8 Optimising reinsurance contracts for a group of independent risks based on minimising the variance of the direct insurer’s net profit – finding the optimal relative retentions
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Risk sharing – reinsurance and deductibles
compound Poisson distribution, so let S i ∼ CP(λi , F Xi ), i = 1, . . . , n, where Xi
is the claim amount variable for risk i. Let Xi have probability density function
fi and distribution function F i , where Fi (x) = 0 for x ≤ 0 and Fi (x) < 1 for all
finite x (that is, the claim amount is not bounded above).
For each risk i, assume there is an excess of loss reinsurance arrangement
with retention Mi , i = 1, . . . , n. For risk i, let the premium charged by the direct
insurer to cover the risk, the direct insurer’s payout, the reinsurer’s payout and
the reinsurer’s security loading be denoted Pi , S iI , S iR and ψi respectively. Let
M = (M1 , . . . , Mn ).
The reinsurer’s premium for risk i is given by
(1 + ψi ) × E[S iR ] = (1 + ψi ) × λi E[reinsurer’s payout on claim Xi ]
= (1 + ψi )λi
∞
(x − Mi ) fi (x)dx.
Mi
The direct insurer’s expected payout on risk i is
E[S iI ] = λi E[direct insurer’s payout on claim Xi ]
Mi
= λi
x fi (x)dx + Mi (1 − Fi (Mi )) .
0
The direct insurer’s net profit, say IP(M), is given by
n
IP(M) =
Pi − (1 + ψi )E[S iR ] − S iI .
i=1
We want to minimise Var[IP(M)], subject to the constraint E[IP(M)] = c.
First, we note expressions for the required quantities:
n
E[IP(M)] =
Pi − (1 + ψi )λi
∞
(x − Mi ) fi (x)dx
Mi
i=1
Mi
− λi
x fi (x)dx + Mi (1 − F i (Mi ))
0
and
n
Var[IP(M)] =
Var[S iI ]
i=1
n
=
λi E[(direct insurer’s payout on claim Xi )2 ]
i=1
n
Mi
λi
=
i=1
0
x2 fi (x)dx + Mi2 (1 − Fi (Mi )) .
5.8 Finding the optimal relative retentions
249
We use the method of Lagrange multipliers to perform the constrained
optimisation. Let
h(M) = Var[IP(M)] − γ E[IP(M)] − c .
We set ∂h/∂Mi = 0, i = 1, . . . , n. The derivatives are given by
∂
Var[IP(M)] = λi Mi2 fi (Mi ) + 2Mi (1 − Fi (Mi )) − Mi2 fi (Mi )
∂Mi
= 2λi Mi (1 − F i (Mi ))
and
∂
E[IP(M)] = ψi λi (1 − F i (Mi )).
∂Mi
These expressions yield
∂h
= 2λi Mi (1 − Fi (Mi )) − γψi λi (1 − Fi (Mi )),
∂Mi
and hence
∂h
= 0 ⇔ 2Mi (1 − Fi (Mi )) = γψi (1 − Fi (Mi )).
∂Mi
The claim amount is unbounded, which means that 1 − Fi (Mi )
0. Hence
∂h
= 0 ⇔ Mi = γψi /2 = θψi for some θ (= γ/2).
∂Mi
It is easy to show that ∂2 h/∂Mi2 > 0, i = 1, . . . , n, at the turning point; for
a function of the form of h(·), this is suﬃcient to confirm that we have a
minimum. Hence the optimal relative retentions are given by
Mi = θψi , i = 1, . . . , n , for some θ.
(5.27)
The result is very simple indeed – the relative retentions for the risks are simply proportional to the reinsurer’s security loadings and do not depend on the Pi
or the distributions of the S i . The actual values of the Mi are obtained from the
constraint E[IP(M)] = c, using a specified value for c. We note from the result
Mi ∝ ψi (from (5.27)) that higher reinsurance costs (higher ψi ) correspond to
higher retentions for the direct insurer.
It may be convenient to use an alternative form for IP(M), which comes
from writing S iI = S i − S iR and which gives
n
IP(M) =
Pi − (1 + ψi )E[S iR ] − (S i − S iR ) .
i=1
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Risk sharing – reinsurance and deductibles
From this we get
n
E[IP(M)] =
Pi − E[S i ] − ψi E[S iR ] .
(5.28)
i=1
It may be easier to calculate the value of this expression, which involves E[S iR ],
than the original expression, which also involves E[S iI ].
We note from this expression that if the direct insurer wants a higher
expected profit, this corresponds to having a lower E[S iR ] and hence a higher
E[S iI ], which in turn corresponds to higher retentions, exposure to more risk
and reduced security.
Example 5.13
Consider three risks:
S 1 ∼ CP(100, F X1 ),
S 2 ∼ CP(200, F X2 ),
S 3 ∼ CP(100, F X3 ),
where X1 , X2 and X3 have exponential distributions with means 1, 2 and 3,
respectively. The direct insurers premium is calculated using a 20% security
loading on the expected aggregate claim amount, while the reinsurer’s premiums are calculated using loadings of 30%, 40% and 50%, respectively, on the
reinsurer’s expected payouts for the three risks.
We want to arrange excess of loss reinsurance such that Var[IP(M)] is minimised, subject to the requirement that the direct insurer’s expected profit be
40. We find immediately from the above result that the optimal retentions
satisfy M1 = 0.3θ, M2 = 0.4θ and M3 = 0.5θ, for some θ.
Now (see Example 5.3) for X ∼ Exp(1/μ) (with mean μ), and with retention
M, we have E[Z ∗ ] = μ, Pr(X > M) = e−M/μ and E[Z] = μe−M/μ . Here we have
E[S 1 ] = 100, E[S 2 ] = 400 and E[S 3 ] = 300, and the direct insurer’s premiums
are P1 = 120, P2 = 480 and P3 = 360. We also have
E[S 1R ] = 100 × 1 × e−0.3θ/1 = 100e−0.3θ ,
E[S 2R ] = 200 × 2e−0.4θ/2 = 400e−θ/5 ,
E[S 3R ] = 100 × 3e−0.5θ/3 = 300e−θ/6 .
Hence, using (5.28) we have
E[IP(M)] = (120 − 100 − 0.3 × 100e−0.3θ )
+ (480 − 400 − 0.4 × 400e−0.2θ )
+ (360 − 300 − 0.5 × 300e−θ/6 )
= 160 − 30e−0.3θ − 160e−0.2θ − 150e−θ/6 .
Setting this equal to 40, and solving numerically (by simple computer evaluation and search, or the use of a mathematics package, or a formal iterative
5.8 Finding the optimal relative retentions
251
solving technique), we find θ = 5.455, from which we find the optimal
retention levels to be M1 = 1.64, M2 = 2.18 and M3 = 2.73.
Suppose the direct insurer had required an expected profit of 30 instead of
40. We find θ = 5.030, and the optimal retention levels are lower, at M1 = 1.51,
M2 = 2.01 and M3 = 2.52 (lower profit corresponds to reduced exposure to
risk and greater security).
5.8.2 Optimal relative retentions in the case of
proportional reinsurance
Suppose we have n independent risks, with aggregate claims for risk i being
denoted S i , i = 1, . . . , n. Suppose that for each i there is a proportional reinsurance arrangement in place. For risk i, let the premium charged by the direct
insurer to cover the risk, the retention level (direct insurer’s retained proportion), the direct insurer’s payout, the reinsurer’s payout and the reinsurer’s
security loading be denoted Pi , βi , S iI , S iR and ψi , respectively. Thus we have
S iI = βi S i and S iR = (1 − βi )S i .
The reinsurer’s premium for risk i is
(1 + ψi ) × E[S iR ] = (1 + ψi )(1 − βi )E[S i ].
The direct insurer’s expected payout on risk i is E[S iI ] = βi E[S i ]. Write β for
(β1 , . . . , βn ). Then the direct insurer’s net profit, IP(β) say, is given by
n
Pi − (1 + ψi )(1 − βi )E[S i ] − S iI .
IP(β) =
i=1
We want to minimise Var[IP(β)] subject to the constraint E[IP(β)] = c.
First, we note expressions for the required quantities:
n
E[IP(β)] =
{Pi − (1 + ψi )(1 − βi )E[S i ] − βi E[S i ]}
i=1
n
=
{Pi − (1 + ψi − ψi βi )E[S i ]}
i=1
and
n
Var[IP(β)] =
n
Var[S iI ] =
i=1
β2i Var[S i ].
i=1
We again use the method of Lagrange multipliers to perform the constrained
optimisation. Let
h(β) = Var[IP(β)] − γ E[IP(β)] − c .
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Risk sharing – reinsurance and deductibles
We set ∂h/∂βi = 0, i = 1, . . . , n. The derivatives are given by
∂h
= 2βi Var[S i ] − γψi E[S i ],
∂βi
and hence
∂h
1
E[S i ]
.
= 0 ⇔ 2βi Var[S i ] = γψi E[S i ] ⇔ βi = γψi
∂βi
2
Var[S i ]
Hence the optimal relative retentions are given by
βi = θψi
E[S i ]
, i = 1, . . . , n , for some θ.
Var[S i ]
(5.29)
Clearly ∂2 h/∂β2i > 0, i = 1, . . . , n; this is suﬃcient for a function of the form
of h(·) to confirm that we have a minimum. The actual values of the βi are
obtained from the constraint E[IP(β)] = c, using a specified value for c.
The relative retentions for the risks are proportional to the reinsurer’s security loadings and do not depend on the Pi . We note from (5.29) that higher
reinsurance costs (higher ψi , higher ψi E[S i ]) correspond to higher retentions
for the direct insurer, while higher uncertainty in the aggregate claims (higher
Var[S i ]) corresponds to lower retentions. We note also (from the expression for
E[IP(β)] above) that if the direct insurer wants a higher expected profit, this
corresponds to having higher retentions, and thus exposure to more risk and
reduced security.
Note that it is possible that this approach to optimising the proportions to be
retained by the direct insurer may result in one or more of them turning out to
exceed 1. If we do get β1 > 1 (say) then what we would do in practice is set
β1 = 1 and use this value in the calculations based on E[IP(β)] = c.
Example 5.14 Consider three risks with the same means and variances as
the risks in Example 5.13, so E[S 1 ] = 100, Var[S 1 ] = 200, E[S 2 ] = 400,
Var[S 2 ] = 1600, E[S 3 ] = 300 and Var[S 3 ] = 1800.
Suppose, also as in Example 5.13, that the direct insurers premium is calculated using a 20% security loading on the expected aggregate claim amount,
while the reinsurer’s premiums are calculated using loadings of 30%, 40% and
50%, respectively on the reinsurer’s expected payouts for the three risks.
We want to arrange proportional reinsurance using the above criterion,
and also requiring that the direct insurer’s expected profit is 30. We find
immediately from (5.29) that the optimal retentions satisfy
β1 = 0.3 × (100/200)θ = 3θ/20,
β2 = 0.4 × (400/1600)θ = θ/10,
β3 = 0.5 × (300/1800)θ = θ/12.
Exercises
253
Then
E[IP(β)] = 120 − (1.3 − 0.3 × 3θ/20) × 100
+ 480 − (1.4 − 0.4 × θ/10) × 400
+ 360 − (1.5 − 0.5 × θ/12) × 300
= 33θ − 180.
Setting this equal to 30, we find θ = 70/11, from which we find the optimal
retentions to be β1 = 0.955, β2 = 0.636 and β3 = 0.530.
Suppose the direct insurer had required an expected profit of 40 instead
of 30. We find the optimal retentions are higher, at β1 = 1 (the insurer now
retains 100% of risk 1), β2 = 0.667 and β3 = 0.556 (higher expected profit
corresponds to higher exposure to risk and reduced security).
Exercises
5.1
5.2
5.3
Claim amounts from a general insurance portfolio are lognormally distributed with mean £1000 and standard deviation £2000. Excess of
loss reinsurance with retention level £1750 is in place. Calculate the
probability that the reinsurer is involved in a claim.
Claim amounts have a Pa(5, 4) distribution, and excess of loss reinsurance with retention level M is in place. Calculate the value of M such
that the mean amounts paid by the direct insurer and the reinsurer on a
claim are equal.
Suppose claim amounts are distributed as X ∼ Exp(1/μ) and an excess
of loss contract with retention level M is in place. Let Y and Z be the
amounts paid out on a claim by the direct insurer and the reinsurer,
respectively. Let p = Pr(X > M) = e−M/μ .
(a) Show that the moment generating function MY (t) of Y is given by
MY (t) =
1
{1 − μpte Mt },
1 − μt
and hence, or otherwise, show that E[Y 2 ] is given by
E[Y 2 ] = 2μ{μ(1 − p) − M p}.
(b) Show that E[Z 2 ] is given by
E[Z 2 ] = 2μ2 p.
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5.4
Risk sharing – reinsurance and deductibles
Suppose claim amounts are distributed as X ∼ Pa(α, λ) and an excess
of loss contract with retention level M is in place. Let Y and Z be the
amounts paid out on a claim by the direct insurer and the reinsurer,
respectively.
Show that E[Y 2 ] and E[Z 2 ] are given by
E[Y 2 ] =
2λ
λ(1 − ηα−2 ) − (α − 2)Mηα−1
(α − 1)(α − 2)
and
E[Z 2 ] =
5.5
2λ2 ηα−2
,
(α − 1)(α − 2)
where η = λ/(λ + M).
The claim amounts arising from a risk have a Pa(α, λ) distribution, where
the value of λ is known. The direct insurer has arranged an excess of
loss reinsurance contract with retention level M. In the past year, there
were a total of m reinsurance claims, that is there were m claims which
exceeded M and hence involved the reinsurer. The amounts of these
claims are known to the reinsurer – suppose the amounts paid by the
reinsurer on these claims are r1 , r2 , . . . , rm , and this is all the information
the reinsurer has.
(a) Show that the maximum likelihood estimator of α, based on the
information available to the reinsurer, is given by
⎛ m
⎞−1
⎜⎜⎜
⎟⎟
αˆ = m ⎜⎜⎝ log(λ + M + ri ) − m log(λ + M)⎟⎟⎟⎠ .
i=1
(b) In a particular situation, with λ = 5000 and M = £6000, a total of
176 claims exceeded £6000 last year, and for these claims
176
log(ri + 11 000) = 1693.3,
i=1
5.6
where the {ri } are defined as above.
(1) Show that the maximum likelihood estimate of α is 3.171.
(2) Using the fitted model, calculate the probability that a claim will
involve the reinsurer, show that the mean amount of reinsurance claims is £11 067, and hence, or otherwise, find the mean
amount of claims which do not involve the reinsurer.
An insurer covers an individual loss X with excess of loss reinsurance in
place with retention level M. The insurer pays Y and the reinsurer pays
Z = X − Y. Show that
Exercises
YZ =
0
M(X − M)
255
if X ≤ M
if X > M.
Hence show that E[YZ] = ME[Z] and Cov[Y, Z] ≥ 0. Deduce that
Var[X] ≥ Var[Y] + Var[Z].
5.7
Let X be a random variable with a lognormal distribution with parameters μ and σ and probability density function f (x). Verify the following
result, which is needed for some reinsurance calculations: for any real
number c > 0 and k = 0, 1, 2, . . .
c
0
5.8
xk f (x)dx = exp kμ +
k 2 σ2
log c − μ − kσ2
Φ
.
2
σ
The aggregate claims for a risk S , in units of £1000, has a compound
Poisson distribution S ∼ CP(100, F X ), where X ∼ Pa(5, 4). The direct
insurer has an excess of loss reinsurance contract in place, with retention level M, where M is the upper 5% point of the individual claims
distribution.
(a) Show that M = £3282 and specify the distribution of the amount
paid by the reinsurer on a reinsurance claim (that is, on a claim which
involves the reinsurer).
(b) Specify the distribution of S R , the aggregate annual claim amount
paid by the reinsurer, and find the mean and standard deviation of S R .
(c) Find the mean of S I , the aggregate annual claim amount paid by the
direct insurer.
5.9 Suppose aggregate claims S ∼ CP(100, FX ), where the claim amount
X is measured in units of £1000 and modelled as a Pa(6, 10) random
variable. The direct insurer is considering entering into one of two different types of reinsurance contract – excess of loss with retention M,
and proportional with retained proportion β. The direct insurer wants an
expected payout of £1500 on a claim. The reinsurer’s premium is to be
chosen such that there is a probability of 0.95 that the reinsurer makes a
profit on the business, and you may assume that the reinsurer’s aggregate
payout can be approximated by a normal distribution.
(a) Show that β = 0.75 and M = 3.195.
(b) Calculate the reinsurer’s premium for each arrangement; comment
on the answers.
5.10 The distribution of total annual claims S on a general insurance portfolio
is to be modelled as a compound Poisson distribution S ∼ CP(1000, F X ).
The direct insurer has an excess of loss reinsurance contract in place,
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Risk sharing – reinsurance and deductibles
with retention level £4000 on each claim. Consider the following models
(all with the same mean) for a loss X (in units of £1000).
Model 1: X has an exponential distribution with mean 2.
Model 2: X has a Pareto distribution with parameters α = 4 and λ = 6.
Model 3: X has a lognormal distributon with parameters μ = 0.2877 and
σ = 0.9005.
(a) State, or derive, the distribution of Z ∗ ≡ X −4 | X > 4 using Model 1.
(b) State, or derive, the distribution of Z ∗ ≡ X −4 | X > 4 using Model 2.
(c) Calculate the values of the mean and standard deviation of the total
annual claims paid by the reinsurer under each of the three models.
(Note: the calculations for Model 3 require the result of Exercise 5.7
above and are the most onerous.)
5.11 Consider the aggregate claims model S ∼ CP(λ, F X ), where X = Y + Z
and S = S I + S R , in the usual notation for the amounts paid out by
the direct insurer and the reinsurer, respectively, in the presence of a
reinsurance contract. In particular, suppose there is an excess of loss
reinsurance contract in place with retention level M on each claim. Let
p = Pr(X > M), and let Z ∗ denote the amount paid by the reinsurer on a
reinsurance claim.
From (5.8) we know that E[Z ∗ ] = (1/p)E[Z]. Show that
Var[Z ∗ ] =
1− p
1
Var[Z] −
{E[Z]}2 .
p
p2
5.12 Suppose the annual aggregate claims S from a portfolio of risks has a
compound Poisson distribution, with Poisson parameter 100. Each claim
which arises comes from one of two separate sub-portfolios and is of
one of two types: the amounts of type 1 claims have an exponential
distribution with mean 1, while those of type 2 claims have an exponential distribution with mean 2, where 60% of claims are of type 1. Let
X represent a randomly selected claim, so we have S ∼ CP(100, F X ).
Excess of loss reinsurance is in place, with retention level 1.8 on each
claim. Let X = Y + Z and S = S I + S R in the usual notaion.
(a) By using conditional expectation arguments with X, Y and Z, verify
the following values:
E[X] = 1.4
E[X 2 ] = 4.4,
E[S ] = 140
Var[S ] = 440,
E[Y] = 0.9756 E[Y 2 ] = 1.3727,
E[Z] = 0.4244 E[Z 2 ] = 1.4994,
E[S I ] = 97.56 Var[S I ] = 137.27,
E[S R ] = 42.44 Var[S R ] = 149.94.
Exercises
257
(b) Suppose the heterogeneity of the portfolio is not recognised, and
the distribution of claim amounts is represented by the single variable X which is exponentially distributed with mean 1.4. Recalculate
all the moments in part (a) and comment on the consequences for
the reinsurer of being unaware of the heterogeneity present in the
system.
5.13 Consider a portfolio of private motor policies. In the event of an accident
or other incident covered by the policy, the loss (cost of repairs and/or
replacement of parts or the whole vehicle) has a Pa(α, λ) distribution. A
deductible (excess) of £150 is applied to all losses – no claim is made
if the loss is less than £150; otherwise a claim is always made (for the
loss less the deductible). A sample of 100 claims has mean £1210 and
standard deviation £1790.
(a) Calculate method of moments estimates of α and λ.
(b) Estimate the proportion of losses that do not lead to claims.
(c) Excess of loss insurance is to be arranged with another company so
that the direct insurer’s mean payout on claims is reduced to £1000.
Find the retention limit the direct insurer must set on individual
claims to achieve the required reduction in the direct insurer’s mean
payout.
5.14 The annual aggregate claims S from a risk has a compound Poisson distribution S ∼ CP(200, F X ), where the individual claim X is modelled
crudely as taking the value £1000 or £5000, with probabilities 0.75 and
0.25, respectively. The direct insurer’s premium is calculated using the
expected value principle, with relative security loading factor 0.3.
The direct insurer wants to arrange excess of loss reinsurance for this
risk. The reinsurer’s premiums are calculated using the expected value
principle, but with a loading factor 0.5.
Let IP(M) denote the direct insurer’s annual profit, net of reinsurance
costs, under a contract with retention level for a claim set at M. Suppose
that M is a value between £1000 and £5000 chosen such that the variable
IP(M) has coeﬃcient of variation 1/3, that is
standard deviation(IP(M)) 1
= .
mean(IP(M))
3
(a) Show that IP(M) satisfies
IP(M) = 145 000 + 75M − S I ,
where S I is the aggregate claims paid out by the direct insurer, and
hence show that that M = £3557.
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Risk sharing – reinsurance and deductibles
(b) Using the value of M found in (a), and assuming normality as
required, calculate the approximate probability of each of the following two events, quoting your calculated probabilities to three decimal
places:
(1) the direct insurer’s annual profit is positive;
(2) the reinsurer’s annual profit is positive.
5.15 Consider the aggregate claims model S ∼ CP(λ, F X ). A reinsurance
arrangment is in place, defined at the level of individual claims. Let Y
and S I represent, respectively, the amount of an individual claim and the
aggregate amount, paid by the direct insurer; similarly for Z and S R for
the reinsurer. So X = Y + Z and S = S I + S R .
(a) By considering Var[S I + S R ], show that the covariance between S I
and S R is given by Cov[S I , S R ] = λE[YZ] and hence find an expression for the correlation coeﬃcient between S I and S R in terms of
moments of Y and Z.
(b) Calculate the correlation coeﬃcient between the direct insurer’s and
the reinsurer’s aggregate claim amounts in Example 5.5.
(c) Calculate the correlation coeﬃcient between the direct insurer’s and
the reinsurer’s aggregate claim amounts in Example 5.6.
5.16 A life insurance company covers 1000 lives for one-year term insurance
in amounts (in units of £100 000) as shown below:
Benefit amount
1
2
Number of insured lives 600 400
The insured lives can be assumed to be independent, with a single
probability of a claim of 0.025 applying to all lives.
(a) Find the mean and standard deviation of the total claim amount,
S , and hence calculate the (approximate) probability that the direct
insurer’s total payout on this business exceeds £4.5 million.
(b) A reinsurance contract is arranged with the aim of reducing both
the uncertainty in the direct insurer’s payout and the probability
calculated in (a). The direct insurer sets a retention level of £160 000
and purchases the necessary cover at a cost of £0.0275 per £1
of cover.
(1) Calculate the cost of the reinsurance.
(2) Find the mean and standard deviation of the direct insurer’s
total payout on claims with the reinsurance in place, and hence
calculate the revised (approximate) probability that the direct
insurer’s total payout on this business exceeds £4.5 million;
comment briefly on the eﬀect of the reinsurance.