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RISK AND INSURANCE I. INTRODUCTION People seek security. A sense of security may be the next basic goal after food, clothing, and ... a Porsche and a Toyota.
RISK AND INSURANCEI INTRODUCTIONPeople seek security A sense of security may be the next basic goal after food clothing andshelter An individual with economic security is fairly certain that he can satisfy his needs foodshelter medical care and so on in the present and in the future Economic risk which we willrefer to simply as risk is the possibility of losing economic security Most economic risk derivesfrom variation from the expected outcomeOne measure of risk used in this study note is the standard deviation of the possible outcomesAs an example consider the cost of a car accident for two different cars a Porsche and a ToyotaIn the event of an accident the expected value of repairs for both cars is 2500 However thestandard deviation for the Porsche is 1000 and the standard deviation for the Toyota is 400 If thecost of repairs is normally distributed then the probability that the repairs will cost more than3000 is 31 for the Porsche but only 11 for the ToyotaModern society provides many examples of risk A homeowner faces a large potential forvariation associated with the possibility of economic loss caused by a house fire A driver faces apotential economic loss if his car is damaged A larger possible economic risk exists with respectto potential damages a driver might have to pay if he injures a third party in a car accident forwhich he is responsibleHistorically economic risk was managed through informal agreements within a definedcommunity If someone s barn burned down and a herd of milking cows was destroyed thecommunity would pitch in to rebuild the barn and to provide the farmer with enough cows toreplenish the milking stock This cooperative pooling concept became formalized in theinsurance industry Under a formal insurance arrangement each insurance policy purchaserpolicyholder still implicitly pools his risk with all other policyholders However it is no longernecessary for any individual policyholder to know or have any direct connection with any otherpolicyholderII HOW INSURANCE WORKSInsurance is an agreement where for a stipulated payment called the premium one party theinsurer agrees to pay to the other the policyholder or his designated beneficiary a definedamount the claim payment or benefit upon the occurrence of a specific loss This defined claimpayment amount can be a fixed amount or can reimburse all or a part of the loss that occurredThe insurer considers the losses expected for the insurance pool and the potential for variation inorder to charge premiums that in total will be sufficient to cover all of the projected claimpayments for the insurance pool The premium charged to each of the pool participants is thatparticipant s share of the total premium for the pool Each premium may be adjusted to reflect anyspecial characteristics of the particular policy As will be seen in the next section the larger thepolicy pool the more predictable its resultsNormally only a small percentage of policyholders suffer losses Their losses are paid out of thepremiums collected from the pool of policyholders Thus the entire pool compensates theunfortunate few Each policyholder exchanges an unknown loss for the payment of a knownUnder the formal arrangement the party agreeing to make the claim payments is the insurancecompany or the insurer The pool participant is the policyholder The payments that thepolicyholder makes to the insurer are premiums The insurance contract is the policy The risk ofany unanticipated losses is transferred from the policyholder to the insurer who has the right tospecify the rules and conditions for participating in the insurance poolThe insurer may restrict the particular kinds of losses covered For example a peril is a potentialcause of a loss Perils may include fires hurricanes theft and heart attack The insurance policymay define specific perils that are covered or it may cover all perils with certain namedexclusions for example loss as a result of war or loss of life due to suicideHazards are conditions that increase the probability or expected magnitude of a loss Examplesinclude smoking when considering potential healthcare losses poor wiring in a house whenconsidering losses due to fires or a California residence when considering earthquake damageIn summary an insurance contract covers a policyholder for economic loss caused by a perilnamed in the policy The policyholder pays a known premium to have the insurer guaranteepayment for the unknown loss In this manner the policyholder transfers the economic risk to theinsurance company Risk as discussed in Section I is the variation in potential economicoutcomes It is measured by the variation between possible outcomes and the expected outcomethe greater the standard deviation the greater the riskIII A MATHEMATICAL EXPLANATIONLosses depend on two random variables The first is the number of losses that will occur in aspecified period For example a healthy policyholder with hospital insurance will have no lossesin most years but in some years he could have one or more accidents or illnesses requiringhospitalization This random variable for the number of losses is commonly referred to as thefrequency of loss and its probability distribution is called the frequency distribution The secondrandom variable is the amount of the loss given that a loss has occurred For example thehospital charges for an overnight hospital stay would be much lower than the charges for anextended hospitalization The amount of loss is often referred to as the severity and the probabilitydistribution for the amount of loss is called the severity distribution By combining the frequencydistribution with the severity distribution we can determine the overall loss distributionExample Consider a car owner who has an 80 chance of no accidents in a year a 20chance of being in a single accident in a year and no chance of being in more than one accidentin a year For simplicity assume that there is a 50 probability that after the accident the carwill need repairs costing 500 a 40 probability that the repairs will cost 5000 and a 10probability that the car will need to be replaced which will cost 15 000 Combining the frequencyand severity distributions forms the following distribution of the random variable X loss due to0 10 x 5000 08 x 50000 02 x 15 000The car owner s expected loss is the mean of this distribution E XE X x f x 0 80 0 0 10 500 0 08 5000 0 02 15 000 750On average the car owner spends 750 on repairs due to car accidents A 750 loss may not seemlike much to the car owner but the possibility of a 5000 or 15 000 loss could create real concernTo measure the potential variability of the car owner s loss consider the standard deviation of theloss distributionX2 x E X f x0 80 750 2 0 10 250 2 0 08 4250 2 0 02 14 250 2 5 962 500X 5 962 500 2442If we look at a particular individual we see that there can be an extremely large variation inpossible outcomes each with a specific economic consequence By purchasing an insurancepolicy the individual transfers this risk to an insurance company in exchange for a fixed premiumWe might conclude therefore that if an insurer sells n policies to n individuals it assumes thetotal risk of the n individuals In reality the risk assumed by the insurer is smaller in total than thesum of the risks associated with each individual policyholder These results are shown in thefollowing theoremTheorem Let X1 X 2 X n be independent random variables such that each X i has an expectedvalue of and variance of 2 Let Sn X1 X 2 X n ThenE Sn n E X i n andVar Sn n Var X i n 2The standard deviation of S n is n which is less than n the sum of the standarddeviations for each policyFurthermore the coefficient of variation which is the ratio of the standard deviation to the meanis This is smaller than the coefficient of variation for each individual X iThe coefficient of variation is useful for comparing variability between positive distributions withdifferent expected values So given n independent policyholders as n becomes very large theinsurer s risk as measured by the coefficient of variation tends to zeroExample Going back to our example of the car owner consider an insurance company that willreimburse repair costs resulting from accidents for 100 car owners each with the same risks as inour earlier example Each car owner has an expected loss of 750 and a standard deviation of2442 As a group the expected loss is 75 000 and the variance is 596 250 000 The standarddeviation is 596 250 000 24 418 which is significantly less than the sum of the standarddeviations 244 182 The ratio of the standard deviation to the expected loss is 24 418 75 0000 326 which is significantly less than the ratio of 2442 750 3 26 for one car ownerIt should be clear that the existence of a private insurance industry in and of itself does notdecrease the frequency or severity of loss Viewed another way merely entering into an insurancecontract does not change the policyholder s expectation of loss Thus given perfect informationthe amount that any policyholder should have to pay an insurer equals the expected claimpayments plus an amount to cover the insurer s expenses for selling and servicing the policyincluding some profit The expected amount of claim payments is called the net premium orbenefit premium The term gross premium refers to the total of the net premium and the amount tocover the insurer s expenses and a margin for unanticipated claim paymentsExample Again considering the 100 car owners if the insurer will pay for all of the accidentrelated car repair losses the insurer should collect a premium of at least 75 000 because that isthe expected amount of claim payments to policyholders The net premium or benefit premiumwould amount to 750 per policy The insurer might charge the policyholders an additional 30so that there would be 22 500 to help the insurer pay expenses related to the insurance policiesand cover any unanticipated claim payments In this case 750 130 975 would be the grosspremium for a policyPolicyholders are willing to pay a gross premium for an insurance contract which exceeds theexpected value of their losses in order to substitute the fixed zero variance premium payment foran unmanageable amount of risk inherent in not insuringIV CHARACTERISTICS OF AN INSURABLE RISKWe have stated previously that individuals see the purchase of insurance as economicallyadvantageous The insurer will agree to the arrangement if the risks can be pooled but will needsome safeguards With these principles in mind what makes a risk insurable What kinds of riskwould an insurer be willing to insureThe potential loss must be significant and important enough that substituting a known insurancepremium for an unknown economic outcome given no insurance is desirableThe loss and its economic value must be well defined and out of the policyholder s control Thepolicyholder should not be allowed to cause or encourage a loss that will lead to a benefit or claimpayment After the loss occurs the policyholder should not be able to unfairly adjust the value ofthe loss for example by lying in order to increase the amount of the benefit or claim paymentCovered losses should be reasonably independent The fact that one policyholder experiences aloss should not have a major effect on whether other policyholders do For example an insurerwould not insure all the stores in one area against fire because a fire in one store could spread tothe others resulting in many large claim payments to be made by the insurerThese criteria if fully satisfied mean that the risk is insurable The fact that a potential loss doesnot fully satisfy the criteria does not necessarily mean that insurance will not be issued but somespecial care or additional risk sharing with other insurers may be necessaryV EXAMPLES OF INSURANCESome readers of this note may already have used insurance to reduce economic risk In manyplaces to drive a car legally you must have liability insurance which will pay benefits to a personthat you might injure or for property damage from a car accident You may purchase collisioninsurance for your car which will pay toward having your car repaired or replaced in case of anaccident You can also buy coverage that will pay for damage to your car from causes other thancollision for example damage from hailstones or vandalismInsurance on your residence will pay toward repairing or replacing your home in case of damagefrom a covered peril The contents of your house will also be covered in case of damage or theftHowever some perils may not be covered For example flood damage may not be covered ifyour house is in a floodplainAt some point you will probably consider the purchase of life insurance to provide your familywith additional economic security should you die unexpectedly Generally life insurance providesfor a fixed benefit at death However the benefit may vary over time In addition the length ofthe premium payment period and the period during which a death is eligible for a benefit may eachvary Many combinations and variations existWhen it is time to retire you may wish to purchase an annuity that will provide regular income tomeet your expenses A basic form of an annuity is called a life annuity which pays a regularamount for as long as you live Annuities are the complement of life insurance Since paymentsare made until death the peril is survival and the risk you have shifted to the insurer is the risk ofliving longer than your savings would last There are also annuities that combine the basic lifeannuity with a benefit payable upon death There are many different forms of death benefits thatcan be combined with annuitiesDisability income insurance replaces all or a portion of your income should you become disabledHealth insurance pays benefits to help offset the costs of medical care hospitalization dental careEmployers may provide many of the insurance coverages listed above to their employeesVI LIMITS ON POLICY BENEFITSIn all types of insurance there may be limits on benefits or claim payments More specificallythere may be a maximum limit on the total reimbursed there may be a minimum limit on lossesthat will be reimbursed only a certain percentage of each loss may be reimbursed or there may bedifferent limits applied to particular types of lossesIn each of these situations the insurer does not reimburse the entire loss Rather the policyholdermust cover part of the loss himself This is often referred to as coinsuranceThe next two sections discuss specific types of limits on policy benefitsDEDUCTIBLESA policy may stipulate that losses are to be reimbursed only in excess of a stated thresholdamount called a deductible For example consider insurance that covers a loss resulting from anaccident but includes a 500 deductible If the loss is less than 500 the insurer will not payanything to the policyholder On the other hand if the loss is more than 500 the insurer will payfor the loss in excess of the deductible In other words if the loss is 2000 the insurer will pay1500 Reasons for deductibles include the following1 Small losses do not create a claim payment thus saving the expenses of processing the claim2 Claim payments are reduced by the amount of the deductible which is translated into premium3 The deductible puts the policyholder at risk and therefore provides an economic incentive forthe policyholder to prevent losses that would lead to claim paymentsProblems associated with deductibles include the following1 The policyholder may be disappointed that losses are not paid in full Certainly deductiblesincrease the risk for which the policyholder remains responsible2 Deductibles can lead to misunderstandings and bad public relations for the insurance company3 Deductibles may make the marketing of the coverage more difficult for the insurance4 The policyholder may overstate the loss to recover the deductibleNote that if there is a deductible there is a difference between the value of a loss and theassociated claim payment In fact for a very small loss there will be no claim payment Thus it isessential to differentiate between losses and claim payments as to both frequency and severityExample Consider the group of 100 car owners that was discussed earlier If the policy providesfor a 500 deductible what would the expected claim payments and the insurer s risk beThe claim payment distribution for each policy would now be0 90 loss 0 or 500 y 0f y 0 08 loss 5000 y 45000 02 loss 15 000 y 14 500The expected claim payments and standard deviation for one policy would beE Y 0 90 0 0 08 4500 0 02 14 500 650Y2 0 90 650 2 0 08 3850 2 0 02 13 850 2 5 402 500Y 5 402 500 2324The expected claim payments for the hundred policies would be 65 000 the variance would be540 250 000 and the standard deviation would be 23 243As shown in this example the presence of the deductible will save the insurer from having toprocess the relatively small claim payments of 500 The probability of a claim occurring dropsfrom 20 to 10 per policy The deductible lowers the expected claim payments for the hundredpolicies from 75 000 to 65 000 and the standard deviation will fall from 24 418 to 23 243BENEFIT LIMITSA benefit limit sets an upper bound on how much the insurer will pay for any loss Reasons forplacing a limit on the benefits include the following1 The limit prevents total claim payments from exceeding the insurer s financial2 In the context of risk an upper bound to the benefit lessens the risk assumed by the3 Having different benefit limits allows the policyholder to choose appropriate coverageat an appropriate price since the premium will be lower for lower benefit limitsIn general the lower the benefit limit the lower the premium However in some instances thepremium differences are relatively small For example an increase from 1 million to 2 millionliability coverage in an auto policy would result in a very small increase in premium This isbecause losses in excess of 1 million are rare events and the premium determined by the insurer isbased primarily on the expected value of the claim paymentsAs has been implied previously a policy may have more than one limit and overall there is morethan one way to provide limits on benefits Different limits may be set for different perils Limitsmight also be set as a percentage of total loss For example a health insurance policy may payhealthcare costs up to 5000 and it may only reimburse for 80 of these costs In this case ifcosts were 6000 the insurance would reimburse 4000 which is 80 of the lesser of 5000 and theactual costExample Looking again at the 100 insured car owners assume that the insurer has not onlyincluded a 500 deductible but has also placed a maximum on a claim payment of 12 500 Whatwould the expected claim payments and the insurer s risk beThe claim payment distribution for each policy would now be0 90 loss 0 or 500 y 0f y 0 08 loss 5000 y 45000 02 loss 15 000 y 12 500The expected claim payments and standard deviation for one policy would beE Y 0 90 0 0 08 4500 0 02 12 500 610Y2 0 90 610 2 0 08 3890 2 0 02 11 890 2 4 372 900Y 4 372 900 2091The expected claim payments for the hundred policies would be 61 000 the variance would be437 290 000 and the standard deviation would be 20 911In this case the presence of the deductible and the benefit limit lowers the insurer s expectedclaim payments for the hundred policies from 75 000 to 61 000 and the standard deviation will fallfrom 24 418 to 20 911VII INFLATIONMany insurance policies pay benefits based on the amount of loss at existing price levels Whenthere is price inflation the claim payments increase accordingly However many deductibles andbenefit limits are expressed in fixed amounts that do not increase automatically as inflationincreases claim payments Thus the impact of inflation is altered when deductibles and otherlimits are not adjustedExample Looking again at the 100 insured car owners with a 500 deductible and no benefitlimit assume that there is 10 annual inflation Over the next 5 years what would the expectedclaim payments and the insurer s risk beBecause of the 10 annual inflation in new car and repair costs a 5000 loss in year 1 will beequivalent to a loss of 5000 1 10 5500 in year 2 a loss of 5000 1 10 2 6050 in year 3 and aloss of 5000 1 10 3 6655 in year 4The claim payment distributions expected losses expected claim payments and standarddeviations for each policy arePolicy with a 500 DeductibleExpected Standardf y t 0 80 0 10 0 08 0 02 Amount DeviationLoss 0 500 5000 15 000 750Claim 0 0 4500 14 500 650 2324Loss 0 550 5500 16 500 825Claim 0 50 5000 16 000 725 2568Loss 0 605 6050 18 150 908Claim 0 105 5550 17 650 808 2836Loss 0 666 6655 19 965 998Claim 0 166 6155 19 465 898 3131Loss 0 732 7321 21 962 1098Claim 0 232 6821 21 462 998 3456Looking at the increases from one year to the next the expected losses increase by 10 each yearbut the expected claim payments increase by more than 10 annually For example expectedlosses grow from 750 in year 1 to 1098 in year 5 an increase of 46 However expected claimpayments grow from 650 in year 1 to 998 in year 5 an increase of 54 Similarly the standarddeviation of claim payments also increases by more than 10 annually Both phenomena arecaused by a deductible that does not increase with inflationNext consider the effect of inflation if the policy also has a limit setting the maximum claimpayment at 12 500Policy with a Deductible of 500 and Maximum Claim Payment of 12 500Expected Standardf y t 0 80 0 10 0 08 0 02 Amount DeviationLoss 0 500 5000 15 000 750Claim 0 0 4500 12 500 610 2091Loss 0 550 5500 16 500 825Claim 0 50 5000 12 500 655 2167Loss 0 605 6050 18 150 908Claim 0 105 5550 12 500 705 2257Loss 0 666 6655 19 965 998Claim 0 166 6155 12 500 759 2363Loss 0 732 7321 21 962 1098Claim 0 232 6821 12 500 819 2486A fixed deductible with no maximum limit exaggerates the effect of inflation Adding a fixedmaximum on claim payments limits the effect of inflation Expected claim payments grow from610 in year 1 to 819 in year 5 an increase of 34 which is less than the 46 increase inexpected losses Similarly the standard deviation of claim payments increases by less than the10 annual increase in the standard deviation of losses Both phenomena occur because thebenefit limit does not increase with inflationVIII A CONTINUOUS SEVERITY EXAMPLEIn the car insurance example we assumed that repair or replacement costs could take only a fixednumber of values In this section we repeat some of the concepts and calculations introduced inprior sections but in the context of a continuous severity distributionConsider an insurance policy that reimburses annual hospital charges for an insured individualThe probability of any individual being hospitalized in a year is 15 That is P H 1 0 15Once an individual is hospitalized the charges X have a probability density function p d ff X x H 1 0 1e 0 1x for x 0Determine the expected value the standard deviation and the ratio of the standard deviation tothe mean coefficient of variation of hospital charges for an insured individualThe expected value of hospital charges isE X P H 1 E X H 1 P H 1 E X H 10 85 0 0 15 0 1 x e 0 1xdx 0 15 x e 0 1x0 15 e 0 1x dx0 15 10 e 0 1x 1 5E X 2 P H 1 E X 2 H 1 P H 1 E X 2 H 10 85 02 0 15 0 1 x 2 e 0 1x dx0 15 10 0 1 2 x e 0 1x dx 30The variance is X2 E X 2 E X 30 1 5 2 27 75The standard deviation is X 27 75 5 27The coefficient of variation is X E X 5 27 1 5 3 51An alternative solution would recognize and use the fact that f X x H 1 is an exponentialdistribution to simplify the calculationsDetermine the expected claim payments standard deviation and coefficient of variation for aninsurance pool that reimburses hospital charges for 200 individuals Assume that claims for eachindividual are independent of the other individualsLet S X i ThenE S 200 E X 300 S2 200 X2 5550 andS 200 X 74 50The coefficient of variation isIf the insurer includes a deductible of 5 on annual claim payments for each individual what wouldthe expected claim payments and the standard deviation be for the poolThe relationship of claim payments to hospital charges is shown in the graph belowClaim Payment with Deductible 5Y max 0 X 5Claim Payment Y0 2 4 6 8 10Hospital Charges XThere are three different cases to consider for an individual1 There is no hospitalization and thus no claim payments2 There is hospitalization but the charges are less than the deductible3 There is hospitalization and the charges are greater than the deductibleIn the third case the p d f of claim payments isf y 5 H 1 0 1e 0 1 y 5fY y X 5 H 1 XP X 5 H 1 P X 5 H 1Summing the three casesE Y P H 1 E Y H 1 P X 5 H 1 E Y X 5 H 1 P X 5 H 1 E Y X 5 H 1P H 1 0 P H 1 P X 5 H 1 0 P H 1 P X 5 H 1 E Y X 5 H 10 15 0 1 y e 0 1 y 5dy 0 15 e 0 50 15 e 0 5 10 0 91E Y P H 1 0 P H 1 P X 5 H 1 0 0 15 0 1 y 2 e 0 1 y 5 dy0 15 e 0 5 0 1y 2 e 0 1 y dyY2 18 20 0 91 17 37 and Y 17 37 4 17For the pool of 200 individuals let SY YiThen E SY 200E Y 182 2SY 200 Y2 3474 and SY 200 Y 58 94Assume further that the insurer only reimburses 80 of the charges in excess of the 5 deductibleWhat would the expected claim payments and the standard deviation be for the poolE 80 SY 0 8 E SY 146 80 SY 0 8 SY 2223 and 80 SY 0 8 SY 47 15IX THE ROLE OF THE ACTUARYThis study note has outlined some of the fundamentals of insurance Now the question is what isthe role of the actuaryAt the most basic level actuaries have the mathematical statistical and business skills needed todetermine the expected costs and risks in any situation where there is financial uncertainty anddata for creating a model of those risks For insurance this includes developing net premiumsbenefit premiums gross premiums and the amount of assets the insurer should have on hand toassure that benefits and expenses can be paid as they ariseThe actuary would begin by trying to estimate the frequency and severity distribution for aparticular insurance pool This process usually begins with an analysis of past experience Theactuary will try to use data gathered from the insurance pool or from a group as similar to theinsurance pool as possible For instance if a group of active workers were being insured forhealthcare expenditures the actuary would not want to use data that included disabled or retiredindividualsIn analyzing past experience the actuary must also consider how reliable the past experience is asa predictor of the future Assuming that the experience collected is representative of the insurancepool the more data the more assurance that it will be a good predictor of the true underlyingprobability distributions This is illustrated in the following exampleAn actuary is trying to determine the underlying probability that a 70 year old woman will diewithin one year The actuary gathers data using a large random sample of 70 year old womenfrom previous years and identifies how many of them died within one year The probability isestimated by the ratio of the number of deaths in the sample to the total number of 70 year oldwomen in the sample The Central Limit Theorem tells us that if the underlying distribution has amean of p and standard deviation of then the mean of a large random sample of size n isapproximately normally distributed with mean p and standard deviation The larger the sizeof the sample the smaller the variation between the sample mean and the underlying value of pWhen evaluating past experience the actuary must also watch for fundamental changes that willalter the underlying probability distributions For example when estimating healthcare costs ifnew but expensive techniques for treatment are discovered and implemented then the distributionof healthcare costs will shift up to reflect the use of the new techniquesThe frequency and severity distributions are developed from the analysis of the past experienceand combined to develop the loss distribution The claim payment distribution can then bederived by adjusting the loss distribution to reflect the provisions in the policies such asdeductibles and benefit limitsIf the claim payments could be affected by inflation the actuary will need to estimate futureinflation based on past experience and information about the current state of the economy In thecase of insurance coverages where today s premiums are invested to cover claim payments in theyears to come the actuary will also need to estimate expected investment returnsAt this point the actuary has the tools to determine the net premiumThe actuary can use similar techniques to estimate a sufficient margin to build into the grosspremium in order to cover both the insurer s expenses and a reasonable level of unanticipatedclaim paymentsAside from establishing sufficient premium levels for future risks actuaries also use their skills todetermine whether the insurer s assets on hand are sufficient for the risks that the insurer hasalready committed to cover Typically this involves at least two steps The first is to estimate thecurrent amount of assets necessary for the particular insurance pool The second is to estimate theflow of claim payments premiums collected expenses and other income to assure that at eachpoint in time the insurer has enough cash as opposed to long term investments to make theActuaries will also do a variety of other projections of the insurer s future financial situation undergiven circumstances For instance if an insurer is considering offering a new kind of policy theactuary will project potential profit or loss The actuary will also use projections to assesspotential difficulties before they become significant