Coinsurance, Deductibles, and Limits

FREQUENCY AND SEVERITY WITH COVERAGE MODIFICATIONS

8.5 Coinsurance, Deductibles, and Limits

The final common coverage modification is coinsurance. In this case, the insurance company pays a proportion,𝛼, of the loss and the policyholder pays the remaining fraction.

If coinsurance is the only modification, this changes the loss variable𝑋to the payment variable, 𝑌 = 𝛼𝑋. The effect of multiplication has already been covered. When all four items covered in this chapter are present (ordinary deductible, limit, coinsurance, and inflation), we create the following per-loss random variable:

𝑌^𝐿=

⎧⎪

⎪⎨

⎪⎪

⎩

0, 𝑋 < 𝑑

1 +𝑟, 𝛼[(1 +𝑟)𝑋−𝑑], 𝑑

1 +𝑟 ≤𝑋 < 𝑢 1 +𝑟, 𝛼(𝑢−𝑑), 𝑋≥ 𝑢

1 +𝑟.

For this definition, the quantities are applied in a particular order. In particular, the coinsurance is applied last. For the illustrated contract, the policy limit is𝛼(𝑢−𝑑), the

maximum amount payable. In this definition, 𝑢 is the loss above which no additional benefits are paid and is called themaximum covered loss. For the per-payment variable, 𝑌^𝑃 is undefined for𝑋 < 𝑑∕(1 +𝑟).

Previous results can be combined to produce the following theorem, presented without proof.

Theorem 8.7 For the per-loss variable, E(𝑌^𝐿) =𝛼(1 +𝑟)

[ E

(𝑋∧ 𝑢 1 +𝑟

)

−E

(𝑋∧ 𝑑 1 +𝑟

)].

The expected value of the per-payment variable is obtained as E(𝑌^𝑃) = E(𝑌^𝐿)

1 −𝐹_𝑋(

𝑑 1+𝑟

).

Higher moments are more difficult. Theorem 8.8 gives the formula for the second moment. The variance can then be obtained by subtracting the square of the mean.

Theorem 8.8 For the per-loss variable, E[

(𝑌^𝐿)²]

=𝛼²(1 +𝑟)²{E[(𝑋∧𝑢^∗)²] −E[(𝑋∧𝑑^∗)²]

− 2𝑑^∗E(𝑋∧𝑢^∗) + 2𝑑^∗E(𝑋∧𝑑^∗)},

where𝑢^∗ = 𝑢∕(1 +𝑟)and𝑑^∗ = 𝑑∕(1 +𝑟). For the second moment of the per-payment variable, divide this expression by1 −𝐹_𝑋(𝑑^∗).

Proof:From the definition of𝑌^𝐿,

𝑌^𝐿=𝛼(1 +𝑟)[(𝑋∧𝑢^∗) − (𝑋∧𝑑^∗)]

and, therefore, (𝑌^𝐿)²

[𝛼(1 +𝑟)]² = [(𝑋∧𝑢^∗) − (𝑋∧𝑑^∗)]²

= (𝑋∧𝑢^∗)²+ (𝑋∧𝑑^∗)²− 2(𝑋∧𝑢^∗)(𝑋∧𝑑^∗)

= (𝑋∧𝑢^∗)²− (𝑋∧𝑑^∗)²− 2(𝑋∧𝑑^∗)[(𝑋∧𝑢^∗) − (𝑋∧𝑑^∗)]. The final term on the right-hand side can be written as

2(𝑋∧𝑑^∗)[(𝑋∧𝑢^∗) − (𝑋∧𝑑^∗)] = 2𝑑^∗[(𝑋∧𝑢^∗) − (𝑋∧𝑑^∗)].

To see this, note that when 𝑋 < 𝑑^∗, both sides equal zero; when𝑑^∗ ≤ 𝑋 < 𝑢^∗, both sides equal2𝑑^∗(𝑋−𝑑^∗); and when 𝑋 ≥ 𝑢^∗, both sides equal2𝑑^∗(𝑢^∗−𝑑^∗). Make this substitution and take expectations on each side to complete the proof.² □

2Thanks to Ken Burton for providing this improved proof.

EXAMPLE 8.7

Determine the mean and standard deviation per loss for a Pareto distribution with 𝛼= 3and𝜃= 2,000with a deductible of 500 and a policy limit of 2,500. Note that the maximum covered loss is𝑢= 3,000.

From earlier examples, E(𝑋∧ 500) = 360and E(𝑋∧ 3,000) = 840.The second limited moment is

E[(𝑋∧𝑢)²] =

∫

𝑢 0

𝑥² 3(2,000)³

(𝑥+ 2,000)⁴𝑑𝑥+𝑢²

( 2,000 𝑢+ 2,000

)3

= 3(2,000)³

∫

𝑢+2,000 2,000

(𝑦− 2,000)²𝑦⁻⁴𝑑𝑦+𝑢²

( 2,000 𝑢+ 2,000

)3

= 3(2,000)³ (

−𝑦⁻¹+ 2,000𝑦⁻²− 2,000² 3 𝑦⁻³||

|||

𝑢+2,000

2,000

)

+𝑢²

( 2,000 𝑢+ 2,000

)3

= 3(2,000)³ [

− 1

𝑢+ 2,000 + 2,000

(𝑢+ 2,000)²− 2,000² 3(𝑢+ 2,000)³

]

+ 3(2,000)³ [ 1

2,000− 2,000

2,000² + 2,000² 3(2,000)³

]

+𝑢²

( 2,000 𝑢+ 2,000

)3

= (2,000)²−

( 2,000 𝑢+ 2,000

)3

(2𝑢+ 2,000)(𝑢+ 2,000). Then, E[(𝑋∧ 500)²] = 160,000and E[(𝑋∧ 3,000)²] = 1,440,000, and so

E(𝑌) = 840 − 360 = 480,

E(𝑌²) = 1,440,000 − 160,000 − 2(500)(840) + 2(500)(360) = 800,000

for a variance of800,000 − 480²= 569,600and a standard deviation of 754.72. □ 8.5.1 Exercises

8.16 (*) You are given that𝑒(0) = 25,𝑆(𝑥) = 1 −𝑥∕𝑤,0≤𝑥≤𝑤, and𝑌^𝑃 is the excess loss variable for𝑑= 10. Determine the variance of𝑌^𝑃.

8.17 (*) The loss ratio(𝑅)is defined as total losses(𝐿)divided by earned premiums(𝑃).

An agent will receive a bonus(𝐵)if the loss ratio on his business is less than 0.7. The bonus is given as𝐵 =𝑃(0.7 −𝑅)∕3if this quantity is positive; otherwise, it is zero. Let 𝑃 = 500,000and𝐿have a Pareto distribution with parameters𝛼= 3and𝜃 = 600,000.

Determine the expected value of the bonus.

8.18 (*) Losses this year have a distribution such that E(𝑋∧ 𝑑) = −0.025𝑑²+ 1.475𝑑− 2.25for𝑑 = 10,11,12,…,26. Next year, losses will be uniformly higher by 10%. An insurance policy reimburses 100% of losses subject to a deductible of 11 up to a maximum reimbursement of 11. Determine the ratio of next year’s reimbursements to this year’s reimbursements.

8.19 (*) Losses have an exponential distribution with a mean of 1,000. An insurance company will pay the amount of each claim in excess of a deductible of 100. Determine the variance of the amount paid by the insurance company for one claim, including the possibility that the amount paid is zero.

8.20 (*) Total claims for a health plan have a Pareto distribution with𝛼= 2and𝜃 = 500.

The health plan implements an incentive to physicians that will pay a bonus of 50% of the amount by which total claims are less than 500; otherwise, no bonus is paid. It is anticipated that with the incentive plan the claim distribution will change to become Pareto with𝛼= 2and𝜃=𝐾. With the new distribution, it turns out that expected claims plus the expected bonus is equal to expected claims prior to the bonus system. Determine the value of𝐾.

8.21 (*) In year𝑎, total expected losses are 10,000,000. Individual losses in year𝑎have a Pareto distribution with 𝛼 = 2 and𝜃 = 2,000. A reinsurer pays the excess of each individual loss over 3,000. For this, the reinsurer is paid a premium equal to 110% of expected covered losses. In year𝑏, losses will experience 5% inflation over yeara, but the frequency of losses will not change. Determine the ratio of the premium in year𝑏to the premium in year𝑎.

8.22 (*) Losses have a uniform distribution from 0 to 50,000. There is a per-loss deductible of 5,000 and a policy limit of 20,000 (meaning that the maximum covered loss is 25,000).

Determine the expected payment given that a payment has been made.

8.23 (*) Losses have a lognormal distribution with𝜇 = 10and𝜎 = 1. For losses below 50,000, no payment is made. For losses between 50,000 and 100,000, the full amount of the loss is paid. For losses in excess of 100,000, the limit of 100,000 is paid. Determine the expected cost per loss.

8.24 (*) The loss severity random variable𝑋has an exponential distribution with mean 10,000. Determine the coefficient of variation of the variables 𝑌^𝑃 and 𝑌^𝐿 based on 𝑑= 30,000.

8.25 (*) The claim size distribution is uniform over the intervals (0,50),(50,100), (100,200), and (200,400). Of the total probability, 30% is in the first interval, 36%

in the second interval, 18% in the third interval, and 16% in the fourth interval. Determine E[(𝑋∧ 350)²].

8.26 (*) Losses follow a two-parameter Pareto distribution with𝛼= 2and𝜃= 5,000. An insurance policy pays the following for each loss. There is no insurance payment for the first 1,000. For losses between 1,000 and 6,000, the insurance pays 80%. Losses above 6,000 are paid by the insured until the insured has made a total payment of 10,000. For

any remaining part of the loss, the insurance pays 90%. Determine the expected insurance payment per loss.

8.27 (*) The amount of a loss has a Poisson distribution with mean𝜆= 3. Consider two insurance contracts. One has an ordinary deductible of 2. The second one has no deductible and a coinsurance in which the insurance company pays𝛼of the loss. Determine the value of𝛼so that the expected cost of the two contracts is the same.

8.28 (*) The amount of a loss has cdf𝐹(𝑥) = (𝑥∕100)²,0≤𝑥≤100. An insurance policy pays 80% of the amount of a loss in excess of an ordinary deductible of 20. The maximum payment is 60 per loss. Determine the expected payment, given that a payment has been made.