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Life Reinsurance Pricing –
Reinsuring The “Guaranteed Minimum Death Benefit”
Attached to Unit-Linked Contracts
Copyright 2005 PartnerRe
96 Pitts Bay Road
Pembroke HM 08, Bermuda
Authors
Frank Pinette, Head of Life
PartnerRe, Paris
[email protected]
Paul-Antoine Darbellay
Pricing Actuary
PartnerRe, Zurich
Photos and graphs
PartnerRe
Keystone
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Revised reprint May 2005
Original publication, December 2000
Introduction
The investment risk can cause
serious concern. Stock markets are volatile and annual
fluctuations of around 40%
are not unrealistic.
This research paper discusses a number of
aspects relating to the “guaranteed minimum
death benefit” in unit-linked contracts. It deals
briefly with the different types of guarantees
found in the market, then examines the forms
of reinsurance that are potentially suitable and
assesses the risks associated with such covers.
Finally, by using the Black & Scholes model, it
analyzes both a pricing method and a method
of reserving that focus on the important differences between the so-called “traditional” and
“financial” risks in reinsurance. These fall into
two key categories: random fluctuation and
systematic risk (accumulation of risk). Therefore
it is in the reinsurer’s best interest, when covering these guaranteed minimum death benefits,
to carefully evaluate its maximum level of liability.
Unit-linked products have become very popular
in the French insurance marketplace over the
last four years. The policyholder premiums are
PartnerRe 2005
Reinsuring the “Guaranteed Minimum Death Benefit”
Attached to Unit-Linked Contracts
invested in investment funds with the policyholder assuming the investment risks. These products are usually sold with an attaching death
benefit.
The most common types of death benefit available in the market are:
- The “guaranteed minimum death benefit” that
protects the beneficiary, in the event of the
insured’s death, against a possible drop in the
value of the unit-linked fund and also guarantees a pre-determined minimum equal to the
total value of the premium paid.
- The indexed (interest-rate) guaranteed death
benefit that provides the beneficiary, in the
event of the insured’s death, with similar protection as above and, in addition, a pre-determined interest-rate minimum guarantee on the
premium paid (for example 3.5%).
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The Different Forms of Reinsurance
Two forms of reinsurance exist:
Individual Method
This method consists of reinsuring, on a weekly or monthly basis, for each individual insured
the difference between the amount guaranteed in the event of death and the surrender
value, when the latter is lower than the guaranteed minimum death benefit. A weekly or
monthly mortality table is applied to this difference. This may be the ideal method but it is
constraining for the ceding company and
requires a sophisticated administrative backup
system.
Global Method
Premium expressed as a percentage of the
reserves.
In this method, the reinsurer prices the death
risk with a premium loading calculated on the
annual mathematical reserve. The reinsurer’s
approach is a global one and the price is
determined according to average age and to
underlying asset volatility. The obvious drawback of this method is that the ceding company
pays a high premium when the sum at risk
reduces (increase in reserves) and conversely.
Premium expressed as a percentage of the
original premium.
As with the method above, this is a global
method whereby a reinsurance premium is
payable on the premium contributed by the
insured. This method is the focus of this paper.
Guaranteed Minimum Death Benefits:
The Reinsurer’s Viewpoint
Firstly, we examine the risks associated with
this guarantee and then the pricing method
covering the assumptions and related problems and finally the issue of reserving.
The Risk
Traditional Life Insurance Contracts
In traditional life insurance contracts, the
investment risk is held by the direct insurer.
The direct insurer assumes both the mortality
and the investment risk. The latter is generally
small as the capital is usually invested in
investment assets with moderately low volatili-
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PartnerRe 2005
Reinsuring the “Guaranteed Minimum Death Benefit”
Attached to Unit-Linked Contracts
ty. The mortality risk can be controlled when
reliable statistics are available for a relatively
substantial insurance portfolio. When dealing
with this type of portfolio, the capability to evaluate significant numbers of similar risks from
many sources reduces the probability of wide
fluctuations and it is possible to forecast the
number of deaths fairly accurately i. e. the law
of large number applies.
Unit-Linked Contracts
In unit-linked contracts, the investment risk is
borne by the insured. The investment risk
comes from stock market fluctuations in the
value of the assets representing the insured’s
assets. However, this no longer holds entirely
true when a guaranteed minimum death benefit is included in the policy providing a death
benefit that would never be less than a predetermined minimum. This means that if, at the
time of the insured’s death, the asset value has
fallen below the level of the guaranteed minimum sum, the insurer will pay the difference.
Therefore there are two essential features to
this type of contract: the mortality risk and the
investment risk. As pointed out above, unlike
the stock market fluctuations, the mortality risk
can be controlled.
The investment risk can cause serious concern. Stock markets are volatile and annual
fluctuations of around 40% are not unrealistic.
The mortality risk in this type of cover is in
essence a minor feature compared with the
investment risk.
There is also an additional risk to be considered: the systemic risk. Stock market fluctuations impact on all contracts of the same type.
There is no spreading of risks between different forms of contract; on the contrary, the
danger posed by accumulation of risk is greater. Furthermore, wide market fluctuations may
have serious impact on both the insurer and
the reinsurer and reduce their capacity to
assume their liabilities.
Pricing Method
Calculating The Premium
A solution for pricing this type of product is to
combine death and survival probabilities (mortality risk) with financial derivatives (investment
risk).
The Black & Scholes Model
Using the Black & Scholes model, we can calculate the price of a European put. The basic
option-pricing formula is as follows:
Specifically,
r
T
X
ST
= The force of interest corresponding to
the risk-free rate
of return
= Time until expiration of the
option contract
= The exercise price (guaranteed
minimum death benefit)
= Price of the underlying stock
at T (random variable)
We can use the following formula, where:
SP = Single reinsurance premium
qx = Probability of death between ages x and
x +1
p
=
Probability for a policyholder aged x to
i x
be alive at age x + i
Pi = Present value of a European put option
exercised at time i
i
= Number of years covered
The first part, multiplying the survival and death
probabilities, is the equivalent of a term assurance without any discount factor (already
included in the put price). This step enables us
to determine the number or percentage of a
group of insureds whose death benefit may
have to be increased if the said benefit were to
be lower than a pre-determined minimum. As
the price of this cover is given by the put, all
that is necessary is to multiply our probabilities
by the put price. Summing over the number of
contract years, we obtain the single reinsurance premium.
The price of a European put dictates for the
buyer the right, not the obligation, to sell
underlying stock at a pre-determined price, the
exercise price, and at a pre-determined date,
the expiry date of the option contract.
We now examine how the put price is obtained
more closely.
PartnerRe 2005
Reinsuring the “Guaranteed Minimum Death Benefit”
Attached to Unit-Linked Contracts
The difference between the exercise price and
the underlying stock price is covered by the
reinsurer if the underlying stock value is lower
than the guaranteed minimum death benefit
(the exercise price). This is the reason why we
apply the maximum of this difference and zero.
We then determine the expected value of this
maximum to obtain the mean reinsurance
coverage that we discount at the risk-free interest rate.
The Black & Scholes model is based on strict
assumptions. It is worthwhile examining these
assumptions that we apply implicitly when we
use this model to evaluate an option or devise
a cover strategy.
The Assumptions
The Black & Scholes model assumes that the
financial markets are efficient, that the interest
rate is constant, and that the option price is
contingent upon the risk-free interest rate, the
underlying security, its lifetime, its volatility and
the exercise price. It also assumes that share
price movements follow a continuous stochastic log-normal distribution.
It is possible to set up a perfect “risk-free”
cover by using this model (as it assumes that
the gains and losses from buying securities
and buying/selling options cannot diverge
from one another for any length of time). The
Black & Scholes model implies that this situation is, at any time and continuously, risk-adjustable, according to the movements in stock
5
prices, ensuring that the cover remains risk
free until the expiry of the option.
true for a call) and that, as a general rule (although this depends on the discount period), it
is an increasing function of the time until expiration (the same as for a call). Furthermore it is
a decreasing function of the risk-free rate (the
opposite is true for a call) and finally, it is a
decreasing function of the underlying stock
price (the opposite applies to a call).
This hypothesis suggests that markets are efficient. In an efficient market, prices reflect,
instantaneously and without bias, all (price)
relevant information relating to the evaluation
of financial instruments, and the market recognizes all information and follows it correctly.
Also, it implies that there are no arbitrage
opportunities (arbitrage being the possibility of
making a riskless profit by, for example, buying
stock on one market and selling it immediately
at a higher price on another market, to outperform the market).
Pricing Problems
How valid are the assumptions of the Black &
Scholes model in the context of insurance?
The duration of the cover and the volatility calculation are undoubtedly the most problematic
assumptions when applying this model to insurance.
The Parameters
Pricing an option using the Black & Scholes
model is derived from the five following parameters:
The Black & Scholes model is normally used
for short intervals of time (2, 3, 6 months) and
not for periods covering several years as is the
case for life insurance. As market conditions
are constantly changing, predicting market
behavior is usually a short-run description, not
extending over a number of years. Thus the
model’s parameters change over time presen-
- Price of the underlying stock
- Exercise price (guaranteed minimum death
benefit)
- Risk-free interest rate
- Duration
- Price volatility of the underlying stock
350
Stock Price
300
250
200
150
100
120
115
110
105
95
100
85
90
80
70
75
65
60
55
45
50
35
40
25
30
20
15
5
0
0
10
50
Time [Days]
Simulation of the stock price using a risk-free interest rate of 3% and a volatility of 25 %.
Thus the price of the option does not depend
on the return from the underlying stock but on
the risk-free rate. This is derived from the
assumptions used in the Black & Scholes
model as it is possible to obtain a perfect riskfree cover. The model does not depend on
expected share returns and ignores any parameter representing market participant preferences. It works whatever the attitude investors
may have towards risks.
It is worth noting that the put is an increasing
function of volatility (the same as for a call)
and also of the strike price (the opposite is
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PartnerRe 2005
Reinsuring the “Guaranteed Minimum Death Benefit”
Attached to Unit-Linked Contracts
ting a completely different price option forecast.
Present market conditions indicate an upward
movement in asset volatility. How should volatility be measured then over relatively long intervals of time?
Note again that in the European-style options
model the underlying stock does not pay dividends so market capitalization is implied.
However, the insurer generally deducts costs
so the total value of the dividends is not reinvested. This creates a bias.
Bearing these problems in mind, it is worthwhile comparing results obtained from the Black &
Scholes model with more pragmatic methods
such as the maximum probable loss or average
mean probable loss or, quite simply, with the
results obtained from using one’s own market
fluctuation predictions.
Reserving
Here we need only revert to the traditional
mathematical reserving formula. The following
is a possible formula.
The present value of future benefits represents the current value of benefits due to the
client in the event of death when the value of
the fund falls below the guaranteed minimum
capital.
Mathematical reserve
=
Present value
of future benefits
–
Present value
of future premiums
If we use the single reinsurance premium
method described above, the formula can be
written as follows:
Mathematical reserve
at time interval t
=
Present value at time interval t
of future benefits
=
This means recalculating the new current value
of future benefits every year. This requires
applying new market measurements and
recalculating the single premium.
We now need to analyze the parameters that
may change from one year to the next more
closely. The price of the underlying security will
obviously vary each year as will the risk-free
rate and the duration. The volatility is readjustable yearly in correlation with empirical data.
Only the exercise price remains constant.
It is quickly apparent that these changes in the
Black & Scholes model parameters may result
PartnerRe 2005
Reinsuring the “Guaranteed Minimum Death Benefit”
Attached to Unit-Linked Contracts
in substantial fluctuations in the evaluation of
the annual mathematical reserves.
Conclusion
We have briefly examined how to reinsure and
reserve guaranteed minimum death benefits
attached to unit-linked contracts. We have
noted that these methods derive from strict
assumptions and that deviations in these
assumptions impact widely on the results
obtained.
Our underwriting methodology is based on the
Black & Scholes model and consequently on
financial derivatives. This means that the reinsurer sells a put anticipating a bullish or flat
stock market whereas the cedant buys a put to
protect against a bearish market, each party
positioning itself accordingly with its market
expectations.
This type of contract requires that we evaluate
the parameters we use as accurately as possible. However, as we have seen, we have to
confront the problem of long-term coverage in
life insurance. This implies that when evaluating the criteria used in pricing this type of risk
we cannot allow for the actual evolution of
market behavior, which will always be different
from what we anticipate today. Furthermore,
there is an important gearing effect in this type
of contract. For example, the total risk exposure could be 40 times the premium.
The reinsurance cover contains an important
financial component. As we have seen, the
reinsurer’s liability is triggered by mortality. The
policyholder needs to die to trigger the reinsurer’s liability but, as we have pointed out, this
mortality risk can be controlled: we can easily
predict the mortality rate of a reasonably sized
portfolio. However, an even greater trigger is
the performance of the stock market at the
time of death. It is obvious that it is the stock
market that governs the reinsurer’s liability.
Furthermore, we have to remember that there
is no spreading of risks between different
forms of contract; on the contrary, the risk is
systemic. As the risk is systematic, it is vital for
the reinsurer to establish the level of potential
loss limits not to be exceeded (accumulation
control). By calculating the maximum amount
to be covered (the total sum at risk, all contracts combined), the reinsurer will be able to
meet all potential liabilities at all times.
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