Month: January 2019

Insurance payment per payment vs payment per loss

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This post picks up where this previous post leaves off. The previous post is the second part of a 3-part discussion on mathematical model of insurance payment. The previous post ends with pointing out the difference between payment per loss and payment per payment. This post focuses on the insurance payment per payment and points out how the two payments are related. Another previous post is on how to calculate the variance of the insurance payment per loss.

Comparing the Two Payments

The insurance coverage being considered here is that losses are paid subject to an ordinary deductible d. If a loss is less than the deductible, the coverage pays nothing and if a loss exceeds the deductible, the coverage pays the loss less the deductible. This is called the insurance payment per loss variable and is 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 L in the subscript of Y_L indicates that the variable is the payment per loss. The variable Y_L includes the probability P(X \le d) as well as the probability P(X > d). Thus the average E(Y_L) is the average over all losses, hence the name payment per loss.

The other payment variable of interest is the payment per payment called Y_P. This variable captures the losses that require a claim payment, i.e. all losses exceeding the deductible. For Y_P, losses below the deductible are not considered and are truncated.

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

The variable Y_P is truncated and shifted. It is truncated below at d and is shifted to the left by the amount d. The expected value E(Y_P) is the average of all losses that require a payment, or the average of all payments that are made, hence the name payment per payment. The average E(Y_P) is also called the mean excess loss function.

(3)……\displaystyle e_X(d)=E(Y_P)=E(X-d \ \lvert X > d )

When the X is understood, we may use the notation e(d) instead of e_X(d). When the random variable X is a distribution for insurance losses, the function e_X(d) is interpreted as mean excess loss – the average portion of a loss that is in excess of the deductible. If X represents a distribution of lifetime, then the expectation is called the mean residual life or complete expectation of life. It is then notated by \overset{\circ}{e}_d or \overset{\circ}{e}(d).

We now compare the probability density functions (pdfs) and the cumulative distribution functions (cdfs) of the two payment variables. Let f(x) and F(x) be the pdf and cdf of the loss random variable X, respectively.

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

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

Both pdfs are a function of the pdf f(x) of the loss X. Assuming that the distribution of X is a continuous distribution, the pdf f_{Y_L}(y) is a mixed distribution (part discrete and part continuous). It has a point mass at y=0 with probability F(d)=P(X \le d), representing the scenario that no payment is made for a small loss. The pdf f_{Y_P}(y) is a continuous one if X is continuous. It is the result of f(x+d) divided by P(X > d) since it is a conditional distribution (only losses exceeding the deductible are considered). The cdfs of the payment per loss Y_L and the payment per payment Y_P are given by the following:

(6)……\displaystyle  F_{Y_L}(y)=\left\{ \begin{array}{ll}                     \displaystyle  0 &\ y<0 \\           \text{ } & \text{ } \\           \displaystyle  F(y+d) &\ y \ge 0           \end{array} \right.

(7)……\displaystyle F_{Y_P}(y)=\frac{F(y+d)-F(d)}{P(X > d)} \ \ \ \ \ \ \ y>0

The cdf F_{Y_L}(y) has a jump at y=0, reflecting the scenario that there is no payment for any loss less than the deductible d. Note that the size of the jump is F_{Y_L}(0)=F(d). The cdf F_{Y_P}(y) has no jump at y=0 since losses below the deductible are not accounted for in the variable Y_P.

Once the pdfs are derived, the calculation of the mean and variance of each of the two variables is possible. Once again, assuming the loss distribution is a continuous one, the following integrals give the calculation. When the loss distribution is discrete, simply replace integrals with summations.

(8)……\displaystyle E(Y_L)=E[(X-d)_+]=\int_0^\infty y \ f_{Y_L}(y) \  dy=\int_0^\infty y \ f(y+d) \  dy

(9)……\displaystyle E(Y_L^2)=E[(X-d)_+^2]=\int_0^\infty y^2 \ f_{Y_L}(y) \  dy=\int_0^\infty y^2 \ f(y+d) \  dy

(10)……\displaystyle Var(Y_L)=E(Y_L^2)-E(Y_L)^2

(11)……\displaystyle e_X(d)=E(Y_P)=\int_0^\infty y \ f_{Y_P}(y) \  dy=\int_0^\infty y \ \frac{f(y+d)}{P[X > d]} \  dy

(12)……\displaystyle E(Y_P^2)=\int_0^\infty y^2 \ f_{Y_P}(y) \  dy=\int_0^\infty y^2 \ \frac{f(y+d)}{P[X > d]} \  dy

(13)……\displaystyle Var(Y_P)=E(Y_P^2)-E(Y_P)^2

Comparing (8) and (11) and comparing (9) and (12) give the following relations.

(14)……\displaystyle e_X(d)=\frac{E(Y_L)}{P(X > d)}……or……\displaystyle E(Y_L)=P(X > d) \ e_X(d)

(15)……\displaystyle E(Y_P^2)=\frac{E(Y_L^2)}{P(X > d)}……or……\displaystyle E(Y_L^2)=P(X > d) \ E(Y_P^2)

The relations (14) and (15) show that only one set of calculation needs to be made. Once the first and second moments of one variable are known, the moments of the other variable are obtained by adjusting with P[X > d], the probability of having a claim.

Demonstrating the Calculation

We work one example to illustrate the above calculation.

Example 1
The loss X has the density function f(x)=\frac{3}{500} \ x (10-x) where 0<x<10. The coverage has a deductible of 3. Compute the mean and variance of the insurance payment per loss. Use the per loss results to obtain the mean and variance of the payment per payment. Write out the cdf of Y_L and the cdf of Y_P.

First, obtain the density function of the payment per loss.

    \displaystyle  f_{Y_L}(y)=\left\{ \begin{array}{ll}                     \displaystyle  F(3)=\frac{108}{500} &\ y=0 \\           \text{ } & \text{ } \\           \displaystyle  f(y+3)=\frac{3}{500} (21+4y-y^2) &\ 0<y<7           \end{array} \right.

The following calculates the mean and variance.

    \displaystyle \begin{aligned} E(Y_L)&=\int_0^7 y \ \frac{3}{500} \ (21+4y-y^2) \ dy \\&=\int_0^7 \frac{3}{500} \ (21 y+4y^2-y^3) \ dy \\&=\frac{4459}{2000}=2.2295  \end{aligned}

    \displaystyle \begin{aligned} E(Y_L^2)&=\int_0^7 y^2 \ \frac{3}{500} \ (21+4y-y^2) \ dy \\&=\int_0^7 \frac{3}{500} \ (21 y^2+4y^3-y^4) \ dy \\&=\frac{21609}{2500}=8.6436  \end{aligned}

    \displaystyle Var(Y_L)=8.6436-2.2295^2=3.6729

The probability of a claim is P(X > 3)=\frac{392}{500}. Dividing the results by this probability gives the following results.

    \displaystyle e_X(3)=\frac{E(Y_L)}{P(X > 3)}=\frac{4459}{2000} \ \frac{500}{392}=\frac{11.375}{4}=2.84375

    \displaystyle E(Y_P^2)=\frac{E(Y_L^2)}{P(X > 3)}=\frac{21609}{2500} \ \frac{500}{392}=\frac{55.125}{5}=11.025

    \displaystyle Var(Y_P)=11.025-2.84375^2=2.9381

To contrast, E(X)=5, the average payment for loss if there is no deductible. By imposing a deductible of 3, on average the amount of 5 – 2.2295 = 2.7705 per loss is shifted to the insured. The average payment per payment 2.84375 is higher than the average payment per loss 2.2295 because the per payment calculation centers on the loss exceeding the deductible.

The cdf of the original loss X is F(x)=\frac{3}{500} \ (5x^2-\frac{1}{3} x^3) where 0 \le x <10. The cdfs of Y_L and Y_P are given by:

    \displaystyle  F_{Y_L}(y)=\left\{ \begin{array}{ll}                     \displaystyle  0 &\ y<0 \\           \text{ } & \text{ } \\           \displaystyle \frac{3}{500} \ \biggl[5(y+3)^2-\frac{1}{3} (y+3)^3 \biggr]  &\ 0 \le y \le 7 \\           \text{ } & \text{ } \\           \displaystyle 1  &\ 7 < y < \infty           \end{array} \right.

    \displaystyle \begin{aligned} F_{Y_P}(y)&=\frac{F(y+3)-F(3)}{P[X > 3]} \\&=\frac{\frac{3}{500} \ \biggl[5(y+3)^2-\frac{1}{3} (y+3)^3 \biggr]-\frac{108}{500}}{\frac{392}{500}} \\&=\frac{3 \ \biggl[5(y+3)^2-\frac{1}{3} (y+3)^3 \biggr]-108}{392} \ \ \ \ \ 0 \le y \le 7  \end{aligned}

With the cdf of Y_P set up, we can determine the probabilities of claim payments. For example, if a claim has to be paid, what is the probability that the claim payment is between 3 and 5?

    \displaystyle \begin{aligned} F_{Y_P}(5)-F_{Y_P}(3)&=\frac{3 \ \biggl[5 \cdot 8^2-\frac{1}{3} \cdot 8^3 \biggr]-108}{392}-\frac{3 \ \biggl[5 \cdot 6^2-\frac{1}{3} \cdot 6^3 \biggr]-108}{392} \\&=\frac{340}{392}-\frac{216}{392} =\frac{124}{392}=0.3163  \end{aligned}

Using Limited Expectation

The above discussion focuses on developing the probability distribution of the payment per loss Y_L and the probability distribution of the payment per payment, as well as how the two distributions relate. Once the distribution of a payment variable is established, we can then evaluate various distributional quantities regarding the payment in question – e.g. mean and variance.

Another way to calculate E(Y_L)=(X-d)_+ is through the limited loss random variable X \wedge u.

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

It is clear that X=(X-d)_+ + X \wedge d. In words, buying a policy with a deductible of d and another policy with a policy limit of d equals full coverage. The expectation E(X \wedge u) is called the limited expectation. Immediately we have the following relationship.

(17)……\displaystyle E(X)=E((X-d)_+)+E(X \wedge d)……or……\displaystyle E(Y_L)=E((X-d)_+)=E(X)-E(X \wedge d )

One advantage of the relationship (17) is that the limited expectation E(X \wedge u ) is provided in a table of distributions (table). For several distributions such as exponential, lognormal, Pareto and a few others, the formulas of E(X \wedge u) are quite easy to implement. When the loss distribution is one of these distributions, (17) provides another way to compute the expected payment per loss. Combining (17) and (14) gives the following formula for the expected payment per payment (or mean excess loss).

(18)……\displaystyle e_X(d)=\frac{E(X)-E(X \wedge d )}{P(X > d)}

Franchise Deductible

The deductible in the above discussion is an ordinary deductible. The key characteristic of ordinary deductible is that the first d dollars are not paid by the insurer. A franchise deductible works like an ordinary deductible except that when the loss exceeds the deductible, the policy pays the loss in full.

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

Instead of building a new set of formulas, we can simply relate the expected payment under the franchise deductible with the expected payment under the ordinary deductible.

(20)……expected payment per loss under franchise deductible = \displaystyle E(Y_L)+d \ P(X > d)

(21)……expected payment per payment under franchise deductible = \displaystyle e_X(d)+d

The payment under a franchise deductible is larger than the payment under an ordinary deductible. By how much? The deductible d. The addition of d \ P[X > d] in (20) reflects the additional benefit when X > d. The expected payment per payment is already conditioned on X > d. Thus d is unconditionally added to e_X(d) in (21). The variance can be calculated using basic principle. Example 2 shows how.

To distinguish between the two deductibles, we adopt the convention that deductible means ordinary deductible. If franchise deductible is used, “franchise” would have to be explicitly stated.

More Examples

Example 2
Work Example 1 assuming a franchise deductible.

To keep things straight, we let Y to denote the payment per loss and Y_{*} to denote the payment per payment (the notation is applicable just for this example). Then \displaystyle E(Y) and \displaystyle E(Y_*) are greater than their counterparts for ordinary deductible according to (20) and (21).

    \displaystyle E(Y)=E(Y_L)+3 \ P(X > 3)=\frac{4459}{2000}+3 \cdot \frac{392}{5000}=\frac{9163}{2000}=4.5815

    \displaystyle E(Y_*)=E(Y_P)+3=\frac{11.375}{4}+3=\frac{23.375}{4}=5.84375

To find the variance, it is helpful to work with the pdfs.

    \displaystyle  f_{Y}(y)=\left\{ \begin{array}{ll}                     \displaystyle  F(3)=\frac{108}{500} &\ y=0 \\           \text{ } & \text{ } \\           \displaystyle  f(y)=\frac{3}{500} (10y-y^2) &\ 3<y<10           \end{array} \right.

For franchise deductible, there is no need to shift the pdf by the deductible. There is a point mass at y=0 like before. The pdf continues with the interval (3, 10). With this pdf, the following gives the calculation of the variance.

    \displaystyle E(Y^2)=\int_3^{10} y^2 \ \frac{3}{500} (10y-y^2) \ dy=\frac{290766}{10000}=29.0766

    \displaystyle Var(Y)=29.0766-4.5815^2=8.0865

We can leverage the per loss variance to obtain the per payment variance.

    \displaystyle E(Y_*^2)=\frac{E[Y^2]}{P[X > 3]}=29.0766 \cdot \frac{500}{392}=37.0875

    \displaystyle Var(Y_*)=37.0875-5.84375^2=2.9381

The cdfs are given by the following:

    \displaystyle  F_{Y}(y)=\left\{ \begin{array}{ll}          \displaystyle  0 &\ y<0 \\           \text{ } & \text{ } \\                       \displaystyle F(3)=\frac{108}{500}  &\ 0 \le y < 3 \\           \text{ } & \text{ } \\           \displaystyle \frac{3}{500} \ \biggl[5 y^2-\frac{1}{3} y^3 \biggr]  &\ 3 \le y \le 10 \\           \text{ } & \text{ } \\           \displaystyle 1  &\ 10 < y < \infty           \end{array} \right.

    \displaystyle  F_{Y_*}(y)=\left\{ \begin{array}{ll}          \displaystyle  0 &\ y<3 \\                      \text{ } & \text{ } \\           \displaystyle \frac{\frac{3}{500} \ \biggl[5 y^2-\frac{1}{3} y^3 \biggr]-\frac{108}{500}}{\frac{392}{500}}=\frac{3 \ \biggl[5 y^2-\frac{1}{3} y^3 \biggr]-108}{392}  &\ 3 \le y \le 10 \\           \text{ } & \text{ } \\           \displaystyle 1  &\ 10 < y < \infty           \end{array} \right.

If the insurer has to pay a claim, what is the probability that it is between 6 and 8?

    \displaystyle  F_{Y_*}(8)-F_{Y_*}(6)=\frac{340}{392}-\frac{216}{392}=\frac{124}{392}=0.3163

Note that this answer is the same one as in Example 1 since there is no shifting with a franchise deductible.

Example 3
The discussion is not complete without mentioning exponential distribution. Suppose that the pdf of the loss X is f(x)=\frac{1}{\theta} \ e^{-x/\theta} where x >0, i.e. the pdf for an exponential distribution with mean \theta. The deductible has an ordinary deductible of d. The expected payment per loss and the variance of payment per loss are:

    \displaystyle E(Y_L)= \theta \ e^{-d/\theta}

    \displaystyle E(Y_L^2)= 2 \ \theta^2 \ e^{-d/\theta}

    \displaystyle Var(Y_L)=2 \ \theta^2 \ e^{-d/\theta}-\theta^2 \ e^{-2d/\theta}

The above results are obtained by using the pdf of the payment per loss Y_L, which can be obtained according to (4). The results for payment per payment are obtained by dividing by P(X > d)=e^{-d/\theta}.

    \displaystyle E(Y_P)= \theta

    \displaystyle E(Y_P^2)= 2 \ \theta^2

    \displaystyle Var(Y_P)=2 \ \theta^2 -\theta^2 =\theta^2

The mean and variance of Y_P are identical to the original exponential loss distribution. This is to be expected. Because the exponential distribution is memoryless, the payment variable Y_P=X-d \ \lvert X > d is identical to the original loss X, an exponential distribution with mean \theta. For models of insurance payments with a deductible, knowing that the loss distribution is exponential simplifies the calculation greatly.

Example 4
This example illustrates how to obtain the mean payment per loss using the table approach – using limited expectation given in this table. Suppose that the loss distribution is a lognormal distribution with parameters \mu=2 and \sigma=2. The coverage has a deductible of 200. Compute the expected payment per loss E(Y_L).

Using the formula for E(X) and the formula for the limited expectation E(X \wedge x) found in the table, the expected payment per loss is given by:

    \displaystyle F(200)=\Phi \biggl( \frac{\ln(200)-5}{2} \biggr)=\Phi(0.15)=0.5596

    \displaystyle 1-F(200)=1-0.5596=0.4404

    \displaystyle E(X)=e^{\mu+\frac{1}{2} \sigma^2}=e^7

    \displaystyle \begin{aligned} E(X \wedge 200)&=e^{\mu+\frac{1}{2} \sigma^2} \ \Phi \biggl( \frac{\ln(200)-\mu-\sigma^2}{\sigma} \biggr)+200 [1-F(200)] \\&=e^{7} \ \Phi \biggl( \frac{\ln(200)-5-2^2}{2} \biggr)+200 [0.4404] \\&=e^7 \  \Phi(-1.85)+88.08 \\&=e^7 \ (1-0.9678)+88.08 \\&=0.0322 \ e^7+88.08 \end{aligned}

    \displaystyle \begin{aligned} E(Y_L)&=E(X)-E(X \wedge 200) \\&=e^7-(0.0322 \ e^7+88.08) \\&=0.9678 \ e^7-88.08=973.2415707  \end{aligned}

Example 5
This example continues Example 4. We now tackle the variance of the payment per loss. For the lognormal loss distribution, it is difficult to find the pdf of the payment per loss Y_L. In the table, there is no direct formula for E[Y_L^2]. However, there is a way to use the limited expectations E(X \wedge d) and E((X \wedge d)^2) to derive E(Y_L^2). First, let’s break down the components that go into E(Y_L^2).

    \displaystyle \begin{aligned} E(Y_L^2)&=\int_d^\infty (x-d)^2 f(x) \ dx \\&=\int_d^\infty x^2 f(x) \ dx-2 \ d \int_d^\infty x f(x) \ dx+d^2 \ P(X > d)  \end{aligned}

The f(x) in above is the pdf of the lognormal distribution. Of course we are not to evaluate the above two integrals directly. However, they can be expressed using limited expectations as follows:

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

    \displaystyle \int_d^\infty x^2 f(x) \ dx=E[X^2]-E[(X \wedge d)^2]+d^2 \ P(X > d)

With respect to the lognormal parameters \mu=2 and \sigma=2 and the deductible 200, the limited expectation E((X \wedge 200)^2) is given by the following. The first limited expectation is already calculated in Example 4.

    \displaystyle E(X^2)=e^{2 \ \mu+\frac{1}{2} \ 2^2 \ \sigma^2}=e^{18}

    \displaystyle \begin{aligned} E((X \wedge 200)^2)&=e^{2 \mu+\frac{1}{2} \ 2^2 \ \sigma^2} \ \Phi \biggl( \frac{\ln(200)-\mu-2 \ \sigma^2}{\sigma} \biggr)+200^2 [1-F(200)] \\&=e^{18} \ \Phi \biggl( \frac{\ln(200)-5-2 \cdot 2^2}{2} \biggr)+200^2 [0.4404] \\&=e^{18} \  \Phi(-3.85)+17616 \\&=e^{18} \ (1-0.9999)+17616 \\&=0.0001 \ e^{18}+17616 \end{aligned}

The calculation continues:

    \displaystyle \begin{aligned}\int_{200}^\infty x f(x) \ dx&=E(X)-E(X \wedge 200)+200 \ P(X > 200) \\&=e^7-[0.0322 \ e^{7}+88.08]+88.08 \\&=0.9678 \ e^7  \end{aligned}

    \displaystyle \begin{aligned}\int_{200}^\infty x^2 f(x) \ dx&=E(X^2)-E((X \wedge 200)^2)+200^2 \ P(X > 200) \\&=e^{18}-[0.0001 \ e^{18}+17616]+17616 \\&=0.9999 \ e^{18}  \end{aligned}

    \displaystyle \begin{aligned} E(Y_L^2)&=\int_{200}^\infty x^2 f(x) \ dx-2 \ 200 \int_{200}^\infty x f(x) \ dx+200^2 \ P(X > 200) \\&=0.9999 \ e^{18}-2 \cdot 200 \cdot 0.9678 \ e^7 +17616\\&=65246490.51  \end{aligned}

    \displaystyle Var(Y_L)=65246490.51-(0.9678 \ e^7-88.08)^2=64299291.36

    \displaystyle \sigma_{Y_L}=\sqrt{64299291.36}=8018.683892

Example 6
This example discusses the Pareto distribution as the loss distribution. The Pareto distribution is a two-parameter family of distributions. The pdf and cdf are given below, along with the mean and the variance.

    \displaystyle f(x)=\frac{\alpha \ \theta^\alpha}{(x+\theta)^{\alpha+1}} \ \ \ \ \ \ \ \ \ \ x>0

    \displaystyle F(x)=1-\biggl(\frac{\theta}{x+\theta} \biggr)^\alpha \ \ \ \ \ x>0

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

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

    \displaystyle Var(X)=E(X^2)-E(X)^2

For the Pareto mean to exist, the parameter \alpha must be greater than 1. For the variance to exist, the parameter \alpha must be greater than 2. When the coverage has an ordinary deductible d, consider the insurance payment per loss Y_L and the insurance payment per loss Y_P. In terms of these two payment variables, the Pareto distribution is very tractable. To find mean payment per loss, either start with the pdf of Y_L or use the limited expectation E(X \wedge d) from the table. Because the payment per payment is also a Pareto distribution, there are shortcuts for finding the variance of payment per loss (see Example 3 here).

When the loss X is a Pareto distribution with parameters \alpha and \theta, the payment per payment Y_P=X-d \ \lvert X>d has a Pareto distribution with parameters \alpha and \theta+d. Let’s examine the expected value of Y_P, or the mean excess loss.

    \displaystyle e_X(d)=E(X-d \ \lvert X>d)=\frac{\theta+d}{\alpha-1}=\frac{\theta}{\alpha-1}+ \frac{1}{\alpha-1} \ d

Note that the mean excess loss e_X(d) is an increasing function of d. This means that the higher the deductible, the larger the expected claim if such a large loss occurs! If a random loss is modeled by such a distribution, it is a catastrophic risk situation. In general, an increasing mean excess loss function is an indication of a heavy tailed distribution. Thus the Pareto distribution is considered a heavy tailed distribution and is a suitable candidate for modeling situations that have potential extreme large losses.

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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.

In contrast, the next post is a discussion on the insurance payment per payment.

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.

In contrast, the next post is a discussion on the insurance payment per payment.

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Practice Problem Set 12 – (a,b,1) class

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The practice problems in this post are to reinforce the concepts of (a,b,0) class and (a,b,1) class discussed this blog post and this blog post in a companion blog. These two posts have a great deal of technical details, especially the one on (a,b,1) class. The exposition in this blog post should be more accessible.

Notation: p_k=P(N=k) for k=0,1,2,\cdots whenever N is the counting distribution that is one of the (a,b,0) distributions – Poisson, binomial and negative binomial distribution. The notation P_k^T is the probability that a zero-truncated distribution taking on the value k. Likewise P_k^M is the probability that a zero-modified distribution taking on the value k.

Practice Problem 12-A
  • Consider a Poisson distribution with mean \lambda=1.2. Evaluate the probabilities P_k where k=0,1,2,3,4,5.
  • Consider the corresponding zero-truncated Poisson distribution. Evaluate the probabilities P_k^T where k=1,2,3,4,5.
  • Consider the corresponding zero-modified Poisson distribution with P_0^M=0.4. Evaluate the probabilities P_k^M where k=1,2,3,4,5.
Practice Problem 12-B

This problem is a continuation of Problem 12-A. The following is the probability generating function (pgf) of the Poisson distribution in Problem 12-A.

    …………….\displaystyle P(z)=e^{1.2 \ (z-1)}
  • Determine the mean, variance and the pgf of the zero-truncated Poisson distribution in Problem 12-A.
  • Determine the mean, variance and the pgf of the zero-modified Poisson distribution in Problem 12-A.
Practice Problem 12-C

Consider a negative binomial distribution with a=2/3 and b=4/3.

  • Evaluate P_0, P_1, P_2 and P_3.
  • Evaluate the mean and variance of the corresponding zero-truncated negative binomial distribution.
  • Evaluate the mean and variance of the corresponding zero-modified negative binomial distribution with P_0^M=0.1.
Practice Problem 12-D

The following is the probability generating function (pgf) of the negative binomial distribution in Problem 12-C.

    …………….\displaystyle P(z)=[1-2 (z-1)]^{-3}
  • Evaluate P'(z), P''(z) and P^{(3)}(z), the first, second and third derivative of P(z), respectively.
  • Evaluate P'(0), \frac{P''(0)}{2!} and \frac{P^{(3)}(0)}{3!}. Compare these with P_1, P_2 and P_3 found in Problem 12-C.
Practice Problem 12-E

This is a continuation of Problem 12-C and Problem 12-D.

  • Using the pgf P(z) in Problem 12-D, find the pgf of the corresponding zero-truncated negative binomial distribution. Call this pgf g(z).
  • Evaluate g'(z), g''(z) and g^{(3)}(z), the first, second and third derivative of g(z), respectively.
  • Obtain P_1^T, P_2^T and P_3^T by evaluating g'(0), \frac{g''(0)}{2!} and \frac{g^{(3)}(0)}{3!}.
Practice Problem 12-F

This problem is similar to Problem 12-E.

  • Using the pgf g(z) in Problem 12-E, find the pgf of the corresponding zero-modified negative binomial distribution. Call this pgf h(z).
  • Evaluate h'(z), h''(z) and h^{(3)}(z), the first, second and third derivative of h(z), respectively.
  • Obtain P_1^M, P_2^M and P_3^M by evaluating h'(0), \frac{h''(0)}{2!} and \frac{h^{(3)}(0)}{3!}.
Practice Problem 12-G

Suppose that the following three probabilities are from a zero-truncated (a,b,0) distribution.

    P_3^T=0.147692308

    P_4^T=0.132923077

    P_5^T=0.111655385

  • Determine the recursion parameters a and b of the corresponding (a,b,0) distribution.
  • What is the (a,b,0) distribution?
  • Evaluate the mean and variance of this (a,b,0) distribution.
Practice Problem 12-H
Consider a zero-modified distribution. The following three probabilities are from this zero-modified distribution.

    P_2^M=0.08669868

    P_3^M=0.162560025

    P_4^M=0.205740032

  • Determine the recursion parameters a and b of the corresponding (a,b,0) distribution.
  • What is the (a,b,0) distribution?
  • Determine P_1^T, the probability that the corresponding zero-truncated distribution taking on the value of 1.
  • Determine P_1^M, the probability that the zero-modified distribution taking on the value of 1.
  • Determine P_0^M.
Practice Problem 12-I
For a distribution from the (a,b,0) class, you are given that

  • a=0.5 and b=1.5,
  • P_5^T=7/60 for the corresponding zero-truncated distribution,
  • P_7^M=0.05 for the corresponding zero-modified distribution for some value of P_0^M.

Determine P_0^M.

Practice Problem 12-J

Generate an extended truncated negative binomial (ETNB) distribution with r=-0.5 and \theta=2. Note that this is to start with a negative binomial distribution with parameters r=-0.5 and \theta=2 and then derive its zero-truncated distribution. The parameters r=-0.5 and \theta=2 will not give a distribution but over look this point and go through the process of creating a zero-truncated distribution. In particular, determine the following.

  • Determine P_k^T for k=1,2,3,4.
  • Determine the mean and variance of the ETNB distribution.

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Problem Answer
12-A
  • Poisson
    • P_0=e^{-1.2}
    • P_1=1.2 \ e^{-1.2}
    • P_2=0.72 \ e^{-1.2}
    • P_3=0.288 \ e^{-1.2}
    • P_4=0.0864 \ e^{-1.2}
    • P_5=0.020736 \ e^{-1.2}
  • Zero-Truncated Poisson
    • \displaystyle  P_1^T=\frac{1.2 \ e^{-1.2}}{1-e^{-1.2}}=0.517215313
    • \displaystyle P_2^T=\frac{0.72 \ e^{-1.2}}{1-e^{-1.2}}=0.310329288
    • \displaystyle P_3^T=\frac{0.288 \ e^{-1.2}}{1-e^{-1.2}}=0.124131675
    • \displaystyle P_4^T=\frac{0.0864 \ e^{-1.2}}{1-e^{-1.2}}=0.037239503
    • \displaystyle P_5^T=\frac{0.020736 \ e^{-1.2}}{1-e^{-1.2}}=0.008937481
  • Zero-Modified Poisson
    • P_0^M=0.4
    • \displaystyle P_1^M=0.6 \ \frac{1.2 \ e^{-1.2}}{1-e^{-1.2}}=0.310329188
    • \displaystyle P_2^M=0.6 \ \frac{0.72 \ e^{-1.2}}{1-e^{-1.2}}=0.186197513
    • \displaystyle P_3^M=0.6 \ \frac{0.288 \ e^{-1.2}}{1-e^{-1.2}}=0.074479005
    • \displaystyle P_4^M=0.6 \ \frac{0.0864 \ e^{-1.2}}{1-e^{-1.2}}=0.022343702
    • \displaystyle P_5^M=0.6 \ \frac{0.020736 \ e^{-1.2}}{1-e^{-1.2}}=0.005362488
12-B
  • Zero-Truncated Poisson
    • mean = \displaystyle E[N_T]=\frac{1.2}{1-e^{-1.2}}=1.717215313
    • second moment = \displaystyle =E[N_T^2]=\frac{2.64}{1-e^{-1.2}}=3.777873688
    • variance = \displaystyle Var[N_T]=E[N_T^2]-E[N_T]^2=0.829045258
    • pgf = \displaystyle P^T(z)=\frac{1}{1-e^{-1.2}}  \ [e^{1.2 (z-1)} - e^{-1.2}]
  • Zero-Modified Poisson
    • mean = \displaystyle E[N_M]=\frac{0.72}{1-e^{-1.2}}=1.030329188
    • second moment = \displaystyle =E[N_M^2]=\frac{1.584}{1-e^{-1.2}}=2.266724213
    • variance = \displaystyle Var[N_M]=E[N_M^2]-E[N_M]^2=1.205145978
    • pgf = \displaystyle P^M(z)=0.4+\frac{0.6}{1-e^{-1.2}}  \ [e^{1.2 (z-1)} - e^{-1.2}]
12-C
  • Negative binomial
    • \displaystyle P_0=\frac{1}{27}
    • \displaystyle P_1=\frac{2}{27}
    • \displaystyle P_2=\frac{8}{81}
    • \displaystyle P_3=\frac{80}{729}
  • Zero-truncated negative binomial
    • \displaystyle E[N_T]=\frac{81}{13}=6.23
    • \displaystyle E[N_T^2]=\frac{729}{13}
    • \displaystyle Var[N_T]=\frac{2916}{169}=17.2544
  • Zero-modified negative binomial
    • \displaystyle E[N_M]=\frac{72.9}{13}=5.607692308
    • \displaystyle E[N_M^2]=\frac{656.1}{13}
    • \displaystyle Var[N_M]=\frac{3214.89}{169}=19.02301775
12-D
  • \displaystyle P'(z)=6 \ [1-2 (z-1)]^{-4}
  • \displaystyle P''(z)=48 \ [1-2 (z-1)]^{-5}
  • \displaystyle P^{(3)}(z)=480 \ [1-2 (z-1)]^{-6}
  • \displaystyle P'(0)=\frac{2}{27}
  • \displaystyle \frac{P''(0)}{2!}=\frac{8}{81}
  • \displaystyle \frac{P^{(3)}(0)}{3!}=\frac{80}{729}
21-E
  • \displaystyle g(z)=P^T(z)=\frac{27}{26}\biggl([1-2 (z-1)]^{-3}-\frac{1}{27} \biggr)
  • \displaystyle g'(z)=\frac{27}{26} \ P'(z)
  • \displaystyle g''(z)=\frac{27}{26} \ P''(z)
  • \displaystyle g^{(3)}(z)=\frac{27}{26} \ P^{(3)}(z)
  • \displaystyle g'(0)=\frac{1}{13}=0.076923077
  • \displaystyle \frac{g''(0)}{2!}=\frac{4}{39}=0.102564103
  • \displaystyle \frac{g^{(3)}(0)}{3!}=\frac{40}{351}=0.113960114
12-F
  • \displaystyle h(z)=P^M(z)=0.1+\frac{24.3}{26}\biggl([1-2 (z-1)]^{-3}-\frac{1}{27} \biggr)
  • \displaystyle h'(z)=\frac{24.3}{26} \ P'(z)
  • \displaystyle h''(z)=\frac{24.3}{26} \ P''(z)
  • \displaystyle h^{(3)}(z)=\frac{24.3}{26} \ P^{(3)}(z)
  • \displaystyle h'(0)=\frac{0.9}{13}=0.069230769
  • \displaystyle \frac{h''(0)}{2!}=\frac{3.6}{39}=0.092307692
  • \displaystyle \frac{h^{(3)}(0)}{3!}=\frac{36}{351}=0.102564103
12-G
  • \displaystyle a=\frac{3}{5} and \displaystyle b=\frac{6}{5}
  • Negative binomial distribution with r=3 and \displaystyle \theta=\frac{3}{2}
  • mean = \displaystyle \frac{9}{2}=4.5 and variance = \displaystyle \frac{45}{4}=11.25
12-H
  • \displaystyle a=-0.5625 and \displaystyle b=7.3125
  • Binomial distribution with n=12 and \displaystyle p=0.36
  • \displaystyle P_1^T=\frac{6.75 \ (0.64)^{12}}{1-0.64^{12}}=0.032027218
  • \displaystyle P_1^M=\frac{P_2^M}{a+\frac{b}{2}}=0.028023158
  • P_0^M=0.125
12-I
  • P_0^M=0.2
12-J
  • ETNB Probabilities
    • \displaystyle P_1^T=\frac{1}{1-3^{0.5}} \biggl(\frac{-1}{3} \biggr) \ 3^{0.5}=0.7886751346
    • \displaystyle P_2^T=\frac{1}{1-3^{0.5}} \biggl(\frac{-1}{18} \biggr) \ 3^{0.5}=0.1314458558
    • \displaystyle P_3^T=\frac{1}{1-3^{0.5}} \biggl(\frac{-1}{54} \biggr) \ 3^{0.5}=0.0438152853
    • \displaystyle P_4^T=\frac{1}{1-3^{0.5}} \biggl(\frac{-5}{648} \biggr) \ 3^{0.5}=0.0182563689
  • ETNB Mean and Variance
    • \displaystyle E[N_T]=\frac{-1}{1-3^{0.5}}=1.366025404
    • \displaystyle E[N_T^2]=\frac{-2}{1-3^{0.5}}
    • \displaystyle Var[N_T]=E[N_T^2]-E[N_T]^2=0.8660254038

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The (a,b,0) and (a,b,1) classes

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This post is on two classes of discrete distributions called the (a,b,0) class and (a,b,1) class. This post is a follow-up on two previous posts – summarizing the two posts and giving more examples. The (a,b,0) class is discussed in details in this post in a companion blog. The (a,b,1) class is discussed in details in this post in a companion blog.

Practice problems for the (a,b,0) class is found here. The next post is a practice problem set on the (a,b,1) class.

The (a,b,0) Class

A counting distribution is a discrete probability distribution that takes on the non-negative integers (0, 1, 2, …). Counting distributions are useful when we want to model occurrences of a certain random events. The three commonly used counting distributions would be the Poisson distribution, the binomial distribution and the negative binomial distribution. All three counting distributions can be generated recursively. For these three distributions, the ratio of any two consecutive probabilities multiplied by integers can be expressed as a linear quantity.

To make the last point in the preceding paragraph clear, let’s set some notations. For any integer k=0,1,2,\cdots, let P_k be the probability that the counting distribution in question takes on the value k. For example, if we are considering the counting random variable N, then P_k=P[N=k]. Let’s look at the situation where the ratio of any two consecutive values of P_k can be expressed as an expression for some constants a and b.

(1)……….\displaystyle \frac{P_k}{P_{k-1}}=a + \frac{b}{k} \ \ \ \ \ \ \ \ \ \ \ \ \ k=1,2,3,\cdots

Multiplying (1) by k gives the following.

(1a)……….\displaystyle k \ \frac{P_k}{P_{k-1}}=a \ k+ b \ \ \ \ \ \ \ \ \ \ \ \ \ k=1,2,3,\cdots

Note that the right-hand side of (1a) is a linear expression of k. This provides a way to fit observations to (a,b,0) distributions.

Any counting distribution that satisfies the recursive relation (1) is said to be a member of the (a,b,0) class of distributions. Note that the recursion starts at k=1. Does that mean P_0 can be any probability value we assign? The value of P_0 is fixed because all the P_k must sum to 1.

The three counting distribution mentioned above – Poisson, binomial and negative binomial – are all members of the (a,b,0) class. In fact the (a,b,0) class essentially has three distributions. In other words, any member of (a,b,0) class must be one of the three distributions – Poisson, binomial and negative binomial.

An (a,b,0) distribution has its usual parameters, e.g. Poisson has a parameter \lambda, which is its mean. So we need to way to translate the usual parameters to and from the parameters a and b. This is shown in the table below.

Table 1

Distribution Usual Parameters Probability at Zero Parameter a Parameter b
Poisson \lambda e^{-\lambda} 0 \lambda
Binomial n and p (1-p)^n \displaystyle -\frac{p}{1-p} \displaystyle (n+1) \ \frac{p}{1-p}
Negative binomial r and p p^r 1-p (r-1) \ (1-p)
Negative binomial r and \theta \displaystyle \biggl(\frac{1}{1+\theta} \biggr)^r \displaystyle \frac{\theta}{1+\theta} \displaystyle (r-1) \ \frac{\theta}{1+\theta}
Geometric p p 1-p 0
Geometric \theta \displaystyle \frac{1}{1+\theta} \displaystyle \frac{\theta}{1+\theta} 0

Table 1 provides the mapping to translate between the usual parameters and the recursive parameters a and b.

Example 1
Let a=-1/3 and b=5/3. Let the initial probability be P_0=81/256. Generate the first 4 probabilities according to the recursion formula (1)

    \displaystyle P_0=\frac{81}{256}

    \displaystyle P_1=\biggl(-\frac{1}{3}+\frac{5}{3} \biggr) \ \frac{81}{256}=\frac{108}{256}

    \displaystyle P_2=\biggl(-\frac{1}{3}+\frac{5}{3} \cdot \frac{1}{2} \biggr) \ \frac{108}{256}=\frac{54}{256}

    \displaystyle P_3=\biggl(-\frac{1}{3}+\frac{5}{3} \cdot \frac{1}{3} \biggr) \ \frac{54}{256}=\frac{12}{256}

    \displaystyle P_4=\biggl(-\frac{1}{3}+\frac{5}{3} \cdot \frac{1}{4} \biggr) \ \frac{12}{256}=\frac{1}{256}

Note that the sum of P_0 to P_4 is 1. So this has to be a binomial distribution and not Poisson or negative binomial. The binomial parameters are n=4 and p=1/4. According to Table 1, this translate to a=-1/3 and b=5/3. The initial probability is P_0=(1-p)^4.

Example 2
This example generates several probabilities recursively for the negative binomial distribution with r=2 and \theta=4. According to Table 1, this translates to a=4/5 and b=4/5. The following shows the probabilities up to P_6.

    \displaystyle P_0=\biggl(\frac{1}{5}\biggr)^2=\frac{1}{25}=0.04

    \displaystyle P_1=\biggl(\frac{4}{5}+\frac{4}{5} \biggr) \ \frac{1}{25}=\frac{8}{125}=0.064

    \displaystyle P_2=\biggl(\frac{4}{5}+\frac{4}{5} \cdot \frac{1}{2} \biggr) \ \frac{8}{125}=\frac{48}{625}=0.0768

    \displaystyle P_3=\biggl(\frac{4}{5}+\frac{4}{5} \cdot \frac{1}{3} \biggr) \ \frac{48}{625}=\frac{256}{3125}=0.08192

    \displaystyle P_4=\biggl(\frac{4}{5}+\frac{4}{5} \cdot \frac{1}{4} \biggr) \ \frac{256}{3125}=\frac{256}{3125}=0.08192

    \displaystyle P_5=\biggl(\frac{4}{5}+\frac{4}{5} \cdot \frac{1}{5} \biggr) \ \frac{256}{3125}=\frac{6144}{78125}=0.0786432

    \displaystyle P_6=\biggl(\frac{4}{5}+\frac{4}{5} \cdot \frac{1}{6} \biggr) \ \frac{6144}{78125}=\frac{28672}{390625}=0.07340032

The above probabilities can also be computed using the probability function given below.

    \displaystyle P_k=(1+k) \ \frac{1}{25} \ \biggl(\frac{4}{5} \biggr)^k \ \ \ \ \ \ \ \ k=0,1,2,3,\cdots

For the (a,b,0) class, it is not just about calculating probabilities recursively. The parameters a and b also give information about other distributional quantities such as moments and variance. For a more detailed discussion of the (a,b,0) class, refer to this post in a companion blog.

The (a,b,1) Class

If the (a,b,0) class is just another name for the three distributions of Poisson, binomial and negative binomial, what is the point of (a,b,0) class? Why not just work with these three distributions individually? Sure, generating the probabilities recursively is a useful concept. The probability functions of the three distributions already give us a clear and precise way to calculate probabilities. The notion of (a,b,0) class leads to the notion of (a,b,1) class, which gives a great deal more flexibility in the modeling counting distributions. It is possible that the (a,b,0) distributions do not adequately describe a random counting phenomenon being observed. For example, the sample data may indicate that the probability at zero may be larger than is indicated by the distributions in the (a,b,0) class. One alternative is to assign a larger value for P_0 and recursively generate the subsequent probabilities P_k for k=2,3,\cdots. This recursive relation is the defining characteristics of the (a,b,1) class.

A counting distribution is a member of the (a,b,1) class of distributions if the following recursive relation holds for some constants a and b.

(2)……….\displaystyle \frac{P_k}{P_{k-1}}=a + \frac{b}{k} \ \ \ \ \ \ \ \ \ \ \ \ \ k=2,3,4, \cdots

Note that the recursion begins at k=2. Can the values for P_0 and P_1 be arbitrary? The initial probability P_0 is an assumed value. The probability P_1 is the value such that the sum P_1+P_2+P_3+\cdots is 1-P_0.

The (a,b,1) class gives more flexibility in modeling. For example, the initial probability is P_0=0.04 in the negative binomial distribution in Example 2. If this P_0 is felt to be too small, then a larger value for P_0 can be assigned and then let the remaining probabilities be generated by recursion. We demonstrate how this is done using the same (a,b,0) distribution in Example 2.

Before we continue with Example 2, we comment that there are two subclasses in the (a,b,1) class. The subclasses are distinguished by whether P_0=0 or P_0>0. The (a,b,1) distributions are called zero-truncated distributions in the first case and are called zero-modified distributions in the second case.

Because there are three related distributions, we need to establish notations to keep track of the different distributions. We use the notations established in this post. The notation P_k refers to the probabilities for an (a,b,0) distribution. From this (a,b,0) distribution, we can derive a zero-truncated distribution whose probabilities are notated by P_k^T. From this zero-truncated distribution, we can derive a zero-modified distribution whose probabilities are denoted by P_k^M. For example, for the negative binomial distribution in Example 2, we derive a zero-truncated negative binomial distribution (Example 3) and from it we derive a zero-modified negative binomial distribution (Example 4).

Example 3
In Example 3, we calculated the (a,b,0) probabilities P_k up to k=6. We now calculate the probabilities P_k^T for the corresponding zero-truncated negative binomial distribution. For a zero-truncated distribution, the value of zero is not recorded. So P_k^T is simply P_k divided by 1-P_0.

(3)……….\displaystyle P_k^T=\frac{1}{1-P_0} \ P_k \ \ \ \ \ \ \ \ \ \ k=1,2,3,4,\cdots

The sum of P_k^T, k=1,2,3,\cdots, must be 1 since P_0,P_1,P_3,\cdots is a probability distribution. The (a,b,0) P_0 is 1/25. Then 1-P_0=24/25, which means 1/(1-P_0)=25/24. The following shows the zero-truncated probabilities.

    \displaystyle P_1^T=\frac{8}{125} \cdot \frac{25}{24}=\frac{8}{120}

    \displaystyle P_2^T=\frac{8}{125} \cdot \frac{25}{24}=\frac{48}{600}

    \displaystyle P_3^T=\frac{256}{3125} \cdot \frac{25}{24}=\frac{256}{3000}

    \displaystyle P_4^T=\frac{256}{3125} \cdot \frac{25}{24}=\frac{256}{3000}

    \displaystyle P_5^T=\frac{6144}{78125} \cdot \frac{25}{24}=\frac{6144}{75000}

    \displaystyle P_6^T=\frac{28672}{390625} \cdot \frac{25}{24}=\frac{28672}{375000}

The above are the first 6 probabilities of the zero-truncated negative binomial distribution with a=4/5 and b=4/5 or with the usual parameters r=2 and \theta=4. The above P_k^T can also be calculated recursively by using (2). Just calculate P_1^T and the rest of the probabilities can be generated using the recursion relation (2).

Example 4
From the zero-truncated negative binomial distribution in Example 3, we generate a zero-modified negative binomial distribution. If the original P_0=0.04 is considered too small,e.g. not big enough to account for the probability of zero claims, then we can assign a larger value to the zero probability. Let’s say 0.10 is more appropriate. So we set P_0^M=0.10. Then the rest of the P_k^M must sum to 1-P_0^M, or 0.9 in this example. The following shows how the zero-modified probabilities are related to the zero-truncated probabilities.

(4)……….\displaystyle P_k^M=(1-P_0^M) \ P_k^T \ \ \ \ \ \ \ \ \ \ k=1,2,3,4,\cdots

The following gives the probabilities for the zero-modified negative binomial distribution.

    \displaystyle P_0^M=0.1 (assumed value)

    \displaystyle P_1^M=0.9 \cdot \frac{8}{120}=\frac{7.2}{120}

    \displaystyle P_2^M=0.9 \cdot \frac{48}{600}=\frac{43.2}{600}

    \displaystyle P_3^M=0.9 \cdot \frac{256}{3000}=\frac{230.4}{3000}

    \displaystyle P_4^M=0.9 \cdot \frac{256}{3000}=\frac{230.4}{3000}

    \displaystyle P_5^M=0.9 \cdot \frac{6144}{75000}=\frac{5529.6}{75000}

    \displaystyle P_6^M=0.9 \cdot \frac{28672}{375000}=\frac{25804.8}{375000}

The same probabilities can also be obtained by using the original (a,b,0) probabilities P_k directly as follows:

(5)……….\displaystyle P_k^M=\frac{1-P_0^M}{1-P_0} \ P_k \ \ \ \ \ \ \ \ \ \ k=1,2,3,4,\cdots

ETNB Distribution

Examples 2, 3 and 4 show, starting with with an (a,b,0) distribution, how to derive a zero-truncated distribution and from it a zero-modified distribution. In these examples, we start with a negative binomial distribution and the derived distributions are zero-truncated negative binomial distribution and zero-modified negative binomial distribution. If the starting distribution is a Poisson distribution, then the same process would produce a zero-truncated Poisson distribution and a zero-modified Poisson distribution (with a particular assumed value of P_0^M).

There are members of the (a,b,1) class that do not originate from a member of the (a,b,0) class. Three such distributions are discussed in this post on the (a,b,1) class. We give an example discussing one of them.

Example 5
This example demonstrates how to work with the extended truncated negative binomial distribution (ETNB). The usual negative binomial distribution has two parameters r and \theta in one version (r and p in another version). Both parameters are positive real numbers. To define an ETNB distribution, we relax the r parameter to include the possibility of -1<r<0 in addition to r>0. Of course if r>0, then we just have the usual negative binomial distribution. So we focus on the new situation of -1<r<0.

Let’s say r=-0.2 and \theta=1. We take these two parameters and generate the “negative binomial” probabilities, from which we generate the zero-truncated probabilities P_k^T as shown in Example 3. Now the parameters r=-0.2 and \theta=1 do not belong to a legitimate negative binomial distribution. In fact the resulting P_k are negative values. So this “negative binomial” distribution is just a device to get things going.

According to Table 1, r=-0.2 and \theta=1 translate to a=1/2 and b=-3/5. We generate the “negative binomial” probabilities using the recursive relation (1). Don’t be alarmed that the probabilities are negative.

    \displaystyle P_0=\biggl(\frac{1}{2}\biggr)^{-0.2}=2^{0.2}=1.148698355

    \displaystyle P_1=\biggl(\frac{1}{2}-\frac{3}{5} \biggr) \ 2^{0.2}=\frac{-1}{10} \ 2^{0.2}

    \displaystyle P_2=\biggl(\frac{1}{2}-\frac{3}{5} \ \frac{1}{2} \biggr) \ \frac{-1}{10} \ 2^{0.2}=\frac{-2}{100} \ 2^{0.2}

    \displaystyle P_3=\biggl(\frac{1}{2}-\frac{3}{5} \ \frac{1}{3} \biggr) \ \frac{-2}{100} \ 2^{0.2}=\frac{-6}{1000} \ 2^{0.2}

    \displaystyle P_4=\biggl(\frac{1}{2}-\frac{3}{5} \ \frac{1}{4} \biggr) \ \frac{-6}{1000} \ 2^{0.2}=\frac{-21}{10000} \ 2^{0.2}

    \displaystyle P_5=\biggl(\frac{1}{2}-\frac{3}{5} \ \frac{1}{5} \biggr) \ \frac{-21}{10000} \ 2^{0.2}=\frac{-399}{500000} \ 2^{0.2}

The initial P_0 is greater than 1 and the other so called probabilities are negative. But they are just a device to get the ETNB probabilities. Using the formula stated in (3) gives the following zero-truncated ETNB probabilities.

    \displaystyle P_1^T=\frac{1}{1-2^{0.2}} \ \biggl( \frac{-1}{10} \ 2^{0.2}\biggr)=0.7725023959

    \displaystyle P_2^T=\frac{1}{1-2^{0.2}} \ \biggl( \frac{-2}{100} \ 2^{0.2}\biggr)=0.1545004792

    \displaystyle P_3^T=\frac{1}{1-2^{0.2}} \ \biggl( \frac{-6}{1000} \ 2^{0.2}\biggr)=0.0463501438

    \displaystyle P_4^T=\frac{1}{1-2^{0.2}} \ \biggl( \frac{-21}{10000} \ 2^{0.2}\biggr)=0.0162225503

    \displaystyle P_5^T=\frac{1}{1-2^{0.2}} \ \biggl( \frac{-399}{500000} \ 2^{0.2}\biggr)=0.0061645691

The above gives the first 5 probabilities of the zero-truncated ETNB distribution with parameters a=1/2 and b=-3/5. It is an (a,b,1) distribution that does not originate from any (legitimate) (a,b,0) distribution.

Practice Problems

The next post is a practice problem set on the (a,b,1) class.

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