Ordinary Deductible

Calculating the variance of insurance payment

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The post supplements a three-part discussion on the mathematical models of insurance payments: part 1, part 2 and part 3. This post focuses on the calculation of the variance of insurance payments.

There are three practice problem sets for the 3-part discussion on the mathematical models of insurance payments – problem set 7, problem set 8 and problem set 9. Problems in these problem sets are on calculation of expected payments. We present several examples in this post on variance of insurance payment. A practice problem set will soon follow.

Coverage with an Ordinary Deductible

To simplify the calculation, the only limit on benefits is the imposition of a deductible. Suppose that the loss amount is the random variable X. The deductible is d. Given that a loss has occurred, the insurance policy pays nothing if the loss is below d and pays X-d if the loss exceeds d. The payment random variable is denoted by Y_L or (X-d)_+ and is explicitly described as follows:

(1)……\displaystyle  Y_L=(X-d)_+=\left\{ \begin{array}{ll}                     \displaystyle  0 &\ X \le d \\           \text{ } & \text{ } \\           \displaystyle  X-d &\ X > d           \end{array} \right.

The subscript L in Y_L is to denote that this variable is the payment per loss. This means that its mean, E[Y_L], is the average payment over all losses. A related payment variable is Y_P which is defined as follows:

(2)……\displaystyle Y_P=X-d \ \lvert X > d

The variable Y_P is a truncated variable (any loss that is less than the deductible is not considered) and is also shifted (the payment is the loss less the deductible). As a result, Y_P is a conditional distribution. It is conditional on the loss exceeding the deductible. The subscript P in Y_P indicates that the payment variable is the payment per payment. This means that its mean, E[Y_P], is the average payment over all payments that are made, i.e. average payment over all losses that are eligible for a claim payment.

The focus of this post is on the calculation of E[Y_L] (the average payment over all losses) and Var[Y_L] (the variance of payment per loss). These two quantities are important in the actuarial pricing of insurance. If the policy were to pay each loss in full, the average amount paid would be E[X], the average of the loss distribution. Imposing a deductible, the average amount paid is E[Y_L], which is less than E[X]. On the other hand, Var[Y_L], the variance of the payment per loss, is smaller than Var[X], the variance of the loss distribution. Thus imposing a deductible not only reduces the amount paid by the insurer, it also reduces the variability of the amount paid.

The calculation of E[Y_L] and Var[Y_L] can be done by using the pdf f(x) of the original loss random variable X.

(3)……\displaystyle E[Y_L]=\int_d^\infty (x-d) \ f(x) \ dx

(4)……\displaystyle E[Y_L^2]=\int_d^\infty (x-d)^2 \ f(x) \ dx

(5)……\displaystyle Var[Y_L]=E[Y_L^2]-E[Y_L]^2

The above calculation assumes that the loss X is a continuous random variable. If the loss is discrete, simply replace integrals by summation. The calculation in (3) and (4) can also be done by integrating the pdf of the payment variable Y_L.

(6)……\displaystyle  f_{Y_L}(y)=\left\{ \begin{array}{ll}                     \displaystyle  0 &\ y=0 \\           \text{ } & \text{ } \\           \displaystyle  f(y+d) &\ y > 0           \end{array} \right.

(7)……\displaystyle E[Y_L]=\int_0^\infty y \ f_{Y_L}(y) \ dy

(8)……\displaystyle E[Y_L^2]=\int_0^\infty y^2 \ f_{Y_L}(y) \ dy

It will be helpful to also consider the pdf of the payment per payment variable Y_P.

(9)……\displaystyle f_{Y_P}(y)=\frac{f(y+d)}{P[X > d]} \ \ \ \ \ \ \ y>0

Three Approaches

We show that there are three different ways to calculate E[Y_L] and Var[Y_L].

  1. Using basic principle.
  2. Considering Y_L as a mixture.
  3. Considering Y_L as a compound distribution.

Using basic principle refers to using (3) and (4) or (7) and (8). The second approach is to treat Y_L as a mixture of a point mass of 0 with weight P(X \le d) and the payment per payment Y_P with weight P(X >d). The third approach is to treat Y_L as a compound distribution where the number of claims N is a Bernoulli distribution with p=P(X >d) and the severity is the payment Y_P. We demonstrate these approaches with a series of examples.

Examples

Example 1
The random loss X has an exponential distribution with mean 50. A coverage with a deductible of 25 is purchased to cover this loss. Calculate the mean and variance of the insurance payment per loss.

We demonstrate the calculation using the three approaches discussed above. The following gives the calculation based on basic principles.

    \displaystyle \begin{aligned} E[Y_L]&=\int_{25}^\infty (x-25) \ \frac{1}{50} \ e^{-x/50} \ dx \\&=\int_{0}^\infty \frac{1}{50} \ u \ e^{-u/50} \ e^{-1/2} \ du \\&=50 \ e^{-1/2} \int_{0}^\infty \frac{1}{50^2} \ u \ e^{-u/50}  \ du \\&=50 \ e^{-1/2}=30.33  \end{aligned}

    \displaystyle \begin{aligned} E[Y_L^2]&=\int_{25}^\infty (x-25)^2 \ \frac{1}{50} \ e^{-x/50} \ dx \\&=\int_{0}^\infty \frac{1}{50} \ u^2 \ e^{-u/50} \ e^{-1/2} \ du \\&=2 \cdot 50^2 \ e^{-1/2} \int_{0}^\infty \frac{1}{2} \ \frac{1}{50^3} \ u^2 \ e^{-u/50}  \ du \\&=2 \cdot 50^2 \ e^{-1/2}  \end{aligned}

    \displaystyle Var[Y_L]=2 \cdot 50^2 \ e^{-1/2}-\biggl( 50 \ e^{-1/2} \biggr)^2=2112.954696

In the above calculation, we perform a change of variable via u=x-25. We now do the second approach. Note that the variable Y_P=X-25 \lvert X >25 also has an exponential distribution with mean 50 (this is due to the memoryless property of the exponential distribution). The point mass of 0 has weight P(X \le 25)=1-e^{-1/2} and the variable Y_P has weight P(X > 25)=e^{-1/2}.

    \displaystyle \begin{aligned} E[Y_L]&=0 \cdot (1-e^{-1/2})+E[Y_P] \cdot e^{-1/2}=50 \ \cdot e^{-1/2}  \end{aligned}

    \displaystyle \begin{aligned} E[Y_L^2]&=0 \cdot (1-e^{-1/2})+E[Y_P^2] \cdot e^{-1/2} \\&=(50^2+50^2) \cdot e^{-1/2} =2 \ 50^2 \ \cdot e^{-1/2}  \end{aligned}

    \displaystyle Var[Y_L]=2 \cdot 50^2 \ e^{-1/2}-\biggl( 50 \ e^{-1/2} \biggr)^2=2112.954696

In the third approach, the frequency variable N is Bernoulli with P(N=0)=1-e^{-1/2} and P(N=1)=e^{-1/2}. The severity variable is Y_P. The following calculates the compound variance.

    \displaystyle \begin{aligned} Var[Y_L]&=E[N] \cdot Var[Y_P]+Var[N] \cdot E[Y_P]^2 \\&=e^{-1/2} \cdot 50^2+e^{-1/2} (1-e^{-1/2}) \cdot 50^2 \\&=2 \cdot 50^2 \ e^{-1/2}-50^2 \ e^{-1} \\&=2112.954696  \end{aligned}

Note that the average payment per loss is E[Y_L]=30.33, a substantial reduction from the mean E[X]=50 if the policy pays each loss in full. The standard deviation of Y_L is \sqrt{2112.954696}=45.97, which is a reduction from 50, the standard deviation of original loss distribution. Clearly, imposing a deductible (or other limits on benefits) has the effect of reducing risk for the insurer.

When the loss distribution is exponential, approach 2 and approach 3 are quite easy to implement. This is because the payment per payment variable Y_P has the same distribution as the original loss distribution. This happens only in this case. If the loss distribution is any other distribution, we must determine the distribution of Y_P before carrying out the second or the third approach.

We now work two more examples that are not exponential distributions.

Example 2
The loss distribution is a uniform distribution on the interval (0,100). The insurance coverage has a deductible of 20. Calculate the mean and variance of the payment per loss.

The following gives the basic calculation.

    \displaystyle \begin{aligned} E[Y_L]&=\int_{20}^{100} (x-20) \ \frac{1}{100} \ dx \\&=\int_0^{80} \frac{1}{100} \ u \ du =32  \end{aligned}

    \displaystyle \begin{aligned} E[Y_L^2]&=\int_{20}^{100} (x-20)^2 \ \frac{1}{100} \ dx \\&=\int_0^{80} \frac{1}{100} \ u^2 \ du =\frac{5120}{3}  \end{aligned}

    \displaystyle Var[Y_L]=\frac{5120}{3}-32^2=\frac{2048}{3}=682.67

The mean and variance of the loss distribution are 50 and \frac{100^2}{12}=833.33 (if the coverage pays for each loss in full). By imposing a deductible of 20, the mean payment per loss is 32 and the variance of payment per loss is 682.67. The effect is a reduction of risk since part of the risk is shifted to the policyholder.

We now perform the calculation using the the other two approaches. Note that the payment per payment Y_P=X-20 \lvert X > 20 has a uniform distribution on the interval (0,80). The following calculates according to the second approach.

    \displaystyle \begin{aligned} E[Y_L]&=0 \cdot (0.2)+E[Y_P] \cdot 0.8=40 \ \cdot 0.8=32  \end{aligned}

    \displaystyle \begin{aligned} E[Y_L^2]&=0 \cdot (0.2)+E[Y_P^2] \cdot 0.8=\biggl(\frac{80^2}{12}+40^2 \biggr) \ \cdot 0.8=\frac{5120}{3}  \end{aligned}

    \displaystyle Var[Y_L]=\frac{5120}{3}-32^2=\frac{2048}{3}=682.67

For the third approach, the frequency N is a Bernoulli variable with p=0.8 and the severity variable is Y_P, which is uniform on (0,80).

    \displaystyle \begin{aligned} Var[Y_L]&=E[N] \cdot Var[Y_P]+Var[N] \cdot E[Y_P]^2 \\&=0.8 \cdot \frac{80^2}{12} +0.8 \cdot 0.2 \cdot 40^2 \\&=\frac{2048}{3} \\&=682.67  \end{aligned}

Example 3
In this example, the loss distribution is a Pareto distribution with parameters \alpha=3 and \theta=1000. The deductible of the coverage is 500. Calculate the mean and variance of the payment per loss.

Note that the payment per payment Y_P=X-500 \lvert X > 500 also has a Pareto distribution with parameters \alpha=3 and \theta=1500. This information is useful for implementing the second and the third approach. First the calculation based on basic principles.

    \displaystyle \begin{aligned} E[Y_L]&=\int_{500}^{\infty} (x-500) \ \frac{3 \cdot 1000^3}{(x+1000)^4} \ dx \\&=\int_{0}^{\infty} u \ \frac{3 \cdot 1000^3}{(u+1500)^4} \ du \\&=\frac{1000^3}{1500^3} \ \int_{0}^{\infty} u \ \frac{3 \cdot 1500^3}{(u+1500)^4} \ du\\&=\frac{8}{27} \ \frac{1500}{2}\\&=\frac{2000}{9}=222.22  \end{aligned}

    \displaystyle \begin{aligned} E[Y_L^2]&=\int_{500}^{\infty} (x-500)^2 \ \frac{3 \cdot 1000^3}{(x+1000)^4} \ dx \\&=\int_{0}^{\infty} u^2 \ \frac{3 \cdot 1000^3}{(u+1500)^4} \ du \\&=\frac{1000^3}{1500^3} \ \int_{0}^{\infty} u^2 \ \frac{3 \cdot 1500^3}{(u+1500)^4} \ du\\&=\frac{8}{27} \ \frac{2 \cdot 1500^2}{2 \cdot 1}\\&=\frac{2000000}{3}  \end{aligned}

    \displaystyle Var[Y_L]=\frac{2000000}{3}-\biggl(\frac{2000}{9} \biggr)^2=\frac{50000000}{81}=617283.95

Now, the mixture approach (the second approach). Note that P(X > 500)=\frac{8}{27}.

    \displaystyle \begin{aligned} E[Y_L]&=0 \cdot \biggl(1-\frac{8}{27} \biggr)+E[Y_P] \cdot \frac{8}{27}=\frac{1500}{2} \ \cdot \frac{8}{27}=\frac{2000}{9}  \end{aligned}

    \displaystyle \begin{aligned} E[Y_L^2]&=0 \cdot \biggl(1-\frac{8}{27} \biggr)+E[Y_P^2] \cdot \frac{8}{27}=\frac{2 \cdot 1500^2}{2 \cdot 1} \ \cdot \frac{8}{27}=\frac{2000000}{3}  \end{aligned}

    \displaystyle Var[Y_L]=\frac{2000000}{3}-\biggl(\frac{2000}{9} \biggr)^2=\frac{50000000}{81}=617283.95

Now the third approach, which is to calculate the compound variance.

    \displaystyle \begin{aligned} Var[Y_L]&=E[N] \cdot Var[Y_P]+Var[N] \cdot E[Y_P]^2 \\&=\frac{8}{27}  \cdot 1687500 +\frac{8}{27} \cdot \biggl(1-\frac{8}{27} \biggr) \cdot 750^2 \\&=\frac{50000000}{81} \\&=617283.95  \end{aligned}

Remarks

For some loss distributions, the calculation of the variance of Y_L, the payment per loss, can be difficult mathematically. The required integrals for the first approach may not have closed form. For the second and third approach to work, we need to have a handle on the payment per payment Y_P. In many cases, the pdf of Y_P is not easy to obtain or its mean and variance are hard to come by (or even do not exist). For these examples, we may have to find the variance numerically. The examples presented are some of the distributions that are tractable mathematically for all three approaches. These three examples are such that the second and third approaches represent shortcuts for find variance of Y_L because Y_P have a known form and requires minimal extra calculation. For other cases, it is possible that the second or third approach is doable but is not shortcut. In that case, any one of the approaches can be used.

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Mathematical models for insurance payments – part 3 – other modifications

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This post is a continuation of the discussion on models of insurance payments initiated in two previous posts. Part 1 focuses on the models of insurance payments in which the insurance policy imposes a policy limit. Part 2 continues the discussion by introducing models in which the insurance policy imposes an ordinary deductible. In each of these two previous posts, the insurance coverage has only one coverage modification. A more interesting and more realistic scenario would be insurance coverage that contains a combination of several coverage modifications. This post is to examine the effects on the insurance payments as a result of having some or all of these coverage modifications – policy limit, ordinary deductible, franchise deductible and inflation. Additional topics: expected payment per loss versus expected payment per payment and loss elimination ratio.

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Ordinary Deductible and Policy Limit

The previous two posts discuss the expectations E[X \wedge u] and E[(X-d)_+]. The first is the expected insurance payment when the coverage has a policy limit u. The second is the expected insurance payment when the coverage has an ordinary deductible d. They are the expected values of the following two variables.

    \displaystyle  X \wedge u=\left\{ \begin{array}{ll}                     \displaystyle  X &\ X \le u \\           \text{ } & \text{ } \\           \displaystyle  u &\ X > u           \end{array} \right.

    \displaystyle  (X-d)_+=\left\{ \begin{array}{ll}                     \displaystyle  0 &\ X \le d \\           \text{ } & \text{ } \\           \displaystyle  X-d &\ X > d           \end{array} \right.

It is easy to verify that X=(X-d)_+ + X \wedge d. Buying a coverage with an ordinary deductible d and another coverage with policy limit d equals full coverage. Thus we have the following relation.

(1) \ \ \ \ E[(X-d)_+]=E[X]-E[X \wedge d]

The limited expectation E[X \wedge d] has expressions in closed form in some cases or has expressions in terms of familiar functions (e.g. gamma function) in other cases. Thus the expectation E[(X-d)_+] can be computed by knowing the original expected value E[X] and the limited expectation E[X \wedge d].

Inflation

Suppose that losses (or claims) in the next period are expected to increase uniformly by 100r%. For example, r=0.10 means 10%. What would be the effect of inflation on the expectations E[X \wedge u] and E[(X-d)_+]?

First, the effect on the limited expectation E[X \wedge u]. As usual, X is the random loss and u is the policy limit. With inflation rate r, the loss variable for the next period would be (1+r)X. One approach is to derive the distribution for the inflated loss variable and use the new distribution to calculate E[(1+r)X \wedge u]. Another approach is to express it in terms of the limited expectation of the pre-inflated loss X. The following is the expectation E[(1+r)X \wedge u], assuming that there is no change in the policy limit.

\displaystyle (2) \ \ \ \ E[(1+r) X \wedge u]=(1+r) \ E \biggl[X \wedge \frac{u}{1+r} \biggr]

Relation (2) relates the limited expectation of the inflated variable to the limited expectation of the pre-inflated loss X. It says that the limited expectation of the inflated variable (1+r) X is obtained by inflating the limited expectation of the pre-inflated loss but at a smaller policy limit \frac{u}{1+r}.

The following is the expectation E[(1+r) (X-d)_+]=E[(1+r) X]-E[(1+r) X \wedge d].

\displaystyle \begin{aligned} (3) \ \ \ \ E[((1+r) X-d)_+]&=E[(1+r) X]-E[(1+r) X \wedge d] \\&=(1+r) E[X]-(1+r) \ E \biggl[X \wedge \frac{d}{1+r} \biggr] \\&=(1+r) \biggl( E[X]- \ E \biggl[X \wedge \frac{d}{1+r} \biggr] \biggr) \\&=(1+r) E \biggl[ \biggl(X-\frac{d}{1+r} \biggr)_+ \biggr] \end{aligned}

Similarly, (3) expresses the expected payment on the inflated loss in terms of the expected payment of the pre-inflated loss. It says that when the loss is inflated, the expected payment per loss is obtained by inflating the expected payment on the pre-inflated loss but at a smaller deductible.

Insurance Payment Per Loss versus Per Payment

The previous post (Part 2) shows how to evaluate the average amount paid to the insured when the coverage has an ordinary deductible. The average payment discussed in Part 2 is the average per loss (over all losses). As a simple illustration, let’s say the amounts of losses in a given period for an insured are 7, 4, 33 and 17 subject to an ordinary deductible of 5. Then the insurance payments are: 2, 0, 28 and 12. The average payment per loss would be (2+0+28+12)/4 = 10.5. If we only count the losses that require a payment, the average is (2+28+12)/3 = 14. Thus the average payment per payment is greater than the average payment per loss since only the losses exceeding the deductible are counted in the average payment per payment. In the calculation discussed here, the average payment per payment is obtained by dividing the average payment per loss by the probability that the loss exceeds the deductible. Note that 10.5/0.75 = 14.

Suppose that the random variable X is the size of the loss (if a loss occurs). Under an insurance policy with an ordinary deductible d, the first d dollars of a loss is responsible by the insured and the amount of the loss in excess of d is paid by the insurer. Under such an arrangement, a certain number of losses are not paid by the insurer, precisely those losses that are less than or equal to d. If we only count the losses that are paid by the insurer, the payment amount is the conditional random variable X-d \lvert X>d. The expected value of this conditional random variable is denoted by e_X(d) or e(d) if the loss X is understood.

Given that P(X>d)>0, the variable X-d \lvert X>d is called the excess loss variable. Its expected value e_X(d) is called the mean excess loss function. Other names are mean residual life and complete expectation of life (when the context is that of a mortality study).

For the discussion in this post and other posts in the same series, we use Y_P to denote X-d \lvert X>d. The P stands for payment so that its expected value would be average insurance payment per payment, i.e. the expected amount paid given that the loss exceeds the deductible. When the random variable X is the age at death, e_X(x) would be the expected remaining time until death given that the life has survived to age x.

The expected value e_X(d) is thus the expected payment per payment (or expected cost per payment) under an ordinary deductible. In contrast, the expected value E[(X-d)_+] is the expected payment per loss (or expected cost per loss), which is discussed in the previous post. The two expected values are related. The calculation of one will give the other. The following compares the two calculation. Let f_X(x) and F_X(x) be the PDF and CDF of X, respectively.

\displaystyle (4) \ \ \ \ E[(X-d)_+]=\int_d^\infty (x-d) \ f_X(x) \ dx

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\displaystyle (5) \ \ \ \ e_X(d)=\int_d^\infty (x-d) \ \frac{f_X(x)}{S_X(d)} \ dx=\frac{E[(X-d)_+]}{S_X(d)}

Note that e_X(d) is calculated using a conditional density function. As a result, e_X(d) can be obtained by dividing the expected payment per loss divided by the probability that there is a payment. Thus the expected payment per payment is the expected payment per loss divided by the probability that there is a payment. This is described in the following relation.

\displaystyle (6) \ \ \ \ e_X(d)=\frac{E[(X-d)_+]}{S_X(d)}=\frac{E[X]-E[X \wedge d]}{S_X(d)}

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Distributions of Insurance Payment Variables

Relation (1) and Relation (4) are used for the calculation of expected insurance payment when the coverage has an ordinary deductible. For deriving other information about insurance payment in the presence of an ordinary deductible, it is helpful to know the distributions of the insurance payment (per loss and per payment).

Let Y_L=(X-d)_+ (payment per loss) and let Y_P=X-d \lvert X>d (payment per payment). We now discuss the distribution for Y_P. First, the following gives the PDF, CDF and the survival function of Y_L.

Payment Per Loss
PDF \displaystyle  f_{Y_L}(y)=\left\{ \begin{array}{ll}                     \displaystyle  F_X(d) &\ \ \ \ y=0 \\           \text{ } & \text{ } \\           \displaystyle  f_X(y+d) &\ \ \ \ y > 0           \end{array} \right.
CDF \displaystyle  F_{Y_L}(y)=\left\{ \begin{array}{ll}                     \displaystyle  0 &\ \ \ \ y<0 \\           \text{ } & \text{ } \\           \displaystyle  F_X(y+d) &\ \ \ \ y \ge 0                      \end{array} \right.
Survival Function \displaystyle  S_{Y_L}(y)=\left\{ \begin{array}{ll}                     \displaystyle  1 &\ \ \ \  y<0 \\           \text{ } & \text{ } \\           \displaystyle  S_X(y+d) &\ \ \ \ y \ge 0                      \end{array} \right.

Note that the above functions have a point mass at y=0 to account for the losses that are not paid. For Y_P=X-d \lvert X>d, there is no point mass at y=0. We only need to consider y>0. Normalizing the function f_X(y+d) would give the PDF of Y_P=X-d \lvert X>d. Thus the following gives the PDF, the survival function and the CDF of Y_P=X-d \lvert X>d.

Payment Per Payment
PDF \displaystyle f_{Y_P}(y)=\frac{f_X(y+d)}{S_X(d)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ y > 0
CDF \displaystyle F_{Y_P}(y)=\frac{F_X(y+d)-F_X(d)}{S_X(d)} \ \ \ \ \ \ \ \ \ \ \ \ \ y > 0
Survival Function \displaystyle S_{Y_P}(y)=\frac{S_X(y+d)}{S_X(d)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ y > 0

The following calculates the two averages using the respective PDFs, as a result deriving the same relationship between the two expected values.

\displaystyle (7) \ \ \ \ E(Y_L)=E[(X-d)_+]=\int_0^\infty y \ f_X(y+d) \ dy

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\displaystyle (8) \ \ \ \ E(Y_P)=e_X(d)=\int_0^\infty y \ \frac{f_X(y+d)}{S_X(d)} \ dy

Relation (1) and Relation (7) are identical. The former is calculated using the distribution of the original loss X and the latter is calculated using the distribution of the payment per loss variable. Similarly, compare Relation (5) and Relation (8). The former is computed from the distribution of the unmodified loss X and the latter is computed using the distribution of the payment per payment variable.

Note that Relation (7) and Relation (8) also lead to Relation (6), which relates the expected payment E[(X-d)_+] and the expected payment e_X(d).

With the PDFs and CDFs for the payment per loss and the payment per payment developed, other distributional quantities can be derived, e.g. hazard rate, variance, skewness and kurtosis.

Loss Elimination Ratio

When the insurance coverage has an ordinary deductible, a natural question is (from the insurance company’s perspective), what is the impact of the deductible on the payment that is made to the insured? More specifically, on average what is the reduction of the payment? The loss elimination ratio is the proportion of the expected loss that is not paid to the insured by the insurer. For example, if the expected loss before application of the deductible is 45 (of which 36 is expected to be paid by the insurer), the loss elimination ratio is 9/45 = 0.20 (20%). In this example, the insurer has reduced its obligation by 20%. More formally, the loss elimination ratio (LER) is defined as:

    \displaystyle LER=\frac{E(X)-E[(X-d)_+]}{E(X)}=\frac{E[X \wedge d]}{E(X)}

LER is the ratio of the expected reduction in payment as a result of imposing the ordinary deductible to the expected payment without the deductible. Though it is possible to define LER as the ratio of the reduction in expected payment as a result of a coverage modification to the expected payment without the modification for a modification other than an ordinary deductible, we do not attempt to further generalize this concept.

Policy with a Limit and a Deductible

In part 1, expected insurance payment under a policy limit is developed. In part 2, expected insurance payment under a policy with an ordinary deductible is developed. We now combine both provisions in the same insurance policy. First, let’s define two terms. A policy limit is the maximum amount that will be paid by a policy. For example, if the policy limit is 10,000, the policy will paid at most 10,000 per loss. If the actual loss is 15,000, then the policy will paid 10,000 and the insured will have to be responsible for the remaining 5,000. On the other hand, maximum covered loss is the level above which no loss will be paid. For example, Suppose that the policy covers up to 10,000 per loss subject to a deductible of 1,000. If the actual loss is 20,000, then the maximum covered loss is 10,000 with the policy limit being 9,000. This is because the policy only covers the first 10,000 with the first 1,000 paid by the insured.

Suppose that an insurance policy has an ordinary deductible d, a maximum covered loss u with d<u and no other coverage modifications. Any loss below d is not paid by the insurer. For any loss exceeding d, the insurer pays the loss amount in excess of d up to the maximum covered loss u, with the policy limit being u-d. The following describes the payment rule more explicitly.

    \displaystyle  (X \wedge u)-(X \wedge d)=\left\{ \begin{array}{ll}                     \displaystyle  0 &\ \ \ \ X \le d \\           \text{ } & \text{ } \\           \displaystyle  X-d &\ \ \ \ d < X \le u \\           \text{ } & \text{ } \\           \displaystyle  u-d &\ \ \ \ X > u           \end{array} \right.

Under such a policy, the maximum amount paid (policy limit) is u-d, which is the maximum covered loss minus the deductible. The policy limit is reached when X > u. When d < X \le u, the payment X \wedge u is X and the payment X \wedge d is d. Then the expected payment per loss under such a policy is:

\displaystyle  (9) \ \ \ \ E[X \wedge u]-E[X \wedge d]

There is not special notation for the expected payment. In words, it is the limited expected value at u (the maximum covered loss) minus the limited expected value at the ordinary deductible d. The higher moments can be derived by evaluating E(Y^k) using the PDF of the loss X where Y=(X \wedge u)-(X \wedge d).

The payment Y=(X \wedge u)-(X \wedge d) is on a per loss basis. It is also possible to consider payment or payment by removing the point mass at zero. The expected payment per payment is obtained by dividing (9) by the probability of a positive payment.

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Franchise Deductible

An alternative to the ordinary deductible is the franchise deductible. It works like an ordinary deductible except that when the loss exceeds the deductible, the policy pays the loss in full. The following gives the payment rule.

    \displaystyle  Y=\left\{ \begin{array}{ll}                     \displaystyle  0 &\ \ \ \ X \le d \\           \text{ } & \text{ } \\           \displaystyle  X &\ \ \ \ X > d           \end{array} \right.

Note that when the loss exceeds the deductible (X > d), the policy with a franchise deductible pays more than a policy with the same ordinary deductible. By how much? By the amount d. Thus the expected payment per loss under a franchise deductible is E[(X-d)_+]+d \ S_X(d). The addition of d \cdot S_X(d) reflects the additional benefit when X > d. Instead of deriving calculation specific to franchise deductible, we can derive the payments under franchise deductible by adding the additional benefit appropriately.

\displaystyle  (10) \ \ \ \ E[(X-d)_+]+d \ S_X(d)

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Benefit Tables

The previous two posts and this post discuss coverage with two types of deductible (ordinary and franchise) as well as coverage that may include a limit. In order to keep the expected payments straight, the following table organizes the different combinations of coverage options discussed up to this point.

A = Ordinary Deductible Only
B = Franchise Deductible Only
C = Ordinary Deductible + Limit
D = Franchise Deductible + Limit

Expected Cost Per Loss Expected Cost Per Payment
A \displaystyle E[(X-d)_+]=E(X)-E(X \wedge d) \displaystyle \frac{E(X)-E(X \wedge d)}{S_X(d)}=e_X(d)
B \displaystyle E(X)-E(X \wedge d)+d \ S_X(d) \displaystyle \frac{E(X)-E(X \wedge d)}{S_X(d)}+d=e_X(d)+d
C \displaystyle E(X \wedge u)-E(X \wedge d ) \displaystyle \frac{E(X \wedge u)-E(X \wedge d)}{S_X(d)}
D \displaystyle E(X \wedge u)-E(X \wedge d )+d \ S_X(d) \displaystyle \frac{E(X \wedge u)-E(X \wedge d)}{S_X(d)}+d

Row A shows the expected payment in a coverage with an ordinary deductible for both per loss and per payment. The expected per loss payment in Row A is the subject of the previous post and the expected payment per payment is discussed earlier in this post. Row B is for expected payments in a coverage with a franchise deductible. Note that the coverage with a franchise deductible pays more than a coverage with an identical ordinary deductible. Thus Row B is Row A plus the added benefit, which is the deductible d.

Row C is for the coverage of an ordinary deductible with a policy limit. The per loss expected payment is \displaystyle E(X \wedge u)-E(X \wedge d ). The per payment expected value is a conditional one, conditional on the loss greater than the deductible. Thus the per payment expected value is the per loss expected value divided by S_X(d).

Note that Row D is Row C plus the amount of d, the additional benefit as a result of having a franchise deductible instead of an ordinary deductible.

Rather than memorizing these formulas, focus on the general structure of the table. For example, understand the payments involving ordinary deductible (Row A and Row C). Then the payments for franchise deductible are obtained by adding an appropriate additional benefit.

The following table shows the expected payments under the influence of claim cost inflation. The maximum covered loss u and the deductible d are identical to the above benefit table. The losses are subject to a (100r)% inflation in the next exposure period. Note that u^* and d^* are the modified maximum covered loss and deductible that will make the formulas work.

E = Ordinary Deductible Only
F = Franchise Deductible Only
G = Ordinary Deductible + Limit
H = Franchise Deductible + Limit

\displaystyle u^*=\frac{u}{1+r}

\displaystyle d^*=\frac{d}{1+r}

Expected Cost Per Loss (with Inflation) Expected Cost Per Payment (with Inflation)
E \displaystyle (1+r) \biggl \{ E(X)-E [X \wedge d^* ] \bigg \} \displaystyle (1+r) \biggl \{ \frac{\displaystyle  E(X)-E [X \wedge d^* ]}{\displaystyle S_{X} (d^*)} \biggr \}
F \displaystyle (1+r) \biggl \{ E(X)-E [X \wedge d^* ]+d^* \ S_{X}(d^* ) \bigg \} \displaystyle (1+r) \biggl \{ \frac{\displaystyle  E(X)-E [X \wedge d^* ]}{\displaystyle S_{X} (d^* )}+d^* \biggr \}
G \displaystyle (1+r) \biggl \{ E [X \wedge u^* ]-E [X \wedge d^* ] \bigg \} \displaystyle (1+r) \biggl \{ \frac{\displaystyle  E [X \wedge u^* ]-E [X \wedge d^*]}{\displaystyle S_{X} (d^* )} \biggr \}
H \displaystyle (1+r) \biggl \{ E [X \wedge u^* ]-E [X \wedge d^* ]+d^* \ S_{X}   (d^* ) \bigg \} \displaystyle (1+r) \biggl \{ \frac{\displaystyle  E[X \wedge u^* ]-E [X \wedge d^* ]}{\displaystyle S_{X} (d^* )}+d^* \biggr \}

There is really no need to mathematically derive the formulas for the second insurance benefit table (Rows E through H). Recall the effect of inflation on the expected payments E[X \wedge u] and E[ (X-d)_+] (see Relation (2) and Relation (3)). Then apply the inflation effect on the first insurance benefit table. Note that the second table is obtained by inflating the first table by 1+r but with smaller upper limit u^* and deductible d^*. Also note the relation between Row E and Row F (same relation between Row A and Row B) and the relation between Row G and Row H (same relation between Row C and Row D).

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Calculation

One approach of modeling insurance payments is to assume that the unmodified insurance losses are from a catalog of parametric distributions. For certain parametric distributions, the limited expectations E[X \wedge d] have convenient forms. Examples are: exponential distribution, Pareto distribution and lognormal distribution. Other distributions do not close form for E[X \wedge d] but the limited expectation can be expressed as a function that can be numerically calculated. One example is the gamma function.

The table in this link is an inventory of distributions that may be useful in modeling insurance losses. Included in the table are the limited expectations for various distributions. The following table shows the limited expectations for exponential, Pareto and lognormal.

Limited Expectation
Exponential E[X \wedge x]=\theta (1-e^{-x/\theta})
Pareto \displaystyle E[X \wedge x]=\frac{\theta}{\alpha-1} \biggl[1-\biggl(\frac{\theta}{x+\theta} \biggr)^{\alpha-1} \biggr], \alpha \ne 1
Lognormal \displaystyle E[(X \wedge x)^k]=\exp(k \mu+k^2 \sigma^2/2) \Phi \biggl(\frac{\log(x)-\mu-k \sigma^2}{\sigma} \biggr)+x^k \biggl[1-\Phi \biggl(\frac{\log(x)-\mu}{\sigma} \biggr) \biggr]

See the table in this link for the limited expectations of the distributed not shown in the above table. Once E[X \wedge d] is computed or estimated, the expected payment for various types of coverage provisions can be derived.

Examples

The first example demonstrates the four categories of expected payments in the table in the preceding section.

Example 1
Suppose that the loss distribution is described by the PDF \displaystyle f_X(x)=\frac{1}{5000} (100-x) with support 0<x<100. Walk through all the expected payments discussed in the above table (Rows A through D) assuming d=12 and u=60.

First, the basic calculation.

    \displaystyle E(X)=\int_0^{100} x \ \frac{1}{5000} (100-x) \ dx=\frac{100}{3}=33.3333

    \displaystyle F_X(x)=\frac{1}{5000} (100x- \frac{1}{2} x^2); \ \ \ 0<x<100

    F_X(12)=0.2256

    S_X(12)=1-0.2256=0.7744

    F_X(60)=0.84

    S_X(60)=1-0.84=0.16

    \displaystyle E(X \wedge 12)=\int_0^{12} x \ \frac{1}{5000} (100-x) \ dx +12 \cdot S_X(12)=\frac{6636}{625}

    \displaystyle E(X \wedge 60)=\int_0^{60} x \ \frac{1}{5000} (100-x) \ dx +12 \cdot S_X(60)=\frac{12864}{625}

Four categories of expected payments are derived and are shown in the following table.

A = Ordinary Deductible Only
B = Franchise Deductible Only
C = Ordinary Deductible + Limit
D = Franchise Deductible + Limit

Expected Cost Per Loss Expected Cost Per Payment
A \displaystyle \frac{100}{3}-\frac{6636}{625}=\frac{42592}{1875}=22.7157 \displaystyle e_X(d)=\frac{42592}{1875} \frac{1}{0.7744}=\frac{10648}{363}=29.3333
B \displaystyle \frac{42592}{1875}+12 \cdot 0.7744=\frac{60016}{1875}=32.0085 \displaystyle \frac{10648}{363}+12=\frac{15004}{363}=41.3333
C \displaystyle \frac{12864}{625}-\frac{6636}{625}=\frac{6228}{625}=9.9648 \displaystyle \frac{6228}{625} \frac{1}{0.7744}=\frac{6228}{484}=12.8678
D \displaystyle \frac{6228}{625}+12 \cdot 0.7744=\frac{12036}{625}=19.2576 \displaystyle \frac{6228}{484}+12=\frac{12036}{484}=24.8678

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Example 2
Suppose that a coverage has an ordinary deductible and suppose that the CDF of the insurance payment per loss is given by:

    \displaystyle  G(y)=\left\{ \begin{array}{ll}                     \displaystyle  0 &\ \ \ \ y<0 \\           \text{ } & \text{ } \\           \displaystyle  1-e^{-\frac{y+20}{100}} &\ \ \ \ y \ge 0                      \end{array} \right.

Determine the expected value and the variance of the insurance cost per loss.

Note that there is a jump in the CDF at y=0. Thus y=0 is a point mass. The following is the PDF of the insurance payment.

    \displaystyle  g(y)=\left\{ \begin{array}{ll}                     \displaystyle  1-e^{-1/5} &\ \ \ \ y=0 \\           \text{ } & \text{ } \\           \displaystyle  \frac{1}{100} e^{-\frac{y+20}{100}}=e^{-1/5} \ \frac{1}{100} e^{-y/100} &\ \ \ \ y > 0                      \end{array} \right.

It is clear that g(y)=f(y+20) where f(y) is the PDF of the exponential distribution with mean 100. Thus g(y) is the PDF of the insurance payment per loss when the coverage has an ordinary deductible of 20 where the loss has exponential distribution with mean 100. We can use g(y) to calculate the mean and variance of Y. Using g(y), the mean and variance are:

    \displaystyle \begin{aligned} E(Y)&=\int_0^\infty y \ e^{-1/5} \ \frac{1}{100} e^{-y/100} \ dy \\&=e^{-1/5} \ \int_0^\infty y \ \frac{1}{100} e^{-y/100} \ dy \\&=e^{-1/5} \ (100)=81.873 \end{aligned}

    \displaystyle \begin{aligned} E(Y^2)&=\int_0^\infty y^2 \ e^{-1/5} \ \frac{1}{100} e^{-y/100} \ dy \\&=e^{-1/5} \ \int_0^\infty y^2 \ \frac{1}{100} e^{-y/100} \ dy \\&=e^{-1/5} \ 2(100)^2 \end{aligned}

    \displaystyle Var(Y)=e^{-1/5} \ 2(100)^2-\biggl( e^{-1/5} \ (100) \biggr)^2=9671.414601

Note that the calculation for E(Y) and E(Y^2) is done by multiplying e^{-1/5} by mean and second moment of the exponential distribution with mean 100.

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Example 3
You are given the following:

  • Losses follow a Pareto distribution with parameters \alpha=3 and \theta=500.
  • The coverage has an ordinary deductible of 100.

Determine the PDF and CDF of the payment to the insured per payment. Determine the expected cost per payment.

Note that the setting of this example is identical to Example 3 in this previous post. Since this only concerns the insurance when the loss exceeds the deductible, the insurance payment is the conditional distribution Y=X-100 \lvert X>100 where X is the Pareto loss distribution.

The following gives the PDF and CDF and other information of the loss X.

    \displaystyle f_X(x)=\frac{3 \ 500^3}{(x+500)^4} \ \ \ \ \ \ \ \ \ \ \ \ \ \ x> 0

    \displaystyle F_X(x)=1-\biggl(\frac{500}{x+500} \biggr)^3 \ \ \ \ \ \ x> 0

    \displaystyle S_X(x)=\biggl(\frac{500}{x+500} \biggr)^3 \ \ \ \ \ \ \ \ \ \ \ \ x> 0

    \displaystyle S_X(100)=\biggl(\frac{500}{100+500} \biggr)^3=\biggl(\frac{500}{600} \biggr)^3=\frac{125}{216}

The following shows the PDF and CDF for the payment per payment variable Y.

    \displaystyle f_Y(y)=\frac{f_X(y+100)}{S_X(100)}=\frac{3 \ 500^3}{(y+100+500)^4} \ \biggl(\frac{600}{500} \biggr)^3=\frac{3 \cdot 600^3}{(y+600)^4} \ \ \ \ \ \ \ \ \ y> 0

    \displaystyle S_Y(y)=\frac{S_X(y+100)}{S_X(100)}=\biggl(\frac{500}{y+100+500} \biggr)^3 \ \biggl(\frac{600}{500} \biggr)^3=\biggl(\frac{600}{y+600} \biggr)^3 \ \ \ \ \ \ y> 0

    \displaystyle F_Y(y)=1-S_Y(y)=1-\biggl(\frac{600}{y+600} \biggr)^3 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ y> 0

Note that these PDF and CDF are for a Pareto distribution with \alpha=3 and \theta=600. Thus the expected payment per payment and the variance are:

    \displaystyle E(Y)=\frac{600}{3-1}=300

    \displaystyle E(Y^2)=\frac{600^2 \cdot 2}{(3-1) (3-2)}=600^2

    Var(Y)=600^2 - 300^2=270000

Example 4
Suppose that losses X follow a Pareto distribution with shape parameter \alpha and scale parameter \theta. Suppose that an insurance coverage pays each loss subject to an ordinary deductible d. Derive a formula for the expected payment per payment e_X(d).

Let Y=X-d \lvert X>d be the payment per payment. In Example 3, we see that the CDF and PDF of Y are also Pareto but with shape parameter \alpha (same as for X) and scale parameter \theta+d (the original scale parameter plus the deductible). Thus the following gives the expected payment per payment.

    \displaystyle e_X(d)=\frac{\theta+d}{\alpha-1}=\frac{\theta}{\alpha-1}+\frac{1}{\alpha-1} d

Note that e_X(d) in this case is an increasing linear function of the deductible d. The higher the deductible, the larger the expected payment. This is a clear sign that the Pareto distribution is a heavy tailed distribution.

Example 5
Suppose that losses follow a lognormal distribution with parameters \mu=5 and \sigma=0.6. An insurance coverage has an ordinary deductible of 100. Compute the expected payment per loss. Compute the expected payment per loss if the deductible is a franchise deductible.

The following calculates E(X) and E[X \wedge 100].

    E(X)=e^{5+0.6^2/2}=e^{5.18}=177.682811

    \displaystyle \begin{aligned} E[X \wedge 100]&=e^{5+0.6^2/2} \ \Phi \biggl(\frac{\log(100)-5-0.6^2}{0.6} \biggr)+100 \ \biggl[1-\Phi \bigg(\frac{\log(100)-5}{0.6} \biggr) \biggr] \\&=e^{5.18} \ \Phi (-1.26)+100 \ [1-\Phi (-0.66 )]\\&=e^{5.18} \ 0.1038+100(0.7454)=e^{5.18} \ 0.1038+74.54=92.98347578 \end{aligned}

As a result, the expected payment per loss is E(X)-E[X \wedge 100]=84.69933522=84.70. If the deductible of 100 is a franchise deductible, the expected payment per loss is obtained by adding an additional insurance payment, which is d \ S(d). The following is the added payment.

    \displaystyle \begin{aligned} \text{added payment}&=100 \ \biggl[1-\Phi \bigg(\frac{\log(100)-5}{0.6} \biggr) \biggr] \\&=100 \ \biggl[1-\Phi (-0.66 ) \biggr] \\&=100 (0.7454) \\&=74.54  \end{aligned}

Thus the expected payment per loss with a franchise deductible of 250 is 84.70 + 74.54 = 159.24.

Practice Problems

The following practice problem sets are to reinforce the concepts discussed here.

Practice Problem Set 7
Practice Problem Set 8
Practice Problem Set 9

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models of insurance payment

ordinary deductible, franchise deductible

maximum covered loss

Daniel Ma Math

Daniel Ma Mathematics

Dan Ma Actuarial

\copyright 2017 – Dan Ma

Mathematical models for insurance payments – part 2 – ordinary deductible

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The post is the second post on models for insurance payments. The focus here is on the insurance payment when the loss is adjusted by a deductible. The first post is on insurance policy with a limit.

Ordinary Deductible

Suppose that the random variable X is the size of the loss (if a loss occurs). Under an insurance policy with a deductible d, the first d dollars of a loss is responsible by the insured and the amount of the loss in excess of d is paid by the insurer. Under this policy provision, what is the expected amount of insurance payment for a given loss? The deductible described here is called an ordinary deductible. The insurance payment (per loss) under the presence of an ordinary deductible d is denoted by (X-d)_+. The idea for this notation is that the insurance payment is X-d as long as X-d is not negative. The following describes the payment rule more explicitly.

    \displaystyle  (X-d)_+=\left\{ \begin{array}{ll}                     \displaystyle  0 &\ X \le d \\           \text{ } & \text{ } \\           \displaystyle  X-d &\ X > d           \end{array} \right.

The goal is to describe the probability model of the insurance payment (X-d)_+. First and foremost, we need to know how much it will cost the insurer on average to pay the insured, which is the mean E[(X-d)_+]. The variance of (X-d)_+ is also important as it is a measure of risk to the insurer. The mean E[(X-d)_+] and the variance Var[(X-d)_+] can be calculated by using the density function of X. At times it will be useful to know the the probability density function (PDF) and cumulative distribution function (CDF) of (X-d)_+ and other distributional quantities.

Moments

First, let’s focus on the moments of (X-d)_+. One way to obtain moments of (X-d)_+ is through the distribution of the unmodified loss X. Let f(x), F(x) and S(x)=1-F(x) be the probability density function (PDF), cumulative distribution function (CDF) and the survival function of the random loss, respectively. Then the moments of (X-d)_+ can be expressed using these distributional quantities of X.

    \displaystyle E[(X-d)_+]=\int_d^\infty (x-d) \ f(x) \ dx

    \displaystyle E[(X-d)_+^k]=\int_d^\infty (x-d)^k \ f(x) \ dx

    \displaystyle Var[(X-d)_+]=E[(X-d)_+^2]-E[(X-d)_+]^2

The above calculation is based on first principle. When the loss X is less than or equal to d, the insurance payment is zero. Note that the zero payment does not have to be explicitly stated in the above integrals. When the loss X exceeds the deductible d, the payment is X-d, the amount of the loss in excess of d. Example 1 and Example 2 below demonstrate how this calculation is performed.

Another way to obtain the moments of (X-d)_+ is obtained by first finding the PDF of (X-d)_+. This is discussed in a section below.

Basic Example

Example 1
Suppose that the loss distribution has PDF f(x)=1/5 (1- x/10) \ \ \ 0<x<10. Evaluate E(X) and E[(X-2)_+]. What is the expected amount of the loss eliminated as a result of imposing the ordinary deductible of 2 (eliminated from the insurer’s perspective)?

    \displaystyle \begin{aligned} E[(X-2)_+]&=\int_2^{10} (x-2) \ \frac{1}{5} (1- x/10) \ dx\\&=\int_2^{10} \frac{1}{5} (\frac{6}{5} x -x^2/10 -2) \ dx\\&=\frac{1}{5} \biggl[\frac{3 x^2}{5}-\frac{x^3}{30}-2x \biggr]_2^{10}=\frac{128}{75}=1.7067  \end{aligned}

From Example 1 in the previous post, E(X)=10/3. Thus when a loss occurs, the expected amount of the loss that is responsible by the insured is

    \displaystyle E(X)-E[(X-2)_+]=\frac{10}{3}-\frac{128}{75}=\frac{122}{75}=1.6267

,which is also the expected amount of the loss eliminated (from the insurer’s perspective) as a result of imposing an ordinary deductible.

Example 2
Continue with Example 1. Calculate the variance of the insurance payment (X-2)_+.

    \displaystyle \begin{aligned} E[(X-2)_+^2]&=\int_2^{10} (x-2)^2 \ \frac{1}{5} (1- x/10) \ dx \\ &=\int_2^{10} \frac{1}{5} \ (x^2-4x+4) \  (1- x/10) \ dx \\&=\int_2^{10} \frac{1}{5} (\frac{7}{5} x^2 -x^3/10-\frac{22}{5} +4) \ dx\\&=\frac{1}{5} \biggl[\frac{7 x^3}{15}-\frac{x^4}{40}-\frac{11x^2}{5} +4x \biggr]_2^{10}=\frac{512}{75}  \end{aligned}

    \displaystyle \begin{aligned} Var[(X-2)_+]&=E[(X-2)_+^2]-E[(X-2)_+]^2 \\ &=\frac{512}{75}-\biggl(\frac{128}{75} \biggr)^2 \\&=\frac{22016}{5625}=3.9140  \end{aligned}

From Example 2 in the previous post, Var(X)=50/9=5.5556. This shows that imposing a deductible that is significant enough has a variance reducing effect on the insurance payment. Had the deductible not present, there would be a greater fluctuation in the payment.

Connection with Policy Limit

The calculation in Example 1 and Example 2 are based on first principle. For many parametric families of distributions, the calculation for E[(X-d)_+] can be done using formulas. To that end, consider summing the payment (X-d)_+ and the payment X \wedge d.

    \displaystyle (X-d)_+ + (X \wedge d)=\displaystyle  \left\{ \begin{array}{ll}                     \displaystyle  0 &\ X \le d \\           \text{ } & \text{ } \\           \displaystyle  X-d &\ X > d           \end{array} \right. +\displaystyle  \left\{ \begin{array}{ll}                     \displaystyle  X &\ X \le d \\           \text{ } & \text{ } \\           \displaystyle  d &\ X > d           \end{array} \right. =X

Summing up each case produces the original loss X. Thus buying a policy with an ordinary deductible d along with a policy with a limit of d equals to full coverage. From a calculation standpoint, E[(X-d)_+] can be computed as follows:

    E[(X-d)_+]=E(X)-E(X \wedge d)

The advantage of using the above idea is that the calculation of E(X \wedge d) is routine for a large number of parametric families of distributions. For examples, most of the distributions in the table in this link have formulas for E(X \wedge x) and E[(X \wedge x)^k]. Then subtracting E(X \wedge x) into E(X) would give E[(X-d)_+]. The following table gives the limited expectation E(X \wedge x) for three distributions, taken from the table in the given link.

Limited Expectation
  • Exponential: E[X \wedge x]=\theta (1-e^{-x/\theta})
  • Pareto: \displaystyle E[X \wedge x]=\frac{\theta}{\alpha-1} \biggl[1-\biggl(\frac{\theta}{x+\theta} \biggr)^{\alpha-1} \biggr], \alpha \ne 1
  • Lognormal: \displaystyle E[(X \wedge x)^k]=\exp(k \mu+k^2 \sigma^2/2) \Phi \biggl(\frac{\log(x)-\mu-k \sigma^2}{\sigma} \biggr)+x^k \biggl[1-\Phi \biggl(\frac{\log(x)-\mu}{\sigma} \biggr) \biggr]

Examples are given to demonstrate how these formulas are used.

Example 3
You are given the following:

  • Losses follow a Pareto distribution with parameters \alpha=3 and \theta=500.
  • The coverage has an ordinary deductible of 100.

For the next loss that will occur, determine the expected amount that will be paid by the insurer.

The following shows the calculation.

    \displaystyle E(X)=\frac{\theta}{\alpha-1}=\frac{500}{2}=250

    \displaystyle \begin{aligned} E[X \wedge 1000]&=\frac{\theta}{\alpha-1} \biggl[1-\biggl(\frac{\theta}{100+\theta} \biggr)^{\alpha-1} \biggr] \\ &=\frac{500}{2} \biggl[1-\biggl(\frac{500}{600} \biggr)^{2} \biggr]=\frac{2750}{36}=76.389  \end{aligned}

    \displaystyle E[(X-1000)_+]=E(X)-E[X \wedge 1000]=250-\frac{2750}{36}=\frac{3125}{18}=173.611

As a result of imposing a deductible of 1000, the payment made by the insurer per loss goes from 250 to 173.611. The reduction of payment of E[X \wedge 100] = 76.389.

Example 4
You are given the following:

  • Losses follow an exponential distribution with mean 2500.
  • The coverage has an ordinary deductible of 1000.

Determine the expected amount per loss that is paid by the insurer.

The following shows the calculation.

    \displaystyle E(X)=2500

    \displaystyle E(X \wedge 1000)=2500 (1-e^{-1000/2500})=2500 (1-e^{-0.4})=824.20

    \displaystyle \begin{aligned} E[(X-1000)_+]&=E(X)-E[X \wedge 1000] \\ &=2500-2500 (1-e^{-0.4})  =2500 e^{-0.4}=1675.80 \end{aligned}

Note that the reduction of payment by the insurer per loss is \displaystyle E(X \wedge 1000)=824.20, which is the amount per loss that has to be met by the insured.

One observation about exponential loss. When the loss X is an exponential distribution, E[(X-d)_+] can actually be directly calculated using first principle since the exponential distribution is very tractable mathematically.

Example 5
You are given the following:

  • Losses follow a lognormal distribution with parameters \mu=6.5 and \sigma=1.75.
  • The coverage has an ordinary deductible of 1000.

Determine the expected amount per loss that is paid by the insurer.

The following shows the calculation.

    \displaystyle E(X)=e^{\mu+(1/2) \sigma^2}=e^{6.5+0.5(1.75)^2}=e^{8.03125}=3075.583751

    \displaystyle \begin{aligned} E[X \wedge 1000]&=e^{6.5+0.5(1.75)^2} \ \Phi \biggl(\frac{\log(1000)-6.5-1.75^2}{1.75} \biggr)+1000 \biggl[1-\Phi \biggl(\frac{\log(1000)-6.5}{1.75} \biggr) \biggr] \\ &=e^{8.03125} \ \Phi (-1.52)+1000 [1-\Phi (0.23) ]  \\ &=e^{8.03125} \ (1-0.9357)+1000 [1-0.5910 ]=606.7600352 \end{aligned}

    \displaystyle \begin{aligned} E[(X-1000)_+]&=E(X)-E[X \wedge 1000] \\ &=e^{8.03125}-606.7600352=2468.82 \end{aligned}

PDF and CDF of Insurance Payment

The preceding section shows that E[(X-d)_+]=E(X)-E(X \wedge d) is a great way to evaluate mean claim cost per loss when the loss distribution is from this list of distributions. For the distributions in this list, the limited expectation E(X \wedge d) is readily available. To calculate the higher moments of (X-d)_+, it is helpful to know more about its distribution. To that end, we derive its PDF and CDF and the survival function.

Before deriving the PDF and CDF, note that the payment variable (X-d)_+ is a censored and shifted random variable. It is censored from below and is shifted to the left by the amount of the deductible d. As far as payment is concerned, any loss that is below the deductible is considered zero. In effect, any loss in the interval (0,d) is recorded as zero. The insurance policy only pays for losses that are in excess of the deductible d. Hence the positive payments are derived by shifting losses in the interval (d,\infty) to the left by d. The resulting distribution for the payment (X-d)_+ is a mixed random variable. It has a point mass at y=0 with probability F_X(d) where F_X(x) is the CDF of the loss X. On the interval (0,\infty) or some appropriate interval (0,M), the density curve of (X-d)_+ is continuous and is the resulting of shifting the density curve of X to the left by the amount d. Let Y=(X-d)_+. Let f_X(x) and F_X(x) be the PDF and CDF of X, respectively. The following is the PDF of Y.

    \displaystyle  f_Y(y)=\left\{ \begin{array}{ll}                     \displaystyle  F_X(d) &\ \ \ \ y=0 \\           \text{ } & \text{ } \\           \displaystyle  f_X(y+d) &\ \ \ \ y > 0           \end{array} \right.

Note the point mass at y=0, which is the probability that the loss is less than or equal to the deductible. The curve f_X(y+d) is the density curve f_X(y) shifted to the left by the amount d. The following is the CDF and the survival function of Y=(X-d)_+.

    \displaystyle  F_Y(y)=\left\{ \begin{array}{ll}                     \displaystyle  0 &\ \ \ \ y<0 \\           \text{ } & \text{ } \\           \displaystyle  F_X(y+d) &\ \ \ \ y \ge 0                      \end{array} \right.

    \displaystyle  S_Y(y)=\left\{ \begin{array}{ll}                     \displaystyle  1 &\ \ \ \  y<0 \\           \text{ } & \text{ } \\           \displaystyle  S_X(y+d) &\ \ \ \ y \ge 0                      \end{array} \right.

Now that we have a description of the model for Y=(X-d)_+, moments and other distributional quantities can be derived. Consider the following two examples.

Example 6
You are given the following:

  • Losses follow a uniform distribution on the interval (0, 10).
  • The coverage has an ordinary deductible of 4.

Determine the mean and variance of Y=(X-4)_+.

The following shows the PDF, CDF and survival function of the loss X.

    \displaystyle f_X(x)=\frac{1}{10} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0<x<10

    \displaystyle F_X(x)=\frac{x}{10} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0<x<10

    \displaystyle S_X(x)=1-\frac{x}{10} \ \ \ \ \ \ \ \ \ \ \ \ \ 0<x<10

The point mass is F_X(4)=0.4. The following is the PDF f_Y(y).

    \displaystyle  f_Y(y)=\left\{ \begin{array}{ll}                     \displaystyle  0.4 &\ \ \ \ y=0 \\           \text{ } & \text{ } \\           \displaystyle  \frac{1}{10} &\ \ \ \ 0<y < 6           \end{array} \right.

Note that the PDF f_X(x) over the interval (4,10) is shifted to the interval (0, 6). The following gives the CDF and survival function of Y.

    \displaystyle  F_Y(y)=\left\{ \begin{array}{ll}                     \displaystyle  0 & \ \ \ \ y < 0 \\           \text{ } & \text{ } \\           \displaystyle  \frac{y+4}{10} & \ \ \ \ 0 \le y < 6 \\            \text{ } & \text{ } \\           \displaystyle  1 & \ \ \ \ y \ge 6            \end{array} \right.

    \displaystyle  S_Y(y)=\left\{ \begin{array}{ll}                     \displaystyle  1 & \ \ \ \ y < 0 \\           \text{ } & \text{ } \\           \displaystyle  1-\frac{y+4}{10}=\frac{6-y}{10} & \ \ \ \ 0 \le y < 6 \\            \text{ } & \text{ } \\           \displaystyle  0 & \ \ \ \ y \ge 6            \end{array} \right.

The moments of Y=(X-4)_+ can be can be obtained by integrating x^k with the PDF f_Y(y).

    \displaystyle  E(Y)=\int_0^6 \frac{1}{10} \ y \ dy=\frac{9}{5}=1.8

    \displaystyle  E(Y^2)=\int_0^6 \frac{1}{10} \ y^2 \ dy=\frac{36}{5}=7.2

    Var(Y)=7.2-1.8^2=3.96

Note that the variance of the unmodified loss is Var(X)=100/12=8.33. In this case, imposing a deductible of 4 reduces the variance by a little more than half.

Example 7
Consider the loss distribution and the coverage in Example 3. Determine the PDF, CDF and the survival function of Y=(X-100)_+. Then find the mean and variance of Y=(X-100)_+.

The loss X has a Pareto distribution with parameters \alpha=3 and \theta=500. The following shows its PDF, CDF and survival function.

    \displaystyle f_X(x)=\frac{3 \cdot 500^3}{(x+500)^4} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x>0

    \displaystyle F_X(x)=1-\biggl(\frac{500}{x+500} \biggr)^3 \ \ \ \ \ \ \ \ x>0

    \displaystyle S_X(x)=\biggl(\frac{500}{x+500} \biggr)^3 \ \ \ \ \ \ \ \ \ \ \ \ \ x>0

The point mass is F_X(100)=91/216. The following is the PDF f_Y(y).

    \displaystyle  f_Y(y)=\left\{ \begin{array}{ll}                     \displaystyle  \frac{91}{216} &\ y=0 \\           \text{ } & \text{ } \\           \displaystyle  \frac{3 \cdot 500^3}{(y+600)^4} &\ y > 0           \end{array} \right.=  \left\{ \begin{array}{ll}                     \displaystyle  \frac{91}{216} &\ y=0 \\           \text{ } & \text{ } \\           \displaystyle  \biggl(\frac{500}{600} \biggr)^3 \ \frac{3 \cdot 600^3}{(y+600)^4} &\ y > 0           \end{array} \right.

Note that the part for y > 0 is S_X(100) times the Pareto density function for parameters \alpha=3 and \theta=600. Thus the moments of Y=(X-1000)_+ can be obtained by multiplying S_X(1000) with the moments of this new Pareto distribution. Since the point mass is for y=0, we can ignore it.

    \displaystyle E(Y)=E[(X-100)_+]=\biggl(\frac{500}{600} \biggr)^3 \ \frac{600}{3-1}=\frac{3125}{18}=173.611

    \displaystyle E(Y^2)=E[(X-1000)_+^2]=\biggl(\frac{125}{216} \biggr) \ \frac{600^2 \ 2}{(3-1)(3-2)}=\frac{625000}{3}

    \displaystyle Var(Y)=E(Y^2)-E(Y)^2=\frac{625000}{3}-\biggl(\frac{3125}{18}\biggr)^2=178192.5154

    \sigma_Y=\sqrt{178192.5154}=422.129

To complete the example, the following gives the CDF and survival function.

    \displaystyle  F_Y(y)=\left\{ \begin{array}{ll}                     \displaystyle  0 & \ \ \ \ y < 0 \\           \text{ } & \text{ } \\           \displaystyle  1-\biggl( \frac{500}{600+y} \biggr)^3 & \ \ \ \ 0 \le y < \infty                      \end{array} \right.

    \displaystyle  S_Y(y)=\left\{ \begin{array}{ll}                     \displaystyle  1 & \ \ \ \ y < 0 \\           \text{ } & \text{ } \\           \displaystyle  \biggl( \frac{500}{600+y} \biggr)^3 & \ \ \ \ 0 \le y < \infty                      \end{array} \right.

Insurance Payment Per Loss versus Per Payment

Note that the E[(X-d)_+] is calculated over all losses (whether they are below or above the deductible). The expected value E[(X-d)_+] reflects the probability P(X \le d) that is assigned to the payment of zero (the point mass), and the probabilities that are applied to the payment x-d over all x in the interval (d,\infty) (the integral). Thus E[(X-d)_+] is the average payment per loss (or over all possible losses). Another average payment to consider is the average payment over all payments, i.e. over all losses larger than the deductible. This topic is continued in the next post.

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\copyright 2017 – Dan Ma