Is Fair Pricing Possible? An Analysis of Participating Life Insurance Portfolios
Working Paper by C. Orozco-Garcia and H. Schmeiser
Universität Hamburg, 11/2015
November 2015Seite 2 Structure
1. Introduction
2. Motivation
3. Model Description
4. Numerical Illustration
5. Conclusion
November 2015Seite 3 1. Introduction
Risk-neutral valuation is the standard procedure in order to derive “fair” premiums inparticipating life insurance contracts, to analyze the solvency of an insurer, or toobtain combinations of equity- and debt capital that do not lead to wealth transfersbetween shareholders and debtholders (policyholders)
Cf. Briys & de Varenne (1997), Grosen & Jorgensen (2002), Ballotta et al. (2006),Ballotta (2009), Bacinello (2001), Tanskanen & Lukkarinen (2003), Schmeiser &Wagner (2014), and many more
The “fair” single upfront premium P0 for the savings part of the contract is given by
𝑃𝑃0 = 𝑃𝑃𝑃𝑃 𝐿𝐿𝑇𝑇 = 𝐸𝐸0ℚ 𝐿𝐿𝑇𝑇 = 𝐸𝐸0
ℚ min(𝑃𝑃𝑇𝑇 ,𝐴𝐴𝑇𝑇)
November 2015Seite 4 1. Introduction
If fair pricing takes place, the shareholders will receive a risk-adequate return on theirinitial contributions (net present value NPV = 0)
𝐸𝐸𝐶𝐶0 = 𝑃𝑃𝑃𝑃 max(𝐴𝐴𝑇𝑇 − 𝑃𝑃𝑇𝑇 , 0) = 𝐸𝐸0ℚ max(𝐴𝐴𝑇𝑇 − 𝑃𝑃𝑇𝑇 , 0)
The solvency level of the insurer can be measured by the default put option
𝐷𝐷𝑇𝑇 = max(𝑃𝑃𝑇𝑇 − 𝐴𝐴𝑇𝑇, 0)
The policyholders receive a yearly interest rate rt on their initial contribution P0 inform of a cliquet-style option
𝑟𝑟𝑡𝑡 = max 𝑔𝑔,α(At/At−1 − 1)
November 2015Seite 5 1. Introduction
Hence, we receive an infinite amount of fair equity / premium combinations betweenthe two stakeholder groups that lead to no wealth transfer (measured by the NPV)
EC0
P0
Fair premium without default
PV(DT)
November 2015Seite 6 Structure
1. Introduction
2. Motivation
3. Model Description
4. Numerical Illustration
5. Conclusion
November 2015Seite 7 2. Motivation
In insurance practice we typically have
… regular premium payments
… different contract starting points / maturity dates and
… the contracts are typically pooled in one legal entity
There is very little research so far that tries to capture these point (cf. Hansen &Miltersen (2002), Gerstner et al. (2008), Ibragimov et al. (2010), Doskeland & Nordahl(2008), Gollier (2008))
The following two research questions have – at least to our knowledge – not beenfocussed so far
November 2015Seite 8 2. Motivation
Research question 1:
Is it possible to simultaneously charge fair premiums (NPV = 0) to all policyholders?
NPV in formal terms:
and
(Present value of the policyholder account (policyholder generation i) minus presentvalue of the first premium paid by policyholder i)
November 2015Seite 9 2. Motivation
If only fair terms for the portfolio can be provided,
…. who pays more? Who pays less?
PV premiums
PV payoff
Net present value
T T+1 T+2 Portfolio
> 0 >= 0 <= 0 = 0
November 2015Seite 10
Premiums
Default
Risk level
T T+1 T+2
Low Medium High
2. Motivation
Research question 2:
Are all policyholders encountering the same default risk?
Is it possible to ensure the same value for the default put option (ratio) to allpolicyholders / generation of policyholders?
November 2015Seite 11
Summary:
Remark
FAIR PREMIUM FOR ALL
GENERATIONS
SAME DEFAULT RISK
LEVELFOR ALL
GENERATIONS
2. Motivation
November 2015Seite 12 Structure
1. Introduction
2. Motivation
3. Model Description
4. Numerical Illustration
5. Conclusion
November 2015Seite 13
Basic framework
Participating life insurance contract with cliquet-style interest rate guarantee
Surrender options and mortality risk are not taken into account
Regular premium payments
Fixed time to maturity for all contracts
Two asset classes: Risky assets and riskless rate of return
Risky assets are modelled through a GBM
Risk management measures that can be taken by the insurer:
Adjusting equity capital at discrete point in time (issuance date of each contract) andchoosing the asset allocation in t = 0
3. Model description
November 2015Seite 14
I. Basic model
II. Model variant with early default (accounting model)
+ 𝑃𝑃1 + 𝐸𝐸𝐶𝐶1
Not Default+(𝑃𝑃2 + 𝐸𝐸𝐶𝐶2)
Not Default+(𝑃𝑃3 + 𝐸𝐸𝐶𝐶3)
…
Not Default−𝐿𝐿1,𝜏𝜏1+𝑇𝑇
Not Default−𝐿𝐿2,𝜏𝜏2+𝑇𝑇
Not Default−𝐿𝐿3,𝜏𝜏3+𝑇𝑇
Default−𝐴𝐴𝜏𝜏3+𝑇𝑇Default
−𝐴𝐴𝜏𝜏2+𝑇𝑇Default−𝐴𝐴𝜏𝜏1+𝑇𝑇
…Default−𝐴𝐴𝜏𝜏3Default
−𝐴𝐴𝜏𝜏2
+ 𝑃𝑃1 + 𝐸𝐸𝐶𝐶1 +(𝑃𝑃2 + 𝐸𝐸𝐶𝐶2) +(𝑃𝑃3 + 𝐸𝐸𝐶𝐶3) …
Not Default−𝐿𝐿1,𝜏𝜏1+T
Not Default−𝐿𝐿2,𝜏𝜏2+𝑇𝑇
Not Default−𝐿𝐿3,𝜏𝜏3+𝑇𝑇
Default−𝐴𝐴𝜏𝜏3+𝑇𝑇Default
−𝐴𝐴𝜏𝜏2+𝑇𝑇Default−𝐴𝐴𝜏𝜏1+T
𝜏𝜏1= 0 𝜏𝜏2 𝜏𝜏3 … 𝜏𝜏1 + 𝑇𝑇 𝜏𝜏2 + 𝑇𝑇 𝜏𝜏3 + 𝑇𝑇
𝜏𝜏1= 0 𝜏𝜏2 𝜏𝜏3 … 𝜏𝜏1 + 𝑇𝑇 𝜏𝜏2 + 𝑇𝑇 𝜏𝜏3 + 𝑇𝑇
Pension scheme
Life insurer
3. Model description
November 2015Seite 15
Formal structure:
Policyholder account of the i-th policyholder generation
Evolution of the assets
3. Model description
November 2015Seite 16
Basic model (I):
Insurer pursued its activities as long as
holds at maturity.Thereby,
denotes the accumulated liabilities of the insurer. In case of a default, the cost ofinsolvency correspond to the difference between the accumulated liabilities and theavailable assets
3. Model description
November 2015Seite 17
Basic model (II):
Proportional insolvency costs for the i-th policyholder generation
Payoff when taken default risk into account
3. Model description
November 2015Seite 18
Basic model (III):
Fair valuation (risk neutral measure)
Dynamics of the rate of return of the assets
Present value of the contracts of the i-th policyholder generation
Values at issuance date τi
3. Model description
November 2015Seite 19
Basic model (IV):
Accumulated value of the portfolio
Value of the default put option for the contracts of the i-th policyholder genera-tion
NPV of the contracts
3. Model description
November 2015Seite 20
Basic model (V):
Fair pricing takes place if the net present value is zero (for both stakeholder groups)
Default put option ratio is defined as
Thereby,
denotes the value of the contract(s) without default
Accounting model: Please cf. working paper
3. Model description
November 2015Seite 21 Structure
1. Introduction
2. Motivation
3. Model Description
4. Numerical Illustration
5. Conclusion
November 2015Seite 22 4. Numerical Illustration
November 2015Seite 23 4. Numerical Illustration
Basic model (I):
It is not always possible to achieve a fair price for the portfolio as a whole
November 2015Seite 24 4. Numerical Illustration
Basic model (II):
Fair pricing and equal default ratios are possible (for the portfolio as a whole and foreach generation) only in the case without default
November 2015Seite 25 4. Numerical Illustration
Basic model (III):
Fair pricing and equal default ratios are possible (for the portfolio as a whole and foreach generation) only in the case without default
November 2015Seite 26 4. Numerical Illustration
Basic model (IV):
The portfolio can be fairly priced, but NPVs for different policyholder generationsand corresponding default ratios vary
November 2015Seite 27 4. Numerical Illustration
Accounting model (I):
Again, we can derive fair pricing – for the portfolio as a whole and for eachgeneration – and equal default ratios in the case without any default risk
November 2015Seite 28 4. Numerical Illustration
Accounting model (II):
November 2015Seite 29 4. Numerical Illustration
Accounting model (III):
The portfolio can be fairly priced, but NPVs for different policyholder generationsand corresponding default ratio vary
November 2015Seite 30 4. Numerical Illustration
Accounting model (IV):
For γ = 9.87 % we get the same results
For higher shares of risk assets, the required equity capital in the accounting model issubstantially lower compared to the basic model
In the accounting model later generations are better of
Default ratios develop differently, NPV discrepancies are smaller compared to thebasic model
In general, one combination of fair pricing and positive but equal default ratios exist(solvency requirements may not allow such combinations)
However, pension schemes are typically not based on the accounting modelframework
November 2015Seite 31 4. Numerical Illustration
Accounting model (V):
November 2015Seite 32 Structure
1. Introduction
2. Motivation
3. Model Description
4. Numerical Illustration
5. Conclusion
November 2015Seite 33 4. Numerical Illustration
In practice, we will face some wealth transfer within policyholder generations andbetween shareholders and policyholders
In addition, default ratios are typically different for each generation
Basic model: Fair pricing for the portfolio may be possible, but default ratios (if > 0)vary – NPVs for each generation differ too –.“Early” generations are better off
Accounting model: A combination of equity contributions and asset allocation mayexist, where fair pricing is possible and equal default ratios can be provided to eachpolicyholder generation
However, even for the accounting model, it is not possible to fix a desired or requiredsafety level for all policyholder generations and obtain fair pricing
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