Fair Valuation and Hedging of Participating
Life-Insurance Policies under Management Discretion
Torsten Kleinow
Department of Actuarial Mathematics and Statistics and the Maxwell
Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh
March 30, 2007
Address: Actuarial Mathematics and Statistics
School of Mathematical and Computer Sciences
Heriot-Watt University
Edinburgh
EH14 4AS
United Kingdom
e-mail:
t.kleinow@ma.hw.ac.uk
Abstract
The fair valuation of participating life insurance policies, also known
as With-Profit policies, is considered. Such policies can be seen as Eu-
ropean path-dependent contingent claims whose underlying security is
the investment portfolio of the insurance company that sold the policy.
The fair valuation of these policies is studied under the assumption
that the insurance company has the right to modify the investment
strategy of the underlying portfolio at any time. Furthermore, it is
assumed that the issuer of the policy does not want or can not setup a
separate portfolio to hedge the risk associated with the life-insurance
policy. Instead, we assume that the management of the insurance
company that sold the policy will use its discretion about the invest-
ment portfolio to choose the portfolio strategy such that the risk is
eliminated or reduced as far as possible. In that sense, the insurer’s
investment portfolio serves simultaneously as the underlying security
and as the hedge portfolio. This means that the hedging problem
can not be separated from the valuation problem. We will show how
a risk-neutral value of such policies can be calculated and how the
management of the insurer can use its discretion about its investment
strategy to hedge the contract if the financial market satisfies some
assumptions.
Keywords: Participating Life Insurance Policy, With-Profits policy, Risk-
Neutral Valuation, Hedging
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1
Introduction and Literature Review
Participating Life insurance contracts, also called With-Profits contracts,
have been issued over the past decades by many insurance companies through-
out the world. Although the details of the contracts vary significantly be-
tween insurer’s there are some common features that all With-Profits con-
tracts share.
In exchange for regular premiums the policyholder receives payoffs at
death, surrender of maturity. Since the payoffs of these contracts are based
on the performance of a particular investment fund, the risk associated with
these contracts is not only mortality risk, but also financial risk. We want
to concentrate on the financial risk and, we therefore ignore mortality and
surrender, and we assume that all contracts reach maturity. Furthermore,
we assume that their is only one premium to be paid by the policyholder at
the time the contract is issued.
As we mentioned above the several mechanism to calculate the final payoff
to policyholders across the insurance industry. To get an overview about
contracts which are common in Europe and the United States see Cummins,
Miltersen & Persson (2004).
Since we ignore mortality, buying a With-Profits contract can be seen
as an investment by the policyholder into a fund, often called the With-
Profits fund, which is managed by the insurance company. In contrast to
standard investment funds, With-Profits contracts provide some protection
against low or negative returns. Instead of receiving the final value of the
fund, the policyholder receives the final balance of an account, called the
policyholder’s account.
The initial balance of this account is a constant
depending on the particular contract. At the end of every year the growth
rate of the policyholder’s account is calculated by the insurance company
based on the return of the With-Profit fund during the year. The balance of
the account at the end of the year is the balance at the beginning of the year
accumulated by the growth rate. At any time before maturity, this balance
does not represent a value in an economic sense. The interpretation of that
account is, that the balance of the account at the end of any year would be
paid to the policyholder if the contract matured immediately.
The calculation of the balance at the end of a particular year is sometimes
based on the performance of the fund over the last few years. However, we
want to consider contracts in which the growth rate of the policyholder’s
account only depends on the performance of the With-Profits fund during
the current year.
In the literature we find two approaches to calculate fair market-consistent
values of With-Profit contracts and derive hedging strategies that the insur-
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ance company can apply to protect itself against the risk associated with the
minimum guaranteed rate of growth of the policyholder’s account.
In the first approach authors treat the With-Profit fund as a fixed invest-
ment portfolio. The price process of the fund is therefore a given stochastic
process. The payoff at maturity, the final value of the policyholder’s account,
is then a path-dependent European contingent claim. This approach allows
us to directly apply methods known from Financial Mathematics. The first
authors to use these methods to price life insurance contracts were Bren-
nan & Schwartz (1976, 1979). These authors considered unit-linked contract
for which the payoff to the policyholder is indeed a contingent claim with a
payoff depending on a unit, which is an externally given reference portfolio.
Since then, Market-consistent valuation of participating insurance contracts
has been studied by a number of authors. Among these are Persson & Aase
(1997), Miltersen & Persson (1999), Miltersen & Persson (2003), Ballotta
(2005). Overviews about the available literature can be found in Kleinow
& Willder (2007) and Bauer, Kiesel, Kling & Ruß (2005) and the references
therein. A very detailed discussion about the approaches and the results of
different authors was carried out by Willder (2004).
A second approach to price and hedge With-Profits funds is based on the
assumption that the management of the insurance company has full discre-
tion about the composition of the With-Profits fund. This means that the
insurer can change its investment strategy in the With-Profits fund at any
time which in contrast to the fixed strategy applied when a reference portfo-
lio is considered. In particular, the insurer can use this discretion to reduce
or increase the volatility of the With-Profits fund by changing the propor-
tion of money invested into equity shares and the proportion invested into
fixed-income securities. Since the With-Profits fund is the underlying secu-
rity of the contingent claim that represents the payoff to the policyholder,
any change in the With-Profits fund will result in change of the value of the
With-Profits insurance contract.
Hibbert & Turnbull (2003) where the first to address this issue. They
consider an insurance company in which the management have limited dis-
cretion in choosing the assets by applying a fixed rule to increase or decrease
the equity exposer of the With-Profits fund depending on the value of the
insurer’s assets and the guarantees already declared. They calculate the fair
value of the With-Profits contract is this situation.
Kleinow & Willder (2007) consider a more realistic setting by assuming
that the growth rate of the policyholder’s account depends on the actual
investment portfolio of the insurer and that the management of the insurer
has the right to change their investment strategy whenever and however they
want to. Any change in this portfolio strategy will lead to a change in the
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underlying price process that is used to calculate the growth rate of the
policyholder’s account. On the other hand, the insurer is not allowed to set
up a separate portfolio to hedge the risks associated with the contract, but
whenever the management of an insurance company wishes to hedge the risk
associated with a contract it would change the investment portfolio of the
company. As this portfolio is the underlying price process for the calculation
of the growth rate of the policyholder’s account any attempt to hedge the
risk associated with the contract leads to a change of the underlying price
process of the contract. The hedging and pricing of participating contracts
in this setting is a non-standard problem in financial mathematics since the
portfolio of the insurer serves simultaneously as the underlying process and
the hedge portfolio of the contract.
Kleinow & Willder (2007) seem to be the first to address this problem
in a systematic way. They assume a discrete time model for the financial
market in which the short interest rate as well as the risky asset follow a
binomial tree model.
Although this is a relatively simple model for the
dynamics of interest rates and risky assets these authors are still able to
draw some interesting economic conclusions. First of all they find that the
insurance company can perfectly hedge the risk associated with the contract
by setting up an appropriate investment strategy. Furthermore, they show
that the investment portfolio of the company will only consist of one-year
and two-year zero-coupon bonds.
These results have been generalized to a continuous time financial market
model by Kleinow (2006). We found that a prefect hedge is not possible
unless the market consisting of a particular zero-coupon bond and the bank
account is complete. However, even in an incomplete market we were able
to derive a fair price for the contract by showing that the contract can be
replicated by a sequence of particular contingent claims. These claims were
priced using risk-neutral valuation.
In the current paper we want to generalize the results found by Kleinow
& Willder (2007) and Kleinow (2006) further. We will particularly emphasis
the relationship between hedging and valuation.
The paper is organized as follows. In Section 2 we introduce the financial
market model. The contract is described in Section 3. The pricing and hedg-
ing problem is introduced in the same Section. In Section 4 we give a precise
formulation of the hedging and valuation problem, define the fair value and
the risk-neutral value of a participating policy and show some properties of
these values. We also investigate the relationship between fair/risk-neutral
valuation and hedging in both, complete and incomplete markets. Finally,
we provide a summary and some suggestions for further research in Section
5.
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2
The Financial Market Model
Let (Ω, F , P) denote a probability space, and let F = (Ft)t∈[0,T] for T ∈ N be
a right-continuous and complete filtration defined on this space. We assume
that F0 is the trivial σ-field.
The tradable assets are a bank account and d ∈ N further risky assets. We
assume that the value S0(t) of the bank account is predictable with respect
to F and S0(0) = 1.
The price processes of the d risky assets are denoted by S1, . . . , Sd. We
assume that these are semimartingales with respect to F under P.
The value of the issuer’s assets at time t is denoted by V (t). The issuer
can invest into all existing assets and the value of his portfolio at any time
t ∈ [0, T ] is therefore
d
V (t) =
ξi(t)Si(t)
(1)
i=0
where ξ = (ξ0, ξ1, . . . , ξd) is predictable with respect to F. The (d + 1)-
dimensional process ξ is called the portfolio strategy. ξ0(t) represents the
number of units of the bank account and ξi(t) for i = 1, . . . , d is the number
of risky assets that belong to the insurer’s portfolio at time t.
A portfolio strategy is self-financing if
t
V (t) = V (0) +
ξ(u)dS(u)
0
d
t
= V (0) +
ξi(u)dSi(u)
∀ t ∈ [0, T ].
i=0
0
We assume that our model is arbitrage-free, and therefore the set of equiv-
alent martingale measures
Q = {Q ∼ P : Si/S0 is a Q-martingale ∀ i = 1, . . . , d} .
(2)
in not empty. Using the notation
D(u, t) = S0(u)/S0(t)
0 ≤ u ≤ t ≤ T
for the discount factor, we obtain for the price of any risky asset
S(u) = EQ[D(u, t)S(t)|Fu] for u < s, ∀ Q ∈ Q
and if the portfolio of the insurance company is self-financing during a time
period [u, t], we obtain a similar property for V ,
V (u) = EQ[D(u, t)V (t)|Fu]
∀ Q ∈ Q
(3)
for all u and t with 0 ≤ u ≤ t ≤ T .
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3
The Contract
We consider an insurance contract with maturity T . As mentioned in the in-
troduction the payoff to the policyholder at maturity depends on the success
of the insurer’s investment strategy and some exogenously given contingent
claim.
More precisely, for a positive real valued function H : R+ → R+ and a
constant x0 ∈ R+ we consider the discrete-time process {X(t), t = 0, . . . , T }
which is given by X(0) = x0
V (t + 1)
X(t + 1) = X(t)H
V (t)
t
V (s + 1)
= x0
H
for t = 0, . . . , T − 1
(4)
V (s)
s=0
where V is the value process of the insurer’s portfolio as defined in (1).
Economically, X represents the balance of an account that the insurance
company holds on behalf of the policyholder. The policyholder can access
this account at maturity only. Since X(T ) is FT -measurable, it can be seen as
a European contingent claim with underlying security V . X(t) for all t < T
can be interpreted as the payoff to the policyholder if the contract were to
mature at time t instead of T . Therefore, X(t) should not be interpreted as
a value in an economic sense for any t < T . Instead, it should be seen as the
”intrinsic value“ at time t of a European contingent claim with maturity T
and payoff X(T ).
We call the function H contract function or bonus distribution mechanism
since H describes how X changes depending on the performance of V .
To generalize the payoff to the policyholder we assume that the policy-
holder receives a further derivative at maturity. A typical example for such
a derivative is a Guaranteed Annuity Option that allows the policyholder to
use the final balance X(T ) of his account to buy an annuity for a price which
was fixed at time 0. In general, the inclusion of such a derivative means that
the final payoff to the policyholder is
XT Γ where Γ is a FT measurable random variable.
To summarize these arguments we define a participating life insurance
policy in the following way.
Definition 1 Let x0 > 0 be a constant, H : R+ → R+ and Γ ∈ FT . We
call the triplet (x0, H, Γ) a participating life insurance policy. The payoff to
the policyholder at maturity is X(T )Γ where X(T ) is the final value of the
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process X defined in (4). The constant x0 is called nominal value, H is called
contract function, and Γ is called terminal option of the contract.
We assume that the contract function H fulfills the following conditions
for all ˜
v ∈ R+:
Assumption 1
(A1) H(y) is continuous and non-decreasing in y with H(0) > 0
(A2) y/H(y) is strictly increasing in y.
A typical example for a contract function H considered by Kleinow &
Willder (2007) and Kleinow (2006) is
H(y) = max eγ, yδ
(5)
where γ and δ ∈ (0, 1) are constant. The growth rate of X during [t, t + 1] is
then
rX(t + 1) = log X(t + 1) − log X(t) = max {γ, δrV (t + 1)}
where rV (t + 1) = log V (t + 1) − log V (t) is the return on V during the
same period. The growth rate rX(t + 1) of X is therefore the maximum of
a guaranteed rate γ and the return rV (t + 1) of the portfolio V during this
period multiplied by a participation rate δ.
We assume that the policyholder does not have a surrender option and
that there is no mortality risk. Therefore, the policyholder will receive X(T )Γ
at maturity T . Some remarks about a surrender option can be found in
Kleinow (2006).
Since X(T )Γ is FT -measurable, the fair price of the contract is given by
EQ[S0(T )−1X(T )Γ|F0]. However, the calculation of this expectation requires
knowledge about the investment strategy ξ of the insurer. We will assume
here that the insurance company has the right to decide about this strategy
and to change their portfolio at any time t ∈ [0, T ].
On the other hand, the insurance company wishes to hedge the risk as-
sociated with the contract (x0, H, Γ), i.e. the European contingent claim
X(T )Γ. We assume that V represents the entire portfolio of the insurer.
This means the insurer can not set up a separate hedge portfolio to hedge
the payoff X(T )Γ since this portfolio would become part of the company’s
assets and therefore, would be part of V which, in turn, changes the law of
the underlying value process V .
However, the insurance company might be able to choose the portfolio
strategy ξ such that the final value V (T ) of its portfolio at maturity is equal
to the value of its liabilities X(T )Γ, in which case the risk associated with
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the contract (x0, H, Γ) is perfectly hedged. In contrast to the hedging of an
option with a payoff depending on a given stochastic process, the portfolio
V serves simultaneously as underlying process and hedge-portfolio.
4
Hedging and Risk-neutral Valuation
Since there is a close relationship between risk-neutral valuation and hedg-
ing of participating contracts we study complete and incomplete markets
separately.
4.1
Complete Financial Markets
In a complete financial market model our first aim is to find an initial capital
V (0) and a portfolio strategy ξ such that V (T ) = X(T )Γ P-almost surely,
that is the value V (T ) of the insurer’s assets is equal to the value X(T )Γ of
the insurer’s liabilities at maturity T . Since the market is complete there is a
unique martingale measure Q ∼ P, and it is sufficient to construct a process
V such that V /S0 is a Q-martingale and V (T ) = X(T )Γ almost surely. Once
the process V is constructed, the completeness of the market ensures that
there exists a portfolio strategy ξ with value process V .
We first define the fair value and the fair relative value of the policy
(x0, H, Γ).
Definition 2 We consider a complete financial market and a policy (x0, H, Γ).
1. The process V = {V (t), t ∈ [0, T ]} is called fair value process of the
policy (x0, H, Γ) if
(a) V /S0 is a Q-martingale and
T
V (t)
(b) P[V (T ) = X(T )Γ] = 1 with X(T ) = x0
H
V (t − 1)
t=1
2. If V (t) is the fair value of (x0, H, Γ) then C(t) = V (t)/X(t) is called
the fair relative value of (x0, H, Γ).
This definition is inline with the risk-neutral valuation approach in finance.
Furthermore, assuming a complete financial market, the existence of a fair
value process is equivalent to the existence of a replicating strategy ξ. We
will show how to construct this strategy after we have studied properties of
the fair value process and discussed its existence and uniqueness.
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To find a fair value process V we start with the last period and obtain
that V (T ) = X(T )Γ if and only if Γ = C(T ) = V (T )/X(T ) and therefore
V (T )
V (T )
C(T ) =
=
(6)
X(T )
X(T − 1)H V (T )/V (T − 1)
= g X(T − 1), V (T − 1), V (T )
(7)
with
v
g(x, v0, v) =
.
xH v/v0
Note that g has the property
g(x, v0, v) = g(αx, αv0, αv)
∀ α = 0.
(8)
We obtain from assumption (A2) that for any x and v0 = 0 there exists the
inverse function g−1(x, v0, c) of g with
c = g(x, v0, v) ⇐⇒ v = g−1(x, v0, c).
It follows that (6) holds almost surely if and only if
V (T ) = g−1(X(T − 1), V (T − 1), C(T )) a.s.
(9)
From (8) we obtain that g−1(x, v0, .) has the property
αg−1(x, v0, c) = g−1(αx, αv0, c)
∀ α = 0
and, in particular, for x = 0 and α = 1/x, g satisfies
1
v
g−1 (x, v
0
0, c) = g−1
1,
, c .
x
x
Using Definition 2 we obtain for the fair value process V and the fair
relative value process C that C(T ) = V (T )/X(T ) = Γ almost surely and
V (T − 1)
C(T − 1) =
(10)
X(T − 1)
V (T )
= EQ D(T − 1, T )
FT−1
(11)
X(T − 1)
V (T − 1)
= EQ D(T − 1, T )g−1 1,
, Γ
FT−1
(12)
X(T − 1)
V (T − 1)
= EQ D(T − 1, T )g−1 1,
, C(T )
FT−1 (13)
X(T − 1)
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Using Definition 2 again we conclude that the fair relative value C(T − 1) of
the policy at time T − 1 must be the solution of the equation
C(T − 1) = EQ D(T − 1, T )g−1 1, C(T − 1), C(T )
FT−1
with C(T ) = Γ, and the fair value at time T − 1 of the policy (x0, H, Γ) is
V (T − 1) = X(T − 1)C(T − 1) which fulfills the equation
V (T − 1) = EQ D(T − 1, T )g−1 X(T − 1), V (T − 1), C(T )
FT−1 .
Using backward induction we obtain the following theorem. The proof can
be found in the appendix.
Theorem 1 A stochastic process V is the fair value process of the policy
(x0, H, Γ) if and only if the corresponding fair relative value process C(t) =
V (t)/X(t) has the following two properties
(1) C(T ) = Γ a.s. and
(2) C(t) = EQ D(t, t + 1)g−1 1, C(t), C(t + 1)
Ft ∀ t = 0, . . . , T − 1.
Let us mention that the fair relative value process C is not a martingale
under Q. However, the fair value process V is a Q-martingale.
We now come to a critical point in our analysis. For general financial
market models (Ω, F , P, F) and general participating policies (x0, H, Γ) we
can not ensure that the fair relative value process exists. We will therefore
assume in the following that it does exist.
Assumption 2 There exists a stochastic process C that has the properties
(1) and (2) in Theorem 1.
Kleinow (2006) shows that the fair relative value process C exists for the
participating policy (x0, H, Γ) with Γ ≡ 1 and H given in (5) if the probability
space (Ω, F , P, F) fulfills some assumptions.
The existence of the fair relative price process C ensures that there is a
process V , the fair price process, such that V /S0 is a Q-martingale and V (T )
is equal to the payoff X(T )Γ of the participating policy. Since the market is
complete we can apply the Martingale Representation Theorem to conclude
that there exists a portfolio strategy ξ such that
t
V (t) = V (0) +
ξ(u)dS(u)
∀ t ∈ [0, T ].
0
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Document Outline

• Introduction and Literature Review
• The Financial Market Model
• The Contract
• Hedging and Risk-neutral Valuation
  • Complete Financial Markets
  • Incomplete Financial Markets
• Summary and Further Research

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