A Risk-Theory Model for Analyzing Surplus

by

Michael A. Clipson, A.S.A.

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Table of Contents

I. Introduction . . . . . • . . • . . • • • • . • • • • . • • . • • • • . •

II. Line-of-Business Surplus Model

Definition of the Line-of-Business Model

Constants • . • •

Random Variables

Assumptions ...

Computation of the Surplus Means and Variances

Estimating the Parameters .....••.•..

III. The Aggregate Surplus Analysis Model ..

Definition of the Aggregate Surplus Model

Computation of the Surplus Means and Variances

1

4

5

5

5

6

7

11

12

12

13

IV. Studies and Results. . • • . • • • • . . • • • • • . • • . • . • • •. 15

Appendix A . . . . . • • • . • • . . . • • • . • • • • • . • . • • • • .• 17

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I. Intrc>duct1on

The insurance coverage issued by the CUNA Mutual Insurance Group is dominated by group term 1 i fe and health insurance, wi th half of our premi um income attributed to credit insurance coverages. Thus except for our pension and individual life insurance lines of business, the C-2 risk is a much larger concern to us than is the C-3 risk. Because the characteristics of CUNA Mutual's business are so different from those of insurers issuing primarily permanent life insurance, we have rejected the simple surplus formulas in use by companies such as Lincoln National 1 in favor of a risk-theory approach.

In the development of our ri sk-theory model, we began with the di screte time model presented in section 12.4 of the Actuarial Hathematics 2 textbook. In following the classical ruin theory approach, we start by defining Sj, j - 0, 1, 2, ... , to be the surplus level at the end of year j and let s • So be the initial surplus level. Let's also define Gj as the statutory gain between times (j-I) and j. The model is then based on the following assumptions:

Sn . s + GI + G2 + G3 + . .. + Gn . (1.1 )

The Gj'S are independent, identically-distributed random variables, and (1. 2)

Il - E[Gj] > o . (1.3)

Now, let the time of ruin, T, be defined as the minimum value of n for which Sn is less than or equal to 0 (with T - ~ if Sn > 0 for all n), and let

;(s) - Pr{ T < ~ } (1.4)

denote the probabil ity of eventual ruin. Given the above definitions and assumptions, the usual bound on the probability of ruin is given by

(1.5)

lRobertson, Richard S., "Managing Life Insurance Company Surplus on a Formula Basis,· The Interpreter (February, 1985), p. 12.

2Bowers, Gerber, Hickman, Jones and Nesbitt, Actuarial Hathematics, [52-1-82], Society of Actuaries.

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where R is the so-called adjustment coefficient. In the special case where the common distribution of the Gj'S is normal with mean # and variance a2 ,

(1.6)

Even if the gains are not normally distributed, equation 1.6 is approximately true for other distributions as well.

This method is certainly very quick and easy to apply, given the first two moments of the distribution of the Gj's. Even though equation 1.5 only gives us an upper bound on ;(s), it has been shown 3 that e-Rs is actually very close to ;(s). In spite of these advantages of the classical ruin theory approach, however, CUNA Mutual's C-2 and C-3 Risk Task Force identified several shortcomings which we sought to overcome:

1. The theory does not tell the actuary how to estimate the mean and variance of the distribution of the Gj's. This is not really a problem with the c 1 as sica 1 ru i n theory approach - - we simply wanted to augment the model with a little more detail that would help the actuary analyze the distribution of the Gj's.

2. In trying to apply the theory to a particular line of business, we found that a common distribution for all of the Gj'S was not always practical. Often the immediate outlook fpr a particular line of business is very different from the long-term outlook. For example, if a line of business is currently in a loss cycle and we extrapolate those results indefinitely into the future, no amount of surplus would be adequate to cover all of those losses.

3. Surplus generates investment income and that should be reflected in the subsequent gains. Thus it is i nappropri ate to assume that the G j' s are independent and identically-distributed random variables.

4. We wanted a theory that would work equally well on the 1 ine-of-business 1 eve 1 and the company 1 eve 1 . Thus we needed to work out some deta i 1 s on

3For a 1 ist of references on this result, you may wish to consult the paper by Allan Brender, RRequired Surplus for the Insurance Risk for Certain Lines of Group Insurance, R Transactions of the Society of Actuaries, XXXVI (1984), p. 14 .

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how the line-of-business models could be combined to form an aggregate model representing the entire company.

5. Finally, we felt that an unbounded time frame was not necessary for our analysis. If we have enough surplus to survive 5, 10, or 20 years with a high probability, then we will probably have enough surplus to survive indefinitely. We decided that, given the fast-changing, short-term nature of our business, we should have enough surplus on hand to survive 5 years with a 99.9% confidence leve1 4.

Given the above objectives, we set out to develop a surplus model that would help the various line-of-business actuaries analyze the surplus needs of their particular lines of business and also to develop a method for combining the line-of-business models into one company-wide model, taking into consideration the correlations between lines of business.

Once the theory behind the line-of-business surplus model was complete, a spreadsheet program was developed to calculate the probability of ruin, given certain assumptions about the line of business and an initial surplus level s. Then the line actuaries made their best estimates of the necessary assumptions for each line of business and entered them into the spreadsheet. Finally, the initial surplus needed to lower the probability of ruin to 1 chance in 1,000 was determined by trial-and-error for each line of business.

After the line-of-business models were developed, the line actuaries were again solicited to estimate correlation factors between the various lines of business. These factors were used in the construction of the aggregate model. Again, the final estimates were entered into a spreadsheet and the required surplus was calculated by trial-and-error.

The next two sections describe the line-of-business model and the aggregate model. Following that is a discussion of how the theory was applied and a brief description of the results.

4Initially, we settled on a 5-year probability of ruin of 1 in 1,000 to calculate required surplus, but once the companion AIDS study was done, we also calculated required surplus using probabilities of ruin of .5, .1, and .01.

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II. Line-of-Business SUrplus Model

As stated in the introduction, CUNA Mutual's C-2 and C-3 Risk Task Force initially decided that it was sufficient to analyze our surplus needs for a period of five years with a probability of ruin no greater than 1 chance in 1,000. Stated in terms of survival, we wish to determine the ~fnimum initial surplus level s such that the probability of survival for five years is a least .999, or, in the language of mathematics,

Pr{ T > 5 I So - s } ~ .999 , (2.1)

where T represents the time of ruin. Now, let us define ;(s,n) to be the probability of ruin within n years, given the initial surplus level s, and let the complement of ;(s,n) be denoted by H(s,n). Then we can write:

;(s,n) - 1 - H(s,n) . (2.2)

H(s,n) - Pre T > 5 I So - s }

z Pre S5 > 0 T > 4 }.Pr{ T > 4 I So as}

Pre S5 > 0 IT> 4 }.Pr{ S4 > 0 IT> 3 } ••• Pre SI > 0 I So as}. (2.3)

Let's next examine the conditional probabilities on the right-hand side of equation 2.3. Observe that the condition of survival to time n-1 only serves to increase the probability of a positive surplus level at time n. That is,

Pre Sn > 0 IT> n-l } ~ Pr{ Sn > 0 }. (2.4)

Using the relationship expressed in equation 2.4 to substitute unconditional distributions for conditional distributions into equation 2.3, we obtain

H(s,n) ~ Pre S5 > 0 }.Pr{ S4 > 0 } ••• Pr{ Sl > 0 } , (2.5)

where it is understood that So a s. Finally, if we have the density function, f(sn)' defined for each Sn' then we can calculate the probabil ities on the right-hand side of equation 2.5 in the usual way:

(2.6)

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Hence if we can find an expression for each of the density functions f(sn)' for n • 1, 2, 3, 4, and 5, then the integral in equation 2.6 can be evaluated using numerical methods to give the unconditional probability of survival for each year. The product of those probabil ities gives us a lower bound on the probability of survival for five years (equation 2.5) and that can be used to calculate an upper bound on the probability of ruin (equation 2.2). Although this only gives an upper bound on the probability of ruin, the approximation is good for small probabilities of ruin and the error is on the conservative side.

In order to develop a density function for each Sn' we need to construct a mathematical model for the surplus level at time n. The Task Force sought to develop a model which allows for the Jnput of the actuary's judgment, aids in the analysis of a line of business, and takes into account the uncertainty of the forecasts.

Definition of the Line-of-Business Model

Constants: So • Initial Surplus Level.

Fj • Federal Income Tax Rate for year j. (AI)

Qj • Ratio of Reserves to Earned Premium for year j. (A2)

Random Variables: Pj. Earned Premium for year j.

Ej = Expenses for year j.

Cj = Incurred Claims and Dividends for year j.

Uj = Underwriting Gain for year j.

Yj • Expense Ratio in year j.

Zj = Loss Ratio for year j.

Ij • Net Investment Income for year j.

Rj • Reserves for year j.

Xj = Net Investment Income Rate during year j.

Tj • Federal Income Taxes for year j.

Gj • Statutory Gain (Contribution to Surplus) in year j.

Sj • Surplus at the end of year j.

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Assumptions: Sj - Sj-l + Gj

Gj - Uj + Ij - Tj

Uj - Pj - Ej - Cj

Ij - (Sj-l + Rj)oXj

Ej - YjOPj

Cj - ZjOPj

Rj - QjOPj

Tj K Fjo(Uj + Ij) .

Pj' Xj' Yj' and Zj are mutually independent.

Uj and Sj_IOXj are mutually independent.

The distribution of Sj is normal.

(A3)

(A4)

(A5)

(A6)

(A7)

(AS)

(A9)

(AIO)

(All)

(AI2)

(AI3)

Some comments about the assumptions are in order. First, assumption AI, that the FIT rate in year j is a given constant, is a reasonable assumption and it greatly simplifies the mathematics. Although there is a degree of uncertainty regarding future tax laws, it is probably better to test various tax rate scenarios by doing several runs of the model rather than assume a non-zero variance for the tax rate.

Second, assumptions A3-AIO may not contain all the terms needed for a particular line of business. In that case, the missing quantities should be lumped into one of the terms in assumptions A3-AIO. For example, experience­rated refunds could be deducted from earned premium to arrive at net earned premium. Pj then represents the net earned premium in year j.

Third, although assumptions A2 and A9, that reserves are a fixed percentage of premium, may be a bit simpl istic, we felt that any major fluctuation in the reserves would be caused by a major fluctuation in premium.

Fourth, if there is no tax on realized capital gains (due to a balance of gains and losses) and no surplus tax, then Fj in assumption AIO represents the tax rate on operating gain. Otherwi se, F j wi 11 have to be an adjusted rate that takes the tax on realized capital gains and the surplus tax into account.

Fifth, the independence assumptions All and Al2 are strong assumptions that significantly reduce the complexity of the model. Although the val idity of these assumptions should be examined for each line of business, for our largest

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lines of business, the group credit insurance lines, the correlations between the random variables should be small.

Sixth, although assumption AlZ deems any correlation between underwriting gain and investment income on surplus to be insignificant, we did not rule out the possibility that gains in successive years could be correlated.

Finally, the normality assumption A13 is reasonable, but not mandatory. As we shall see shortly, the probability of ruin can be calculated using any probability distribution as long as the density function is determined by the mean and variance. Thus one should choose the mathematical form that best fits the conditional distribution of surplus.

computation of the Surplus Means and Variances

The main results needed for the calculation of the probability of ruin over several years are the mean and variance of Sj, given the initial surplus level s. The derivations which follow use only the basic identities which are summarized in Appendix A.

Using a top-down approach, the following equations follow directly from assumption A3 and identities IZ, 13, 16, and IS:

Var(Sj) - Var(Sj_l) + Var(Gj) + ZCov(Sj_l,Gj)

j-l

- Var(Sj_l) + Var(Gj) + Z~COV(Gk'Gj) k-l

j-l

(Z.7)

- Var(Sj_l) + Var(Gj) + Z~RhO(Gk'Gj)Jvar(Gk).var(Gj) (Z.8)

k-l

where Rho(Gk,Gj) represents the correlation between gains in years k and j.

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Continuing, we can use assumptions A4 and AIO to derive expressions for the mean and variance of the statutory gain:

- (1 - Fj)o(Uj + Ij) •

- (1 - Fj)o{E[Uj] + E[lj])

Yar(Gj) • (1 - Fj)2Yar(Uj + Ij)

- (1 - Fj)2{Yar(Uj) + Yar(lj) + 2Cov(Uj,lj)}

(2.9)

(2.10)

(2.11)

Assumpt ions AS, A7, and A8 can be used to write a simpl e express ion for the underwriting gain, and then the independence assumption All and identities 17 and 18 can be used to derive expressions for the mean and variance of Uj:

- Pjo(I - Yj - Zj)

- E[Pj]o(I - E[Yj] - E[Zj])

Yar(Uj) • Yar(Pjo(I - Yj - Zj»

- Yar(Pj)o{Yar(Yj) + Yar(Zj)} +

Yar(Pj)o(1 - E[Yj] - E[Zj])2 +

E[Pj]2(Yar(Yj) + Yar(Zj)} •

(2.12)

(2.13)

(2.14)

Similar techniques allow us to derive expressions for the mean and variance of the net investment income:

- (Sj-I + QjOPj)oXj •

- (E[Sj_I] + QjoE[Pj])oE[Xj]

Yar(lj) - (Yar(Sj_l) + Qj 2Yar(Pj)}oYar(Xj) +

(Yar(Sj_l) + Qj2Yar(Pj)}oE[Xj]2 +

(E[Sj_I] + QjoE[Pj])2Yar(Xj) .

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(2.15)

(2.16)

(2.17)

Now, assumption A12 helps us derive a simple expression for the covariance of the underwriting gain and the net investment income:

Cov(Uj,(Sj_l + QjOPj)oXj)

• Cov(Uj,Sj_loXj) + Cov(Uj,QjOPjoXj)

• 0 + Cov(Pjo(l - Yj - Zj),QjOPjoXj)

• E[Pj 2(1 - Yj - Zj)oQjoXj] -

E[Pj]o(l - E[Yj] - E[Zj])oQjoE[Pj]oE[Xj]

• (E[Pj 2] - E[Pj]2)o(1 - E[Yj] - E[Zj])oQjoE[Xj]

- Var(Pj)o(l - E[Yj] - E[Zj])oQjoE[Xj] .

/ \

(2.18)

Now, using the relationships in equations 2.13 2.14,2.16,2.17, and 2.18 to make substitutions in equations 2.10 and 2.11, we have the final formulas for the mean and variance of the statutory gain:

E[Gj] • (1 - Fj)o(E[Pj]o(l - E[Yj] - E[Zj]) +

(E[Sj_l] + QjoE[Pj])oE[Xj])

Var(Gj) = (1 - Fj)2(Var(Pj)(1 - E[Yj] - E[Zj])2 +

(Var(Pj) + E[Pj]2)o(Var(Yj) + Var(Zj» +

(Var(Sj_l) + Qj 2Var(Pj»o(Var(Xj) + E[Xj]2) +

(E[Sj_l] + QjoE[Pj])2Var(Xj) +

2Var(Pj)o(1 - E[Yj] - E[Zj])oQjoE[Xj]

(2.19)

(2.20)

If we wanted to carry this further, we could substitute the right-hand sides of equations 2.19 and 2.20 into equations 2.7 and 2.8, but the resulting equations would be too cumbersome to work with. Examination of those equations, however, reveals that the mean and variance of Sj depends only on the initial surplus

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level s, the correlations between gains, and the actuary's estimates of the following parameters for each year under study:

Description of Variable Expected Value Variance

Federal Income Tax Rate Fj 0

Ratio of Reserve to Premium Qj 0

Earned Premium E[Pj] Var(Pj)

Net Investment Income Rate E[Xj] Var(Xj)

Expense Ratio E[Yj] Var(Yj)

Loss Ratio E[Zj] Var(Zj)

Estimating the correlations between gains can be simplified somewhat by assuming that all gains one year apart have the same correlation coefficient, gains two years apart have another common correlation coefficient, and so on. This way, only four correlation coefficients have to be estimated for each line of business in the 5-year study.

Now, if we use the assumption that the density of Sj is normally distributed, then we can write the density function as follows:

1 (Sj - E[Sj])2 } Ex -i ----­

Var(Sj) (2.21)

Finally, the density function 2.21 is used in equation 2.6 to calculate the probability of a positive surplus for each year, and then an upper bound on the probability of ruin is calculated using equations 2.5 and 2.2. Through a process of tria1-and-error, different initial surplus levels are tested until the desired bound on the probability of ruin is obtained. In actual practice, the assumptions and formulas are easy to program into a simple spreadsheet.

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Estimating the Parameters

The line-of-business model developed in the preceding sections depends heavily on the parameter estimates supplied by the line actuaries. Such estimates can be difficult to determine since the model requires estimates of Fj and Qj as well as the means and variances of Pj' Xj' Yj' and Zj for each year in the study. Estimates also have to be given for correlations between gains. Nevertheless, any worthwhile surplus analysis Method requires some judgment and this model provides a structure to help the line actuary analyze each line of business.

The estimates of the model parameters are not just simple averages of historical results. Rather, they the line actuaries' best guesses of future performance. Thus some natural sources for the means of the random variables are the projections that are done periodically for our Planning Committee or for our annual budgeting process.

The estimates of the variances of the random variables represent the degree of confidence in the forecasts. A small variance shows a high degree of confidence in the precision of the forecasts, a large variance shows a low degree of confidence. In estimating variances, recall that Chebyshev's inequality states that at least 75% of any distribution lies within two standard deviations of the mean. If one assumes that the parameters are normally distributed, more than 95% of the distribution lies within two standard deviations of the mean. Thus if one can choose a symmetric interval about the estimate of the mean which encompasses a 75% confidence interval (or 95% with the assumption of a normal distribution), then the estimate of the variance can be obtained by dividing the range by four and squaring the result. For example, suppose that the expected loss ratio for year j is 65% and that the actual loss ratio will fall between 55% and 75% with a 95% certainty. Then the estimates would be E[Zj] •. 65 and Var(Zj) • [(.75 - .55) + 4]2 •. 0025.

Initially, three scenarios were studied using the line-of-business surplus model: 1) a scenario based on current pricing assumptions, 2) an influenza epidemic scenario modeled after the 1917-1918 outbreak, and 3) an AIDS scenario that assumes claims continue to increase. Each of the scenarios tested were coordinated through the Task Force so that all actuaries were using the same premises. As new information emerged on the AIDS epidemic, however, we decided to give special attention to the AIDS scenario. The AIDS study is described in more detail in Dave Creswell's companion paper "Required Surplus with Emphasis on the C-2 Risk."

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III. The Aggregate Surplus Analysts Model

The aggregate model is a simpl ification of the line-of-business model. In order to ease the task of combining the assumptions of many disparate lines of business, the earned premium, expenses and incurred claims from the line-of­business model were combined into one quantity., the underwriting gain. Then each year's underwriting gains and reserves for all lines of business were combined to form the aggregate estimates of the underwriting gain and reserves. In arriving at the aggregate estimates, the means of the 1 ine-of-business variables could be added together, but the variances had to be combined using correlation coefficients representing the interdependency of the various lines of business. The two correlation matrices (one for the underwriting gain, one for the reserves) were arrived at by consensus among the responsible product actuaries.

Definition of the Aggregate Surplus Model

The aggregate surplus model is based on most of the same assumptions used in the line-of-business model. Of course, the presence of 13 lines of business complicates the model somewhat, but the mathematics is still very much the same as in the line-of-business model. Now, if we let Uij represent the underwriting gain for line of business i in year j, then we can write:

- E[Pij]·(l - E[Yij] - E[Zij]) •

Var(Uij) - (Var(Yij) + Var(Zij»(Var(Pij) + E[Pij]2)

+ (1 - E[Yij] - E[Zij])2Var(Pij)

U.j

E[U.j] - E[Ul,j] + E[U2,j] + E[U3,j] + ••• + E[U13,j]

Var(U.j) - Var(Ul,j) + ••• + Var(U13,j) + 2~COV(Uij'Ukj) i<k

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(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

As a practical matter, the covariance terms in equation 3.6 can be calculated by estimating the correlation coefficients, Rho(Uij,Ukj), for each combination of two lines of business and then using the relationship

(3.7)

Equations analogous to equations 3.4 through 3.7 can also be written for the aggregate reserves, R.j. The mean and variance of Rij are calculated from the line-of-business model as follows:

E[Rij]

computation of the Surplus Means and Variances

(3.8)

(3.9)

Then, using the same definitions as in the line-of-business model, we can write equations for the mean and variance of Sj:

E[Sj] .. E[Sj-l] + E[Gj] . (3.10)

j-l

Var(Sj) = Var(Sj_l) + Var(Gj) + 2~RhO(Gk'Gj)Jvar(Gk)oVar(Gj), (3.11)

k-l

where now Rho(Gk,Gj) represents the correlation of aggregate gains between years k and j. The simplification of the model described above naturally leads to simpler expressions for the mean and variance of the statutory gain:

Var(Gj) - (1 - Fj)2{Var(U.j) +

(Var(Sj_l) + Var(R.j»o(Var(Xj) + E[Xj]2 +

(E[Sj_l] + E[R.j])2Var(Xj)} .

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(3.12)

(3.13)

The aggregate model, then, depends on the initial surplus level s, estimates of the correlation of gains between years, and the following parameters:

Description of Variable Expected Value Variance

Federal Income Tax Rate Fj 0

Underwriting Gain E[U.j] Var(U.j)

Reserves E[R.j] Var(R.j)

Net Investment Income Rate E[Xj] Var(Xj)

Thus the aggregate model looks simpler than the line-of-business model, but it involves an extra step. The aggregate underwriting gain and aggregate reserves are calculated from the line-of-business models using estimated correlations between lines of business. With 13 lines of business, this requires a 13x13 symmetric matrix of correlations for the underwriting gain, and another 13x13 matrix for the reserves.

The correlations of the aggregate gains between years also need to be estimated. In our case, we used a weighted average of the line-of-business estimates of the correlations in the gains between years. The weight was assigned according to each line's standard deviation of the statutory gain.

The remainder of the development of the aggregate model is completely analogous to the development of the line-of-business model. The combination of the 13 lines of business using equation 3.6 was done in a separate program, and then the aggregate parameter estimates were entered into a spreadsheet program.

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IV. Studies and Results

The line-of-business models provided us with an organized, systematic procedure for pooling the product li ne actuari es' knowl edge. The 1 i ne actuari es were provided with blank spreadsheets, into which they entered their best estimates of the required parameters, for all U.S. business written directly by the Madison-based companies, except for the pension and individual life insurance lines of business5. These estimates were then reviewed and discussed by the rest of our actuarial community until a consensus was reached. Estimates of the correlation coefficients between lines of business were developed in a simil ar way.

Following the development of the line-of-business models, the aggregate model was run nine times, using three correlation assumptions:

1. The lines of business are perfectly correlated. This is the most conservative assumption possible since the aggregate variance is at its maximum.

2. The product actuaries' best guess of the correlation coefficients. As always, the results shoul d not be construed to have greater precision than the input assumptions.

3. The lines of business are perfectly independent. This is the most liberal assumption tested, but not the most liberal assumption possible. A more liberal assumption would have the lines of business negatively correlated.

and three different scenarios:

1. Pricing assumptions. This assumes that business proceeds as usual wi th no unusual catastrophes other than those allowed for in our pricing.

2. Influenza epidemic. This scenario expects a sharp increase in claims during the second year of the study with a sudden return to

5The C-2 and C-3 risks were analyzed separately for the pension and individual life insurance lines since it was felt that the group insurance surplus model would not be appropriate for analyzing the C-3 risk in those lines of business.

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normal mortal ity after the second year. It is IIIOdeled after the 1917-1918 influenza epidemic.

3. AIDS epidemic. This scenario assumes a steady increase in claims. At the time the first study was done (1986), the extent of the AIDS risk was not well understood, so the 5-year scenlrio seemed adequate at the time. later studies considered longer time frames.

Although the actuaries' best-guess estimates of the correlation coefficients are subjective, they almost certainly should lie somewhere between 0 and 1. (Although negative correlations between 1 ines of business are possible and quite advantageous, we haven't been that clever in developing our business portfolio.) Hence we can use the required surplus calculated under the two extreme correlation assumptions as upper and lower bounds with respect to any error introduced by the best-guess estimates of the correlation coefficients.

The results of the first round of studies showed that Simply adding the line­of-business required surplus results together is much too conservative. This is because, when considered in the aggregate, the lines of business have the opportunity to offset gains and losses, which brings down the variance in the aggregate gain. The effect is most pronounced with the assumption of perfect independence. The required surplus calculated under the actuaries' best-guess correlations was closer to that obtained with the perfect independence assumption than to that calculated with the assumption of perfect correlation.

The model calculated required surplus levels which appear quite reasonable. As an added check, we applied the classic ruin theory equations 1.5 and 1.6 to the gain in the fifth year of the study and calculated probabilities of eventual ruin that were, in most cases, much less than 1%. The worst case, the AIDS scenario with perfect correlations, had a long-term chance of ruin of 2.3%.

In 1987, we modified the model to its present form and added a more extensive study of the AIDS risk (please see the companion paper by Dave Creswell). The long-term nature of the AIDS risk suggests that a 5-year study is no longer adequate and probabilities of ruin less than .001 are virtually impossible to attain with our current business plan. As a result, we have relaxed our probabil ity of survival requirement to .99 and we are extending the model to study 20 years. The extension of the model to 20 years is more a practical problem than a theoretical one since the equations will stay the same.

MAC 04/27/88

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Appendix A

Sole Basic Identities

For any random variable X, we havp. the following relationship:

Var(X) (II)

For any two random variables X and Y with correlation coefficient PXY, we have

E[X+Y] - E[X] + E[Y] .

Var(X+Y) - Var(X) + Var(Y) + 2Cov(X,Y)

Cov(X,Y) - E[XY] - E[X].E[Y]

Cov(X,Y) - PxyJvar(x).var(Y)

For any three random variables X, Y, and Z, we can write

cov[(x+y),Z] K E[(X+Y).Z] - E[X+Y].E[Z]

- E[XZ+YZ] - E[X].E[Z] - E[Y].E[Z]

K E[XZ] - E[X].E[Z] + E[YZ] - E[Y].E[Z]

- Cov(X,Z) + Cov(Y,Z) .

For any two independent random variables X and Y, it is well known that

E[XY] - E[X].E[Y] .

Var(XY) - E[X2y2] - E[Xy]2

- E[X2].E[y2] - E[X]2.E[y]2

- (Var(X) + E[X]2).(Var(Y) + E[y]2) - E[X]2.E[y]2

- Var(X).Var(Y) + Var(X).E[y]2 + E[X]2.Var(Y) .

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(12)

(13)

(14)

(15)

(16)

(17)

(18)