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Actuary /
CExamBasicsActuary.CExamBasics HistoryHide minor edits - Show changes to output Changed line 37 from:
---->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) to:
---->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). Changed line 49 from:
{$$f_{Y^P}(y) = F_x(d) { at } y=0, \qquad f_X(y) \text{ for } y > d \text{ (a mixed discrete and continuous distribution,)}$$} to:
{$$f_{Y^P}(y) = F_x(d) \text{ at } y=0, \qquad f_X(y) \text{ for } y > d \text{ (a mixed discrete and continuous distribution,)}$$} Added lines 68-73:
->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 {$$P_{N^p}(z)= P_{N^L}(1+v(z-1)).$$} Added line 67:
->The expected cost per payment is obtained by dividing by {$S(d/(1+r)$}, provided the value is well defined. Changed line 62 from:
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 to:
->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 Changed lines 65-66 from:
to:
->if the inflation is {$1+r$}, then the expected cost per loss is {$$(1+r)\left( E(X) - E\left(X\wedge \frac{d}{1+r} \right)\right).$$} Added lines 62-63:
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 {$$\frac{E(x) - (E(x)-E(x\wedge d))}{E(x)}= \frac{E(X\wedge d)}{E(X)}. $$} Changed lines 56-58 from:
----> ----> to:
---->Franchise deductible: expected cost per loss = {$E(X) - E(X\wedge d) + d(S(d))$}. ---->Franchise deductible: expected cost per payment = {$\frac{E(X) - E(X\wedge d)}{S(d)}+d$}. Added lines 53-58:
--->How all of this affects expectations: ---->Ordinary deductible: expected cost per loss = {$E(X) - E(X\wedge d)$}. ---->Ordinary deductible: expected cost per payment = {$\frac{E(X) - E(X\wedge d)}{S(d)}$}. ---->Ordinary deductible: expected cost per loss = {$E(X) - E(X\wedge d) + d(S(d))$}. ---->Ordinary deductible: expected cost per loss = {$\frac{E(X) - E(X\wedge d)}{S(d)}+d$}. Changed line 31 from:
{$$Y^P = X-d \text{ if } X>d, \qquad \text{ undefined if } X leq d.$$} to:
{$$Y^P = X-d \text{ if } X>d, \qquad \text{ undefined if } X \leq d.$$} Changed line 35 from:
{$$Y^P = X-d \text{ if } X>d, \qquad 0 \text{ if } X leq d.$$} to:
{$$Y^P = X-d \text{ if } X>d, \qquad 0 \text{ if } X \leq d.$$} Changed line 40 from:
{$$Y^P = X \text{ if } X>d, \qquad \text{ undefined if } X leq d.$$} to:
{$$Y^P = X \text{ if } X>d, \qquad \text{ undefined if } X \leq d.$$} Changed line 43 from:
{$$S_{Y_P}(y) = 1 \text{ for } 0 \leq y \leq d, \qquad to:
{$$S_{Y_P}(y) = 1 \text{ for } 0 \leq y \leq d, \qquad S_X(y)/S_X(d) \text{ for } y>d, $$} Changed line 46 from:
{$$Y^P = X-d \text{ if } X>d, \qquad 0 \text{ if } X leq d.$$} to:
{$$Y^P = X-d \text{ if } X>d, \qquad 0 \text{ if } X \leq d.$$} Changed line 50 from:
{$$S_{Y_P}(y) = S_X(d) \text{ for } 0 \leq y \leq d, \qquad to:
{$$S_{Y_P}(y) = S_X(d) \text{ for } 0 \leq y \leq d, \qquad S_X(y) \text{ for } y>d, $$} Changed lines 30-32 from:
---->a)''per-payment'' (it's an excess loss variable): {$Y^P = X-d to:
---->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 {$y=0$}. Changed lines 34-36 from:
---->b)''per-loss'' (it's a left censored and shifted variable): {$Y^P = X-d to:
---->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 {$y=0$}. Changed lines 39-40 from:
---->a)''per-payment'': {$Y^P = X to:
---->a)''per-payment'': {$$Y^P = X \text{ if } X>d, \qquad \text{ undefined if } X leq d.$$} Changed lines 45-47 from:
---->b)''per-loss'' : {$Y^P = X-d to:
---->b)''per-loss'' : {$$Y^P = X-d \text{ if } X>d, \qquad 0 \text{ if } X leq d.$$} ---->Probability is {$F_X(d)$} at {$y=0$}. Changed lines 34-45 from:
to:
--->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 {$$f_{Y^P}(y) = F_x(d) { at } y=0, \qquad f_X(y) \text{ for } y > d \text{ (a mixed discrete and continuous distribution,)}$$} {$$S_{Y_P}(y) = S_X(d) \text{ for } 0 \leq y \leq d, \qquad, S_X(y) \text{ for } y>d, $$} {$$h_{Y_P}(y) = 0 \text{ for } 0<y<d, \qquad h_X(y) \text{ for } y>d \text{ (undefined at }d).$$} Changed lines 30-32 from:
---->a)''per-payment'' to:
---->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)}. Added line 27:
-->Start with an unmodified RV {$X$}. Added lines 29-32:
--->1)''Ordinary deductible'', labeled {$d$}: ---->a)''per-payment'': {$Y^P = X-d {\text if } X>d, \qquad \text{ undefined if } X leq d.$} ---->The key functions are {$f_{Y^P}(y) = f_X(y+d)/S_X(d)$}, ---->b)''per-loss'': {$Y^P = X-d {\text if } X>d, \qquad 0 \text{ if } X leq d.$} Changed line 22 from:
{$$\frac{S-E(S)}{\frac{\sigma(S)}{\sqrt{n}}$$} to:
{$$\frac{S-E(S)}{\frac{\sigma(S)}{\sqrt{n}}}$$} Changed lines 21-24 from:
to:
->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 {$n$}. Changed lines 34-36 from:
->1. Calculate VaR, and TVaR and explain their use and limitations. to:
->1. Calculate VaR, and TVaR and explain their use and limitations. Added lines 16-19:
->Writing {$X$} instead of {$X_j$} because all the means and variances are the same. {$$E(S) = E(N)E(X);$$} {$$Var(S) = E(N)Var(X) + Var(N) E(X)^2.$$} Added lines 3-14:
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''. Changed lines 2-4 from:
1. Compute relevant parameters and statistics for collective risk models. 2. Evaluate compound models for aggregate claims. 3. Compute aggregate claims distributions. to:
->1. Compute relevant parameters and statistics for collective risk models. ->2. Evaluate compound models for aggregate claims. ->3. Compute aggregate claims distributions. Changed lines 7-12 from:
1. Evaluate the impacts of coverage modifications: a) Deductibles b) Limits c) Coinsurance 2. Calculate Loss Elimination Ratios. 3. Evaluate effects of inflation on losses. to:
->1. Evaluate the impacts of coverage modifications: -->a) Deductibles -->b) Limits -->c) Coinsurance ->2. Calculate Loss Elimination Ratios. ->3. Evaluate effects of inflation on losses. Changed line 15 from:
1. Calculate VaR, and TVaR and explain their use and limitations. to:
->1. Calculate VaR, and TVaR and explain their use and limitations. Added lines 1-13:
C. Aggregate Models 1. Compute relevant parameters and statistics for collective risk models. 2. Evaluate compound models for aggregate claims. 3. Compute aggregate claims distributions. D. For severity, frequency and aggregate models 1. Evaluate the impacts of coverage modifications: a) Deductibles b) Limits c) Coinsurance 2. Calculate Loss Elimination Ratios. 3. Evaluate effects of inflation on losses. E. RiskMeasures 1. Calculate VaR, and TVaR and explain their use and limitations. |