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Policy Limits

A policy limit is the maximum amount that insurance will pay for a single loss. The limited loss variable \(X\wedge u\) in terms of the loss \(X\) is:

\(X\wedge u=\left\{ \begin{align} & X,X<u \\ & u,X\ge u \\ \end{align} \right.\), or \(Max(0,u-X)=X\wedge u\)

\(E[{X}^k]\)

\(E[X\wedge u]\) is called the Limited Expected Value, which is called Limited Average Severity (LAS) in Introduction to Ratemaking.

For a continuous random variable X for which \(Pr(X<0)=0\), the definition of moments is:

\(E[{X}^k]=\int_{0}^{\infty }{x^kf(x)dx}\) or \(E[{X}^k]=\int_{0}^{\infty }{kx^{k-1}S(x)dx}\)

For limited expected moments,

\(E[{(X\wedge u)}^k]=\int_{0}^{u}{x^kf(x)dx}+u^kS(u)\)

\(E[{(X\wedge u)}^k]=\int_{0}^{u}{kx^{k-1}S(x)dx}\), only when the lower bound is \(0\).

When \(k = 1\),

\(E[X\wedge u]=\int_{0}^{u}{xf(x)dx}+uS(u)\)

\(E[X\wedge u]=\int_{0}^{u}{S(x)dx}\), only when the lower bound is \(0\).

Combined with Inflation

Suppose \(Y=(1+r)X\), where \(X\) is the original variable and \(Y\) is the inflated variable, then:

\(E[Y\wedge u]=E[(1+r)X\wedge u]=(1+r)E[X\wedge \dfrac{u}{1+r}]\)

Deductibles

Introduction

Fixed percentage deductibles. For each claim, a certain percentage is not paid.
Disappearing deductibles. For a loss of \(d\), nothing is paid. For a loss of \(u\), where \(u\) is a specific number greater than \(d\), the full amount \(u\) is paid. For amounts between \(d\) and \(u\), the deductible is arrived at by linear interpolation between \(d\) at \(d\) and 0 at \(u\).
Fixed dollar deductible per calendar year. Common on health insurance. This contrasts with an ordinary deductible, which is per loss rather than per year.
Elimination periods. Common on disability insurance. Disability benefits are not paid for the first \(n\) days, where n may be 30, 90, 180, or some other number. Disability is usually considered a life/health coverage rather than a property/casualty coverage, but it is part of workers compensation. If the length of time needed to quality for retroactive payment equals the elimination period, the elimination period functions like a franchise deductible.

Ordinary and Franchise Deductibles

Ordinary deductibles (Fixed dollar deductibles). For each claim, the first \(d\) dollars are not paid.
\(Y=\left\{ \begin{align} & 0,X<d \\ & X-d,X>d \\ \end{align} \right.\)
Franchise deductibles. Pay nothing for losses up to and including \(d\), but pay the full loss for losses greater than \(d\).
\(Y=\left\{ \begin{align} & 0,X<d \\ & X,X>d \\ \end{align} \right.\)

Payment Per Loss with Deductible

Let \(X\) be the random variable for loss size. The random variable for the payment per loss with a deductible \(d\) is \(Y^L=(X-d)_+\)

CDF

\({F_{Y^L}}(x)=\Pr (Y^L\le x)=\Pr (X-d\le x)=\Pr (X\le x+d)=F_X(x+d)\)

Expectation

\(E[(X-d)_+]=\int_{d}^{\infty }{(x-d)f(x)dx}\) or \(E[(X-d)_+]=\int_{d}^{\infty }{S(x)dx}\)

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

Payment Per Payment with Deductible

The random variable for payment per payment on an insurance with an ordinary deductible is the payment per loss random variable conditioned on \(X>d\), or \({Y^P}=(X-d)_+|X>d\)

CDF

\(F_{Y^P}(x)=\Pr (X-{{d}^P}\le x|X>d)=\dfrac{F_X(x+d)-F_X(d)}{1-F_X(d)}\)

\(S_{Y^P}(x)=\dfrac{S_X(x+d)}{S_X(d)}\)

Expectation

\(E[Y^P]=e_X(d)=\dfrac{E[(X-d)_+]}{S(d)}=\dfrac{\int_{d}^{\infty }{(x-d)f(x)dx}}{S(d)}=\dfrac{\int_{d}^{\infty }{S(x)dx}}{S(d)}\)

\(E[X]=E[X\wedge d]+{{e}_{X}}(d)S(d)\)

Special Cases

- Exponential:
  If \(X\sim Exp(\theta)\), \(e(d)=\theta\)
- Uniform:
  If \(X\sim U(0,\theta]\), then \(Y^P\sim{\ }U(0,\theta-d)\) and \(e(d)=\dfrac{\theta-d}{2}\)
- Beta:
  If \(X\sim Beta(a,b,\theta)\), then \(Y^P\sim Beta(a,b,\theta-d)\) and \(e(d)=(\theta-d)\dfrac{a}{a+b}\)
- 1-Pareto:
  If \(X\sim 1-Pareto(\alpha,\theta)\) and \(d>\theta\), then \({{Y}^{P}}\sim 1-Pareto(\alpha,\theta-d)\)
  If \(X\sim 1-Pareto(\alpha,\theta)\) and \(d<\theta\), then \({{Y}^{P}}\) is a shifted 1-parameter Pareto distribution shifted by \(-d\), then \(e(d)=\dfrac{\alpha (\theta -d)+d}{\alpha -1}\)
- 2-Pareto:
  If \(X\sim 2-Pareto(\alpha,\theta)\) and \(d>\theta\), then \({{Y}^{P}}\sim{\ }2-Pareto(\alpha,\theta+d)\) and \(e(d)=\dfrac{\theta +d}{\alpha -1}\)

Loss Elimination Ratio

The loss elimination ratio is defined as the proportion of the expected loss which the insurer doesn’t pay as a result of an ordinary deductible.

\(LER(d)=\dfrac{E[X\wedge d]-E[X\wedge B]}{E[X]-E[X\wedge B]}\) for an ordinary deductible \(d\) and basic deductible \(B\)

Increased Limits Factors and Indicated Deductible Relativities

Increased Limits Factors (ILF)

The lowest policy limit is the basic limit, and the premiums for the other limits are expressed as multiples of the premiums for the basic limit. The multiples are called “increased limits factors”, or ILFs.

If \(B\) is the base limit and \(U\) is the policy limit, the \(ILF\) for \(U\) is:

\(ILF(U)=\dfrac{E[X\wedge U]}{E[X\wedge B]}\)

Limited Average Severity (LAS)

In practice the company would not know the sizes of losses above the limit. Thus for each ILF calculation, you should use data only from policies with the policy limit of interest or a higher policy limit. Since there will be different numbers of policies used in the calculation of limited losses, you should use limited average severity (LAS) rather than the gross amounts of losses in the calculation.

Risk Load

Higher policy limits result in greater volatility of payments. A company may add a risk load to compensate for the additional volatility, then the \(ILF\) for limit \(U\) with respect to basic limit B is computed as

\(ILF(U)=\dfrac{LAS(U)+Risk_{Load}(U)}{LAS(B)+Risk_{Load}(B)}\)

Indicated Deductible Relativities

Some insurance coverages are offered with several deductible options. The deductible relativity is the ratio of the premium for a policy with a deductible to the premium for a policy with the basic deductible. We make the same simplifying assumptions for deductibles as we made for policy limits in the previous section. With those assumptions, the indicated deductible relativity is the ratio of payment per loss with the deductible \(d\) to the payment per loss with the basic deductible \(B\):

\(IDR(d)\) \(=\dfrac{E[X]-E[X\wedge d]}{E[X]-E[X\wedge B]}\)

In practice losses below deductible may not be available since the losses wouldn’t be reported. Thus relativities between deductibles can only be computed using data for deductibles less than or equal to the lower deductible.

The indicated deductible relativity for deductibles of \(d_1\) and \(d_2\) relative to the basic deductible of \(B\) means:

- Use data only from policies with the policy limit \(B\) to calculate \(IDR(d_1)=\dfrac{E(X-d_1)_+}{E(X-B)_+}\)
- Use data only from policies with the policy limit \(B\) and \(d_1\) to calculate \(IDR(d_2)=(\dfrac{E(X-d_2)_+}{E(X-d_1)_+})\dfrac{E(X-d_1)_+}{E(X-B)_+}\)
  1. The indicated deductible relativity for deductibles of \(d_2\) relative to the basic deductible of \(d_1\): \(\dfrac{E(X-d_2)_+}{E(X-d_1)_+}\)
  2. \(IDR(d_2)=\dfrac{E(X-d_2)_+}{E(X-B)_+}=(\dfrac{E(X-d_2)_+}{E(X-d_1)_+})\dfrac{E(X-d_1)_+}{E(X-B)_+}\)

LER, IDR and ILF

	Increased Limit Factor	Loss Elimination Ratio	Indicated Deductible Relativities
Increased Limit Factor	\(ILF(U)=\dfrac{E[X\wedge U]}{E[X\wedge B]}\)	–	–
Loss Elimination Ratio	–	\(LER(d)=\dfrac{E[X\wedge d]-E[X\wedge B]}{E[X]-E[X\wedge B]}\)	\(IDR(d)=1-LER(d)\)
Indicated Deductible Relativities	–	\(IDR(d)=1-LER(d)\)	\(IDR(d)\) \(=\dfrac{E[X]-E[X\wedge d]}{E[X]-E[X\wedge B]}\)

Reinsurance

Functions of reinsurance

Capital relief. By reinsuring, the primary insurer does not have to hold reserves and other funds for the reinsured policies.
Increased underwriting capacity. Since the primary insurer lowers its risk by reinsuring, it can sell more insurance.
Catastrophe protection.
Stabilization of loss experience. Companies try to avoid large volatility of earnings, even if they don’t bankrupt the company.
Diversification of risk. Excess risk concentration can be reinsured.
Technical expertise. Reinsurers have wide experience and may be in a better position to assess risk.
Withdrawal from a market or line of business. It is possible to reinsure an entire block of business if the primary company does not want to be in that market any more.

Reinsurance Types

Reinsurance can be treaty or facultative:

Treaty reinsurance is automatic; the treaty specifies how much of each future policy that the primary insurer sells once the contract is effective gets reinsured. The amount of the policy that is not reinsured is called the primary insurer’s retention.
Facultative reinsurance is not automatic. The primary insurer s elects which policies it wants to reinsure, and the reinsurer then underwrites those policies and decides whether to reinsure them. The reinsurance agreement specifies the reinsurance premium that the primary insurer pays. This type of reinsurance is used for large complicated cases.

Treaty Reinsurance

Treaty reinsurance is of the following types:

Proportional (or pro rata) reinsurance. Proportional reinsurance can be quota share or surplus share.
- Quota share means that a fixed proportion of each policy is reinsured.
- Surplus share means that a fixed percentage of each policy above the primary insurer’s retention is reinsured.
The reinsurer pays the primary insurer a ceding commission when it assumes the risk. This reimburses the primary insurer for its sales and underwriting expenses.
Excess of loss reinsurance. The reinsurer pays the excess of “loss” over a specified amount, and the specified amount is called the attachment point. The word loss was put in quotes because there are several bases that may be used:
- It may be per claim: one covered claim from one policy.
- It may be per occurrence.
- It may be per period, usually per year. Stop-loss reinsurance reimburses a company for aggregate losses over a year in excess of the retention limit.

Pricing Reinsurance

Proportional Reinsurance

To price proportional reinsurance, the reinsurer would collect historical data excluding catastrophic cases, develop and trend it, compute a loss ratio (based on the primary insurer’s premium rate, since that is what the reinsurer gets), add a provision for catastrophes, and compute an expense ratio. The reinsurer would add the loss ratio and the expense raho to obtain the combined ratio. The reinsurer would then adjust the ceding commission to meet its pricing goals.

Excess of loss reinsurance

For excess of loss treaties, experience rating and exposure rating are the available methods.

- Experience Rating: The reinsurer uses the primary insurer’s experience, trended and developed, to compute the loss ratio of the layer that is being reinsured.
- Exposure Rating: The reinsurer uses industry data to develop a distribution of claim severity, and then the expected losses in the layer are estimated using increased limits factors or simulation.

Increased Limits Factors for the proportion of losses in each layer

The \(ILF\) is the ratio of the expected loss at the limit of interest to the expected loss at the basic level. Thus the proportion of losses in layer \((a,b)\) is \(\dfrac{ILF_b-ILF_a}{ILF_{\infty }}\). If all losses are below \(U\), we can replace \(ILF_{\infty}\) with \(ILF_U\).

Risk Measures and Tail Weight

The risk measures we are interested in are measures for the solvency of a company, and a risk measure is a real-valued function of a random variable. We use the letter \(\rho\) for a risk measure; \(\rho(X)\) is the risk measure of \(X\).

Moments. \(E[X]\), \(Var(X)\), etc.
Percentiles. For example, the median is a real valued function of X.
Premium principles. For example, the premium may be set equal to the expected loss plus a constant times the standard deviation of a loss, or \(\rho(X)=\mu x+k\sigma x\). This is called the standard deviation principle.

Coherent Risk Measures

Translation Invariance. \(\rho (X+c)=\rho (X) + c\)
Positive Homogeneity. \(\rho(cX)=c\rho(X)\)
Subadditivity. \(\rho(X+Y)=\le\rho(X)+\rho(Y)\)
Monotonicity. \(\rho(X)\le\rho(Y)\), if \(X\le Y\)

Risk measures satisfying all four of these properties a re called coherent.

Value-at-Risk (VaR)

The Value-at-Risk at security level \(p\) for a random variable \(X\), denoted \(VaR_{p}(X)\), is the \(100p^{th}\) percentile of \(X\):

\(Va{{R}_{p}}(X)={{\pi }_{p}}={{F}_{X}}^{-1}(p)\)

\(VaR\) for Normal Distribution

If \(X\sim N(\mu ,{{\sigma }^{2}})\), \(VaR_{p}(X)=\mu +z_{p}\sigma \)

\(VaR\) for Lognormal Distribution

If \(X\sim LogN(\mu ,{{\sigma }^{2}})\), \(VaR_{p}(X)={e^{\mu +z_{p}\sigma }}\)

Tail-Value-at-Risk (TVaR)

The tail-value-at-risk of a continuous random variable \(X\) at security level \(p\), denoted \(TVaR_{p}(X)\), is the expectation of the variable given that it is above its \(100p^{th}\) percentile:

\(TVaR_p(X)=E[X|X>VaR_p(X)]=\dfrac{\int_{VaR_p(X)}^{\infty }{xf(x)dx}}{(1-F(VaR_p(X)))}=\dfrac{\int_{F_{X}^{-1}(p)}^{\infty }{xf(x)dx}}{(1-p)}=\dfrac{\int_{p}^{1}{VaR_{y}(X)dy}}{(1-p)}\)

For random variables \(X\) following other distributions for which the tables give \(E[X \wedge x]\), the above equation can be translated into:

\(TVaR_p(X)=VaR_{p}(X)+e_{X}(VaR_{p}(X))=VaR_p(X)+\dfrac{E(X)-E[X\wedge VaR_p(X)]}{1-p}\)

\(TVaR_p(aX+b)=a\times TVaR_p(X)+b\) since \(TVaR\) is coherent

Risk Measures	\(VaR\)	\(TVaR_p\)
Translation Invariance	√	√
Positive Homogeneity	√	√
Subadditivity	×	√
Monotonicity	√	√
Coherent?	×	√

\(TVaR\) for Lognormal Distribution

If \(X\sim LN(\mu ,\sigma )\), \(TVaR_p(X)=E[X](\dfrac{\Phi (\sigma -z_p)}{1-p})\), where \(z_p=\Phi^{-1}(p)\)

\(\begin{align} \text{TVaR}_{p}(Y) &= \text{E}\!\left[Y\mid Y>\text{VaR}_{p}(Y)\right] \\ &= \text{E}\!\left[Y-\text{VaR}_{p}(Y)\mid Y>\text{VaR}_{p}(Y)\right]+\text{VaR}_{p}(Y) \\ &= \dfrac{\text{E}\!\left[\left(Y-\text{VaR}_{p}(Y)\right)_{+}\right]}{\Pr\left(Y>\text{VaR}_{p}(Y)\right)}+\text{VaR}_{p}(Y) \\ &= \dfrac{\text{E}\!\left[Y\right]-\text{E}\!\left[Y\wedge \text{VaR}_{p}(Y)\right]}{\Pr\left(Y>\text{VaR}_{p}(Y)\right)}+\text{VaR}_{p}(Y) \end{align}\)

\(TVaR\) for Normal Distribution

If \(X\sim N(\mu ,\sigma )\), \(TVaR_p(X)=\mu +\sigma (\dfrac{\phi (z_p)}{1-p})\), where \(\phi(z_p)=\dfrac{e^{-x^2/2}}{\sqrt{2\pi}}\)

Bonuses

If the loss ratio is less than \(c\text{%}\), an insurance agent will receive a bonus equal to \(k\text{%}\) of the difference between its loss ratio and \(c%\). If the earned premium is \(P\), then:

\(B=k\text{%}\times P\times Max(0,c\text{%}-\dfrac{X}{P})=k\text{%}\times Max(0,c\text{%}\times P-X)=k\text{%}\times (c\text{%}\times P-Min(X,c\text{%}\times P)\)

\(E[B]=k\text{%}\times (c\text{%}\times P-E[X\wedge c\text{%}\times P]))\)

Discrete Distributions

The \((a,b,0)\) Class

Let \(p_{n}=Pr(N=n)\), \(\dfrac{p_n}{p_{n-1}}=a+\dfrac{b}{n}\) for \(k=2,3,4,…\)

Distribution	\(p_{n}\)	\(a\)	\(b\)	\(E[X]\)	\(Var[X]\)
Poisson	\({p_{n}}={{e}^{-\lambda }}\dfrac{{{\lambda }^{n}}}{n!}\)	\(0\)	\(\lambda \)	\(\lambda \)	\(\lambda \)
Binomial	\(p_{n}={{m}\choose{n}}{q}^{n}{(1-q)}^{m-n}\)	\(-\dfrac{q}{1-q}\)	\((m+1)\dfrac{q}{1-q}\)	\(mq\)	\(mq(1-q)\)
Negative Binomial	\(p_{n}={{n-1+r}\choose{n}}{(\dfrac{1}{1+\beta })}^{r}{(\dfrac{\beta }{1+\beta })}^{n}\)	\(\dfrac{\beta }{1+\beta }\)	\((r-1)(\dfrac{\beta }{1+\beta })\)	\(r\beta \)	\(r\beta (1+\beta )\)
Geometric	\(p_n=\dfrac{\beta^n}({1+\beta)^{n+1}}\) \(Pr(N\ge n)=(\dfrac{\beta }{1+\beta })^{n}\)	\(\dfrac{\beta }{1+\beta }\)	\(0\)	\(\beta \)	\(\beta (1+\beta )\)

Moments

\(\mu_(j)=\dfrac{(aj+b){{\mu }_{(j-1)}}}{1-a}\)

When \(j=1\), \(E[N]=\dfrac{a+b}{1-a}\), \(Var[N]=\dfrac{a+b}{{(1-a)}^{2}}\)

The \((a,b,1)\) Class

In addition to modifications of \((a,b,0)\) distributions, the \((a,b,1)\) class includes an extended negative binomial distribution with \(-1<r<0\); the negative binomial distribution only allows \(r>0\). This extended distribution is called the ETNB (extended truncated negative binomial), even though it may be zero-modified rather than zero-truncated.

Zero-Truncated Distributions

Often, special treatment must be given to the probability of zero claims, and the three discrete distributions of the \((a,b,0)\) class are inadequate because they do not give appropriate probability to \(0\) claims,, The \((a,b,1)\) class consists of distributions for which \(p_0\) is arbitrary, but the \((a,b,0)\) relationship holds above 1. One way to obtain a distribution in this class is to take one from the \((a,b,0)\) class and truncate it at 0, i,e., make the probability of 0 equal to zero, and then scale all the other probabilities so that they add up to 1.

\({p_0}^T=0\), \({p_n}^T=\dfrac{p_n}{1-p_0}\), \(n>0\)

Zero-Modified Distributions

\(p_{0}^M=1-c\), \(p_{n}^M=\dfrac{1}{1-p_0}{p_n}\), \(n>0\)

\(P_n^M\) and \(P_n^T\)

\(p_{n}^M=(1-p_{0}^M)p_{n}^T\)

Mean and Variance

For \(X\) a zero-modified random variable,

\(E[X]=cm\) and \(Var(X)=c(1-c)m^2+cv\), where

- - - \(c=1-P_0^M\)
    - \(m\) is the mean of the corresponding zero-truncated distribution.
    - \(v\) is the variance of the corresponding zero-truncated distribution.

Poisson/Gamma

The negative binomial can be derived as a gamma mixture of Poisson’s:

Assume that in a portfolio of insureds, loss frequency \(N|\lambda\sim Poisson(\lambda)\) and \(\lambda \sim Gamma(\alpha,\theta)\), then the unconditional loss frequency for an insured picked at random is a negative binomial: \(X\sim NB(r=\alpha,\beta=\theta )\)

Multiple Exposures

Since the sum of negative binomials with parameters \(r_i\) and \(\beta\) is: \(X\sim NB(r=\sum{r_i},\beta)\), if the portfolio of insureds has \(n\) exposures, the distribution of total number of claims for the entire portfolio will be \(X\sim NB(nr,\beta)\)

For example,

If an event:
- \(N|\lambda\sim Poi(\lambda)\) per minute;
- The parameter \(\lambda\) varies by day;
- \(\lambda\sim \Gamma(\alpha,\theta)\);
- Events occurs in \(k\) minutes on a random day;

then the number of events \(N\sim NB(r=\alpha,\beta=k\times \theta)\)

If an event:
- \(N|\lambda\sim Poi(\lambda)\) per minute;
- The parameter \(\lambda\) varies by miniute;
- \(\lambda\sim \Gamma(\alpha,\theta)\);
- Events occurs in \(k\) minutes on a random day;

then the number of events \(N\sim NB(r=k\times r,\beta=\theta)\)

Pages: 1 2 3

Policy Limits

\(E[{X}^k]\)

Combined with Inflation

Deductibles

Introduction

Ordinary and Franchise Deductibles

Payment Per Loss with Deductible

CDF

Expectation

Payment Per Payment with Deductible

CDF

Expectation

Special Cases

Loss Elimination Ratio

Increased Limits Factors and Indicated Deductible Relativities

Increased Limits Factors (ILF)

Limited Average Severity (LAS)

Risk Load

Indicated Deductible Relativities

LER, IDR and ILF

Reinsurance

Functions of reinsurance

Reinsurance Types

Treaty Reinsurance

Pricing Reinsurance

Proportional Reinsurance

Excess of loss reinsurance

Increased Limits Factors for the proportion of losses in each layer

Risk Measures and Tail Weight

Coherent Risk Measures

Value-at-Risk (VaR)

\(VaR\) for Normal Distribution

\(VaR\) for Lognormal Distribution

Tail-Value-at-Risk (TVaR)

\(TVaR\) for Lognormal Distribution

\(TVaR\) for Normal Distribution

Other Topics in Severity Coverage Modifications

\(Y^{L}\)

\(E[Y^{L}]\)

\(Var[Y^{L}]\)

\(Var[Y^P]\)

Alternative Methods

Bonuses

Discrete Distributions

The \((a,b,0)\) Class

Moments

The \((a,b,1)\) Class

Zero-Truncated Distributions

Zero-Modified Distributions

\(P_n^M\) and \(P_n^T\)

Mean and Variance

Poisson/Gamma

Multiple Exposures