Option Valuation in Life Insurance

The consequences of the tax change in 2005 for implicit options and bonus payments as a marketing tool to attract customers

Master's Thesis, 2004

53 Pages, Grade: 2 (B)




The impact of the German Tax Raises on the Importance of Options

A. The Role of Bonus Payments in Capital Forming Life Insurance Contracts

B. A Model of Bonus Payments to attract New Customers
1. Legal Framework
2. Model

C. The influence of the change in law on the profit sharing in 2005

D. Numerical results
Implicit Options in Life Insurance

E. Options in Financial Theory

F. Implicit options

G. Classification of options in life insurance according to the risk type
1. Risk-free Options
2. Actuarial Risks
3. Financial Risks

H. Classification of the Most Common Options
1. Risk-free Options
2. Actuarial Options
3. Financial Options

Basic Models of Option Valuation

I. Equity Option Models
1. Black-Scholes Model
2. Binomial Tree Model of Cox, Ross and Rubinstein
3. Risk neutral valuation

J. Interest Rate Option Models

1. Hull-White Model

Models of the Valuation of the American Options in Life Insurance

K. Tribinomial tree for the interest rate and asset process

L. Valuation of the Surrender Option in Capital Forming Life Insurance

1. The model
2. Numerical Results

M. Grosen-Jorgensen Approach

Option Valuation in Life Insurance Master Thesis by Ekaterina Avershina

N. Longstaff and Schwartz Approach





Table 1: Mean Value and Standard Deviation of Returns on Investment

Table 2: Returns on investment by life insurance companies in 2002

Table 3: Life Insurance Reserves excluding Unrealised Capital Losses (rounded to million euro)

Table 4: Bonus Rates for 2005 with respect to different discount rates

Table 5: The price of the surrender option with respect to different guaranteed insurance sum levels (source: Dillmann, 2002, p. 183)


Figure 1: Display of the cost/revenue mechanism regarding the bonus payment (source: own figure)

Figure 2: Binomial Tree of Cox, Ross and Rubinstein for the stock price process (source:Dillmann, 2002, p.150)

Figure 3: Branching approaches by the interest rate trinomial tree model (source: Dillmann, 2002, p.143)

Figure 4: Trinomial tree of Hull and White for the interest rate process (source: Dillmann,l2002, p.145)

Figure 5: Branching approaches in tribinomial tree model (source: Dillmann, 2002, p.158) .

Figure 6: Tribinomial tree of Dillmann (source: Dillmann, 2002, p.160)


The capital forming life insurance appears currently to be in a very vulnerable state. It was usually an attractive investment opportunity with stable returns comparable to other investment opportunities. In 2000-2002 it was difficult for the life insurance companies to overcome the consequences of the stock market crises, the losses of the insurance companies were enormous. Today there is another challenge for the insurance companies to overcome - the end of the tax privilege starting in 2005.

These events bring our attention to the problem of profit sharing. In this paper I show that the changes in the tax law related to the life insurance profits in Germany lead to an increased competition for new customers in 2004 by paying maximum possible bonus rates and to the drastic decrease of it in 2005 which will force the insurers to look for alternative methods to attract new customers like implicit options embedded in the insurance contracts.

Such options are liabilities to the issuer, they also constitute a potential danger to the company’s solvency. Therefore, they should be properly valued. Historically that has not been done which turned out to be a disaster for some companies.

In the first chapter of this work I introduce the mechanism of profit sharing, its legal framework, the changes in the tax law crucial for the insurance companies and my own model describing how the insurer actually chooses the bonus rate of the insurance contract. Furthermore, the predictions about bonus rates in 2005 and its signification for the options will be made. The second chapter is devoted to the definition, classification and the examples of the most common implicit options on the German life insurance market. The third chapter shows the most common models of the valuation of interest rate and asset options. The tree models will be described particularly in detail. The fourth chapter is dedicated to the models of valuation of the non-European options in life insurance contracts.

The impact of the German Tax Raises on the Importance of Options

I will argue in this chapter that the coming tax raises for capital forming life insurance contracts in Germany will make options in these contracts even more important. The line of argumentation will be the following: The end of the tax exemption will lead to much lower bonus payments (Überschussbeteiligungen) in the future, starting in 2005. The mechanism which translates the higher taxes into lower bonus payments will be laid out in my own model. These dramatically decreased bonus payments will then lead to other characteristics of insurance contracts becoming more important, for example, options. As customers will expect lower bonus payments and trust companies’ bonus promises less, they will look for contracts that give them more flexibility. Insurers will have to focus more on which options to offer and how to price them externally and internally.

A. The Role of Bonus Payments in Capital Forming Life Insurance Contracts

Traditionally, profit sharing contracts that involve bonus payments by an insurer have been very popular in Germany, they still dominate the market. These contracts are also called “participating” or “with profit” contracts. They are characterised by the fact that the insurer shares her profits with a policyholder. The policyholder pays a single or an annual premium which is invested in the capital market. A minimum return is guaranteed to the policyholder. As the return of the investment regularly exceeds the guaranteed interest payment, a part of this excess return (Überschuss) is also allocated to the contract holder’s benefits (Überschussbeteiligung) - which I will call “bonus payment” hereafter. These bonus payments are credited to the mathematical reserves of the contract at the end of each year and this implies the “purchase” of additional insurance (Bacinello, 2001b, p.2). The bonus payment and the minimum interest add up to the total interest of the insurance contract (Gesamtverzinsung).

As it is common that the insurer allocates substantial bonus payments, this makes life insurance contracts competitive with regard to other investment opportunities on the market. For the customer, the total interest that she receives is comparable to that of the underlying portfolio. It can even be shown that the customer’s return bears less variance (Grosen and Jorgensen, 2000, p.41-47), so less risk than an underlying portfolio.

From the insurer’s point of view, the profit sharing is a free-will payment to which the customer is not legally entitled - a gift which represents additional costs to it. Klaus Math (2002, p.143) argues that this situation is positive for an insurer, because decreasing the bonus payment below what had been promised is a valuable option that could be exercised in the future.

The larger part of the literature in this area is preocupied with the fair valuation of life insurance liabilities represented, for example, by options and guarantees embedded in the insurance contract. These authors assume that the minimum interest rate is guaranteed to a policyholder and that some extra bonus dependent on the performance of the reference portfolio is paid - in participating or with-profit contracts. Some of the authors derive an explicit expression of the bonus rate, for example, Grosen and Jorgensen (2000, p.44) and Bacinello (2003, p.8; 2001a, p.6; 2001b, p.8). Others, for example Dillmann (2002, p.134) state a direct dependency of the bonus rate on the performance of a reference portfolio.

Norberg (1999) develops a „Theory of bonus in life insurance“: He sees the bonus system as a possibility for the insurer to safeguard against risks that are indeversifiable, like variations in overall mortality:

„ The only way the insurer can safeguard against this kind of risk is to build into the premium a safety loading that makes it cover, on the average in the portfolio, the benefits under (in principle) any possible economic-demographic development. Such a safety loading will typically create a systematic surplus, which by statute is the property of the insured and has tobe repaid in the form of so-called bonus ( ‘ good ’ in Latin). “ (p. 374)

Norberg tries to find a model that not only hypothetically predicts the different risky parameters (what he calls first-order - technical - basis), but can incorporate the realizations of these paramters from the so-called second-order (experience) basis and use these realizations to fine-tune the parameter’s distributions.

According to Grosen and Jorgensen (2000, p. 43-44) the bonus rate is given by

illustration not visible in this excerpt

where P(t) is the policy reserve at t, B(t) is the bonus reserve at t, a is the distribution ratio, g is the target buffer ratio which must be reached in order to pay any additional bonus, rG is the minimum guaranteed rate. a and g are determined by the insurance company, they are the main parameters of the bonus policy of an insurer. In this model they are taken as exogenous variables, in other words, they are taken as given.

In the model of Anna Rita Bacinello (2003, p.8) the bonus rate is given by

illustration not visible in this excerpt

where h is the participation coefficient, i is called the technical rate, can also be interpreted as the minimum guaranteed rate, gt is the rate of return of the reference portfolio during the t -th year of the contract.

Bacinello (2001a, p.13-14) derives the expression which defines a trade-off1 between any pair of “control” variables of an insurance company, which are the participation level and the technical or minimum guaranteed rate, as well as the riskiness of the investments composing the reference portfolio, measured by the volatility coefficient:

illustration not visible in this excerpt

where c is the time 0 value of a European call option on a non dividend paying stock with initial price equal to 1, option with maturity 1 and strike price equal to 1+i/ h. From this equation any of the control variables can be calculated if other variables are fixed.

These models are based on the “fair valuation” approach where the liability side represented by the policy reserve and the bonus reserve, and the asset side represented by the market value of the assets backing the contract of the insurer’s balance must be equal. The insurer is seen as always paying the excess return she earns minus costs (like administration) to the customer. This view would lead to the prediction that insurers pay a high bonus in years when they have earned a high return on their capital and pay a lower bonus when the return was low. This is not what we observe: Insurers have paid stable bonus payments over the years that did not go up and down with their yearly returns (see also table 1).

Although under perfect competition, a “zero profit condition” does certainly exist, it might be another condition than the one cited above. We have to take into account the specific reasons why the insurance companies spend money on these gifts: to attract new customers. We will find a slightly different “zero profit condition” that holds under perfect competition. Especially a changing environment - like higher taxes in the German case - will lead to new conlusions with respect to the bonus payments.

B. A Model of Bonus Payments to attract New Customers

My aim is to explain why the insurer is willing to share her profits in addition to the minimum guaranteed rate and to which extent she is willing to do that, in other words how the insurer can cover the “profit sharing costs”. Last but not least the forecast of the bonus rate for the year 2005 and following years will be made.

The profit sharing is costly for an insurer so the main question is why she is actually sharing the profits. The main reason is the attraction of new customers. As virtually all insurers stick to the legally fixed minimum guaranteed interest rate in their contracts, the main differentiation between insurers is their allocated bonus payment. In Germany, insurers usually announce in November or December how much they will allocate in the following year. With these figures they calculate their return promises when offering customers new contracts. Although customers should be aware of the fact that the bonus payments are not guaranteed, it is clear that actual and past bonuses provide a clue on how much the company might pay in the future.

So the bonus policy can be considered as a marketing tool to attract new customers. The insurance market is highly competitive and profit sharing is a very effective weapon against rivals. Although in his model, Klaus Math (2002, p.146) does not take the competition into account, he agrees that the decision to decrease a bonus rate by the insurer depends on the current situation of competition on this market2.

For standard goods it is common that the marketing costs of the producer are actually carried by the customers because these costs are included in the price of the product. In life insurance market it is not like that. To attract new customers the insurer pays extra bonus to her existing policyholders, in other words, the existing customers even profit from marketing costs for the new customers.

The next question to answer is to which extent the insurer is ready to share her profits. I argue that the bonus payment is bounded by two factors:

- first, an insurer needs to have enough means to pay the bonus rate. The source of these means is the so-called bonus reserve, which is part of the liability side of the insurer’s balance, these reserves are filled at good years and they are used in bad years.
- Second, an insurer must adjust what she shares today with what she will earn tomorrow. This is one of the innovations of this model: the costs of profit sharing that an insurer bears today are actually an investment in future profits from new policyholders.

In the figure below we take the insurance term equal to 30 years because the average term in Germany, according to Dillmann (2002, p. 14) is 27,9 years.

illustration not visible in this excerpt

Figure 1: Display of the cost/revenue mechanism regarding the bonus payment (source: own figure)

The most important part of the analysis is the influence of the changes in the legal framawork for capital forming life insurance in Germany. The tax exemption will disappear on 1st of January 2005. According to this law all contracts that are closed after 31.12.04 are subject to income tax. This concerns life insurance contracts that are used as an investment opportunity instead of retirement provisions. To a policyholder who reaches the age of full 60 years whose contract was closed more than 12 years ago the so-called half-gain approach (Halbeinkünfteverfahren) will be applied: only half of the contract earnings are taxed. In other cases the whole earnings are taxed. This means a very large loss for the life insurance companies, these changes make life insurance contracts less attractive compared to other market investment opportunities. The fact that the gains of participating contracts that are closed before 01.01.05 are still tax free makes this market even more competitive than ever, this is so to say the last chance to get many new clients, that is why insurance companies do their best to attract them. The main conclusion that can be made is that in 2005 the profit sharing rate will be probably rather small.

The insurance analyst Marco Metzler from Fitch Ratings argues that the promised bonus rate increases for next year can be considered as marketing activity and that not all of the insurance companies will do it or will be able to do that. The insurance expert Manfred Poweleit agrees that the companies that increase bonus rates look forward to represent themselves the best in the competitive environment (Der Tagesspiegel, 03.09.043 ).

1. Legal Framework

By German law4, the insurer is required to pay to the policyholder 90 % of the net return (return minus administration costs) that she has earned with the assets invested for the policyholder’s insurance sum. So the total interest of the insurance (Gesamtverzinsung) has to be at least 90 % of this net return. Although this might seem like tying the insurer’s hands, especially in the current situation and in the short run it is not. The reason is that the net return is calculated according to the profit the insurer shows in her balance sheet. This profit can easily be increased or decreased by using hidden reserves (stille Reserven) or hidden obligations (stille Lasten):

- Hidden reserves: The insurer is required to account assets with their purchase value if the current value is larger. The difference between the current value and the current book value (purchase value) is a hidden reserve that exists in reality, but not in the insurer’s balance sheet. The insurer can decide to use this reserve at any time by simply selling the asset and therefore having to account the realized profit.
- Hidden obligations: If the current value of an asset is lower than its purchase value, the insurer can choose to account it either with its current value or still with its higher purchase value5. If the current value is chosen and the value increases again, the book value has to be adjusted accordingly (but not higher than the original purchase value).

As the life insurance market in Germany has suffered from very bad returns in the years 2000-2002, insurers have dissolved hidden reserves and built up hidden obligations. According to the Versicherungsjournal (26.5.2003)6, these losses added up to the astonishing sum of more than 100 billion euros.7 Therefore it can be safely assumed that in the coming years, if the insurers want to decrease their accounted profit, they can simply dissolve some of their hidden obligations or build up new hidden reserves. In the long run, however, they will finally have to pay out 90 % of their net return again. As this analysis tries to predict the bonus payments for the very next years, this legal requirement can be assumed as not binding in the short-run.

2. Model

To introduce the model the following notation will be used:

t: actual time

Ns: number of new contracts in the last years s=0,1, … ,t

Nt+1: number of potential policyholders for the next year

rB(s): bonus rate (profit sharing rate) for s=0,1, … ,t defined with respect to the

policyholder’s account balance: bonus= rB*P

rG: minimum guaranteed rate

rP(s): interest rate credited to the policy in year s=0,1, … ,t (total interest)

P(s): policyholder’s account balance at s=0,1, … ,t

B(s): buffer (bonus reserve) at s=0,1, … ,t

p Nt+1: expected gross profit from new policies: discounted difference between earned interest on the market and the guaranteed rate

rN(t+1): expected excess return on the reference portfolio of the new policies at t+1 (the difference between the interest rate earned on the market and the minimum guaranteed rate).

The bonus rate is determined similar to Grosen and Jorgensen (2000, p.41): rB(s)= rP(s) - rG, which is non-negative for all s. For simplicity I assume that the account balance is the same for all policyholders and all times: P(s)=P for all s.

1. The bonus paid by an insurer to all existing policyholders should be covered by the expected profits from potential clients who are attracted by the profit sharing rate of the insurer, in other words the costs of a bonus payment should not be larger than the revenues of a bonus payment:

illustration not visible in this excerpt

where EQ is the expected value in the risk-neutral world (all securities traded on the market have the same expected interest rate r(s)) and

illustration not visible in this excerpt

is the present value of the expected payoff of the new policies over T years, the duration of the contract, at time t+1; r is the discount factor. For simplicity we assume that the expected excess return on the reference portfolio, rN, is the same for all years. Accordingly, the present value of the profit as a percentage of the account balance, Vv, is also constant, Vv=V=const for all v=t+1,t+2, … .

2. At the same time the bonus payments are limited by the reserves that an insurance company has, so the “budget” restriction is given by

illustration not visible in this excerpt

We know that the capital forming life insurance market is highly competitive so we can assume that in equilibrium the bonus payment to all policyholders is equal to the expected gross profit of the new contracts. In other words we assume perfect competition on this market so the zero-profit condition should be fullfilled, which means equation 1.1 is binding given that the reserves are large enough (equation 1.2):

illustration not visible in this excerpt

In equilibrium all insurance companies will pay the same bonus. If an insurer pays less than other insurers she will not be able to compete with others for new customers. An insurance company could get more profit by increasing the bonus attracting new customers. If an insurer pays more bonus than other insurers she may bear losses.

If the “budget restriction” (equation 1.2) is not binding, the allocated bonus is independent of the earned interest on the market given future profit expectations (equation

1.1). This fits the observations of the past, bonus payments have been very stable as we can see from the table represented by Heinrich R. Schradin (1999, p.14-16):

illustration not visible in this excerpt

Table 1: Mean Value and Standard Deviation of Returns on Investment

This table represents the comparison analysis of different forms of capital investments. For the life insurance branch the average over the 30 biggest life insurance companies is taken. Mean value (MV) and standart deviation (STD) are calculated for the one year period and in the average over 5-, 10- and 15-year period. As mentioned above the table shows that capital forming life insurance performance is very stable even over one- year period (STD low). This leads also to the conclusion that the bonus rates are stable as well. This anables the smoothing of the rates of return of life insurance policies which leads to stable total returns over years.

illustration not visible in this excerpt

C. The influence of the change in law on the profit sharing in 2005

The change in capital forming life insurance legislation concerns all contracts closed after 31.12.04. The profits of the contracts that are closed before that date remain tax free, it makes the competition between insurance companies tougher, their efforts to attract as many new policyholders as possible together with the policyholders rational expectations allow to assume that the number of new contracts closed in 2004 will be higher than the average. By the same reason and because of the increased number of new contracts this year the expected number of new contracts in 2005 is relatively low, i.e. below the average. The bonus rate for 2004 should be and probably is rather high to be able to attract as many new clients as possible, while in 2005 it will be relatively low for two reasons. First, less (below the average) new policyholders are expected in 2005, so according to the equation (1.1) the payoff from these contracts will not be enough to cover high costs of profit sharing. Second, this payoff is not enogh to cover the costs of profit sharing because the number of current policyholders is increased, more than average. This hypothese will be proved by numerical results of the model (shown in the example).

D. Numerical results

All contracts have the same account balance P and the same durability of 30 years. Assume that the number of new contracts every year is Ns=100 for s=0,1, … ,t-1, so the insurer has 3000 current contracts every year. Let us set t at 2004 and assume that this year there will be Nt=400 new customers, whereas the number will drop to Nt+1=55 in 2005.

illustration not visible in this excerpt

Assume that the “budget” restriction is fullfilled so that condition (1.1) is binding. In this case as we can see from the last two equations, the bonus rate in 2004 is eight times larger than the bonus rate in 2005. If we assume that the profit sharing rate in 2004 is 3,2%, then the profit sharing rate in 2005 is only 0,4%.

But the facts rather show in the other direction: Since the stock market crash in 2000, insurers’ reserves have dropped dramatically. Returns on capital investment were decreased drastically: from 7,5% to 4,14%. Subtracting the unrealised capital losses it is even lower: 2,1% (VersicherungsJournal, 03.09.038 ). The following table shows the returns on capital investment by life insurance companies in (VersicherungsJournal, 26.05.03):

illustration not visible in this excerpt

Table 2: Returns on investment by life insurance companies in 2002

The total loss for the life insurance branch in the period from 2000 to 2003 was about 104 billion euro, in particular the loss of reserves about 59 billion euro (VersicherungsJournal, 26.05.039 ). By the estimation of market research service (Marktforschungsdienst) the total amount of reserves at 2003 available in life insurance branch was about 5,1 billion euro. The following table shows that the reserves of particular German life insurance companies were drastically decreased from 2001 to 2002 (VersicherungsJournal, 28.05.0310 ):

illustration not visible in this excerpt


1 „If the minimum interest rate guaranteed is high the insurance company cannot afford to set a high participation level. In good years it has to put aside a sufficient amount of non-distributed funds in order to be able to fulfil the minimum guarantee promise in bad years. A highly volatile reference portfolio can produce high returns as like as heavy losses...Therefore in this case, to protect itself, the insurance company must keep the technical rate and/or participation level down“.

2 “… Vernachl ä ssigt man die Situation, dass eine Senkung der Gewinnbeteiligung aufgrund einer unzureichenden Finanzierbarkeit erfolgen muss, so h ä ngt die Entscheidung f ü r eine Senkung in der H ö he wie im Zeitpunkt der Durchf ü hrung sicher stark von der jeweiligen Wettbewerbssituation ab ”.

3 http:// www.tagesspiegel.de

4 § 81c „Versicherungsaufsichtsgesetz“ and § 3 of „Verordnung über die Mindestbeitragsrückerstattung in der Lebensversicherung“, see internet link „Allianz“ in the appendix.

5 If the lower value is considered to be only „transient“ (§ 253 Abs. 2 Satz 3 1. Halbsatz HGB, changed with the „Versicherungskapitalanlagen-Bewertungsgesetz“ on 26.03.2002, Bundesgesetzblatt I S. 1219 ff).

6 http://www.versicherungsjournal.de/archiv

7 The bad situation led to the increased flexibility in accounting law cited in the footnote above.

8 http://www.versicherungsjournal.de/archiv

9 http://www.versicherungsjournal.de/archiv

10 http://www.versicherungsjournal.de/archiv

Excerpt out of 53 pages


Option Valuation in Life Insurance
The consequences of the tax change in 2005 for implicit options and bonus payments as a marketing tool to attract customers
LMU Munich  (Seminar for Insurance Studies)
2 (B)
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ISBN (eBook)
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1296 KB
Option, Valuation, Life, Insurance, Untertitel
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Ekaterina Avershina (Author), 2004, Option Valuation in Life Insurance, Munich, GRIN Verlag, https://www.grin.com/document/30754


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