WHY INSURANCE? 1.1 THE EVOLUTION OF INSURANCE

Transcription

1 WHY INSURANCE?. THE EVOLUTION OF INSURANCE Humas have strived for security sice the begiig of their existece. At its earliest poit, security existed if there was a assurace of food, warmth, ad shelter. The Bible relates the story of how, i aciet Egypt, Joseph set aside part of the crop i good years i a attempt to cover the expected shortfall i years of drought. The World Bak has recetly idetified casualty (or geeral) isurace as a critical elemet for the developmet of emergig ecoomies. This is oly the latest recogitio of the importace of casualty isurace to ecoomic developmet. The roots of isurace ca be traced back to Babyloia, over four thousad years ago, whe traders developed markets to isure the goods o their caravas agaist loss o the hazardous trade routes. Without this form of property isurace, traders would have bee reluctat, or fiacially uable, to egage i the trade that led to this ascet wester civilizatio. Recogized as the oldest brach of isurace, marie isurace was developed i aciet Greece ad eabled trade to occur ad civilizatio to flourish. Agai, forms of casualty isurace were the essetial igrediets to ecoomic developmet. The lack of life isurace o the captai, or a pesio system for the sailors, did ot stop ships from sailig. But without isurace o the ships ad cargo, trade stopped. As society developed ad the roles of idividuals withi the ecoomic framework became more specialized, the eed for ecoomic security icreased. Ecoomic security is the opposite of ecoomic risk, which we will refer to simply as risk. Risk derives from variatio from the expected, ot from probability. For example, o a cloudy morig we may be told there is a From Steve D Arcy, CAS Presidet, The Actuarial Review, Nov 2005, p. 7.

2 2 Chapter risk of rai. What is meat, more correctly, is that there is a high probability of rai. The variatio associated with the weather forecast could be just as high or higher o a suy morig. A moder idustrial society provides may examples of risk. A homeower faces a large variatio associated with the potetial ecoomic loss caused by a house fire. A driver faces a similar, though less variable, potetial ecoomic loss if his or her car is damaged. A larger possible ecoomic loss would be associated with the ijury of a third party i a car accidet for which you are resposible. Examples of early iformal isurace arragemets ca be foud i the cooperatives ad fraterals that existed i Europe over 400 years ago. For example, the farmers i a certai area would agree, usually iformally, that if oe farmer s bar was destroyed, the commuity would see that it was rebuilt. If the breadwier i a family uit died, the commuity would pass the hat to establish a fud for the survivig depedets. I this iformal arragemet, each perso s ecoomic risk was shared or pooled amog the members of the commuity. These iformal systems proved to be adequate for several hudred years. At the time of the idustrial revolutio, however, the eed for a more formal system arose. Because of the rapid urbaizatio of the populatio, it became true that oe s eighbor could be a strager with whom oe had o iterests i commo. Hece, it was o loger sufficiet to expect a commual or cooperative respose whe oe family uit met with a ecoomic reversal. It was perfectly atural that the poolig cocept of the existig cooperatives ad fraterals became formalized i the ew isurace idustry. Uder the ew formal arragemet, each policyholder still implicitly pooled his or her risk with all other policyholders. However, it was o loger ecessary for ay idividual policyholder to kow or have ay coectio with ay other policyholder.

3 Why Isurace? 3.2 HOW INSURANCE WORKS If we look at the risk profile of a idividual, we see that there is a extremely large variatio of possible outcomes, each with a specific ecoomic cosequece. Thus, ay idividual is exposed to a sigificat amout of risk associated with perils like death, fire, disability, ad so o. By purchasig a isurace policy, a idividual (the isured) ca trasfer this risk, or variability of possible outcomes, to a isurace compay (the isurer) i exchage for a set paymet (the premium). We might coclude, therefore, that if a isurer sells policies to idividuals, it assumes the total risk of the idividuals. I fact, the isurer, through careful uderwritig ad selectio will ed up with a average risk that is relatively smaller compared to the origial risk to idividual policyholders. The explaatio of this surprisig result is a priciple called the law of large umbers, which states that as the umber of observatios icreases, the differece betwee the observed relative frequecy of a evet ad the true uderlyig probability teds to zero. Similarly, the differece betwee the observed average severity of a evet (the average size of a loss) ad the expected severity teds to zero as the umber of observatios icreases. So, accurate predictio of outcomes is much easier with may separate (idepedet) risks tha with oly oe or two. Here is aother way to see the reduced variability of outcomes based o larger samples. At a certai age, the probability of death withi oe year is.000, or 0 i 0,000. If we have a sample of 0,000 lives, we ca predict with 95% probability that the umber of deaths will be betwee 4 ad 6, a rage of 6 away from the mea of 0. If we have a sample of,000,000 lives, the 95% cofidece iterval is (938, 062), a rage of 62 away from the mea of 000. But we observe that the variability is 60% of the mea i the first case, but oly 6.2% of the mea i the case with the larger sample. As log as the idividuals beig isured are idepedet risks (i.e., a claim from oe policyholder does ot icrease the probability of a claim from ay other policyholder), the the larger the sample size, the smaller the variace of the average claim, ad, hece, the smaller the risk. Thus, through the isurace mechaism, idividuals ca trasfer their risks to a

4 4 Chapter isurer without havig the isurer takig o a umaageable level of risk i total. I life isurace, the risk is associated with the variability i the umber of death claims, which is modeled by a probability frequecy distributio. I most property/casualty lies of isurace (e.g., auto), ot oly is there a frequecy distributio for umber of claims, but there is also a severity (or loss) distributio for size of claim, from which variability also arises. That is, give that a claim has occurred, the size of the loss paymet is still highly variable. By buyig isurace, the idividual policyholder trasfers his or her risk to the isurer, but, because of the law of large umbers, the isurer eds up with a total risk that is maageable. This is illustrated i Figures.a ad.b, showig the risk profiles for the idividual ad the isurer, respectively. Probability of a Loss of L 0 L Figure.a For the idividual, the probability is very high that there will be o loss at all from the defied evet, but there is a o-zero probability of a sigificat loss. We deote the expected value of the loss to the policyholder by ph, ad the variace of the loss to the policyholder by 2 ph. If the isurer selects idetical ad idepedet policyholders, each with the same risk profile as that illustrated i Figure.a, the the loss distributio for the isurer ca be illustrated by Figure.b. For the isurer, the probability of o loss at all, give policyholders, will be virtually zero if is large, ad the rage of possible losses per policy is much smaller tha for the idividual policyholder.

5 Why Isurace? 5 Probability of a Loss of L per Policy Figure.b If the isurer selects idetical ad idepedet policyholders, the expected value of the average loss per policy is ph, the same as for the idividual policyholder, but the variace of the average loss per policy is 2 ph or, equivaletly, a stadard deviatio of 0 L ph These results are derived i the followig example.,. EXAMPLE. Give idepedet policyholders with idividual loss radom variables X, X,..., X, such that the expected value of ay policyholder s loss is 2 ph ad the variace is 2 ph, show that for the isurer providig these policyholders with isurace, the expected value of the isurer s average loss per policy is ph, ad the variace of the average loss per policy is 2 ph. Solutio Let S X X2 X.

6 6 Chapter Let The ad But X S ( X X2 X ). E[ X] E[ S ] ph p, 2 2 ph Var( S ) Var( X X X ). Var( X ) Var S 2 ph Var( S ) ph. Hece we ca see that the risk to the isurer, measured by the variace of the average loss, is oly th of the risk to the idividual policyholder..3 INSURANCE AND UTILITY It should be clear that the existece of a private isurace idustry, of ad by itself, will ot decrease claim frequecies or loss severities. Viewed aother way, merely by eterig a isurace cotract a perso s expectatio of loss does ot chage. Thus, with perfect iformatio, the et premium for ay policyholder would have to be the expected value of loss. But the policyholder would have to pay a gross premium i excess of the et premium so as to cover the expeses of sellig ad servicig the cotract. Why would someoe pay a gross premium for a isurace cotract that must exceed the expected value of the loss? The aswer lies i a priciple called the decreasig margial utility of moey. Accordig to this

7 Why Isurace? 7 priciple, as extra uits of wealth or icome are added, the utility derived from such uits decreases. This is displayed i the graphs that follow. Total Utility Margial Utility Figure.2a Figure.2b As a example, with early dollars of icome we buy food, clothig, ad shelter, which represet high utility. With later dollars of icome, we buy items such as a stereo for the jacuzzi, which is of lower utility. The priciple of decreasig margial utility of moey applies to ayoe who is a risk averse, which is the case for most people. There are some people who are risk seekers, for whom the priciple of decreasig margial utility does ot apply. Such a perso, for example, could be expected to forgo basic eeds, such as food or shelter, to gamble o a chace for large wealth. The examples that follow assume that the purchaser of isurace is a risk avoider. EXAMPLE.2 A prospective purchaser of isurace has 00 uits of wealth. He faces a situatio whereby he could icur a loss of Y uits, where Y is a radom loss with a uiform distributio betwee 0 ad 36. This perso has a persoal utility curve give by u( x) x. What maximum gross premium would this perso be willig to pay for isurace? Solutio Wealth Wealth Note that for this idividual u( x) 0, so that u icreases with x, ad u ( x) 0, so that each additioal uit of x brigs less tha oe additioal uit of utility, u. Hece this prospective policyholder is a risk avoider, sice the law of decreasig margial utility applies. (A risk seeker would have a icreasig margial utility curve.)

8 8 Chapter Further, otig that the p.d.f. for the radom loss is f( y) 36, we ca fid E[ Y] y f ( y) dy 0 36 y dy y 72 8, 0 so the expected value of the loss is 8. The isurer must therefore charge a gross premium i excess of 8 to cover sales commissios ad admiistratio costs. Why would a policyholder pay more tha 8 to buy isurace whose expected value is 8? The aswer lies i the margial utility curve for this policyholder illustrated i the followig figure. Margial Utility G Wealth Figure.3 The policyholder will pay a gross premium of G for the isurace, so he loses G whether or ot the loss occurs, leavig him with 00 G uits of wealth. Without isurace, however, the policyholder faces a possible loss of 36 uits of wealth, which is 36% of his total wealth. If the policyholder buys isurace, the resultig wealth positio is certai; it will be 00 G, with utility value 00 G. If he does

9 Why Isurace? 9 ot buy isurace, the resultig wealth positio is probabilistic, give by 00 Y, ad the expected utility value of the resultig wealth positio ca be calculated as 0 36 E[ U] u(00 y) f ( y) dy y dy (00 y ) The policyholder should be willig to pay a premium G that equates the expected utility values of the resultig wealth positios with or without isurace. Thus we fid G such that 00 G , which results i G Thus the policyholder will pay up to 8.33 for this isurace, which exceeds its expected value of 8, ad if the isurer ca charge a premium less tha 8.33, the isurace purchase will be made. Give this or a similar utility fuctio, we ca see why it may ot make sese to isure agaist small losses (e.g., theft of goods worth less tha $200). I this case, the utility value of the gross premium will exceed the expected utility value, because we have ot moved far eough i the decreasig margial utility curve to overcome the expese elemet iheret i the gross premium. EXAMPLE.3 You are tryig to decide whether to ivest i Compay A or B. For this ivestmet, the utility profile ca be measured by the fuctio where P represets profit. 3/2 u( P) P 00, P 00, 36 0

10 0 Chapter (a) Show that this is the utility fuctio of a risk avoider. (b) Give the followig iformatio, determie your ivestmet strategy based o (i) expected moetary value, ad (ii) expected utility value. Profit Probability Compay A Compay B Ecoomy Advaces Ecoomy Stagates Solutio (a) Give that the ad u( P) P 00, u( P) ( P00) 2 /2 3/2 u ( P) ( P00). 4 This shows that ad so the ivestor is risk averse. u ( P) 0, for P00, u ( P) 0, for P00, (b) The followig table shows the moetary payoffs ad their associated utilities. Profit Probability Compay A Compay B Ecoomy Advaces (62.45) 2800(5.96) Ecoomy Stagates (0.00) 400(7.32) (i) Expected moetary value: E(Compay A).40(4000).60(200) 720 E(Compay B).40(2800).60(400) 360 Ivest i Compay A.

11 Why Isurace? (ii) Expected utility value: E(Compay A).40(62.45).60(0.00) E(Compay B).40(5.96).60(7.32) 3.8 Ivest i Compay B. EXAMPLE.4 A idividual faces the followig possible losses: Loss Size Probability $ If the utility fuctio of a potetial purchaser of isurace is: u() x (a) Show that this perso is risk averse. (b) Calculate the maximum premium this idividual would pay for isurace give the above loss distributio ad iitial wealth of $2000. Solutio (a) 0.6 x 0.6 u( x) x 0 if x u( x) 0.6x 0 if x 0.4 u( x) 0.24x 0 So we have decreasig margial utility, which idicates the idividual is risk averse. (b) With isurace that costs $G, the outcome is kow ad 0.6 equals 2000 G with utility (2000 G). Without isurace, we have a loss distributio with three possible outcomes ad resultig expected utility ($000).00($900).899($2000)

12 2 Chapter 0.6 So set (2000 G) G $.22 Note: EL [ ].00($000).00(00) $.00 So G E[ L]..4 WHAT MAKES A RISK INSURABLE We have show i the previous sectios that a idividual will see the purchase of isurace as ecoomically advatageous if the priciple of decreasig margial utility applies (i.e., the idividual is a risk avoider). O the other had, the isurer will agree to isure a prospective policyholder if the law of large umbers ca be applied to the risk pool to which the prospective policyholder wishes to belog. With these priciples i mid, what makes a risk isurable? () It should be ecoomically feasible. If we do ot move far eough o the utility fuctio, the the utility gaied by isurig will ot be eough to cover the utility of the cost of the isurace mechaism (e.g., sales commissios ad head office expeses). (2) The ecoomic value of the isurace should be calculable. A example of where this criterio holds is auto collisio isurace. Here a large umber of small losses are experieced. We ca get a lot of data o collisio experiece ad, through the law of large umbers, ca calculate a expected premium with a high degree of cofidece. Isurig a uclear reactor agaist meltdow is a example of where this criterio does ot hold. Such a policy ca be issued by usig a risk-sharig arragemet amog may isurers so that the exposure to risk for ay oe compay is maageable. (3) The loss must be defiite. This criterio is meat to guard agaist policyholder maipulatio ad moral hazard. Moral hazard occurs whe the isured is able to icrease the value of the isurace beyod that expected i the price or premium. A car accidet with police documetatio is defiite. Death is defiite. What is ot so defiite, but still isured, is disability. Whe is a isured well eough to retur to work? How do you guard agaist maligerig? (4) The loss must be radom i ature. Agai we wish to have the isured evet beyod the cotrol of the policyholder. The presece of criteria three ad four allow the actuary to assume radom samplig i

13 Why Isurace? 3 the projectios of future claim activity. That is, there is o statistical bias i the selectio of oe isurace uit versus aother. (5) The exposures i ay rate class must be homogeeous. This meas that, before the fact, the loss expectatio for ay uit i a class must be the same as for ay other uit i the class. I terms of radom samplig, this is aalogous to each elemetary uit havig the same probability of beig draw. Through ati-selectio by policyholders, this criterio might ot be satisfied. Ati-selectio occurs whe the policyholder has more iformatio tha the isurer, ad the policyholder uses that extra iformatio to gai a price/rate loss advatage. (6) Exposure uits should be spatially ad temporally idepedet. I terms of radom samplig, this implies that selectio of oe elemetary uit does ot affect the probability of drawig ay other elemetary uit. I more practical terms, we wish to avoid ay catastrophic exposure to risk. We would ot, for example, isure all the stores i oe retail area, sice oe fire or oe riot could result i a huge loss. I isurace terms, the fact that oe isured has a claim should ot affect whether aother isured has a claim. These criteria, if fully satisfied, mea that the risk is defiitely isurable. The questios of risk classificatio ad price still follow. O the other had, the fact that a potetial risk exposure does ot fully satisfy the criteria does ot ecessarily mea that isurace will ot be issued. Some special care or risk sharig i these circumstaces (e.g., reisurace) may be ecessary. I property/casualty isurace, rarely does a isurable risk meet all of the listed criteria..5 WHAT INSURANCE IS AND IS NOT There is ofte cofusio i the mids of cosumers ad regulators as to the purposes ad itet of isurace. The isurace mechaism is used to trasfer risk from the idividual policyholder to the pooled group of policyholders represeted by the isurace corporatio. If the isured pool is a large collectio of idepedet policyholders the the per-uit risk will be greatly reduced ad will be maageable for the isurace compay. The isurace compay admiisters the pla, ivests all fuds, pays all beefits, ad so o. The isurace compay ca oly pay out moey that comes from the pooled fuds. If claims rise, so too must premiums.

14 4 Chapter From the policyholders viewpoit, isurace is available oly for pure risks; that is, where the outcome is either loss or o loss. The policyholder caot profit from buyig isurace. I speculatio, there is also a trasfer of risk, i that a idividual ca trasfer a uwated risk to a speculator. The motive for the speculator is the chace to make a profit. A good example of how speculatio ca be used to trasfer risk is the futures market. Suppose a farmer plats a field of witer wheat i October. He will deliver this wheat i July. This farmer is risk averse ad does ot wish to speculate o what the price of grai might be i July. The farmer goes to the futures exchage ad sees that it is possible to sell the grai i October to a speculator for $4 a bushel with delivery i July. I July, grai is actually sellig for $3.50 a bushel. The farmer delivers the grai as agreed ad is paid $4 a bushel. The speculator must ow realize the loss of $0.50 a bushel. Had grai prices rise to $4.75 a bushel (e.g., i a dry summer) the speculator would have made a profit of $0.75 a bushel. By takig o this risk, the speculator does two positive thigs. First, the risk of fluctuatig prices is removed from the risk averse farmer ad assumed by the speculator (who hopes to make a profit). To the extet that the speculator is correct i his/her projectios, prices are stabilized. Note, however, that the risk has oly bee trasferred; it has ot bee reduced or removed. There are two key differeces betwee speculatio ad isurace. The first is the profit motive behid speculatio. There is o profit motive o the part of the policyholder i eterig a isurace agreemet (the isurer, however, hopes to make a profit). Secod, the isurace process sigificatly reduces total risk through the Law of Large Numbers. Speculatio trasfers risk, but does ot reduce it. I gamblig, risk is created where oe existed ad oe eeded to exist. I terms of utility, gamblig works i a fashio opposite of isurace. People sped early ad high utility dollars i the hopes of gaiig large wealth that has lower utility value. Overall, gamblig decreases societal utility by redistributig icome i a o-optimal fashio. Some theorize that gamblers have utility curves that explai their actios, i.e., both u( x) ad u ( x) would be positive.

15 Why Isurace? 5 If the profits from the gamblig process (e.g., a state or provicial lottery) are spet o high utility eeds (e.g., a hospital), the it is possible for the fial result of this process to icrease total societal utility. Otherwise gamblig decreases total utility ad is a waste of huma resources..6 RISK, PERIL, AND HAZARD Risk is a measure of possible variatio of ecoomic outcomes. It is measured by the variatio betwee the actual outcome ad the expected outcome. Peril is used as a idetifier of a cause of risk. Examples iclude fire, collisio, theft, earthquake, wid, illess, ad so o. The various cotributig factors to the peril are called hazards. There are physical hazards such as locatio, structure, ad poor wirig, ad there are moral hazards such as dishoesty, egligece, carelessess, idifferece, ad so o. A example might help. Mr. Rich ows a cabi cruiser. Hazards whe sailig are egligece o the part of the captai, rocks, shoals, ad so o. These are cotributig factors. Perils would be thigs like fire or collisio (i.e., cause of risk) which may or may ot cause a fiacial loss, which is risk. I coclusio, a isurace cotract will reimburse the policyholder for ecoomic loss caused by a peril covered i the policy. Thus the policyholder trasfers this risk to the isurace compay..7 PURCHASE OF INSURANCE: OTHER REASONS While utility theory provides a uderlyig ecoomic ratioale for the decisio to purchase isurace, quite ofte some other practical reasos are preset: () Legal requiremets. Most jurisdictios have fiacial resposibility laws that apply to all licesed motor vehicles. The licesee must show that he or she ca satisfy judgmets redered as a result of accidets resultig from operatio of the vehicle. The most popular way of satisfyig this requiremet is through isurace. There are other laws ad regulatios that require isurace before a licese to egage i certai busiesses is issued.

16 6 Chapter (2) Leders requiremets. Whe a idividual takes out a mortgage o property or takes out a loa to purchase a vehicle, the leder almost always requires isurace o the property or vehicle up to the amout of the loa (this to protect the leder s isurable iterest i the property). This is also commo for commercial loas, which are secured by property. (3) Commercial requiremets. I the course of busiess trasactios, oe party will ofte obligate itself i some measure to perform a service, to deliver a product, etc. It is commo that isurace is purchased to compesate the ijured party if the service is ot performed or the goods are ot delivered. Such busiess arragemets are ofte cotiget o the performig party obtaiig isurace. (4) Special expertise. The isurace compay may provide a service o a more cost-effective basis tha the isured ca do o its ow. The most obvious example is adjustmet of claims. Isurers have large, experieced, claim departmets. A example of this would be usig a isurace compay to admiister the paperwork of a large detal isurace program. Some compaies also see value i havig a third party, the isurer, hadle claims made by its customers. Other services iclude boiler ispectios, ad loss cotrol audits. (5) Taxatio. If a compay i the Uited States or Caada self-isures its exposures, it ca oly claim a tax deductio for losses as they are paid. I cotrast, the cost of isurace is expesed immediately sice the premium is paid up frot. Thus i log-tailed lies such as product liability, the deductio for icome tax purposes ca be accelerated by may years ad provide a real ecoomic beefit.

17 Why Isurace? 7.8 EXERCISES Sectio.2. (a) State the law of large umbers. (b) Explai the importace of the law of large umbers to the isurace mechaism. Sectio.3.2 Cofirm that the utility fuctio log, for u( x) k log x, ad k 0 x 0, is the utility fuctio of a decisio maker who is risk averse..3 Which of the followig two proposals i the table below would a risk avoider choose? Proposal A Proba- Proposal B Proba- Outcome Payoff Utility bility Payoff Utility bility O 80, , O 0, , O 30, , Two busiessme view the followig proposals. X Y Success Failure Success Failure Profit 50,000 20,000 5,000 5,000 Probability Their respective utility schedules for the project are as follows. Busiessma x A B 20, , , , What decisios would they make based o: (a) expected moetary value, ad (b) expected utility value?

18 8 Chapter.5 Assume the maagemet of a ivestmet firm has utility fuctio, for ay project, U( P) P 000, where P represets profit. (a) Cofirm that maagemet is risk averse. (b) Cosider the followig two proposals, below: Proposal A Proposal B Profit Probability Profit Probability Which proposal would maagemet choose based o: (i) expected moetary value, ad (ii) expected utility value?.6 A market gardeer faces the possibility of a early frost that would destroy part of his crop. He ca buy crop isurace. This creates four possible outcomes, which are preseted, i the followig table. Profit Freeze No Freeze No Isurace 0,000 30,000 Isurace 20,000 25,000 (a) Based o expected moetary value, what probability must the farmer attach to early frost to make buyig isurace a wise decisio? (b) Give his existig wealth, the farmer has the followig utility profile. Profit Utility 0, , , , Based o expected utility value, what probability must the farmer attach to a early frost to make buyig isurace a wise decisio?

19 Why Isurace? 9.7 You are subject to the utility fuctio x.9 u( x), where x is 0,000 wealth. Your curret wealth is 50,000. What is the maximum premium you would pay to isure agaist a loss that is uiformly distributed betwee 0 ad 30,000?.8 You follow the utility fuctio ux x ( ) exp, where x is 00,000 wealth. Your curret wealth is 20,000. What is the maximum amout you would pay to take part i a fair coi toss where you have.5 probability of wiig 0,000? If you wi you do ot receive a retur of your wager..9 A perso has a utility fuctio, over the relevat rage, give by 2 u( x) 0,000 x x, where x is wealth. Her curret wealth is What is the maximum wager she would make i a game where there is a 30% chace of wiig 2000 plus the retur of her wager?.0 You are give the followig iformatio. (i) The gross premium for isurace is (ii) The idividual kows he will have, 2, or 3 losses with equal probability. (iii) Each loss will cost (iv) u /6 measures the loss of utility for the idividual, where u is a measure of utility, is the expected value of loss, ad is the stadard deviatio of loss. Uder these coditios, determie whether the prospective policyholder will buy isurace. Why?

20 20 Chapter.. Mr. Smith has a total wealth of 525,000 ad his utility of wealth is u( x) l( x). He ows a sports car worth 50,000. The isurace o his sports car is due for reewal. Based o Mr. Smith s drivig record, the risk of damage to his car i the ext year is as follows. Amout of Damage Probability , , ,000.0 Sectio.4 Mr. Smith s isurace compay charges premiums for all its policies equal to the expected value of its claim paymets uder the policy plus 0% of this expected value as a loadig. (a) Should Mr. Smith fully isure his car at the isurace compay s premium? Explai why or why ot. (b) As a alterative to its full coverage policy, the isurace compay is offerig a ew policy that will pay 50% of all damage amouts for accidets greater tha or equal to 20,000. All other damage amouts are paid by the isured. Should Mr. Smith isure his car with this ew policy?.2 It is commo for successful race horses to be sold for stud (breedig purposes) at the ed of their racig careers. Not all such horses are successful. Should it be possible to buy isurace to idemify you for loss if a race horse you buy is ot a successful breeder?

21 Why Isurace? 2.3 The XYZ Isurace Compay has bee asked to issue a 2-year term isurace policy o a specially traied dog that is goig to star i a movie. If the dog dies i year oe, 8000 will be paid at the ed of year oe. If the dog dies i year two, 5000 will be paid at the ed of year two. If the dog lives to the start of year three, o paymet is made ad the cotract eds. The dog is ow age x, ad the isurace compay develops the followig survivorship data based o kow mortality experiece of dogs of the give age ad breed. x x x2 x3 x Sectio.5 (a) Is this a isurable risk? (b) If i = 0%, determie the et sigle premium for the cotract. (c) Calculate the associated variace..4 From a ecoomic viewpoit, compare ad cotrast gamblig ad isurace. Briefly explai why isurace is more acceptable. Sectio.6.5 (a) Differetiate amog risk, peril, ad hazard. (b) Give a example of each.