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C. Aggregate Models

1. Compute relevant parameters and statistics for collective risk models.

The collective risk model is {$$S=X_1+\ldots+X_N,$$} where a. Conditional on {$N=n$}, the {$X_j$}'s are i.i.d. random variables. b. Conditional on {$N=n$}, the common distribution of the {$X_j$}'s does not depend on {$n$}. c. The distribution of {$N$} does not depend on the values {$X_j$}. In practice, {$N$} is the number of claims and {$X_j$}'s are the amount paid on the claims

{$N$} is called the claim count random variable or just claims; or frequency. {$X_j$}'s are called (single-)loss random variables, or severity. {$S$} is called aggregate loss random variable, or total loss random variable.

2. Evaluate compound models for aggregate claims.

Writing {$X$} instead of {$X_j$} because all the means and variances are the same.

{$$E(S) = E(N)E(X);$$} {$$Var(S) = E(N)Var(X) + Var(N) E(X)^2.$$}

3. Compute aggregate claims distributions.

Once we have mean and standard deviation for the aggregate claim, we can use normal approximation to compute the number of claims needed to make the tail ends of {$S$} small: Use the fact that

{$$\frac{S-E(S)}{\frac{\sigma(S)}{\sqrt{n}}}$$}

is approximately a standard normal distribution, for large {$n$}.

D. For severity, frequency and aggregate models

1. Evaluate the impacts of coverage modifications:

Start with an unmodified RV {$X$}.

a) Deductibles

1)Ordinary deductible, labeled {$d$}:

a)per-payment (it's an excess loss variable):

{$$Y^P = X-d \text{ if } X>d, \qquad \text{ undefined if } X \leq d.$$}

No probability at {$y=0$}.

The key functions are {$f_{Y^P}(y) = f_X(y+d)/S_X(d)$}, {$S_{Y_P}(y) = S_X(y+d)/S_X(d)$}, {$h_{Y_P}(y) = h_X(y+d)$}.

b)per-loss (it's a left censored and shifted variable):

{$$Y^P = X-d \text{ if } X>d, \qquad 0 \text{ if } X \leq d.$$}

Probability is {$F_X(d)$} at {$y=0$}.

The key functions are {$f_{Y^P}(y) = f_X(y+d)$}, {$S_{Y_P}(y) = S_X(y+d)$}, {$h_{Y_P}(y) = h_X(y+d)$} (this one is undefined at 0).

2)Franchise deductible, also labeled {$d$} here:

a)per-payment:

{$$Y^P = X \text{ if } X>d, \qquad \text{ undefined if } X \leq d.$$}

The key functions are:

{$$f_{Y^P}(y) = f_X(y)/S_X(d) \text{ for } y > d \text{ (undefined elsewhere)},$$} {$$S_{Y_P}(y) = 1 \text{ for } 0 \leq y \leq d, \qquad S_X(y)/S_X(d) \text{ for } y>d, $$} {$$h_{Y_P}(y) = 0 \text{ for } 0<y<d, \qquad h_X(y) \text{ for } y>d \text{ (undefined at }d).$$}

b)per-loss :

{$$Y^P = X-d \text{ if } X>d, \qquad 0 \text{ if } X \leq d.$$}

Probability is {$F_X(d)$} at {$y=0$}.

The key functions are

{$$f_{Y^P}(y) = F_x(d) \text{ at } y=0, \qquad f_X(y) \text{ for } y > d \text{ (a mixed discrete and continuous distribution,)}$$} {$$S_{Y_P}(y) = S_X(d) \text{ for } 0 \leq y \leq d, \qquad S_X(y) \text{ for } y>d, $$} {$$h_{Y_P}(y) = 0 \text{ for } 0<y<d, \qquad h_X(y) \text{ for } y>d \text{ (undefined at }d).$$}

How all of this affects expectations:

Ordinary deductible: expected cost per loss = {$E(X) - E(X\wedge d)$}.

Ordinary deductible: expected cost per payment = {$\frac{E(X) - E(X\wedge d)}{S(d)}$}.

Franchise deductible: expected cost per loss = {$E(X) - E(X\wedge d) + d(S(d))$}.

Franchise deductible: expected cost per payment = {$\frac{E(X) - E(X\wedge d)}{S(d)}+d$}.

b) Limits

c) Coinsurance

2. Calculate Loss Elimination Ratios.

The loss elimination ratio is the ratio of the decrease in expected payment with an ordinary deductible to the expected payment without the deductible. So it is

{$$\frac{E(x) - (E(x)-E(x\wedge d))}{E(x)}= \frac{E(X\wedge d)}{E(X)}. $$}

3. Evaluate effects of inflation on losses.

if the inflation is {$1+r$}, then the expected cost per loss is

{$$(1+r)\left( E(X) - E\left(X\wedge \frac{d}{1+r} \right)\right).$$}

The expected cost per payment is obtained by dividing by {$S(d/(1+r)$}, provided the value is well defined.

How do deductibles affect claim frequency:

Setup: Let {$X_j$} be the severity, representing the ground-up loss on the j-th loss when there's no coverage modification. Given a deductible {$d$}, let {$v= Prob(X>d)$}. Let {$N^P$} and {$N^L$} denote the the number of payment and number of losses, respectively. Then we can show that the # of payments is affected from the deductible by

{$$P_{N^p}(z)= P_{N^L}(1+v(z-1)).$$}

E. Risk Measures

1. Calculate VaR, and TVaR and explain their use and limitations.