# Tag Archives: Negative Binomial

## Frequency with Respect to Exposure and Coverage Modifications

In a portfolio of risks, there are two types of modifications which can influence the frequency distribution of payments.

1. Exposure Modification (not in syllabus) — increasing or decreasing the number of risks or time periods of coverage in the portfolio
2. Coverage Modification — applying limits, deductible or adjusting for inflation in each individual risk
EXPOSURE MODIFICATION
If there is an exposure modification, you would adjust the frequency distribution by scaling the appropriate parameter to reflect the change in exposure.  The following list provides the appropriate parameter to adjust for each distribution:
1. Poisson:  $\lambda$
2. Negative Binomial:  $r$
(Geometric is a Negative Binomial with $r = 1$)
3. Binomial:  $m$
(Only valid if the new value remains an integer)
Example 1:  You own a portfolio of 10 risks.  You model the frequency of claims with a negative binomial having parameters $r = 2$ and $\beta = 0.5$.  The number of risks in your portfolio increases to 15.  What are the parameters for the new distribution?
Answer:  The frequency distribution now has parameters $r = 3$ and $\beta = 0.5$  Note that since the mean and variance are $r\beta$ and $r\beta(1+\beta)$ respectively, the new mean and variance are multiplied by 1.5.
COVERAGE MODIFICATION
Coverage modifications shift, censor, or scale the individual risks, usually in the presence of a deductible or claim limit, and they change the conditions that trigger a payment.  For example, adding a deductible $d$ is considered a coverage modification and this changes the condition for payment because any losses below the deductible do not qualify for payment.  If the risk is represented by random variable $X$, then adding a deductible would change the random variable to $(X-d)_+$. Scaling in the presence of a deductible or claim limit also affects the frequency distribution.  The following lists parameters affected by coverage modifications:
1. Poisson:  $\lambda$
2. Negative Binomial:  $\beta$
3. Binomial:  $q$
4. Geometric:  $\beta$
These parameters are scaled by the probability that a payment occurs.
Example 2:  The frequency of loss is modeled as a Poisson distribution with parameter $\lambda = 5$.  A deductible is imposed so that only 80% of losses result in payments.  What is the new distribution?
Answer:  It is Poisson with $\lambda = 0.8(5) = 4$.
Example 3:  The frequency of payment $N$ is modeled as a negative binomial with parameters $r = 3$ and $\beta = 0.5$.  Losses $X$ are pareto distributed with parameters $\alpha = 2$ and $\theta = 100$.  The deductible is changed from $d=10$ to $d=15$.  What are the new parameters in the frequency distribution?
Answer:  Firstly, $N$ is the frequency of payment.  So it reflects the current deductible.  If you wanted the distribution of $N$ without deductible, you would divide $\beta$ by $\Pr{(X>10)}$.  Now to find the distribution of $N$ with the deductible of 15, you multiply $\beta$ by $Pr(X>15)$.  To summarize:
$\begin{array}{rll} \beta_{new} &=& \displaystyle \beta_{old}\times\frac{\Pr(X>15)}{\Pr(X>10)} \\ \\ &=& \displaystyle 0.5\times \frac{0.75614367}{0.82644628} \\ \\ &=& 0.45746692 \end{array}$

## Frequency Models

Frequency models count the number of times an event occurs.

1. The number of customers to arrive each hour.
2. The number of coins lucky Tom finds on his way home from school.
3. How many scientists a Tyrannosaur eats on a certain day.
4. Etc.
This is in contrast to a severity model which measures the magnitude of an event.
1. How much a customer spends.
2. The value of a coin that lucky Tom finds.
3. The number of calories each scientist provides.
4. Etc.
The following distributions are used to model event frequency.  For notation, $p_n$ means $Pr(N=n)$.

## Poisson:

$\begin{array}{lr}\displaystyle p_n = e^{-\lambda} \frac{\lambda^n}{n!} & \lambda > 0 \end{array}$
Properties:
1. Parameter is $\lambda$.
2. Mean is $\lambda$.
3. Variance is $\lambda$.
4. If $N_1, N_2, ..., N_i$ are Poisson with parameters $\lambda_1, \lambda_2, ..., \lambda_i$, then $N = N_1 + N_2 + ... + N_i$ is Poisson with parameter $\lambda = \lambda_1 + \lambda_2 + ... + \lambda_i$.

## Negative Binomial:

$\begin{array}{lr} \displaystyle p_n = {{n+r-1}\choose{n}}\left(\frac{1}{1+\beta}\right)^r\left(\frac{\beta}{1+\beta}\right)^n & \beta>0, r>0 \end{array}$
Properties:
1. Parameters are $r$ and $\beta$.
2. Mean is $r\beta$.
3. Variance is $r\beta\left(1+\beta\right)$.
4. Variance is always greater than the mean.
5. Is equal to a Geometric distribution when $r=1$.
6. If $N_1, N_2, ..., N_i$ are negative binomial with parameters $\beta_1 = \beta_2 = ... = \beta_i$ and $r_1, r_2, ..., r_i$, then the sum $N = N_1 + N_2 + ... + N_i$ is negative binomial and has parameters $\beta = \beta_1$ and $r = r_1+r_2+...+r_i$.  Note: $\beta$‘s must be the same.

## Geometric:

$\begin{array}{lr} \displaystyle p_n = \frac{\beta^n}{\left(1+\beta\right)^{n+1}} & \beta>0 \end{array}$
Properties:
1. Parameter is $\beta$.
2. Mean is $\beta$.
3. Variance is $\beta\left(1+\beta\right)$.
4. If $N_1, N_2, ..., N_i$ are geometric with parameter $\beta$, then the sum $N = N_1+N_2+...+N_i$ is negative binomial with parameters $\beta$ and $r = i$.

## Binomial:

$\displaystyle p_n = {{m} \choose {n}}q^n\left(1-q\right)^{m-n}$
where $m$ is a positive integer, $0.
Properties:
1. Parameters are $m$ and $q$.
2. Mean is $mq$.
3. Variance is $mq\left(1-q\right)$.
4. Variance is always less than mean.
5. If $N_1, N_2, ..., N_i$ is binomial with parameters $q$ and $m_1, m_2, ..., m_i$, then the sum $N=N_1+N_2+...+N_i$ is binomial with parameters $q$ and $m = m_1+m_2+...+m_i$.

## The (a,b,0) recursion:

These distributions can be reparameterized into a recursive formula with parameters $a$ and $b$.  When reparameterized, they all have the same recursive format.
$\displaystyle p_k = \left(a+ \frac{b}{k}\right)p_{k-1}$
It is more common to write
$\displaystyle \frac{p_k}{p_k-1} = a+\frac{b}{k}$
The parameters $a$ and $b$ are different for each distribution.
1. Poisson:
$a = 0$ and $b =\lambda$.
2. Negative Binomial:
$\displaystyle a = \frac{\beta}{1+\beta}$ and $\displaystyle b = \left(r-1\right)\frac{\beta}{1+\beta}$.
3. Geometric:
$\displaystyle a = \frac{\beta}{1+\beta}$ and $\displaystyle b = 0$.
4. Binomial:
$\displaystyle a = -\frac{q}{1-q}$ and $\displaystyle b = \left(m+1\right)\frac{q}{1-q}$.
Pop Quiz!
1. A frequency distribution has a = 0.8 and b = 1.2.  What distribution is this?
Answer: Negative Binomial because both parameters are positive.
2. A frequency distribution has mean 1 and variance 0.5.  What distribution is this?
Answer: Binomial because the variance is less than the mean.