Valuation of guarantees on maturity values for unit linked life insurance policies with future payments

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values for unit linked life insurance

policies with future payments

K.M.C. Dijkshoorn

Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam

Faculty of Economics and Business Amsterdam School of Economics Author: K.M.C. Dijkshoorn Student nr: 5886708

Email: kellydijkshoorn@hotmail.com Date: December 16, 2014

Supervisor: Prof. Dr. Ir. M.H. Vellekoop

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Abstract

For valuating unit linked life insurance policies with guarantees on maturity values literature agrees that the Black-Scholes Option Pricing Model is a useful valuation method if no future payments are involved. If there are periodic premiums, literature states that the guarantees at maturity can be defined as a series of Asian put option. In this case the Edgeworth Series Approximation is a useful valuation method. Literature considers the valuation of policies with premiums for death risk independent of the fund value. However, no literature considers the valuation of policies with fund value dependent risk premiums. Since the risk premiums are an important part of the contract considered in this study, it is not possible to valuate these policies with an analytical approach. However, the Edgeworth Series Approximation (Jori, 2008) could be further examined, despite the difference in dealing with the risk premiums.

A specific valuation formula considered by a Dutch insurer is based on applying the Black-Scholes formula on the current fund value and every future payment separately. It turns out that this formula has several disadvantages. Since the formula values future payments separately, there is an overestimation of the expected loss on maturity. Furthermore, this formula is not able to deal with risk premiums that depend on the fund value. The considered bad fund return paths in case of Monte Carlo simulates make that the expected loss based on Monte Carlo simulations is higher if the risk premiums are based on a constant death benefit, or lower if the risk premiums are based on a variable death benefit. This results for the considered valuation formula in an underestimation or

overestimation of the expected loss, respectively. Another shortcoming of the considered valuation formula is that it cannot value negative payments which can occur. Therefore this study concludes that the only useful alternative valuation method for the policies considered is Monte Carlo simulations. However, this numerical approach can be time consuming.

Keywords Black-Scholes, future payments, guarantees, life insurance policies, maturity values, Monte Carlo, periodic premiums, risk premium, valuation, unit linked

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Preface

On June 10th, 2014, I started with my internship at a.s.r. Here I’ve been working on my thesis for over half a year. I had my ups and downs, but thanks to the following persons it has been a journey I will never forget.

First of all I thank my supervisors Rob Bruning, Frans Pommer and Michel Vellekoop for their patience, knowledge and support. I also thank Francis Hendriks, Jantine Clement and Vladimir Kulikov for helping me with my thesis. I thank a.s.r. and Rob Kaas for the great internship and I thank all my colleagues for the safe and friendly atmosphere they provided.

Furthermore I thank my parents and sisters for always supporting me. I thank my friends Louana and Ineke, with whom I shared my thesis experiences. Finally, I thank my love, Joey, for always being there for me.

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Introduction

1.1 Background

Option pricing has always been of great interest both to the academic and financial world. Many models are built in the past decades. One well-known model became a breakthrough in the field of accurate option pricing: the Black-Scholes Option Pricing model. This model is frequently used, because of the advantages it offers. However, one should always be careful in applying the Black-Scholes model in specific situations. One of the major insurers in the Netherlands (a.s.r.) has unit linked life insurance policies with a guarantee on the maturity benefit and future payments (UL policies) in its portfolio. The future payments include premiums, costs and fund value dependent risk premiums. The UL policies offer a minimum guaranteed amount on maturity. To value the guarantee, Monte Carlo simulations are possible, but the insurer considers an analytical formula based on the Black-Scholes model. If there are no future payments involved, applying this valuation formula does not give any problems. In reality there are future payments, i.e. (risk) premiums and expenses, between the valuation date and the maturity date of the contract. However, it looks like the formula does not function well if future payments are involved. Because of the possible shortcomings of this valuation formula, this study examines this formula, as well as alternative valuation methods to value the UL policies.

1.2 Research Question

In this study the following research question is examined:

”What is the best method to value the guaranteed amount on maturity of unit linked life insurance policies if future payments are involved?”

To answer this question the following sub-questions are answered as well:

1) What does literature say about valuation of policies with a guaranteed amount? 2) Does literature provide useful alternative valuation methods to value policies if future payments are involved?

3) What are the specifications of the unit linked life insurance policies and, in particular, the nature of the guarantee examined in this study?

4) Is the valuation formula the insurer considers sufficient to value the guarantee on the unit linked life insurance policies, especially if future payments are involved?

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2 K.M.C. Dijkshoorn — Valuation of guarantees on maturity values

1.3 Towards a Solution

The answer on the research question of this study is based on a literature review and on the results from the valuation formula the insurer considers.

The literature is reviewed that describes valuation methods for pricing insurance con-tracts with a guaranteed amount. Of the unit linked life insurance policies considered in this study the specifications are given, as well as the details on the nature of the guaranteed amount.

The losses on maturity are based on the maximum of the difference between the guaranteed amount and the fund value on maturity, and zero. The establishment of the fund value for each month is described in detail.

To test if the considered valuation formula is sufficient to value the UL policies, the present value of the losses based on this formula are compared to the present value of the losses based on Monte Carlo simulations. The application of the considered valuation formula and of the Monte Carlo simulations are described in detail. The considered valuation formula needs to give approximately the same results as the Monte Carlo simulations in order to be sufficient to value the UL policies.

For the results several tests are analyzed. Each test is based on a different selec-tion of variables (premiums, expenses). This way the results show the influence of each part of the unit linked life insurance policy on the performance of the considered valu-ation formula. The numerical results of the study are supported by the findings of the literature.

1.4 Outline of the Thesis

Chapter 2 contains a literature review. The insurer considers to value its UL policies with analytical formula based on the Black-Scholes model. To get a better understanding of this considered valuation formula, in the first two sections the history of the Black-Scholes model is described as well as the working of the model. Alternative valuation methods for valuing UL policies with single or periodic premiums are given in the last section of this chapter. This section ends with the discounting of the losses on maturity if Monte Carlo simulations are used, i.e. with deterministic or stochastic discount factors. The first section of Chapter 3 describes the details of the UL policies of the insurer and gives the assumptions of the used data. The second section describes the formula to calculate the fund value on a monthly basis. To determine the loss on the maturity date of the policy, the fund value is compared to the guaranteed amount on maturity date. Chapter 4 evaluates the used method in this study. The first section focuses entirely on the application of the analytical formula the insurer considers to value the guarantee on maturity of UL policies. The second section describes the application of Monte Carlo simulations, which are used to examine if the considered valuation formula is sufficient to valuate the UL policies. Chapter 5 continues with the analysis and the results of the valuation of the UL policies, with the focus on the valuation of future payments. Several tests are performed, each based on a different selection of variables (premiums, expenses). To each new test more variables are added, which makes the valuation more complex per test. Section 5.1 gives the specifications of each test and gives an explanation of the representation of the results. Section 5.2. gives the results of the considered valuation formula compared to the results of Monte Carlo simulations, based on the several tests. For the test where periodic premiums are involved, the Monte Carlo simulations are adapted to the considered valuation formula, to get a better understanding of the impact of the considered valuation formula. Section 5.3 gives a summary of the results. Chapter 6 gives the summary of the research, including the conclusions and the answers on the sub-questions and research question.

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Literature Review

This study focuses on the valuation of unit linked life insurance policies with guarantees on maturity values and future payments. As stated in the Introduction, the insurer considers to valuate the guarantee on the UL policies with an analytical formula based on the Black-Scholes model. To give a better understanding of the Black-Scholes formula in general, the following two sections describe the history and the specifications of the Black-Scholes model. The last section gives valuation methods for policies with a guaranteed amount, based on a single premium or periodic premiums.

2.1 History Black-Scholes Model

Over a century ago the foundation of the Black-Scholes model was already visible. For decades many academics have contributed to the eventual form of the Black-Scholes model.

It all started in 1900, when the French mathematician Louis Bachelier, at the time a postgraduate student (Vassiliou, 2010), introduced with his PhD thesis “Th´eorie de la Sp´eculation” (Bachelier, 1900) a stochastic process to evaluate stock prices. Based on this stochastic process and the Central Limit Theorem, Bachelier (1900) assumes that the price of an underlying asset is normally distributed. This assumption arises since Bachelier does not take into account the effect of compounding on the distribution of stock returns (Kritzman, 2000). Although normally distributed underlyings allow nega-tive asset prices, Bachelier’s stochastic process, now known as the arithmetic Brownian motion, is still useful in specific situations (Dowd, Blake, Cairns & Dawson, 2006). The findings of Bachelier are remarkable and make Bachelier the pioneer of Mathematical Finance (Vassiliou, 2010).

Unfortunately for Bachelier, after successfully defending his PhD thesis, he didn’t get the recognition he deserved (Bernstein, 1992; Ruppert, 2004). None of his superiors knew where to place his findings, since the covered topic by Bachelier was so different from the topics studied by other mathematicians. It took over 50 years before Bachelier’s work was discovered accidentally by the mathematical statistician Jimmy Savage in the University of Chicago library. He shared his discovery with, among others, Paul Samuelson, Professor of Economics of the Massachusetts Institute of Technology (MIT). Samuelson was at the time really interested in the valuation of warrants and options and was impressed by Bachelier’s work. He suggested MIT PhD student Paul Cootner, who was working on an edited volume of worthy original finance papers (Cootner, 1964), to add the English translation of Bachlier’s thesis (Ruppert, 2004; Davis & Etheridge, 2006).

Before the publication of Samuelson (1965), two other mathematicians published interesting studies concerning warrant and option pricing. Sprenkle (1961) assumes in his study a log-normal distribution for the price of the underlying asset, instead of the

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normal distribution of Bachelier. He includes a term for the growth rate of the price of the underlying asset (Kritzman, 2000). However, the derived formula of Sprenkle (1961) still contains some arbitrary parameters, which he wasn’t able to estimate (Bellalah, 2010). Boness (1964) discovered the importance of the time value of money. To get the present value of the underlying asset, he discounted the value on maturity with a discount factor based on the expected return of the underlying asset (Kritzman, 2000). Samuelson (1965)1 who based his study on the findings of Bachelier, went a step further than Boness (1964). His option formula allows the expected return on the un-derlying asset and the expected return on the option to vary (Kritzman, 2000). He assumed that both expected returns were known and constant trough time. However, in real life the assumed expected returns are unknown and most unlikely to be constant (Mehrling, 2005).

In a following paper (Samuelson & Merton, 1969), that Samuelson wrote together with his student Robert Merton, the option price is treated as a function of the stock price (Bellalah, 2010). This paper makes the assumption that the option value depends on the risk preference of the investor (Kritzman, 2000).

The young researchers Fischer Black and Myron Scholes read the paper of Samuelson & Merton (1969). First they didn’t tell Samuelson and Merton about their own progress. However, in the autumn of 1970, Scholes talked with Merton about his work with Black. Merton recognized the importance of their work and he was the one who eventually named the model the “Black-Scholes Option-Pricing Model” (Mackenzie, 2006; )

In 1973 Black & Scholes published their paper (Black & Scholes, 1973). They used the empirical formula for warrants presented by Thorp and Kassouf (1967) to develop their model. This formula was used to calculate “the ratio of shares of stock to options needed to create a hedge position”(Black & Scholes, 1973, p. 640). However, Thorp and Kassouf (1967) didn’t pay attention to the equilibrium condition, which says that “in equilibrium, the expected return on such a hedged position must be equal to the return on a riskless asset” (Black & Scholes, 1973, p. 640). Black & Scholes (1973) use this equilibrium condition for the derivation of their theoretical valuation method, i.e. the Black-Scholes Option Pricing Model. Since this valuation method of Black & Scholes is based on important findings of Merton as well, this famous method for the pricing of options is known as the Black-Scholes-Merton Option Pricing Model as well (Mackenzie, 2006).

2.2 Specifications Black-Scholes Model

With the Black-Scholes model designed by Black & Scholes (1973) the goal was to let the option value solely depend on the price of the underlying asset, with the other variables constant and known. To establish this, certain important assumptions were necessary, the so-called “ideal conditions” (Black & Scholes, 1973, p. 640) in the market for the underlying asset and option (see for details Appendix A).

The Black-Scholes model is based on the current price of the underlying asset2 St, time t, volatility σ of the price of the underlying asset and the risk-free rate of interest, which are all independent of risk preferences. It is used to value European call and put options, which are options that can only be exercised on the expiration or maturity date T (see Appendix A). This date is specified in the option contract, just as the exercise or strike price K, the price for which the option can be exercised on maturity (Hull, 2012).

1

Samuelson won in 1970, as the first American, the Nobel Prize for economics (Nobel Prize, 1970)

2

Note that the value of the underlying asset is never negative. If the value is zero no investor wants to buy the stock, even if the price is extremely low.

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2.2.1 Derivation of Prices of European Options

A European call option, with payoff C, gives the right to buy an asset on maturity date for the strike price as stated in the option contract. It should only be exercised on maturity if the price of the underlying asset is above the strike price:

C = max(ST − K, 0) (2.1)

A European put option, with payoff P, gives the right to sell an asset on maturity date for the strike price as stated in the option contract. It should only be exercised on maturity if the price of the underlying asset is below the strike price:

P = max(K − ST, 0) (2.2)

The prices of European call (c) and put (p) options based on the Black-Scholes model are calculated as follows (Hull, 2012, 313):

c = S0N (d1) − Ke−rTN (d2) (2.3)

p = Ke−rTN (−d2) − S0N (−d1) (2.4) with e-rT the continuously compounded discount rate on maturity T based on the con-tinuously compounded risk-free rate r.

N(d), with argument d1 or d2, is the probability that a variable with a standard normal distribution (i.e. mean 0 and standard deviation 1) will be less than d, i.e. N(d) is the area under the standard normal density function from -∞ to d, with N(-∞) = 0, N(0) = 1₂ and N(∞) = 1, : N (d) = √1 2π Z d −∞ e−12s2ds (2.5)

The arguments d1 and d2 are calculated as follows:

d1 = ln( S0 K) + (r + σ2 2 ) ∗ T σ√T (2.6) d2 = ln( S0 K) + (r − σ2 2 ) ∗ T σ√T = d1 − σ √ T (2.7)

To assure that no arbitrage opportunity (i.e. possibility to make a risk-less profit) is possible, the call and put prices of a European option got upper and lower bounds.

The holder of a call option has the right to buy one share of the underlying asset for a specific price, the strike price K. Therefore the option value can never be higher than the underlying asset. Hence the upper bound of a call option is the price of the underlying asset S0.

A lower bound for the call price of a European option on a non-dividend-paying stock is S0 - Ke-rT (see Hull, 2012, p. 220). The worst scenario is that the call option expires worthless, which gives a price of 0. Therefore the bounds of the price of a call option are as follows:

max(S0− Ke−rT, 0) ≤ c ≤ S0 (2.8) The holder of a put option has the right to sell one share of the underlying asset for a specific price, the strike price K. Therefore the option value on maturity can never be higher than K. Hence the upper bound of a put option is the present value of K.

A lower bound for the put price of a European option on a non-dividend-paying stock is Ke-rT _{- S}

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expires worthless, which gives a price of 0. Therefore the bounds of the price of a put option are as follows:

max(Ke−rT − S0, 0) ≤ p ≤ Ke−rT (2.9) An important relationship between the call and put prices of European options, which have the same strike price K and time to maturity T, is the put-call parity (see Hull, 2012, p. 222):

c + Ke−rT = p + S0 (2.10)

The put-call-parity shows that the price of a European call option and the price of a European put option can be derived from each other, if the have the same specific strike price and maturity date.

2.2.2 Stochastic Process used for the Black-Scholes Derivation

A stochastic process which can be used if the price of an underlying asset is based on a log-normal distribution is the geometric Brownian motion (Dowd, Blake, Cairns & Dawson, 2008). The geometric Brownian motion is the basis of the derivation of the Black-Scholes model (Brewer, Feng & Kwan, 2012). The discrete-time version of the geometric Brownian motion looks as follows (Hull, 2012):

∆S S = µ∆t + σ √ ∆t (2.11) or ∆S = µS∆t + σS √ ∆t (2.12)

with epsilon i.i.d. Gaussian stochastic variables

√

∆t = ∆W (2.13)

such that the variable W follows, in the limit for ∆t to zero, a Wiener process or Brownian motion. The ∆S is the change of the price of the underlying asset S in a small time interval ∆t, µ the expected rate of return per unit of time of the underlying asset, σ the volatility of the price of the underlying asset and is a random drawing from a standard normal distribution, i.e. with a mean of zero and a standard deviation of 1 (Hull, 2012).

2.3 Valuation of Policies with a Guaranteed Amount

In literature there are several studies that describe valuation methods for pricing insur-ance contracts with an asset value guarantee. A selection of these studies is discussed in the following subsections.

2.3.1 Single Premium

On maturity the policyholder receives an uncertain benefit which is at least the guaran-teed amount. An objective of the study of Brennan and Schwartz (1976) is to determine the equilibrium price to be charged by the insurer, to meet the benefit on maturity. The study shows that the present value of the uncertain benefit can be written in two ways, as the sum of:

1) The present value of the guaranteed amount (known) and the present value of an immediately exercisable call option

2) The present value of the investments to be made in the investment fund or reference portfolio (known) and the present value of an immediately exercisable put option

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Both call and put option have as exercise price the guaranteed amount and is the option to purchase or sell the reference portfolio, respectively. To provide the guarantee on maturity the insurer charges an extra amount, the guarantee premium, which can be invested in the reference portfolio. Combining the two ways stated above (see Formula 2.10) gives the guarantee premium, which is equal to the price of a put option on the reference portfolio. The put option depends on the guarantee and investments, which are both known, and the call option. Only the call option has to be valuated. Brennan and Schwartz (1976) show that in case of a single premium contract without any contributions after the start of the contract (except for the purchase price), the call option can be valuated with the Black-Scholes model as a call option on a non-dividend paying stock on valuation date. So Brennan and Schwartz (1976) show that the Black-Scholes model is suitable when valuating policies with a guaranteed amount if a single premium is involved. However, the insurer is interested in an analytical method to value unit linked life insurance policies with a guaranteed amount and future payments.

2.3.2 Periodic Premiums

In case of periodic premiums, Brennan and Schwartz (1976) state that no known ana-lytical solution exists. They offer a numerical method including examples, to determine the total annual premium. This premium is the part of the premium invested in the reference portfolio plus an annual put premium. The approach is only slightly different from the one in this study, since this study is interested in the expected loss based on a definite premium, while Brennan and Schwartz (1976) determine an equilibrium price to be charged by the insurer, in order to avoid a loss. Unfortunately, a numerical method can be time consuming, which is expensive for the insurer.

Bacinello (2009) presents the basic principles with respect to the valuation of com-plex life insurance contracts. Periodic premium contracts require periodic investments in the reference asset. Hence the value of the reference portfolio and therefore of the benefit become path-dependent3_{. European options are used to replicate the benefit (see}

Formula 2.11 and 2.12). These options are Asian4-like if periodic premiums are involved. Bacinello (2009) does not provide a computing solution for this type of options.

Jori (2008) studies the valuation of guaranteed unit linked contracts. The specified contract is very similar to the contract valuated in this study. Jori (2008) states more explicitly than Bacinello (2009) that the guarantees on maturity can be defined as a series of Asian put options. Unfortunately “there are no closed analytical formulas for the value of that kind of options” (Jori, 2008, p. 96). Finding a realistic upper bound analytically is not possible. No unique method exists giving good results for all types of policies, based among others on different durations or guarantees. However, Jori (2008) offers an approximation for the value of the Asian put option.

The payoff of Asian options “depends on the finite sum of correlated log-normal variables, which is not log-normal and for which there is no recognizable probability function” (Milevsky & Posner, 1998, p. 2). To determine the unknown distribution, Jori (2008) uses the Edgeworth Series Approximation (Jori, 2008, p.99 – 101). Jori (2008) compares four different Edgeworth’s expansions, based on one or more correction terms. For the guaranteed percentage and volatility used in this study, the results of Jori (2008) show that the most useful approximation for valuating UL policies is the Edgeworth’s expansion including only the first term. The derivation of this approximation is given in a summary of Jori (2008) (see Appendix B).

Fusai & Roncoroni (2008) state that Turnbull & Wakeman (1991) and Levy (1992) approximate the unknown distribution with a log-normal distribution. Turnbull & Wake-man (1991) use the Edgeworth’s expansion including the first four terms (Fusai &

Ron-3

Not only the final value, but the whole followed path is of interest (Hull, 2012)

4

Hull (2012) defines an Asian option as “an option with a payoff dependent on the average price of the underlying asset during a specific period”

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coroni, 2008). However, Jori (2008) shows that for the policies considered in this study the Edgeworth’s expansion including only the first term is useful. Therefore the approx-imation used in the study of Turnbull & Wakeman (1991) is not applicable. Levy (1992) approximates the unknown distribution with a log-normal distribution, where the first two moments are fitted. The formula for the price of an Asian call option is given by a modified Black-Scholes formula. Levy’s approach is simple and therefore popular (Fusai & Roncoroni, 2008).

Zhang & Mraovic (2014) examines Levy’s approach by comparing the results with the results based on Monte Carlo simulations. They conclude that “Levys analytic so-lution tends to overestimate Asian option values when volatility is constant, but under-estimates under the scenario of having stochastic volatility” (Zhang & Mraovic, 2014, p.2). The insurer is interested in an accurate analytical formula, where over- or under-estimation is negligible. Therefore Levy’s approach is not useful in this case.

Until now the Edgeworth Series Approximation, including only the first term (Jori, 2008), is the only useful approximation method to value the considered UL policies. As mentioned before, the contract used in Jori (2008) is very similar to the contract valuated in this study. A difference is that Jori (2008) includes a premium for expenses and death and withdrawal benefits, defined as the total premium minus the premium invested in the fund. Here the savings premiums are independent of the fund value. Therefore Jori (2008) considers a fixed premium for death risk. However, the UL policies considered in this study includes risk premiums, which are fund value dependent and therefore depend on the fund return rates. The Edgeworth Series Approximation cannot be used in this case. However, for further research the Edgeworth Series Approximation could be examined to see if this approximation is applicable to the policies considered in this study.

There are no studies available offering an accurate analytical formula to value policies with a guaranteed amount and future payments, where fund value dependent risk premiums are involved.

2.3.3 Discounting Future Payments

This study examines a valuation formula considered by the insurer. To decide if this formula is sufficient to valuate UL policies, the results of this method are compared to the results of Monte Carlo simulations. For these simulations two possibilities can be considered with respect to discounting the expected loss in each simulation-path. The first possibility is that in each simulation discounting is based on other fund return rates, the rates corresponding to a different fund return path. This is called stochastic discounting. The other possibiliy to discount losses is with the fixed risk-free curve on valuation date, which is called deterministic discounting. The studies of Brennan and Schwartz (1976) and Jori (2008) base there research on deterministic discounting.

The best possibility to discount each loss on maturity date, i.e. with stochastic or deterministic discount factors, depends on the correlation between the development of the fund return and the interest rate. When investing in a deposit fund, consisting of fixed rate short term loans, it can be considered as a cash-account with approximately a risk-free return. The fund volatility will generally be very low (for example 3%) and similar or close to the interest rate volatility. In such cases there is a high correlation between the development of the fund return and the development of the interest rate. Consequently, discounting with stochastic discount factors is appropriate. In case of Monte Carlo simulations, these stochastic discount factors, based on the fund returns, will be close to the corresponding discount factors from the risk-free yield curves that correspond to that fund-return-path. Hence, each simulation has got approximately a fitting risk-free discount curve.

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When investing in stocks, the fund volatility might be very high (for example 20%) and very different from the interest rate volatility. In these situations there is most likely a low correlation between the development of the fund return and the development of the interest rate. The problem is now how to determine the risk-free discount factor for each fund-return-path in the simulations. Therefore discounting with deterministic discount factors, i.e. discount factors based on the Solvency II risk-free yield curve on valuation date might be the (second-)best solution.

For the purpose of the UL policies considered in this study the insurer invests in a mix fund. This fund is a combination of, for example, 60% stocks, 30% bonds and 10% short term time deposits. The considered valuation formula is based on deterministic discount factors and therefore probably sufficient if investments are completely in stocks. When investing in a deposit fund this formula is no longer an adequate valuation method. In this case an adjusted or alternative valuation method is necessary, based on stochastic discounting. Since the insurer invests in a mix fund for the UL policies, the question remains if valuating losses on maturity date by the considered valuation formula is sufficient.

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Chapter 3

Insurance Product

The purpose of this study is to examine valuating unit linked life insurance policies with guarantees on maturity values and future payments, or UL policies. To analyze if the valuation formula considered by the insurer (CVF) is a proper valuation method for this specific situation, the results of the CVF are compared with the present values of the losses based on Monte Carlo simulations. The losses are based on the fund values and guaranteed amounts on maturity. For the policies fund values and guaranteed amounts are calculated based on several assumptions, product data specifications and policy data specifications. These are stated in the first section below. The second section describes the formula to calculate the fund value.

3.1 Unit Linked Life Insurance Policies with Guarantee

The data of the unit linked life insurance policies are obtained from the insurer (see the introduction). The data, which are a subset of the total portfolio, are divided in three subsections: product specifications, policy specifications and assumptions. The three subsections are followed by a subsection of the formula to calculate the fund value at the end of each month.

3.1.1 Product Data Specifications

The product data specifications consist of general product-specific variables, like policy expenses, acquisition expenses and risk premiums. Appendix C gives a complete overview, including the symbols, signs (if applicable), values and sources of the

product data specifications.

3.1.2 Policy Data Specifications

The policy data specifications consist of several policy-specific variables, like

specifications of the death benefits and important dates. This information, together with the general product specifications, is needed to determine the capitals of risk, risk premiums and the releases due to death and surrender per month. Information about the premium and guaranteed amount is available as well. Appendix D gives a complete overview, including the values and sources of these considered policy data

specifications. Section 3.2 contains the formulas of some of these variables.

The UL policies examined in this study contain a guarantee on the benefit on maturity date. This guaranteed amount is based on a fixed fund return of g_%. The policyholder receives at least the guaranteed amount on maturity. If on maturity the fund value is lower than the guaranteed amount, the insurer suffers a loss. There are some additional conditions (see Appendix E) that have to be met, but the insurer does not consider these

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details when calculating the fund value. Therefore this information is not included in the model.

Among the policies there are four different types of death benefits. The most common death benefit is the one where the capital is a fixed amount over time. Other possible death benefits contain premium restitution, increasing premium restitution, or the death benefit is a certain percentage of the fund value, for example 110%, which is the WTV-percentage. Before the starting date of the policy the height of the death benefit is determined. See Appendix F for an overview of the distribution of the death benefit types per policy, in the subset of policies considered in this study.

There are two types of premium: monthly or yearly. A yearly premium is only paid in the month corresponding to the month of the starting date of the policy. So in this case only 1 out of 12 months a premium is paid.

3.1.3 Assumptions and Calculation Specifications

Assumptions are made with respect to surrender and mortality rates, which are used to calculate the release of fund value:

1) Surrender rates, which depend on the policy year

2) Mortality: Prognosis table AG 2012-2062, adjusted for mortality experience, which depends on the gender and age of the insured

Specific calculation choices are as follows:

1) Start of the valuation (k = 0) is on the 1st of July, 2014 2) Calculations are on a monthly basis

3) Day of birth of the insureds is on the first day of their month of birth.

3.2 Fund Value Formula

The development of the fund value consists of several positive and negative variables. These variables, which are stated in Subsection 3.1.1, are multiplied by purchase or distributions expenses, depending on their sign (see Appendix C and D).

The positive variables, which are the gross premium payment and the compensation risk premiums1, are multiplied by the purchase expenses (pc%), while the acquisition and policy expenses are multiplied by the distribution expenses (dc_%), just as the risk premiums. The exact formulas are given below.

To calculate the fund value Vk+1 at the end of each month, all variables, except for the release due to death and surrender, are calculated at the start of the (k+1)thmonth, for k = 0, 1, ..., n - 1. After that the corresponding monthly yield, multiplied by the sum of the variables and the fund value at the start of the (k+1)th month, is added. At last the release, due to surrender and mortality, at the end of the month is subtracted, which results in the fund value at the end of the (k+1)th month. The release is based on the release probability, which is calculated from the cumulative probabilities of staying (i.e. alive and no surrender), multiplied by the start fund value, all variables and the part based on the monthly yield. See the formulas below. The fund value at the end of the kth month is the same as the fund value at the start of the (k+1)th (i.e. following) month.

Each calculation starts at the start of the (k+1)th month. The first calculation, to calculate V1, takes place on moment k = 0. The last calculation, of Vn, is on the maturity date (n) of the contract, where k = n - 1. The fund value is only calculated

1

The compensation risk premiums or negative risk premiums are in case the insured death benefit is lower than the fund value

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if the valuation date is on or before the maturity date of the policy, since after this date the fund value is zero. All variables in the Fund Value Formula (see Formula 4.3) already include the purchase and distribution expenses.

The variables that have a positive influence on the fund value are the premium Pk and the compensation risk premiums CRPj,k, with j = 1,2 for the first (1) or second (2) insured (see Appendix C and D). If only one insured is involved, CRP2,k = 0 for all k. Because of the positive influence on the fund value these variables get a ”plus sign” (+) in the Fund Value Formula.

At the start of the (k+1)th month the corresponding fund return rate fk over all variables, including the fund value at the start of the (k+1)th month, results in the fund return with symbol Fk. Since the fund return rates can be positive or negative, the variable Fk will get a ”plus sign” (+) in the Fund Value Formula as well.

The variables with a negative influence on the fund value are the policy expenses PEkand the risk premiums RPj,k, with j = 1,2 for the first (1) or second (2) insured (see Appendix C). If only one insured is involved, RP2,k = 0 for all k. The other variables with a negative influence are the fixed acquisition expenses AEk, with k ≤ 120 for the product considered in this study, and the release Rk due to surrender and mortality. All these variables get a ”minus sign” (-) in the Fund Value Formula, which looks as follows:

Vk+1 = Vk+ Pk− P Ek− RP1,k− RP2,k+ CRP1,k+ CRP2,k− AEk+ Fk− Rk (3.1) The fund value on the maturity date (Vn) is compared to the guaranteed amount on the maturity date (Gn). The insurer only suffers a loss if the necessary guaranteed amount on the maturity date is higher than the fund value on the maturity date. The variables of the Fund Value Formula are based on the following underlying variables:

Variable Formula or Restriction Description

n n > 0 Total duration in

months of the contract.

nP nP ≤ n Total duration in

months of the premium payments Pk.

j j = 1,2 First (1) or second (2)

insured.

q_j,k,% k = 0, 1,.., n-1 Monthly mortality rate

with mortality experi-ence, for the jth insured in the (k+1)th month.

surrender_k,% k = 0, 1,.., n-1 Monthly surrender rate

in the (k+1)th month. S_%,k S_%,k-1*(1-q_1,k-1,%)(1 - q_2,k-1,%)(1 - surrender_k-1,%)

for k = 0, 1,.., n with S%,0 = 100% and S%,n the probability of staying till maturity

Probability of staying until the beginning of the

(k + 1)th _month.

R%,k 1

-S%,k+1

S%,k for k = 0, 1,.., n-1 Probability of release.

DBj,k k = 0, 1,.., n-1 Death benefit, depends

on the death benefit type (Appendix F).

WTV_%,j,k k = 0, 1,.., n-1 WTV-percentage,

de-pends on the WTV-date (Appendix D).

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Table 3.1 - Underlying Variables of the Fund Value Formula Variables

An overview of the formulas of the variables of the Fund Value Formula (see Formula 4.3) is given below:

1) The fund value on valuation date V0 is based on the amount of units in the mix fund U0 and their value UV0 (see Appendix C and D)

V0 = U V0∗ U0 (3.2)

2) The gross premium Pgross is the premium paid by the policyholder. The premium invested into the mix fund at time k is Pk. This premium decreases per month, since it is based on the probability of staying. Since the premium is a positive variable, it depends on the purchase expenses (see Appendix C)

Pk= Pgross∗ S_%,k∗ (1 − pc_%) (3.3) 3) The policy expenses variable PEk differs per month, since it is based on the

probability of staying. Furthermore it depends on a fixed amount2 and, since it is a negative variable, it depends on the distribution expenses (see Appendix C)

P Ek= 5 ∗ S%,k∗ (1 + dc%) (3.4) 4) The risk premium differs per policyholder j, since it is based on a risk premium specific mortality rate (qRPj,k) and capital of risk (RCj,k, see Table 3.1). It differs per

month, since it is based on the probability of staying. Furthermore it could contain a medical raise in case of the first policyholder (j = 1). Since the risk premium is a negative variable, it depends on the distribution expenses (see Appendix C)

RPj,k = (

qRPj,k ∗ RCj,k∗ (1 + M ED%,j=1) ∗ S%,k∗ (1 + dc%), if RCj,k > 0

0, otherwise (3.5)

5) The compensation risk premium only takes place if there is a negative capital of risk. It differs per policy, since it is based on a risk premium specific mortality rate (qRPj,k) and capital of risk (RCj,k, see Table 3.1). It differs per month, since it is based

on the probability of staying. Furthermore it could contain a medical raise in case of the first policyholder (j = 1). Since the compensation risk premium is a positive variable, it depends on the purchase expenses (see Appendix C)

CRPj,k= (

qRPj,k∗ RCj,k∗ (1 + M ED%,j=1) ∗ S%,k∗ (1 − pc%), if RCj,k < 0

0, otherwise (3.6)

6) The acquisition expenses are paid during a fixed amount of months (maximum of 120 months, i.e. k ≤ 120), which can differ per policy. For this time period the gross acquisition expenses remain constant. The acquisition expenses differs per month, since it is based on the probability of staying. Since the acquisition expenses variable is a negative variable, it depends on the distribution expenses (see Appendix C)

AEk= AEgross,k∗ S_%,k∗ (1 + dc_%) (3.7) for k ≤ 120.

7) The fund return Fk is based on the monthly fund return rate fk, multiplied by the sum of the fund value and the variables (2) till (6) starting at the (k+1)th month (see Formula 3.3 - 3.7)

Fk= fk∗ (Vk+ Pk− P Ek− RP1,k− RP2,k+ CRP1,k+ CP R2,k− AEk) (3.8)

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8) The release Rk ending at the (k+1)th month is based on the release probability, multiplied by the sum of the fund value and the variables (2) till (7) starting at the (k+1)th month (see Formula 3.3 - 3.8)

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Valuation Methods

It looks like the considered valuation formula (CVF) of the insurer to value the guarantee on maturity of UL policies, does not work well if future payments are involved. The first section gives further notes on the assumptions with respect to the method. In the second section the CVF is introduced. To examine if the CVF is sufficient to value the UL policies, the results are compared to the results of Monte Carlo simulations (MC). The application of MC is stated in the third section. Both CVF and MC are based on the fund value on the maturity date of the policy, compared to the necessary guaranteed amount on the maturity date. The results of both methods show if the insurer on maturity suffers a loss. The calculations are performed in Microsoft Excel 2010 (Excel). The third and fourth sections describe the implementation of the CVF and MC in detail.

4.1 Assumptions

Assumptions about the interest rate, (guaranteed) fund returns and volatility are as follows:

1) The interest rates rk, based on the Solvency II yield curve, are monthly forward rates in the (k+1)th month:

1 + rt= (1 + rk)12 (4.1)

with rt the 1-year forward rates in the (t+1)th year and rk the corresponding monthly forward rates. It follows that

rk= (1 + rt)

1

12 − 1 (4.2)

2) Fixed yearly guaranteed fund return rate g%

3) The fund return rates fk for the CVF are based on the Solvency II yield curve 4) The fund return rates fi,k for MC depend on the ith simulated fund return path 5) Volatility σ of the CVF is derived from historical fund growth

6) Volatility of MC is similar to the volatility of the CVF

For the purpose of the unit linked policies with guarantees on maturity values, the insurer invests in a mix fund, which is a combination of, for example, 60% stocks, 30% bonds and 10% short term time deposits. In the market the volatility of this type of funds is not available. Therefore the volatility of this mix fund is based on the historical fund growth of the mix fund (see Appendix G).

4.2 Application Considered Valuation Formula (CVF)

To value the guarantee on maturity of UL policies, the insurer considers an analytical valuation formula, based on the Black-Scholes model (see Chapter 2). The guarantee

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on maturity date on these policies can be considered as a put option (see Formula 2.4). On maturity the policyholder receives the maximum of the guaranteed amount and the fund value on maturity.

The CVF consists of two parts. The first part is the valuation of the present

(CVFpresent), the value of the guarantee that is assigned to the current or present fund value. The second part consists of the valuation of the future parts per month, where each future part is discounted to the valuation date (CVFfuture). The discount factors (DF(k)) are based on the Sovency II risk free yield curve (see Section 4.1)

DF (k) = DF (k − 1)

1 + rk (4.3)

with DF(0) = 100% and with rk the monthly yield (see Formula 4.2 for details on rk). The sum of CVFpresent and CVFfuture is multiplied by the probability of staying till maturity (S%,n), which gives the total expected loss on the valuation date of the CVF (L0,CVF).

L0,CV F = (CV Fpresent+ CV Ff uture) ∗ S%,n (4.4) It is assumed that the guaranteed amount on maturity Gn of an UL policy is based on a fixed fund return of g_% per year, so Gn is fixed. Since the valuation with the CVF is based on two parts, the guaranteed amount Gn is divided between these two parts. One part is with respect to the fund value (Gpresent), the other part is with respect to the future premiums and expenses (Gfuture), where

Gpresent= Gn− Gf uture (4.5)

Gn depends on the selection of variables that is included in the model. For example, if only premiums are included, the guaranteed amount on maturity is higher than if expenses are included as well. Gfuture depends on the future variables, excluding the fund value on valuation date (V0), based on a fixed fund return of g% per year.

The variable ALk is the allocation in the mix fund of the premium paid by the policyholder at the start of the (k+1)th month, less deductions for expenses and risk premiums plus compensation risk premiums (if applicable). Since each month is valuated separately, the Gfuture is divided in n different parts (GALk), so each month

has its accompanying future guaranteed amount on maturity, i.e. at the end of the nth month: GALk = ALk∗ ((1 + g%) n−k 12 − 1) (4.6) Gf uture= n−1 X k=0 GALk (4.7)

Gpresent is the remainder and therefore the guaranteed amount completely related to the fund value, excluding all other variables. So CVFpresent is based on Gpresent and CVFfutureis based on Gfuture. Subsection 4.2.1. describes how the CVF works in general. Subsection 4.2.2 till 4.2.4 describe CVFpresent, CVFfutureand L0,CVF in more detail.

4.2.1 The CVF in General

There is a loss at maturity n if the guarantee Gn is bigger than the fund value at maturity (Vn), i.e. max(Gn - Vn, 0). The insurer considers the analytical CVF, which is based on the Black-Scholes model (see Section 2.2).

Let V0= S0, Gn = K and Gn*DF(n) = Ke-rn, then the amount of the loss valued at k = 0, is max(Ke-rn - S0, 0), i.e. the loss can be seen as a put option with strike price K (see Formula 2.2). The general shape of the CVF is the same as the Black-Scholes formula for a put option (see Formula 2.4, 2.6 and 2.7):

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CV F (S0, Ke−rn, nσ2) = Ke−rn∗ N (ln( Ke−rn S0 ) + 1 2nσ 2 √ nσ2 ) − S0∗ N ( ln(Ke_S−rn 0 ) − 1 2nσ 2 √ nσ2 ) (4.8) For the σ2 the insurer uses the variance of the natural log of the last 10 yearly returns on the mix fund involved (see Appendix G for details on the value of σ2_{). For the} values of S0and Ke-rn the assumption is made that these are at least 1. Consequently the formula, based on fractions and natural logarithms, never gives an error.

The general shape of the CVF (see Formula 4.8) is used to value the present and the future, where different variables are assigned to S0 and Ke-rn. The following subsections describe the valuation of the present and future in detail.

4.2.2 The Present Valuation (CVFpresent)

The valuation of the present is based on the CVFpresent formula. Here S0 = V0, the fund value on the valuation date and Ke-rn = Gpresent*DF(n), the guaranteed amount discounted with the risk free yield curve, which is the ”present part” of the guaranteed amount on the valuation date (see Formula 4.5). So this gives the following formula:

CV Fpresent = CV F (V0, Gpresent∗ DF (n), nσ2) (4.9) When multiplied by S_%,n this is the first part of L0,CVF.

4.2.3 The Future Valuation (CVFfuture)

To valuate the future, the insurer considers to us the CVFfutureformula, which consists of n future monthly parts, all discounted to k = 0. Therefore this formula is the sum of n separate, discounted CVFfuture parts. As stated before, the CVF valuates the future months independently of each other:

CV Ff uture= n−1 X

k=0

CV Fk(S0, Ke−r∗(n−k), (n − k) ∗ σ2) ∗ DF (k) (4.10)

Here S0 = ALk, the allocation in the mix fund, where k is the start of the (k+1)th month, for k = 0, 1, 2, ..., n - 1. To obtain Ke-r*(n-k), each GALk (see Formula 4.6) is

discounted to the start of moment k, since each part of CVFfuture is valuated in the (k+1)th month. So Ke-r*(n-k) = GALk_{DF (k)}∗DF (n). This gives the following formula:

CV Ff uture= n−1 X k=0 CV Fk(ALk,GAALk∗ DF (n) DF (k) , (n − k) ∗ σ 2_{) ∗ DF (k)} _(4.11)

Or, in a more short way

CV Ff uture= n−1 X

k=0

CV Fk∗ DF (k) (4.12)

When multiplied by S_%,n this is the second part of L0,CVF.

4.2.4 The Total Valuation (L0,CVF)

When combining the two parts of the CVF as described in subsection 4.2.1 and 4.2.2, L0,CVF looks as follows: L0,CV F = (CV Fpresent+ n−1 X k=0 CV Fk∗ DF (k)) ∗ S%,n (4.13) This formula is considered to valuate the policies and determine for the insurer the expected loss on the valuation date.

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4.3 Application Monte Carlo Simulations

Another method to value the policies is by means of Monte Carlo simulations (MC). With this valuation method a good impression of reality is given. The more

simulations, the more accurate the results are. To limit the necessary simulation time, a maximum amount nsim of 10,000 simulations is chosen. With Visual Basic for Applications (VBA) in Excel the average loss (L0,MC, based on nsim different fund return paths, is calculated for all policies

L0,M C = 1 nsim nsim X i=1 max(Gn∗ S%,n− Vn,i, 0) ∗ DF (n) (4.14)

with Vn,i the fund value on maturity n based on the ith fund return path. This fund value is based on the Fund Value Formula (see Formula 3.1), where for each (k + 1)th month the fund value is multiplied by the corresponding probability of staying S%,k for k = 0, 1, ..., n-1, n. DF(n) is the discount factor on maturity based on the Solvency II yield curve. The next subsection describes the determination of the fund return paths.

4.3.1 Fund Return Paths

For MC nsim fund return paths are used (see Section 4.3), which are based on generated cash flows in Excel. The basis for each ith cash flow is the variance σ2 (see Appendix G) and the Solvency II yield curve (see Section 4.1), just as in case of the CVF. An amount of nsimcash flows, with a validity of 50 years, is calculated as follows

CFi,t = ( CFi,t−1∗ ert− σ2 2 +ν∗σ, if 0 < t ≤ 50 0, otherwise (4.15)

with the start fund value CFi,0 = 1,000 for all i and ν ∼ N(0,1) based on the

Brownian motion1 (see Section 2.2). The yearly rates rt, with t the time in years, are based on the Solvency II yield curve and determined as follows

rt= ln(

DF (t − 1)

DF (t) ) (4.16)

with DF(0) = 100% and r0 = 0. The nsim generated cash flows are converted to yearly fund return rates:

fi,t =

CFi,t− CFi,t−1 CFi,t−1

(4.17) and monthly fund return rates:

fi,k = (1 + fi,t)

1

12 − 1 (4.18)

are used for MC to calculate the fund values.

1

In Excel ν = STAND.NORM.INV(ASELECT()), where ASELECT() gives an evenly distributed real number c, with 0 ≤ c < 1. Every time the Excel worksheet is calculated, ASELECT() gives another random real number. The STAND.NORM.INV() calculates the inverse of the cumulative standard normal distribution (N(0,1)). The argument of this function is the probability that corresponds to the normal distribution.

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Analysis and Results

This chapter shows the analysis and results of the valuation of the UL policies, with the focus on the valuation of future payments. Several tests are performed, each based on a different selection of variables of the Fund Value Formula (see Formula 3.1). To each new test more variables are added, like the premiums or expenses, which makes the valuation more complex per test. Section 5.1 gives the specifications of each test and gives an explanation of the representation of the results. Section 5.2. gives the results of the CVF compared to the results of MC, based on the several tests. The results are compared with the findings of the literature review. For the test where periodic premiums are involved, MC is adapted to the CVF, to get a better

understanding of the impact of the considered valuation formula. Section 5.3 gives a summary of the results.

5.1 The Specifications of the Tests

Of the UL policies an average policy is examined, policy1 (see Appendix H for details). Several tests are performed, to show the effect on the valuation if future payments are involved. These tests are given in Table 5.1 below. Since the tests are based on

different selections of variables of the Fund Value Formula (see Formula 3.1), the fund value Vk and the guaranteed amount on maturity date Gn differ for each test.

Test Variables Description

Test 1 V0 and Fk Start Fund Value and Fund Return

Test 2 Test1 and Pk Added Premiums

Test 3 Test2 and PEk and AEk Added Policy and Acquisition Expenses

Test 4 Test3 and Rk Added Releases

Test 5 Test4 and RPj,kand CRPj,k Added (Compensation) Risk Premiums based on death benefit DBj,k, which is a constant amount (see Appendix H) Test 6 Test4 and Alternative RPj,k

and Alternative CRPj,k

Added (Compensation) Risk Premiums based on a variable death benefit of 110% of the fund value, i.e. DBj,k = 110%*Vk Table 5.1 - The different tests with the selected variables of the Fund Value Formula Test1 is the most basic test. Only the start of the fund value and the fund returns are included in the Fund Return Formula. No future payments (premiums or expenses) are considered. Test2 is Test1 with the premiums added. This means that future payments are involved. These payments have a positive influence on the fund value. Test3 is Test2 with policy and acquisition expenses added. These extra future

payments have a negative influence on the fund value. Test4 is Test3 with the release 19

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added. The release is based on the death and surrender rates. Therefore the

probability of staying S_%,k for k = 0, 1, ..., n, is included. The only difference between Test4 and Test3 is that the losses based on both methods are a factor S%,k smaller. This does not change the outcome of the difference between both methods, apart from possible rounding and the influence of the maximum amount of simulations. Test5 is Test4 with the (compensation) risk premiums added, based on a constant death benefit. This test includes all variables of the Fund Value Formula and represents the complete UL policy. Therefore this test is the only test applied to all available UL policies. Test6 is similar to Test5, only with the added (compensation) risk premiums based on a variable death benefit. This death benefit is 110% of the fund value, with 110% based on the WTV% before the WTV date (see Appendix H).

For each test the loss based on MC is the expected loss at valuation date. The loss based on the CVF shows how much the CVF overestimates or underestimates the loss based on MC. In the second section the results of the methods examined are shown in tables. The column names of these tables are explained below.

Column Name Description Remark

Test (#) States which test is applied

Gn Corresponding guaranteed amount on maturity date

Differs per test diff% Difference in percentage between

the examined methods

L0,MC Total loss on valuation date based on Monte Carlo simulations (see Formula 4.14)

CVFpresent Valuation with CVF of the current fund value V0 (see Formula 4.9)

Only influenced by the current fund value V0. Therefore it gives the same results for each test. The tests are all based on the current fund value and the fund return rates (Test1) CVFfuture Valuation with CVF of the future

payments (see Formula 4.11)

Zero for Test1, since this test does not con-tain future payments S%,n Probability of staying till maturity For Test1 till Test3 this

is 100%, since here no death or surrender risk is considered

L0,CVF Total loss on valuation date based on the CVF (see Formula 4.13)

Table 5.2 - The description of the column names of the tables with results The diff% is calculated as follows:

dif f_%= L0,M C− L0,CV F

L0,CV F ∗ 100% (5.1)

If diff% << 0 the CVF overestimates the expected loss. If diff% >> 0 the CVF underestimates the expected loss.

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5.2 Results of the CVF versus the Results of MC

The results1 of the CVF are compared to the results of MC with a maximum of nsim simulations (see Section 4.3). Table 5.3 shows the results, based on the several tests. In the following subsections per test the results are discussed. For each test the influence is described of the added variables on the results of the CVF and on the results of MC. The first subsection starts with the basic Test1 where no future payments are involved. As stated in the literature the valuation does not give any problems. The subsections following are based on the more complex tests, which are based on future payments. Literature shows that valuation is difficult when future payments are involved.

Test (#) Gn diff% L0,MC CVFpresent CVFfuture S%,n L0,CVF

Test1 15,395 -0.42 2,423 2,433 0 100% 2,433 Test2 59,313 -14.71 5,873 2,433 4,453 100% 6,886 Test3 55,578 -14.73 5,505 2,433 4,023 100% 6,456 Test4 55,578 -14.72 2,013 2,433 4,023 36.56% 2,361 Test5 24,382 40.83 1,896 2,151 1,531 36.56% 1,346 Test6 53,541 -16.95 1,927 2,478 3,867 36.56% 2,320 Table 5.3 - Results of policy1 based on the CVF and MC with 10,000 simulations

5.2.1 No Future Payments - Basic Test (Test1)

As stated in the literature review Brennan and Schwartz (1976) state that the Black-Scholes model is a suitable valuation method when no future payments are involved. Since the CVF is based on the Black-Scholes model, it is a useful analytical formula for valuating policies without future payments. For the basic test (Test1) there are no deductions from the fund (due to expenses or mortality risk premiums) or other future payments involved. The difference between the CVF and MC is

approximately zero (-0.42) as expected2(see Table 5.3)).

5.2.2 Future Payments - Premiums (Test2)

When future payments are involved the valuation is more complex. In the case of periodic premium Brennan and Schwartz (1976) state that no known analytical formula exist. Bacinello (2009) and Jori (2008) give the same conclusion.

The CVF, which is an analytical formula, overestimates the expected loss on maturity with almost 15% if premiums are involved (Test2). The overestimation is due to the way the CVF valuates the future payments. When valuating the total guarantee, the CVF divides the guaranteed amount to all separate payments and the current fund value. The CVF values the guarantee allocated to the fund value and the guarantees allocated to every future payment separately. Each month the CVF is based on the payment and guaranteed amount in that specific month. The formula checks if on maturity there is a loss or not, independently of all other past and future payments. However, this is not what happens in reality. In reality on maturity the total of all payments is compared to the total guaranteed amount. So the past and future payments influence each other on aggregate level. Consequently, for the future valuation the CVF is much more complicated.

Example 1 below shows a sample calculation to illustrate this shortcoming of the CVF. It shows that valuating UL policies with the CVF can give a structural

overestimation of the expected loss when considering future payments, since the possible compensation between the payments is not taken into account.

1

For Test2 till Test4 the results of diff%are slightly different due to rounding. 2

For Test1 diff% slightly differs from zero due to rounding and the chosen maximum amount of

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Example 1

Consider a policy with a guaranteed fund return of g_% = 1.5% per year and a

duration of 2 years. Suppose a single fund return scenario where for the first year the fund return f%,0 is -2% and for the second year the fund return f%,1 is 5%. The start fund value V0 is 1,000 and premiums P0and P1, in the first and second year are 100. There are two ways to calculate the expected loss on maturity. The first one resembles the real nature of the guarantee of the policy (reality). Here the total fund value on maturity is compared to the total guaranteed amount. The second one is based on the CVF. Here the expected loss on maturity is calculated per payment and aggregated. 1) BASED ON REALITY

Guaranteed amount on maturity:

G2 = (V0+ P0) ∗ (1 + g%)2+ P1∗ (1 + g%) = 1, 234.748 Fund value on maturity:

V2 = (V0+ P0) ∗ (1 + f%,0) ∗ (1 + f%,1) + P1∗ (1 + f%,1) = 1, 236.900 Loss on maturity:

L2= max(G2− V2, 0) = 0 So in reality there is no loss at all.

2) BASED ON THE CVF Guaranteed amount on maturity:

G2 = G2,V 0+ G2,P 0+ G2,P 1= 1, 234.748 G2,V 0 = V0∗ (1 + g_%)2 = 1, 030.225

G2,P 0= P0∗ (1 + g_%)2 = 103.023 G2,P 1= P1∗ g_%= 101.500 Fund value on maturity:

V2 = V2,V 0+ V2,P 0+ V2,P 1= 1, 236.900 V2,V 0 = V0∗ (1 + f%,0) ∗ (1 + f%,1) = 1, 029.000 V2,P 0= P0∗ (1 + f%,0) ∗ (1 + f%,1) = 102.900 V2,P 1 = P1∗ (1 + f%,1) = 105.000 Loss on maturity: L2= L2,V 0+ L2,P 0+ L2,P 1= 1.347 L2,V 0 = max(G2,V 0− V2,V 0, 0) = 1.225 L2,P 0= max(G2,P 0− V2,P 0, 0) = 0.122 L2,P 1 = max(G2,P 1− V2,P 1, 0) = 0

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A Better Understanding of the Impact of the CVF

The results so far show that the CVF valuates in a different way than MC. To fully understand what the CVF does when valuating UL policies and to test the

calculations and the CVF formula, MC is adapted to the CVF. This is called the Alternative Monte Carlo (MCA). MCA is based on a present part and a future part, just as the CVF. First MC is written in the shape of CVF, i.e. based on two parts. The valuation on maturity, corresponding to the ith fund return path, looks as follows:

ΦM C_i,n = max G ∗ S%,n− (Vi,0∗ (1 + f ci,0) ∗ S%,n+ n−1 X k=0 ALi,k∗ (1 + f ci,k) ∗ S%,k), 0 ! (5.2) where fci,k is the cumulative fund return rate from k (not necessarily zero) till

maturity n, corresponding to the ith fund return path: f ci,k =

n Y

l=k

(1 + fi,l) (5.3)

The valuation by MC on valuation date can be obtained by

ΦM C_i,0 = ΦM C_i,n ∗ DF (n) (5.4) The CVF, however, valuates every month independently, so no compensation between different moments can take place. Therefore the results of the CVF are compared to the results based on MCA, of which the valuation on maturity looks as follows: ΦM CA_i,n = max (Gpresent− Vi,0∗ (1 + f ci,0), 0)+

n−1 X

k=0

max GALi,k − ALi,k∗ (1 + f ci,k), 0

(5.5) ΦM CA_i,0 = ΦM CA_i,n ∗ DF (n) ∗ S_%,n (5.6) where the total guaranteed amount on maturity Gn = Gpresent +Pn−1k=0GALk (see

Formula 4.5, 4.6 and 4.7).

Test (#) Gn diff% L0,MCA CVFpresent CVFfuture S%,n L0,CVF Test2 59,313 -1.40 6,790 2,433 4,453 100% 6,886 Table 5.4 - Results of policy1 based on the CVF and MCA with 10,000 simulations Table 5.4 shows the difference for Test2 between the CVF and MCA based on nsim simulations (see Section 4.3). The results of policy1 show a difference in the direction of zero.

5.2.3 Future Payments - Expenses (Test3)

If policy and acquisition expenses are added, Table 5.3 shows that the difference between the two methods does not change. The premiums and expenses act the same. The only difference is the sign of these future payments. This as no influence on diff% as long as the total payments are positive.

The CVF cannot value negative payments, due to the natural logarithms it contains (see Formula 4.8). However, the UL policies involve expenses (and risk premiums). It could occur that there are no premium payments anymore to compensate for these negative future payments. This could be the case for policies where the maturity date of the premium payments is before the maturity date of the contract or when the policy is based on a single premium payment. A valuation method is necessary which is able to valuate negative future payments.

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5.2.4 Future Payments - Release (Test4)

For Test4 the difference between the two methods does not change as compared to Test3. Only the losses for both methods are multiplied with the factor S%,n of 36.56%, i.e. the probability of staying till maturity.

5.2.5 Future Payments - (Compensation) Risk Premiums (Test5)

(Based on a Constant Death Benefit)

As stated in the Literature Review (Section 2.3) Jori (2008) examines a policy with a premium for death benefits. However this premium is independent of the fund value. Therefore no valuation methods are available to value policies with a guaranteed amount at maturity and periodic premiums where fund value dependent risk premiums are involved.

The (compensation) risk premiums are based, among others, on the capital of risk (see Formula 3.5 and 3.6). Table 3.1 shows that the capital of risk depends on the fund value. Here a problem is apparent, since the CVF valuates the payments separately (see Section 5.2.2). However, the fund value depends on the past payments, which depend on the fund return rates. Therefore the CVF is not able to value the policies if (compensation) risk premiums are involved.

There are risk premiums involved when the policy has a positive capital of risk, for example when a constant death benefit is involved. In this situation the CVF can give a structural underestimation, especially when the involved constant death benefit is a large amount. This is the case for policy1 (see Appendix H). The results in Table 5.3 show that the CVF gives an underestimation of over 40%. This underestimation is explained as follows:

A loss on maturity happens when the fund return rates are low. This means that for MC only the bad fund return paths are taken into account when valuating the UL policies. In these situations the risk premiums, based on a high constant death benefit, are higher to compensate for the bad fund returns. Consequently less premium is available to invest in the mix fund. This increases the probability that the fund value on maturity is lower than the guaranteed amount, i.e. a higher loss on maturity is expected when based on MC.

The higher the capital of risk (due to the high constant death benefit), the bigger the impact on the risk premiums, which results in a higher underestimation. For the compensation risk premiums (or negative risk premiums), it is the other way round, since an overestimation can occur.

Results of All Available Unit Linked Life Insurance Policies

Test5 includes all variables of the Fund Value Formula and represents the complete UL policy. Therefore this test is applied to all available unit linked life insurance policies (see Appendix F). These policies depend on different policy data characteristics and product data characteristics (see Section 3.1). Table 5.5 shows for Test5 for all policies the maximum (max), minimum (min) and average differences between the CVF and MCA based on nsim simulations (see Section 4.3).

Test (#) Policies (#) max diff% min diff% average diff%

Test5 200 73.60 -19.35 0.61

Table 5.5 - Results of all policies based on the CVF and MCA with 10,000 simulations Despite the average difference is close to zero (see Table 5.5), the CVF is not accurate. The results in Table 5.5 show a large range of the differences between the results based on the CVF and MCA. On the one hand, the extremely high underestimation of

Valuation of guarantees on maturity values for unit linked life insurance policies with future payments

values for unit linked life insurance

policies with future payments

K.M.C. Dijkshoorn

Contents

Preface

Introduction

1.1

Background

1.2

Research Question

1.3

Towards a Solution

1.4

Outline of the Thesis

Literature Review

2.1

History Black-Scholes Model

2.2

Specifications Black-Scholes Model

2.3

Valuation of Policies with a Guaranteed Amount

Chapter 3

Insurance Product

3.1

Unit Linked Life Insurance Policies with Guarantee

3.2

Fund Value Formula

Valuation Methods

4.1

Assumptions

4.2

Application Considered Valuation Formula (CVF)

4.3

Application Monte Carlo Simulations

Analysis and Results

5.1

The Specifications of the Tests

5.2

Results of the CVF versus the Results of MC