, poten- the ULG-products. There is no general rule that will give advanced indi- tially using non-linear hedging instruments, would be used, decreasing the cation as to whether the approach works reliably. However, using the hedge slippage considerably. But on the other hand, the hedge slippage in techniques presented in this article, it should be possible to decide this example was calculated without taking any volatility risk into account, whether the approaches work and how accurate they are once they have which would increase the hedge slippage. been applied. The advantage is that the replication portfolio approach is much simpler to The advantages of these approaches are so huge that the possibility that handle sensitivities – that is, greeks can be determined based on closed-form they could work always justifies exploring whether they are applicable. solutions within mere fractions of a second, in comparison to the full sto- chastic approach, “bumping the model”, which needs hours, if not days, of Summary computation time to determine the greeks. Implementing a dynamic hedging approach for unit-linked life insurance This approach in determining the error of the replication portfolio approach products with investment guarantees presents large technical challenges. has the following advantages: The determination of the sensitivities and greeks can be especially time- It measures the impact of a potential deviation in terms of risk and not consuming and involve considerable estimation error. To perform the the sensitivity itself; tasks implied by a dynamic hedging scheme, it is suggested that a replica- It compares the replication approach error to other sources of risk, which tion portfolio approach is used. This not only increases speed and are usually much bigger and thus prevents overly complex approaches reliability of the computations but also increases accuracy and removes delivering spurious accuracy; calibration error. L&P It is a method to perform a proper hedge assessment without the need of nested stochastic calculations, saving an immense amount of computation Tigran Kalberer is a principal at Towers Perrin. time. E-mail [email protected]

Appropriate experience with this approach is required to determine the can- I would like to thank Manuel Sales and Carole Bozkurt for their review, and didate assets and to prevent nasty surprises concerning approximation errors. Jo Oechslin for his review, excellent input and the very valuable discus- However, as demonstrated above, this approach is worthwhile to consider, as sions about the contents of this paper. the actual hedging application can be simplified and sped up significantly, allowing for additional checks and risk-measurements on a daily basis.

Applications of the replication portfolio approach References The approach can be used for three different purposes: Kalberer T (2006) To determine the daily sensitivities and greeks based on one stochastic Market consistent valuation of insurance liabilities – Basic approaches and tool-box base-run, by performing the base-run for liabilities and all candidate Der Aktuar 12 (2006) Heft 1 assets; determining the replication portfolio; determining the sensitivi- Oechslin J et al (2007) ties based on the replication portfolio; and increasing the accuracy of the Replicating embedded options base-run using the replication portfolio approach. As previously men- Life & Pensions, February 2007 tioned, this approach should work in most circumstances and there are Hull J good error bounds. Options, futures and other derivatives To perform stochastic base-runs only if necessary and use the replication Prentice-Hall, 2000 portfolio in the meantime. This would include the following factors: The daily hedging routine will be based on the replication portfolio as as the risk factors do not move too far away from the base-case used for determining the stochastic simulation; Large deviations to the base-

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