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Linked Life Insurance Contracts with Guarantees CUTTING EDGE Guaranteed links to the life market This article addresses a frequent problem The approach requires the determination of the sensitivity of the option relevant for dynamically hedging unit- value to small changes in a range of risk factors. These sensitivities are called linked life insurance contracts with ‘greeks’ 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-volatility 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 financial market 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 short 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 capital market 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 www.life-pensions.com December 2007 39 technical_december.indd 39 5/12/07 10:41:08 CUTTING EDGE 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 moneyness (‘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. 40 Life & Pensions 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.
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