TITLE : OFT= USE OF INSLJRANCE, MSS EREVENTION, LOSS REDUCTION AND SELF INSLJRAhCE AUTHOR: Martin Rosenberg Martin Rosenberg received BA and MA degrees in Economics from Queens College of the City University of New York, where he also lectured in Economics. He is a Fellow of the C&3 (May, 1981), and a Member of the American Academy of Actuaries (1979). In addition to the current paper, Martin submitted the paper, Selection of the Optimum Asset Portfolio to Satisfy Cash Needs to the CAS 1981 Call Paper Program, Inflation Implications for Property-Casualty Insurance. Martin serves on the CM Education and Examination Committee. He has been known to exercise by jogging around the home office building of Reliance Insurance in Philadelphia. Mr. Frederick 0. Kist Mr. Kist is a Vice President and Consulting Actuary for Tillinghast, Nelson & Warren, Inc. He received his FCAS in 1979 and is a member of the American Academy of Actuaries. Mr, Kist attended Northwestern University and earned a B.A. degree in mathematics in 1973. Mr. Kist s insurance experience includes seven years with CNA Insurance, the last two of which were in the National Accounts Division where Re was directly involved with the pricing and underwriting of large national accounts. - 307 -
Introduction An asset owner such as a mercantile business can use various methods to protect Its assets from potential financial loss due to fortuitous occurrences, Three Important methods of asset protection will be discussed In this paper. They are 1) Insurance, 2) prevention, and 3) reduction. These are explained as follows: 1) Insurance of the loss. The purchase of an Insurance policy by a business Involves the business paying a premium to an Insurance company In return for which the Insurance company reimburses the business for all or part of a loss to assets. 2) Prevention of the loss. Loss prevention means reducing the probability that a loss will arise. For example, the use of locks, safes, and armed guards will help to prevent a property loss due to theft, 3) Reduction of the loss. Loss reduction means reducing the severity of the loss once It occurs. For example, lnstall- atlon of a sprinkler system In a building will reduce the. damage done by a fire, If a fire occurs. A business may use no asset protection methods to protect Its assets from all or part of a potential loss. For the purpose of this paper, 'self Insurance' refers to the situation where a business uses no asset protection methods to protect against all or part of a potential loss. One purpose of this paper 1s to show how an asset owner can select how much of each asset protection method to use to mlnlmlze the expected reduction in assets, Refer to the combin- - 308 -
ation of asset protection methods that minimizes the expected reduction In assets as the 'optimum mix' of asset protection methods. It will be shown that the optimum mix of asset protection methods depends on the relative benefit and cost of each method. An Iterative process is used to Illustrate how to determine the optimum mix. The analysis of the selection of the optimum mix of asset protection methods has important implications for the pricing of Insurance as follows, I) The analysis of the optimum mix of asset protection methods can be an Important complement to the actuarial analysis of the adequacy of insurance rates. This is because one goal of setting a price for insurance coverage Is to provfde the insurer with enough money for claims, expenses, profit, and contingencies. However, in the market place, the price at which Insurance can be sold may differ from the price the insurer would like to receive. This may occur If other asset protection methods are relatively less expensive than lnrmrance, Therefore, for example, the insurer may not be able to receive the fill amount of contingency loading that the Insurer wants to Include in the rate, 2) The analysis of the optlmum mix of asset protection methods can be useful as a planning tool. By making estim- ates of the various parameters that affect the optimum mix, an estimate can be made of hon much insurance can be sold for each price of insurance. Therefore, for example, If the Insurer's goal is to write a certain number of premtum dollars, this model can be used to select the pride at - 309 -
which this goal may be obtained. 3) Investment income that an insurance company earns will depend, In part, on the number of dollars of premiums that are written because, plainly, the greater the premium wrlt$ngs, the greater the number of dollars the insurer has to Invest. Based on the analysis of the optimum mix of asset protection methods, and using the economic concept of 'elasticity of demand;' the model can help to determine the price of insurance that maximizes the premium writings, and therefore maximizes the number of dollars available to an insurance company for investment. IhIs analysis Is also useful in the pricing of insurance when both under- writing profit and investment income are considered ln detennln- ing the price. In addition to the pricing of Insurance, the model has lmpllca- tlons for other aspects of the insurance market, as follows. I) 'Ibis model has Implications for the effect of regulation on the insurance business. For example, If regulators require the purchase of at least minimum amounts of insurance (for example, auto llablllty insurance), then this will affect the selection of asset protection methods, 2) The effect of inflation on the insurance market can be analyzed by assuming an inflationary effect on various parameters and then determining the resulting lmpact on the optimum mix of asset protection methods. Very importantly, the size of the business which is selecting asset protection methods can have Important meaning for analysis of all of the above considerations. This 1s because the values - 310 -
of the parameters of the model can depend on the she business. of the we non introduc8 the model to be used to select th8 optimum mix Of ass8t'pz'otection methods and use this mod81 to alialy'z8 Some of the financial implications to insurance companies of the selection. - 311 -
General Description of the Model As discussed above, the potential for loss of assets to a business can be reduced using asset protection methods. We anticipate that the greater the amount of asset protection that a business purchases, the loner the expected value of loss to assets. lhe definition of the 'amount' of asset protection depends on the form of each asset protection method. For example, Insurance contracts can pro- vide for various deductibles (e.g. straight deductible,i.disappear-. ing deductible, franchise deductible, etc.) and for various limits of Insurer's liability for loss. The type of the deductible and limit of liability would, In part, determine what is meant by the 'amount' of asset protection provided by an insurance policy. Below, ne will use the model in terms of specific forms of asset protection. For non, we wish to develop the model in general terms. Therefore, we use symbols to represent the amount of asset protection provided by each asset protection method, In this regard let I = amount of Insurance coverage p= amount of loss prevention R = amount of loss Consider the mathematical function where (1) L = L(I,P,R) reduction L = number of dollars of expected loss of assets This function shows the number of dollars of expected loss for each combination of asset protection methods. We assume that the number of dollars of expected loss decreases as the amount of each asset protection method increases. In math- ematical terms, this means the partial derivatives of L with respect to I, P, and R are negative. That Is - 312 -
Using discrete In w~rds,~~is notation, (3) g(o ; g+-<o * &(O ' AR the decrease in expected loss 86 the amount Of in- surance fs increased by $1. s is the decrease in expected loss as the probability of loss Is reduced by.ol.%k is the decrease In expected loss as the amount of each potential loss is decreased by 10% of its full value. If asset protection methods were costless to a business, then the business would use as great sn amount of asset protection methods as it could obtain. This is because we assume that, everything else constant, the lxsiness prefers to have as ion a level of expected loss as possible. However, asset protection methods are not costless. By purchasing asset protection the asset ounlng business gives up some assets which could have besn used instead to produce income for the business. For example cash spent by a business on installation and maintenance of smoke detectors to reduce the loss frvm f%re could have been used Instead to purchase inventories to process and sell at a profit, Therefore, protecting assets from loss is itself another source of reduction 5.n assets. Continuing with the mathematical development of the model to take into account expenditures on asset protection methods, consider where (4) E = E(I,P,R) E= the number of dollars spent by the asset owner on asset protection methods - 313 -
I, P, R are as defined above. This function shows the number of dollars spent on asset protection for each combination of I, P, and R. We assume that the number of dollars Spent on asset protection methods increases as the amount of each asset protection method, which is purchased, increases. In mathematical terms, the partial derivatives of E with respect to I, P, and R are positive. That Using discrete notation, be In words, az is the increase In the number of dollars spent on asset protection methods for a $1 Increase In the amount of ln- sumnce. is the increase in the number of dollars spent on kb asset protection as the probability of loss is decreased by.ol. $1~ the increase in the number of dollars spent on asset protection as the value of each loss Is reduced by.l of its full value, We assume that the goal of an asset owning business In purchasing and using asset protection methods is to minimize the expected reduction In assets. SinC8 both th8 POtential for 1066 due to fortuitous occurrences and expenditures to protect against the loss have the effect of reducing assets, we assume that the goal of the asset owner 5.6 to minimize the sum (7) S = L(I,P,R) * E(I,P,R). Assume the function S Is mathematically well behaved. Then the - 314 -
'first order' conditions from elementary calculus for S to be a minimum are that the partial derivatives of S with respect to I, P, and R are equdl to %8I O. That is, 03) + +de,o d1 (9) *-t+$- 50 (10) *+$+o In discrete notation, (11) g-- q" g = 0 (12) ++ $$- z 0 03) *+A&-0 AR - The purpose of stating t?ondltlons (8) through (13) is to be able to easily Identify and locate the point of minimum expected re- duction to assets. To continue the development of the model, repeat that L = expected value of loss to assets Further, E= number of dollars spent on asset protection methods let q = probability that a loss occurs pi = conditional probability that a loss is of size I, given that a loss has occurred. In the absence of the use of any asset protection (14) L = q( P,(i) + P&2)+...Pk(k)+.,.) (15) E = 0 methods, - 315 -
Specific Discussion, Assuming Insurance is the Only Asset Protection Method Available We will now consider and discuss a specific use of the model. Assume that the only asset protection method available is in- surance. Assume that the insurance policy work8 as follows. The insurance policy carries a limit of coverage of K dollars. This means the insurance company reimburses the asset owner in full for a loss of up to and including K dollars. For a loss above K dollars, the insurance company reimburses the asset owner only K dollars. For example, if a loss of (K-l) occurs, the loss to the asset owner is 0. If a loss of (K+l) occurs, the loss to the asset owner is 1. If a loss of (K+2) occurs, the loss to the asset owner is 2. Therefore, in this example, the limit of coverage is the amount of insurance the asset owner purchases, Let, Then Lk 5 the expected value of the loss to the asset owner that has an insurance policy with limit of coverage of K dollars (16) ~~ = qupk+l )(l> + tp,+,)t2) + (p,,,)(3) + l **) Now assume that the limit of liability were (K+l) Instead of K. Then 117) $+I = q(( k+&( ) + (p,,,>(1) + tp,+,)t2) + l **) The change in expected loss to the asset owner from Increasing the limit of coverage $1 from K to (K+l)ls (la) $+I - Lk = -q( k+l + pk+2 + P k+3 +... 1-316 -
An equivalent way of writing (18) Is 09) $+I - Lk = -q Jj$pk+j Equation (19) shows the change in expected loss to the asset owner as the amount of Insurance is increased by $1. There- fore, In terms of the notat2n introduced above (see(3)), (20) $& = -q)p Jzf k+j Now consider expenditures on insurance. Assume that the price the asset owner pays for insurance is b per dollar of cov8rag8. L8t Then % = the number of dollars spent for Insurance with a limit of liability of K dollars. (a) E,/ Kb Fox a limit of liability of (K+i) dollars, (22) Ewi = (k+l)b Then the change in expenditure on insurance as the amount of insurance. I.6 increased by $1 from K to (K+l) 1s (23) s+l - Ek = (K+l)b - Kb = b In terms of notation Introduced above (see (6)), (24) E - b since insurance is the only asset protection method avail- able, in this example, then the point at which the reduction to assets is a minimum is where (11) repeated ~+&o* Substituting (20) and (24) into (ii), we get (25) 'qspk+j + b = 0, Or (26) q <'k+j = bt Of: b (27)?Pdj,l k+j = F - 317 -
Equation (2'7) determines the limit of coverage K which mini- mizes the reduction in assets from expscted losses and expen- ditures on insurance. This limit of coverage is where the cumulative conditional probability of loss above the limit of coverage equals the ratio of the cost per dollar of coverage to the probability that a loss ocours. Note that If b,q, no in- surance is purchased, and all losses are self insured. Before proceeding, it should be pointed out that in the above, the price 4per dollar of insurance coverage Is assumed to be constant (and equal to b). This assumption is mad8 in order to keep the example which follows uncomplicated. In fact, In the real world, there are reasons to believe that an insurer would charge a decreasing rate per dollar of coverage as the limit of coverag is increased by $1. If we used a decreas- ing rat8 per dollar of coverage, equation (26) would become (28) 9ck+j =&b(k) where Ah(K) represents the decrease in the cost per dollar of coverage as the limit of coverage is Increased from K to (K+l). A minimum point of reduction to assets uould then exist if 'second order conditions were satt.sfied. The second order conditions are essentially th:t q> is abovehb(k) for,,=d k+j small values of K and then qxp declines faster than &b(k), d:, k+j as K Increases. We will continue the discussion with a numerical example of equation (27) and assuming a constant value of b, for the sake of simplicity. - 318 -
NumerlcalExample I In order to give a numerical example of using equation (27) to determine th8 optimum amount of insurance, assume the following. The asset owner believes that the probability of a loss occuring, q =.25. The asset owner believes that the conditional probability of loss follows a normal distribution with mean =A= 100 standard deviation =V= 50. Now assume that the insurer, using Its own assumptions about the probability of loss and the distribution of losses that may occur, sets a price of b =.05 per dollar of insurance coverage. Then, using equation (27).,05 (29) Xl'= $===,210 Equation (29) indfoates that the optimum limit of liability for the asset owner is where the cumulative probability of loss above the limit of liability Is.20. Since the conditional probability of loss that the asset owner assumes is normal, we can locate th8 optimum amount of insurance using 'z* values from a table of the standard normal distribution. As you know, The 'z' value corresponding to a cumulative probability of 0.20 is approximately z =.84. We axe given thata = 100, and v = 50. Substituting these values into (3) we get - 319 -
(3).84 = ;y 9 oj.- (32) X = 100 + (9)(.84) = 142. We have found that the optimum amount of insurance that the asset owner will purchase is 142. The portion of any loss above 142 is self Insured. If the asset owner were to buy sn amount of ln- surance greater than 142, the decrease in expected loss due to having more insurance than 142 would be more than matched by the additional cost of the extra coverage, If the asset owner uere to hy an amount of insurance less than 142, the decrease In cost from buying less insurance than 142 would bs more than matched by the additional expected loss from having less Insurance. There- fore, an insurance policy with a limit of coverage of 142 mini- mizes the sum of the cost of insurance plus the expected loss to the asset owner, - 320 -
The Relationship Among Price of Insurance, Premiums Written, and Insurer's Profit Ue now discuss some of the financial implications of the asset owner's choice of the amount of insurance to purchase and the amount of potential loss to self Insure. A key equation of the model is (27) repeated $Pk+j = $ Equation (27) can be used to find K, the optimum amount of in- surance coverage, as Illustrated above, The optimum amount of insurance coverage depends, in pert, on the price the insurance company charges. This is because a given value of b in equation (27) determines the cumulative probability above K. The higher the value of b, the higher th8 cumulative probability, Therefore, the higher the value of b, the lower the amount of insurance that will be purchased. %bl8 I shows the amount of insurance that will be purchased for each value of b. These values have been determined using the pro- cedure followed in kun8rlcal Example I, above. As you can see the figures in the table follow the pattern described above, namely, that at higher values of b, lower amounts of insurance are purchased. The data contained in Table I, showing the price of Insurance and the corresponding amounts that are purchased, is known in economic jargon as a 'demand schedule' for in- surance. An insurance company is typically interested in the premium volume thatlit writes or that it oan write. Table II shows the insurance company premium volume for each price of insur- ante. The premium volume for each policy (co3umn 3 of Table II) is simply the price per dollar of coverage (column 1 of Table II) - 321 -
times the number of dollars of coverage purchased (column 2 of Table II). We fkrther assume for the sake of exposition that there are 100 identical asset owners, and that each selects the same amount of insurance coverage. Therefore, the total premium volume (column 4 of Table II) Is simply 100 times the premium volume for each policy. It is obvious from Table II that the insurer cannot select both the price the Insurer will charge and the premium volume it will write. For example, if the insurer decides to charge a price of.23 then the insurer cannot expect to write more than 690 in premiums. If the insurer wants to write, for example, 902 in premiums, It is necessary to lower the price to.22. In fact, a price of.07 will also bring &out a premium volume of just above 900. In order to further examine the relationship between price and premium volume as well as other financial relationships, we now Introduce the economic concept of 'elasticity of demand.' Table I shows the relationship of price (b) to the amount of insurance purchased (K) for the specific parameters given in Numerical Example I, above. Consider the more general situation where K is some continuous function of b, i.e. (33) K = K(b) The elasticity of demand, ED, is defined as, In words, the elasticity of demand shows the proportional change in the amount of insurance as a ratio to the proportional hang8 in the price of insurance. In order to use the discrete data In Table I to determine the elasticity of demand at various points on the demand schedule, we us8 a discrete approxiamtion to equation (34) as follows. - 322 -
Let bo = a given price of Insurance = the next higher price above b,, bl = the next lower price below bo b-l Kg SKI ek,l = the correspondingamounts Then the discrete approximation is of Insurance (35) ED = For example, using the values In Table I, the elasticity of demand at b=.o6, 12s -I% ED,o6 =- Tr.a-,-.05 - =.289-06 Ihe elasticities of demand fox the remaining prices are shonn in Table III. We non introduce some Important financial relationships which can be explained in terms of the elasticity of demand. The first concept is the relationship of elasticity of demand to total premium volume. It Is shown in standard economic theory that revenue (I.e. premium volume for an Insurance company) is maximized at the point where elasticity of demand equals I. It Is of importance to an insurance company to know the price of insurance which will produce the maximum premium volume when an insurer's operating goal is to write as large a market share as possible. It Is also of importance to an insurer when the Insurer wishes to maximize premium volume so as to have as much money as possible to invest, The fact that maximum premium volume occurs where elasticity of - 323 -
demand equals 1 Is derived In Appendix I. 1Je can show that elasticity of demand equals 1 at the point of maximum premium volume for the data we have developed above. Table Iv shows the price of insurance, the corresponding total premium, and the corresponding elasticity. As you can see, the maximum premium volume of 1312 occurs at b =.16. At b-=.i6,elasticity = 1.073, which Is the elasticity value closest to 1 on Table IV. There is no elasticity value of precisely 1 because the data in Table IV is discrete and not continuous, Therefore, In our example, an Insurer wishing to maximize premium volume would charge a price of b =.16. As we have indicated above, knowledge of the price might be important to an insurer seeking to maximize premium volume in order to have the maximum amount of money available from premium writings for investment, This is characteristic of the current pricing environment where some Insurers have been seeking to maximize premium volume to take advantage of the very, very high interest return available from even relatively secure high grade bonds, The concept of elasticity of demand can give further insight into the relationship between premium volume and the profit- ability of the business that the Insurer writes. Introduce the following notation in order to discuss profitability, Let P= number of dollars of profit K = amount of coverage written, b = price of Insurance, as above as above i = rate of return on insurance company Investments n = number of dollars paid by the insurer for all expenses and claims, per dollar of coverage - 324 -
Then define profit as (36) P = Kb(l+i) - nk We show in Appendix II that the elasticity of demand at the price which maximizes the number of dollars of profit 16 (37) EDpfmax= b b- f+ It follows from equation (a) that the price at which profit is maximized will not equal the price at which written premiums are maximized. Ihe price at which premiums are maximized is where elasticity equals I, as indicated above. For ED prof max to equal 1, n must equal 0. We do not expect n to equal 0 because this would mean that no dollars are need- ed for losses and expenses, Therefore, unless the demand I schedule does not have a point where elasticity equals I, the price that wfll maximize premiums is different from the price that will maximize profits, This insight has some importance for describing insurance company pricing and marketing activities during the current underwriting cycle, The recent economic environment has been one of unprecedented high rates of interest. This has given insurers an incentive to obtain as much cash as possible from premium writings in order to Invest the cash at the high interest rates. Therefore, insurers have. been pricing insurance policies so as to generate maximum premium nrltlngs. In terms of our elasticity analysis, insurers are pricing to reach the point on the demand schedule where elasticity is I. However, using the elasticity analysis developed above, we realize that it Is inevitable that the point of maximum - 325 -
premium nrltings (where elasticity is 1) cannot be the point of maximum profits. Therefore, we can predict that insurers will adjust their prices so as to move away from the point of maximum premium writings once they realise that it Is not the point of maximum profits. Note that we are not suggesting that those people who are responsible for pricing.insurance policies are aware of the demand schedule for insurance coverage. Nor are we suggesting that insurance prices are consciously changed In response to calculations of elasticities of demand. Nevertheless, the elasticity model can provide the knowledge that the point of maximum premium volume is not the point of maximum profit. Therefore, whether the Insurance company is aware of the concept of elasticity or not, we can use the concept tc predict some aspects of insurance company pricing behavior, lbat the premium maximizing price is not equal to the profit maximizing price Is illustrated numerically using data from our Numerical Example I. In order to use the profit equation (361, use the values for K and b from Table I. Further assume, i =.13 n =.1? Then determining the number of dollars of profit by sub- stituting these values of K, b, i, and n into equation (36) we get the values shown In Table Y. As you can see from Table V (column 5), profit Is maximized at 337 where b =.21. Let's see if we can derive the profit maximizing price from elasticity considerations alone by using equation (3). Consider b =.18, The actual elasticity at b =.18 is 1.34 (see Table III). According to equation (y), the elasticity of - 326 -
demand required for b =.18 to bs the profit maximizing price IS b ED profmax= = 6.090 b-3 Since the required elasticity (6.090) is nowhere near the actual elasticity of 1.394, then b =.18 cannot be the profit maximizing price. Now consider the price b =.Zl. The actual elasticity of demand at b =.Zi is 3.570 (see Table III). For the profit maximizing price to be b =.2l,the elasticity of demand at the point should be (using (p)), ED profmax= The actual elasticity of 3.570 is very close to the required elasticity of 3.526. Therefore, the priceofb=.zl is the price at which profit is maximized. As above, the small difference exists because the data in our example Is discrete data. We have seen therefore that the data In our Numerical Example conforms to the conclusions reached above. - 327 -
Specific Discussion, Assuming All Asset Protection Methods Are Available We now consider the case where the three asset protection methods described above (lnsurance,preventlon, and reduction) are avail- able. The criterion to be used to determine the selection of how much of each of these methods to use is the simultaneous solution of equations (Ii), (12), and (13). above. In the follow- ing example the solution is approached through an iterative process as follows. In the first step, it is assumed #at only in surance Is purchased, No prevention or reduction is used, Then the decrease In losses from using a small amount of prevention and/or reduction of loss- es Is compared to the corresponding increase in expenses from using the prevention and/or reduction. If the decrease in losses is greater than the Increase In expenses of using a small amount of asset protection,,then the asset protection is used, Successively larger amounts of asset protection methods are tested tc see if their use decreases losses more than they increase expenses, Asset protection methods are used in the amounts up to the point where the decrease In losses is just equal to the Increase in expenses for all the asset protection methods. This is the point at which equations (ii), (12), and (13) are satisfied. - 328 -
Numerical Example II We assumed in Numerical Example I, above, that the probability of a loss is q = A?5, and that the dlstritition of losses, given that a loss occurs, Is normal nith mean of 100 and standard deviation of 50. Given the price of insurance to be b=.05, we showed that the optimum amount of insurance is I = 142. Now assume that it is possible to decrease the probability that a loss occurs. That is, the value of q can bs decreased. For the sake of illustration, assume the following function show the expenditure, E(q), required to decrease the prob- ability of loss, q, to the value ~0. (38) E(q) = 1000(.25-90)~ for q.25 For example, to reduce the probability of loss to q. =.20 requires spending.l25. (If the figures are In terms of hundreds of dollars, this becomes a more realistic $12.50) Also, assume that it Is possible to reduce the size of a loss, once a loss occurs, For bthe sake of illustration, assume that losses can be reduced to fraction r of their value by expending E(r) on loss reduction methods where (39) E(a) = 90(1 - d3 For example, reducing losses to 8% of their value requires spending,400. (Again, if the figuresare in terms of hundreds of dollars, this becomes a more realistic $40) Note that equations (9) and (3) are, obviously, not intended to describe any actual asset protection situation. Rather these equations have been fixed so as to Illustrate an increasing cost, which increases at an increasing rate, as - 329 -
as the probability and/or the amount of loss is decreased. Equations showing increasing cost are used because we expect that decreasing the probability and/or amount of loss becomes more and more expensive as the loss is decreased further and further. Repeating, the price of insurance is b =.05 and the costs of prevention and reduction are given by equations (38) and (3). How do we select the optimum amounts of each of these to use? Table VII shows the changes in losses and expenses for alternative amounts of insurance, prevention, and reduction. As you can see, If r were held fixed at r = I, the decrease of q to.20 (with the corresponding change In I) produces the minimum value of losses and expenses. This is because the decrease in q from.25 to.20 entails an expense 0f.125. However, at q =.20, the optimum amount of lnsurante Is decreased from 142 to 134, causing a decrease in expense of &O, and an increase in expected loss from 1.33 to 1.49. The net effect of all of these Is a decrease in loss plus expense of..llf, Decreasing q tc q =.lj (with r still at I) would mean an increase in expense on prevention of.875 (from,125 to 1.000). The amount of i.nsurance coverage would decline in response to the decrease In q to I = 122 This would mean a decrease in expenditure on insursuce of 0.6. However, expected losses would Increase from I.49 tc 1.3. Therefore, the net effect of decreasing q to.i5 from.20 would bs an increase In expense and expected loss of.365 Therefore, of the values shown in the table with r = 1, decreasing the probability of loss to q =.20 is the optlmm level of prevention. - 330 -
Now consider loss reduction, I.e. the posslbity of decreasing r. htq-,20, reducing losses to 90% of their value means a reduction In expenses on insurance and expected losses of 0.75 (= 8.19-7.44). Note that we are assuming that expenditures on prevention, E(q), are not affected by use of loss reduction methods. The cost of reducing losses to 90$ of their value is only.05. Therefore the net gain of reducing losses to 90% is 0.70. Will further loss reduction (a lower r) improve the situation? Let's see. Reducing losses to 8% of their value from 9@% means an increase in expenditures on loss reduction of.35o (i.e. from.oso to,400). However, expenditures on insurance plus losses decrease by -8.50 (1.8, from 7.44 to 6.59). There- fore the reduction In losses from 9C$ to 80% of their value proves tc bs an optimizing move. Will further loss reduction cause improvement? Let's see. Reducing losses from 80% to 70% of their value means an ln- crease in expenditures on loss reduction of,950 (1.8. from.400 to 1.350). However the decrease in expenditure on In- surance plus expected loss is only.86o(l,e. from 6.59 to 5.741. Therefore, the reduction from 80$ to 70% is not an optimlz5ng move. Based on the above discussion, the optimum situation is where the probability of loss is decreased to q =.20, and the value of losses Is reduced to 8096 of their value, The resulting optimum amount of insurance coverage is I = 107. The asset owner self Insures the part of any loss over 107. - 331 -
The point is pursuing thisnumerlcalexample is to show explicitly that an asset owner can use several means to protect assets, only one of which is insurance. The extent to which each asset protection method is used depends on its impact on losses as well as its cost. - 332 -
Concluding Remarks This paper has Illustrated the point that the amount of insurance that will be sold depends on the price of insurance as well as the relative price of other asset protection methods. The paper has also considered some of the financial effects on insurance company profits of the price that an insurer charges for insurance coverage. It Is felt that an analysis of these items can serve as a comple- ment t.c the various rate making techniques which have been devel- oped. Rate making techniques typically concern themselves with setting a price that will cover insurance company claims and ex- penses. This paper considered aspects of the reaction of potential Insurance buyers tc the price the insurer sets. An under- standing of both how tc deve1op.a price to cover claims and expenses and the reaction of potential buyers to the price set by an insurer is necessary to fully understand what is going on in the insurance market place. - 333 -
TABLE! I (1) Price Per Dollar of Insurance Coverage (= b).ol l 02 903.04.05.06.07.08.09.I0.I1.I2.I3.I4 915.I6.17.I8.19.20.2l.22 023.24 (2) Amount of Insursnce Coverage Purchased (= K) 188 170 159 19 142 135 129 124 118 113 108 103 97 92 87 82 76 71 65 9 50 41 30 12 Notes: I. see text - 334 -
TABLE II (1) Price Per Dollar of Insurance Coverage (2) Amount of Ineurance Coverage Purchased ' (3) (4) Premium Per Policy Total PremiuQ.Ol 188.02 170.o3 159.04 150 *OS 142.06 135.07 129.08 124.09 118.I0 113.I1 108.12 103 013 97.I4 92.15 87.16 82.I7 76.I8 71.19 65.20 58.2l 50.22 41 023 30.24 12 la2 1883 3.40 w 4.77 477 6.00 600 7.10 710 8.10 810 9.03 903 9.92 992 10.62 1062 il.30 1130 11.88 1188 12.36 1234 12.61 1261 12.88 1288 13.05 1305 13.12 1312 12.92 1292 12.78 12?8 12.35 1235 11.60 1160 10.50 1050 9.02 902 6.90 690 2.88 288 @ I - 335 -
Notes (TABLE XI) 1. See Table I 2. column (3) = CO~UQXI (1) X Column (2). Here, 1.68 =.Ol X 188 3. Column (4) = Column (3) X 100 because we assume there are 100 identical insured6 who each select an identical amount of coverage Here 188 = 1.88 X 100-336 -
TABLE III 11) Price Per Dollar of Insurance Coverage (2) Amount of Insursncei Coverage Purchased (3) Elasticity.Ol 188.02 170.03 159.04 150 905 142.06 135.07 129.08 124.09 118.I0 113.I1 108.12 103 013 97.I4 92 015 87.I6 82.I7 76 918 71.I9 65.20 58.2l 50.22 41-23 30.24 12,171,189 l 227.264 339 298-355.419,442 a9.641.737.,761.862 1 l 073 2 1.230 1.394 1.900 2.586 3.570 5.366 11.117-337 -
Notes (TABLE III) 1. See Table I 2. Elasticity =.- 76-87 a?#.i7-.i5 (- s/6 ) = /,0?3-338 -
TABLE IV (1) PricePerDollarof Insugnce Coverage (2) Total Prdun! (3) Elasticity2.Ol 188.02 340-03 477.04 600.05 710.06 810.07 903,08 992.09 1062 JO 119 *II 1188.I2 1236.13 1261.14 1288.l5 1305.16 1312.I7 1292.18 1278..19 1235.20 1160.2l 109.22 902-23 690.24 288,171.I89 227.264,289 298 * 355.419,442 8509,641 9737.761,862 1.073 1,230 1.394 I.900 2.586 3.570 5.366 11.I17 Notes 1. See Table II 2. See Table III - 339 -
TABLE!' (1) (2) (3) Price Per Amount of Premium Dollar of Infsuxance Per Insurance Coverage l Policy2 Coverage Purchased (4) Profit Per Policy (5) Total Profit.Ol 188 1.88.02 170 3.40.03 159 4.77.04 150 6.00.05 142 7.10.06 135 8.10.07 129 9.03.08 124 9.92.09 118 10.62.I0 113 11.30.I1 108 11.88.12 103 12.46.13 97 12.61.I4 92 12.88.I5 87 13.05.I6 82 13.12.I7 76 12.92.18 71 12.78.19 65 12.35.20 58 11.60.2l 50 10.50 l 22 41 9.02.23 30 6.90.24 12 2.88-29.84-25.06-21.64-18.72-16.12-13.80-11.73-9.87-8.07-6.44-4.94-3.54-2.24-1.09-0.44 + 0.89 + I.68 + 2.373 + 2.9l +.3.25 + 3.37 + 3.22 + 2.70 + I.21-2506 -2164-1872 -1612-1380 -1173-987 - 807-644 - 494-354 - 224-109 -44 + 89 +16&J + x37 4 + 291 + 325 + 337 + 322 + no +12l - 340 -
Notes (TABLE V) 1. See Table I 2. See Table II 3. Pmfit = lx (1 +A) - nk In Table V, i =.13, n =.17 At this point, b =.18 K = 71 Then Profit (.18) (71) (1.13) - (1.7) (71) = 2.37 4. Total Profit = Profit Per Policy X 100 Hem 237 = 2.37 x 100-341 -
TABLE VI (1) Price Per Dollar of In-ce coverage (2) Total P?XilBiUd (3) Total Profit2 (4) ElastiLity?.Ol.02.03.04.05.06.07.08.09.I0.I1.I2.l3.I4.I5.16.I7.18.I9.20.2l.22.23.24 188 340 477 600 710 810 903 992 1062 1130 1188 12% 1261 1288 1305 1312 1292 1278 1235 1160 109 902 690-2984 -2506 464-1872 -1612-1380 -1173-987 - 807-644 - 494-354 - 224-109 -44 + 89 +168 + 237 +291 + 325 + 337 + 322 + 270 +12l.171.I89,227.264 a9 298-355.419 A42 0509,641.737,761.862 I.073 1.230 I.394 I.900 2.586 3.570 5.366 11.117 Notes 1. See Table II 2. SeeTableV 3. See Table III - 342 -
TABLE VII PORTFOLIOS OF ASaT PRO'XECTION MFXHODS AND TNEWEXPECTEDLOSSESANDEWENSES ll4f.0 ql E(q12 I3 E(1) L5 L E(I)+? E(r,8 E(I)+L+E(r)+E(q?, -25 0 142 7.1 1.325 8.425 0 8.425 l 20.I25 134 6.7 1.490 8.190 0 8.315.15 1.000 122 6.1 I.580 7.680 0 8.680.I0 3.375 100 5.0 I.980 6.980 0 10.355.05 8.000 0 0 5.000 5.000 0 13.000 r= 0. Q E(q) I E(I) L E(I)+L E(I)+L+E(r)+E(q) 128 6.4 1.193 7.s93 122 6.1 I.341 7.w 110 5.5 1.422 6.922 90 4.5 1.782 6.282 0 0 4.5oo 4.joo.r = 0.8 E(I)+L+E(r)+E(q) 025 0.20.125-15 I.000 *IO 3.375.o5 8.000 114 107 98 80 0 7.160 7.117 7.564 9.359 12.400 Y=f-l-- 99 5.0 94 4.7 85 4.3 70 3.5 r= 0.7 E(r) E(I)+L+E(r)+E(d I 1 7.278 7.218 7.756 9.611 12.85O
NOES (TABLE VII) I. q = probability that a loss occurs 2. Expendltumson ~duciugqare imumedaccordingto E(q) = 2000 (.25 -.q)3 For example, the expense of reducing q to 30 is E(.20) = 1000 (.05)3 = 125 3. 4. 5. 6. 7. 11s the optlmusamountof iusuranceand is determined as per Numerical Example I, above. E(1) is the expenditure on I and equals.05 I L is the expected value of loss to the asset owner, given the values of q, I, and r (see below). E(I)+L is the sum of E(1) and L r represents the level to which losses have been reduced. An r of 0.90 means losses value have been reduced to qo$ of the they uould have had in the absence of loss reduction methods. 8. Expenditures on reducing losses are incurred according to E(r) = 50 (1.0 - r)3 For exasple, the expense of reducing losses to 90% of their 9. value Is E (.90) = 50 (.20)3 =.OP E(I)+L+E(r)+E(q) represent the sus of losses and expenditures on reducing and preventing losses, It is the swn which the assstownerisassumed to wish tb reduce toamlnimum. - 344 -
Appendix I This appendix shows that the price of insurance at which uritten premiums are maximized is where elasticity equals 1. Let Then In order Then b = price per dollar of coverage K(b) = amount of coverage purchased at price VP= written premium W = H(b) to maximize WP, set d(we) db jfkbjdblbl--+k=o b-j+=+ -++I = o b Since elasticity = - ' dk K db Then elasticity equals 1 at the point of maximum written premiums. - 34s -
Appendix II This appendix shows that at the price at which insurance company profits are maximized, elasticity equals b b- /k I WC) Profit is defined as (36) repeated P = bk(l+i) - nk To maximize P, set or elasticity =-at b-~~ the point of maximum profit - 346 -