Insurance Pricing Basic Statistical Principles

7/29/2019 Insurance Pricing Basic Statistical Principles

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Insurance Pricing:

Basic Statistical Prin cipl es

Michae l R. Pow ers


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Basic Definitions --Forecasting vs. Estimating

Unknown Q uantity:

Any qua ntity about which there is im perfect informa tion.

Why is in format ion imperfect?

Qua ntity has not yet been realized.

Qua ntity has not yet been observed.

Qua ntity cannot be observed.


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Random Variable:

A formal stat istical mode l used to rep resent an unknown quan tity

(usua lly denoted by a letter, e.g.,XorY).

The model is described by :

(i) The set of poss ib le valu es that the unknown quan tity can

assume (samplespace).

(ii) The rela tive likelihoods tha t the quan tity will take on each

of the valu es it can assume (probability distribution function).


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Parameter:

A prop erty of a rando m variable that pr ovide s a convenient way ofexpressing som e informa tion abou t (i) and (ii ).

Examples:

Mean , Median, Mode (Measures of Location).

Variance, Stan dard Dev iat ion, Coefficient of Varia tion(Measu res of Spread).

Percentile s (10th, 25th, 50th, etc.).

If a param eter is un know n, shou ld it be treated as a randomvariable?


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Observation:

The value of a ran dom variable when it has become know n.

If there are man y observations of the sam e random variable,

they are usua lly represented by X1 ,X2 ,X3 , , or Y1 ,Y2 ,Y3 , .

Sample:

A collection of observations.

Usually, the le ttern is us ed to de note the num ber of

observat ions in a samp le .


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Estimate:

A reasonab ly good guess at the value of an unknow n pa ram eter,

based upon a sampl e of one or mo re observa tions.

Often, one beg ins by estimat ing the mean an d standa rd

devia tion of a rand om var iable.

Statistician s use a var ie ty of different methods for estimat ing

pa ram eters (e.g., maximum likelihood, methodofmoments, least

squares, minimum meansquared-error, minimum expectedloss).

Statistician s use a varie ty of different criteria for evalua tin g the

qua lity of an estimat e (e.g., unbiasedness, efficiency, consistency,

robustness).


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Forecast (Prediction):

A reasonab ly good gue ss at the value of an uno bserved rand om

variable , based upon a sam ple of one or m ore observat ions.

A ran dom variable is usual ly forecast by estimat ing its mean ,med ian , or mo de .

Caution: After spend ing much effort developing a good

estimat ion method, it is easy to forge t that the ult ima te goal is toforecast.


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Meas ures of Location/Central Ten dency

Mean:

The expected valu e of a rando m variable , on the average; also

called the expectedvalue or the average.

If a ran dom variable Xcan take on the valuesA ,B, C, ..., then

Mean A PrX A B PrXB C PrX C .

The mean is the mo st commo nly used measu re of what valu e a

rand om variable is like ly to ta ke.


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Me dian:

The smallest value tha t has at l east a 50% chance of being greatertha n the rand om variable ; also called the 50thpercentile.

(This value may not be un iqu e.)

For a rand om variable X, the Med ian is the smallest valu e ofx

such that PrXx 1 2.

The med ian is often used in plac e of the mean when the rand om

variable has askeweddistribution .

This occurs when it is possible for the random variable to ta ke

on extremely large (or smal lbut not both) values with small

probabil ity.


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Mode:

Of all possible values, the on e tha t a ran dom variable is most like lyto take. (This valu e may not be u niqu e.)

For a rand om variable X, Pr X Mode PrX x for anyind ividua l alt ernat ive valuex.

The mode may be useful in short-term, on e-shot plann ing,

when de cisions are based upon cons iderat ion of the most lik ely

outcome.

In comp lex problems, it ma y be easier to calcu lat e the mode

tha n either the mean or the med ian .


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Population vs. Sample Parameters

The three param eters de fined above are all popu lation

pa ram eters.

They are sometimes called thepopulationmean,population

median, and populationmode, respectively .

The word pop ulation refers to the fact that these param eters

are theoretical val ues under lying the distribution of the random

variable , regard less of what values are ultima tely observed in a

samp le .


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Each of the three param eters defined above has a sampl e

counterpa rt that can be calcu lat ed from observat ions of the

rand om var iable, an d used to estimat e the correspon d in gpop ulation parameter.

Given n observations X1 ,X2 ,X3 , ,Xn ,

Sample Mean X1

1

n

X2

1

n

X3

1

n

Xn

1

n

X1 X2 X3 Xnn .

The sample mean is often denoted by X (read X-bar).


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Sample Median Observation Xi that is m id-way up a ran kin gofX1 ,X2 ,X3 , ,Xn .


Sample Mode Observation Xi that occ urs mo re frequ ently

tha n any other ind ividua l observation.



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Meas ures of Spread/Dispersion

Variance:

The expected valu e of the squa red d ifference between a random

variable and its mean , on th e average.

If a ran dom variable Xcan take on the valuesA ,B, C, ..., then

Variance A Mean 2

PrXA

B Mean 2

PrX B C Mean 2 Pr X C .

Even thou gh it ma y be less fami lia r than th e stan dard de viat ion,

the variance is a more primitive param eter (i.e., it mu st be

calculated first).


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Standard De viation:

The square root o f the variance.

The standa rd devia tion is often u sed to cons truct in tervals

where an estimat ed pa rameter or forecast rando m variable is

likely to be (as in p lus or minu s 2 standard deviat ions ).

This app lication is formal ized in the construction ofconfidence

intervals.


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Coefficient of Variation:

The rat io of the standard deviat ion to the mean .

The coefficient of variat ion measu res the spread in th e

d istrib ution of a ran dom variable in rela tion to the mean of the

rand om var iable.

Consider tw o ran dom var iablesXand Ysuch that :

Standard Deviation X Standard Deviation Y , an d

Mean X Mean Y .

Then, Coeff. of Variation X Coeff. of Variation Y .


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SampleParameters

As with the measu res of location, the re are samp le

counterparts to each of the above three popu lation

pa ram eters that can be us ed to est ima te the popu lation

pa ram eters.


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Benefits of Large Sampl e Sizes

Law of Large Numbers:

As the sampl e size (n) increases, the stan dard deviat ion of the

samp le mean (X) becomes smaller and sma ller , an d the samp le

mean gets closer an d closer to th e popu lation mean .

The law of large num ber s provide s the primary supp ort for the

notion that bigger sampl e sizes are better (as long as they are

not much more expensive to comp ile).


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Portfolio

Risk 1 Risk 2 Risk 3 Risk 4 Risk 5 Risk 6 Risk 7 Risk 8 Risk 9 Risk 10 Average

1996 538,633 511,498 554,861 502,400 404,106 480,249 512,676 548,757 512,700 506,100 507,198

1997 397,637 487,200 470,054 521,500 545,700 457,900 517,689 505,112 472,888 548,430 492,411

1998 460,502 440,451 467,068 375,000 478,235 466,317 547,136 504,308 540,812 480,348 476,018

1999 500,253 525,099 464,706 534,700 533,256 524,403 479,853 557,838 485,198 451,369 505,668

2000 523,079 476,300 510,151 487,628 459,689 533,214 427,038 491,862 436,491 459,816 480,527

Sample Statistics

Mea n 48 4,021 48 8,110 49 3,368 48 4,246 48 4,197 49 2,417 49 6,878 52 1,575 48 9,618 48 9,213 492,364

Stand ard Dev. 56 ,539 32,889 39,115 63,6 58 57,5 33 34,3 08 45 ,7 60 29 ,605 39 ,609 39 ,264 14,181

Coeff. Of Var. 0.12 0.07 0.08 0.13 0.12 0.07 0.09 0.06 0.08 0.08 0.03

Exhibit 1c: Total Loss Experience for 10 Risks ( No Trend Effec ts )


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Risk Theory Applications:

(i) Consider a risk poo l with n members, each of which hasidentical exposure to loss. If each member of the pool cedes

its entire exposure to the pool in return for covering 1 n thof the pools losses, then each member reduces the standa rd

devia tion of its loss payment. As n increases, the standa rd

devia tion gets smaller and smal ler , decreasing to 0.

(ii) Consider an insurance compan y that covers n identical

exposu res, and charges premiu ms with a fixed profit

load ing. If the compan ys surp lus remain s prop ortional to

n, then, as n increases, the compan y's probability of

insolvency get s smalle r an d smaller, decreasin g to 0.


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Normal Approximation (Central Limit Theorem):

As the sampl e size (n) increases, the samp le mean (X) takes on a

d istrib ution that is no rma l (i.e., havin g the well-know n bel l-

shap ed cur ve) with a stan dard de viat ion that g ets smal ler and

smaller.


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The n orm al approxim ation provides justificat ion for theassu mpt ions that:

(i) an int erval of plus or minus 2 standard d eviat ions will

capture an estimated param eter (or forecast rand om

variable) abo ut 95% of the time, and

(ii) an int erval of plus or minus 1 standard d eviation will

capture an estimated param eter (or forecast rand om

variable) abo ut 68% of the time.


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For insurance pool with n risks (an d avg. expense load ing ):

Pr insolvency in 1 period Pr 1 E Tot.Losses Tot.Losses 0

Pr Zn

nSD Lossperrisk nE Lossperrisk

Pr Zn

Avg. Risk n


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For insurance company with n risks (an d avg. profit/ expenseload ing ):

Pr insolvency in 1 period

Pr Capitalt 0 1 E Tot.Losses Tot.Losses 0

Pr Zn

Capitalt 0

nE Lossperrisk

nSD Lossperrisk nE Lossperrisk

Pr Zn Avg. Capital n

Avg.Risk n


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1

Avg.Risk n ~ k1 n

Avg. Capitaln ~

k2

n

n


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Pr insolvency of company 0

n

Pr insolvency of pool


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Least Squ ares Regression

Regression

Consider a samp le of observat ions, Y1 ,Y2 , ,Yn , and another

samp le X1 ,X2 , ,Xn .

Assume tha t there is a relationsh ip between each pa ir Yi and Xi ,

so tha t ifXn 1 were known, then Yn1 could be forecast .


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The process of quant ifying the rela tionship between the Ys and

theXs is called regression.

The Ys are called the dependentor targetvar iables, and th eXs are

called t he independentorexplanatory variables.

Often, theXs are fixed po ints in time (e.g., the mid-points or

end s of successive years or mon ths).


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Linear Least Squares Regression

To study the relat ionsh ip b etween the Ys and th eXs, one could

plot the collection of points Xi ,Yi on a graph.

If the points on the graph app ear to follow a straight line with

only random d eviat ions, then it might be reasona ble to assum e

that

Yi aXi b Random Error i ,

where a and b are parameters that must b e estimat ed .


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To estimate a and b usin g the method of l east squa res, on e

would use math emat ical techniq ues to solve for the va lues a

an db that mi nim ize the sum of squa re d errors,

SSE Y1 aX1 b

2

Y2 aX2 b 2

Yn aXn b 2.


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The lin e y ax b is o ften referred to as the lin e of best fit.

To forecast Yn1 given Xn 1, simpl y plu g Xn 1 int o the equa tion

for the lin e of best fit; then:

Yn1 a Xn1 b .


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Simple Log-Linear Mode l

If the points Xi ,Yi app ear to follow an expon entia l cu rve w ithonly ran dom d eviat ions, then it might be reasona ble to assum e

that

ln Yi aXi b Random Error

i ,

where ln denotes the natural logarithm function.

The least-squa res estimates a andb may th en be obtained as in

the linear model. To forecast Yn1 , one mus t first forecast

ln Yn 1 , and then invert the natural logarithm .


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What is the difference be twee n:

Simple Linear Ordinary Least-Squares Regression

and (for example)

Multipl e Logistic Generalized Min.-Abs.-Dev. Regression?

Insurance Pricing Basic Statistical Principles

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