Question

wthout insunnce coverage? The text ook aplans inutvely why te risk-areธe consumer brys nsurance; tis homework assgment uses a simple numerical exampia.) If losses are uitrmy dstrbutad frm 0 to average loss, as we wored out oarier? The actuarial pure prem um is 1% of the awengo loss. This should bo haf the pure premium of the sconato whon al losses are total bsas) Microoconomics, Module 24, Risk and Uncortainty (Chapter 18) Hamowark The atadhed PDFfie has better formating-) Rek Avarson and Insuranan Coverage Η. ,000, what is te actuanal pre premium? (What is the L What is the utity of a consumerwho buys insuance coveraga? (The consumer paysthe premium but pro cedure s he samo as usod abova, but the pun pemium is only haf as J. In practon, insurers dort sal oovarage at teaduatal prepnmum. A sume agar that ail b,05 are s recompensed for any fre loss. The consumers wealth $300,000-PPr, and the consumar's usity is ($300,000- PPr laga.jNo indogration is noedad. Economists atban wow n 뢰 arco tom anskaversion pespectve. Thiss not always raalsta: consumers buy parmanant Ife insuanceforthe taxadwantagadinvestment and auto insuancebecause tisrequirod by he state governments provide tax wto-ofts for group hoalth insurance and morgago landes requre otal lossas. The insurar has exponses and profit equrements hat are included in he gross promium. What & the mamumexpense and proft provision at which the consumerstl buys the solve for the gross premium, GPr,byoquating theconsumer's unitywith we athof $300,000 expoctod unity wh no insurance covaraga. We workod out he axpectod uity wth no insurance coverage aboe. Squan thisnumberand subtract the om $300,000. Given the actuarial pure premium and the maximum goss premium,themomum expenseand ptoft commission is GPr-PPr. The maximumexpense and prof t ratio is 1-PPdGPr.) Insuran㎝ matts depend on many things, Only one of which is risaverson, but for tis homewok assignmentwefocus on tisk avarsion. Washowan equlibtum for insurancemakots based onrisk avesion. The mathematics s simplified, but tha concepts an used by aconomists K. Fnd ha maxmum expanse and proft provision if ho sss are uniformly distributed from zeto t 300,000. Suppose consumers have homes wort $300,000 apiece and no oter weath. The dance of afns 1% each yaar. We use two atem atves tr ta szo ofíoss dasrtuton: tre losses a al S300,000 orfi) distrbutod urifornly fom zaroto$300,000. (In tuh, small bssas ae muchmon frequenthanlarge lossas: owr 90% offro losses aro๒ss t an 10% ofthe home valua. We use askewod dtstrbuton n te last part of his postng.) Quoton Is this how Insurance coverage is prcod? notactualy prced this way, would his be how nsurance covarage shoud be priced if wa knew the bs distrbuton and the utlity function of consumos? Consumas are tsk avase. The consumar's utity the square oot of his or herwealth. Aconsumar wh no monay and no home has utity of o-0. A consumer with a home has a utlity of 300,000-547.72. This udityfuncton &simple to work with forthe homework assignmont it s not used in praction.) Ansvar: lns ance makets are “mpettve, we sad bofore tat in a oompette market, the largerm equibrium prtceis based on the producers cost functions. The price is theminmum long-un avarage cost. Thisdoes nat dapandon he ulitytunctonofthe consumer. Adtuatal pricing dapands onlong-darm expansas provisions of the insurer, not the utity function of the consumer A. A“nsumar has ahome wth a 1% chance of a fre loss. tf to fra loss wal b@ a total loss (the frst assumpton above), whatis he consumer expected woat at ta end ofthe yoar? (99% probatity of Quostion: Doesn1 the unity fundion of the aonsumar afect the demand cuve? 300,000 and 1% prbabityof $0.) B. what tste expected utityof ts consumer at the end of the year? (99% prbatityof 547.72 and 1% probability of zao.) The expectod utity is not the same as heutity of he expected woah, since the Answar: For compet tve makats, the bng-tun equilibtum prce does not depand on the demand curvo o consmor 1tdepards on theoost curves ofte fm. ra corpottve nsurance market, te oquitrum proe does not dapand on consumes utlity curvas. C. Assume that fia bssas are uniformly disrbuted ovr the range 0, $300,000]. What s the consumars L is 1/300,000. We can intograto x300,000 from to 300,000 to find the avarago loss. The intogration gives1300,000at x-300,000 $1 50,000. This exotise s smpla, and wedont nood to integrate,since the avarage loss $150,000 The oopeded weat atte end of ta ysar is a 99% probabity of $300,000 and 1% probablity of axpocted wealh attha and of the yea (Thelikelhoodofa losso Quoston: How would wouse utityfunctions for insurance pricing? Answer:Sometmest emis nooquitnum fortst-dollar insuran㎝ coverage. Suppose, tatconsumersre only sighty nsk ansa; tey buycowngolf the apanse and proft protion s 20% but not if t is 40%. ranrs have a 40%expense and profit provelon, we might car clude t at no oovenge wil be sodintis maket. But a good aduay mayrestudtue the poicy to make it mare desiabla, as shown below. 150,000.) D. fffre losses are disttbuted over the ranga [O,$ whatis tho consumar's axpe ctadutity LPA. Gvan hat a lossoccurs, the kalihood of a lossofsae L is 1/300,000. Forte apedad uityifterets absa, we itagmte 300,000 L) 1300,000 fom zeo to300,000. The expectedutity at the end of the year the waghtad avaraga of tis utity (1% probabity) and te tl 547.72 utity(99% probabity. The rt aton gns x at the and of the yoar? (Wihaloss of size L hounlity is (300 The folowing example s not part of the homework assignmant, since it roquires more mathornatcs hanb approprate for this cousa. 300,000-LPeuatod at L-0. A. Suppose losses have an expor antinl szeofíoss dsrtu如ntom zaro to to wlue of the home E. wo mtum tothe scarat。ofonty a total loss. Iftha insunnoe oownge is sotd atfar odds (at the actuaria pure pramium) whut the pramium? (The actuaral puro promum means no axpansas or ($300,000). Since te exponen tai danturton ubonded, we must retaancethe įkeihood of a b seventy between zro and $300000 to 100%. If te proba blity of a loss S300,000ls P, we mutply to likeihood of each lass by 1P The put, premium, orpPr, is 1 % K$100,000.) F. What is tho utity of a consumerwho buys insurance coverago? (The consumar pays the premium but s rcompansed for anyfire loss The consumar's weath is $300,000-PPt, and the consumar's utity s the square oot of theirwealth. C. D. The probablityof a t oss is 1%. İf insurers offend coverage at te actar i pre prenurn, consumers would buy ticoverage. This s rue as long as consumars are tsk avese G. Woulda risk avarse consumer buy nsurance covarage if t is offared at the actuata pure promum? (s ho utity at the and of he yoarwhinsurance aovanga greator than teexpactod unity at and of tho yoar E. What the maximum expense rato at which consumars buy full coverago? This dapends of the size o loss disthbuton. Hal lbssas are total bssas, consumers buyinsuance covarage evon at high axponse

Answer 1

A.

Chances of fire loss = 1%

Wealth of the property = $300000

Expected wealth of the consumer at the end of the year

E(W) = 1% * 0 + 99% * 300000 = 0.99*300000 = $297000

(Since fire destroys everything, wealth left in case of fire = 0)

B.

Expected utility of the consumer is given as

$E(U) = \sum p_iu(x_i) = 0.01 \sqrt(0) + 0.99 \sqrt(300000)$

$E(U) = 0.99*547.72 = 542.24$

while Utility of the expected wealth $= \sqrt(270000) = 544.98$

These two are not same since the consumer is risk averse.

C.

Assume that the losses are uniformly distributed over the range [0, 300000]

Since probability of loss is 1%, likelihood of loss of size L = 1/300000

Average loss is given by

$AL = \int_{0}^{300000}\frac{x}{300000}dx = \frac{1}{300000}\frac{x^2}{2}|0\rightarrow300000$

$= \frac{1}{300000}\frac{300000*300000}{2} = 150000$

Thus consumer's expected wealth at the end of the year = 0.99*300000 + 0.01*150000 =298500.

D.

If there is a loss of size L, wealth = 300000 - L

and the expected utility is $= \sqrt{300000 - L}$

Since probability of loss is 1%, likelihood of loss of size L = 1/300000

Average expected utility from the loss is

$= \int_{0}^{300000}\frac{1}{300000}\sqrt{300000 - L}dL = \frac{2}{3}\frac{1}{300000}(300000-L)^{3/2}| 0 \rightarrow 300000$

$= \frac{2}{3}\frac{1}{300000}(300000)^{3/2} = \frac{2}{3}\sqrt{300000} =$$365.15$

Thus expected utility at the end of the year

1% * 365.15 + 99%*547.72 = 0.01*365.15 + 0.99*547.72

= 3.6515 + 542.24 = 545.90