# Estimating loss reserves using an actuarial report.

Gregory N. Alff is senior vice president and senior actuary of the
Research and development Division of Corroon & Black Corp. in
Nashville, TN, a subsidiary of Willis Corroon P.L.C.

In today's casualty insurance market, more risk managers are retaining their first-dollar primary layers of risk through a self-insured retention or a high deductible or captive reinsurance arrangement. However this is accomplished, the retention translates into a variable cost and a financial liability in the form of loss reserves.

To establish an appropriate liability for accounting and financial decision-making purposes, it is important to estimate the ultimate cost of claims for the retained limits. Usually, an accrued liability is established by estimating the ultimate cost of claims for damages, then subtracting prior payments to calculate loss reserves. Depending on accounting practices, additional calculations may be needed to discount or calculate the present value of loss reserves. An actuarial reserve analysis calculates loss reserve liabilities and performs discounting calculations if needed.

Most risk managers agree that estimating loss reserves requires special skills, which is why they often call on the expertise of casualty actuaries. But some management may have difficulty evaluating a reserve analysis. That's why it is necessary to understand the key elements involved in actuarial reserve analysis.

Methods of Analysis

Estimating ultimate losses used in the reserve analysis may involve several methods: incurred development, paid development, frequency and severity, projection and completion. Because each method has strengths and weaknesses, it is important to use more than one to confirm estimates or identify the range of probable estimates.

Casualty actuaries often use the incurred development method because it is easily explainable and uses the loss data of a company to the fullest extent possible. Known incurred losses-payments plus outstanding reserves are developed to estimate ultimate losses using incurred loss development factors. This method can be used in most situations, except with exposure having low frequency and high severity claim potential. It is most effective when historical data from a particular exposure is used to calculate loss development factors, since the method assumes that patterns of loss development will be repeated.

The accuracy of incurred development estimates suffers when significant changes have occurred in the company's claim handling procedures or loss reserving philosophies. If current case reserves set by claims examiners are significantly higher than in past years, applying historical loss development factors could result in redundant estimates of ultimate losses. If case reserves are lower than in the past, applying historical development factors could lead to deficient estimates of ultimate losses.

The paid development method also uses historical loss development factors but, unlike incurred development, includes only paid losses. The advantage of this method is that it does not depend on case reserves set by claims examiners. Since only payments are used, altering reserving procedures has no effect on estimates. The drawback to this method is that for early ages of development, the factors required are large. Because of this high leverage effect, the presence or absence of one or two large loss payments may significantly change estimates of ultimate losses generated.

The frequency and severity method may be useful if experience analysis is needed for an acquisition or divestiture. Probable adjustments to estimated ultimate losses should be quantified by looking separately at frequency and severity elements and then recombining the two to arrive at estimated ultimate losses. Analyzing frequency and severity separately makes the analysis more detailed and may uncover important information hidden within aggregate incurred or paid amounts. But unless the volume is great, this method may have limited value because the data could lack stable patterns.

The protection method may be best for estimating losses for the current policy period. This method relies on estimates of ultimate losses for prior historical policy periods, then adjusts those estimates to current year cost levels by considering changes in exposures and average loss costs. The estimated ultimate losses for recent prior years are adjusted to current year cost levels by using trend factors.

Trend factors can be generated from the Consumer Price Index, but should probably come from a more specific index. For example, the medical care component of the Consumer Price Index may be a better choice if most of the losses are attributable to medical expenses. In this method an exposure adjustment should adjust inflation-sensitive exposure bases, such as payroll, to the current level. Indexes compiled by the Bureau of Labor Statistics can provide an appropriate measure of such trends for payroll.

The projection method has the advantage of not being influenced by changes in claim handling and reserve philosophies or the presence or absence of large claim payments in immature current experience. However, this may also be a disadvantage, because important information contained in current loss data could go unused.

The completion method is a hybrid of two or more of the other methods. In the first step expected ultimate losses are estimated based on permissible loss ratios or a loss projection method. A completion factor is then calculated equal to (1-1/LDF), where LDF is the account's incurred loss development factor based on data maturity. Multiplying this completion factor by the expected losses yields the amount of expected loss development.

The completion amount, when added to known losses, yields a new estimate of expected ultimate losses, which is quite viable because it uses known losses. A completion-type method such as the Bornhuetter-Ferguson method can be helpful when dealing with loss frequencies and severities that are unstable from year to year. The completion part of the estimate has the disadvantage of being based on expected losses without regard to known losses.

Applying Professional Judgment

Experienced judgment regarding the strengths and weaknesses of the various methods is often the most important element of actuarial analysis. It is especially critical in the incurred development method when selecting individual age-to-age development factors; selections based on simple mechanical averages may be inappropriate. The example on page 82 explains the basics of loss development methodology and illustrates the importance of informed judgment allied with basic mathematical analysis. It represents the incurred loss development history for XYZ Corp. and the calculated development factors based on that history.

The ultimate cost of claims incurred for a specific time period is usually not known until several years after the close of that period. Loss development factors quantify the late developing aspects of certain losses, such as claims involving medical complications unrecognized in early treatment stages or verdicts for litigated claims that differ from the amount reserved to pay the claims. They also account for losses incurred during the policy period but not reported until a later date, commonly referred to as incurred but not reported losses (IBNRs).

XYZ Corp.'s losses incurred during the 1986 accident year are shown in yearly increments from 12 to 48 months after the beginning of that year. Generally, the value of reported losses increases from one evaluation to the next as a result of developing reported claims and substantiated IBNR losses. Age-to-age factors are the rates by which losses developed from year to year. For example, the 12 to 24 months age-to-age factor for the 1986 accident year is 2.148. This number represents the 24-month incurred amount ($562,996) divided by the 12-month incurred amount ($262,135). The 2.148 age-to-age factor means that the value of losses incurred during the 1986 accident year increased by 114.8 percent during the 12-to-24-months interval. An age-to-age development factor less than 1.0 indicates that the value of reported losses declined, possibly due to claims being settled for less than what was reserved.

The chart also shows three averages of age-to-age factors from one evaluation to the next, with the "average" being the mean of all age-to-age factors in the column; "midaverage," the point reached after excluding the highest and lowest age-to-age factor in each column; and "three-year average," the average of the three most current factors. In the line labeled selected, the average, midaverage and three-year average factors are evaluated, and the factors that best represent XYZ Corp.'s expected loss development are shown. The loss development factors are based on the selected age-to-age factors. The 48-month to ultimate loss development factor, for instance, is found by multiplying the 48-to-60-months age-to-age selected factor by the 60 month to ultimate loss development factor. The loss development factors shown on the chart are used to develop the current amounts of reported losses to their expected ultimate value.

For XYZ Corp., the goal was to select age-to-age factors and calculate loss development factors appropriate to develop losses for 1986, 1987, 1988 and 1989 to their expected ultimate levels. For example, the 1.21 loss development factor for 48 months to ultimate multiplied by 1986 losses of $1,267,484 would yield estimated ultimate losses of $1,534,000. Such estimates of ultimate losses would be input to reserve calculations for those years and provide a basis for a projection method to estimate losses for the 1990 accident year.

Important judgments are made in the age-to-age factors selection for each column. For example, the last factor shown in each column of historical factors is based on the loss development that took place during calendar year 1989. The 2.052 factor for 1988 indicates the increase in incurred losses between the Dec. 31, 1988, and Dec. 31, 1989, evaluation dates. Likewise, each number up the last diagonal, 2.052, 1.721, 1.393, 1.132 and so on, indicates the change in insured losses that took place during 1989. The first prior diagonal, 2.188, 1.616, 1.645 and so on, indicates changes in incurred losses during 1988.

When selecting factors, it is common to emphasize the most recent historical factors in each column. However, the three-year average indicated factors are significantly higher than those indicated by other averages for the yearly periods between 36 and 84 months. The risk manager and claims administrator indicated that significant reserve strengthening had occurred during 1988 and 1989 and cautioned against applying factors based on the last two diagonals.

If the level of reserves had been significantly increased, applying factors generated by those increases would compound them and generate excessive estimates of ultimate losses. This seemed a real possibility. Note that the factors cited, including those in the 24 to 36 months column, appear higher than factors for previous years in the same columns. It was suggested that selected development factors might have to be lower than even longer term historical averages to adjust for reserve strengthening.

High and Low Factors

Although part of this reasoning was solid, further analysis leads to different conclusions. The chart indicates those factors that are too low or too high. It is not unusual to see isolated reversals for a single accident year. For example, in 1978 the 48 to 60 months factor is high and the 60 to 72 months factor is low Conversely, for 1980 the 36 to 48 months factor is low and the 48 to 60 months factor is high. In each case a low factor is immediately offset by a high factor, caused by slight differences in development timing. However, for 1981 to 1985 accident years, something very different and more systematic occurred.

By adding lines to the chart that enclose the groups of low and high factors, a different scenario is defined than the theory of much stronger overall reserves originally presented. High factors in the last two diagonals for 1981 to 1985 appear to be a necessary follow-up to the low factors for the same years found in the previous two diagonals. During 1986 and 1987, which have diagonals marked low, incurred losses did not rise as fast as they had historically or would again in subsequent years. Reserve levels must have been held down by a change in reserve philosophy, claims management or some other action.

The action in 1986 and 1987 was apparently temporary. Note that the age-to-age development factors for 1986 and 1987 (horizontal) are similar to those in 1982. After comparing the factors for 1986, 1987 and 1988 to those for 1979 to 1982, it becomes clear that factors for 1986 and subsequent occurrence years are not high but instead represent a return to normal patterns.

Two diagonals of historical factors have been distorted low, followed by many factors in the latest two diagonals that were distorted high to compensate. As a result, the goal becomes selecting factors that represent more normal development patterns for XYZ Corp. Note that the factors selected are significantly lower than the three-year average for the four periods between 36 and 84 months. However, they are significantly higher than any average for the 24 to 36 months column. In no case is it appropriate to select factors lower than long-term historical averages; doing so may result in underestimating ultimate losses. The indicated loss development factors multiplied by known losses yield XYZ Corp.'s estimates of ultimate incurred losses as shown on page 84.

The large factor applied to 1989 causes any large claim amounts to be highly leveraged in estimating ultimate losses, indicating that the potential range around this estimate is wide and that the results of other methods should be considered. Most loss development patterns are not this difficult to analyze. However, this example of an actual development history shows where thorough analysis by an experienced professional may avoid erroneous conclusions.

Other areas of actuarial analysis where experienced judgment plays an important role includes choosing methods to calculate estimated ultimate losses; limiting the effect of large claims when applying loss development and trend factors; and segregating large insureds' data to account for differences in loss development patterns and/or differing frequency and severity characteristics. Other analytical techniques include adjusting for the effect of any acquisitions or divestitures that have different loss severity or frequency characteristics than the total historical operations, and selecting trend factors based on the entity's own data or from reliable government or industry sources.

Confidence Level

After using two or more methods to independently estimate ultimate expected losses, results will be weighted by the actuary. Those weights will depend on the credence assigned to each estimate after evaluating its strengths and weaknesses. Given the variation in results produced by the individual methods, it is clear that there is a wide range of potential estimates. Indeed, the definition of range may be as important as the estimate itself. The selected estimate is approximately equal to the expected or mean estimate at the 50 percent confidence level, or center of the range. That means there is a 50 percent chance that the ultimate aggregate losses will be less than the selected number and a 50 percent chance they will be more.

A range of results in terms of confidence levels can become an important tool. For mature experience periods, considering the potential for additional development of the few individual claims remaining open will help define a probable range. The experienced actuary will evaluate the IBNR potential in estimating the range around the selected estimated ultimate losses. The actuary may also indicate that he or she is highly confident that the losses will not exceed 1.30 times the expected losses.

This is a reasonable approach. However, defining a range on any individual year is problematical when a single jury verdict could radically change the value of any significant claims remaining open. Therefore, the estimated range for a given year should not be considered individually, but rather as part of an overall estimate of the range around the incurred losses and/or reserves for all experience periods combined.

A mathematical modeling approach can be used to estimate amounts for confidence levels around the estimated ultimate amounts. Because modeling the combined frequency and severity effects is difficult, it is common to examine separately the frequency distribution of claim counts and the severity distribution of amounts of individual losses. These distributions may be estimated based on historical empirical distributions from a corporation's own data. The empirical distributions might then be input to a computer model designed to fit a statistically defined distribution to the empirical frequency distribution and then to the empirical severity distribution. The definition of approximate statistical distributions subsequently allows computer modeling or simulation of the distribution of losses that may ultimately occur.

By random sampling from the fitted statistical distributions, a series of potential frequency and severity outcomes can be simulated or modeled, an approach often referred to as Monte Carlo simulation. If the individual outcomes are then ranked by magnitude, a range of aggregate loss estimates can be defined. By splitting the range into percentiles, confidence levels can be established. One significant problem is the definition of empirical claim distributions with claims developed to ultimate levels. Accurate confidence levels will be established if statistical distributions fit well to the empirical distributions and an appropriate number of simulated random trials are run. The output from such a simulation for XYZ Corp. is shown below.

XYZ Corp.'s aggregate loss probability distribution was created by rank ordering aggregate losses over many simulated years. Its 1988 expected losses of $2,133,000 have a 51.6 percent probability of containing the actual losses. This probability means that approximately 52 times out of 100, losses will fall below $2,133,000. Conversely, 48 percent of the time losses will exceed this figure. As the level of aggregate losses is increased, the probability that losses will be less than the aggregate amount also increases, thus the probability that the losses will exceed that level declines. By moving down the scale, aggregate limits can be selected that have a higher probability of not being exceeded.

With the distribution of expected ultimate losses defined by confidence level, the risk manager and corporate officers can select the level of estimated ultimate losses appropriate to their independent objectives and philosophy. Although this decision-making process has taken place in previous years, it may not have been done with such structured input.

Reserves and Discounting

After selecting the confidence level of expected ultimate losses, loss reserve liabilities proceeds can be calculated by subtracting the amount of prior loss payments from the selected estimated ultimate losses to yield the estimated reserves necessary to pay remaining claims. The reserve number from this equation is a dollar amount that does not recognize the time value of money. Since all unpaid claims will not be paid immediately, loss reserves might be invested to generate income until the claims are paid. The investment income from such a fund might be used initially to provide a buffer fund which, if added to loss reserves, would raise the confidence level that reserves would not eventually be exceeded by payments.

A more aggressive approach is to discount loss reserves to a level where the initial reserve fund plus expected investment income on those funds are sufficient to pay claims. The initial reserve established in this funding approach is known as the discounted or present value loss reserve. The term "discounted" refers to reserves discounted by an interest rate from the time of expected payment to the present. The term "present value" means the amount that at the evaluation date would be sufficient if invested at interest to pay all loss amounts as they become due. To make these discounting or present value calculations, the probable timing of future loss payments and the interest rate are the data required in addition to the estimated loss reserves.

A conservative approach should be used to determine the discount rate. There is variability in the estimate of ultimate losses, in the potential timing of payment and in potential reinvestment rates. Because of the risk created by this variability, a discount rate should be selected at, or slightly less then, current short-term interest rates.

The decision to include only discounted or present value reserves in a corporate financial statement should be carefully weighed and discussed with external auditors before being implemented. Without external auditors' prior approval, a conditional or qualified statement might be given by the auditors. If a corporation retains any primary risk, potential balance sheet liability should be seriously considered. Large retained risks and long time periods between accident and claim payment dates make professional actuarial input important.

In today's casualty insurance market, more risk managers are retaining their first-dollar primary layers of risk through a self-insured retention or a high deductible or captive reinsurance arrangement. However this is accomplished, the retention translates into a variable cost and a financial liability in the form of loss reserves.

To establish an appropriate liability for accounting and financial decision-making purposes, it is important to estimate the ultimate cost of claims for the retained limits. Usually, an accrued liability is established by estimating the ultimate cost of claims for damages, then subtracting prior payments to calculate loss reserves. Depending on accounting practices, additional calculations may be needed to discount or calculate the present value of loss reserves. An actuarial reserve analysis calculates loss reserve liabilities and performs discounting calculations if needed.

Most risk managers agree that estimating loss reserves requires special skills, which is why they often call on the expertise of casualty actuaries. But some management may have difficulty evaluating a reserve analysis. That's why it is necessary to understand the key elements involved in actuarial reserve analysis.

Methods of Analysis

Estimating ultimate losses used in the reserve analysis may involve several methods: incurred development, paid development, frequency and severity, projection and completion. Because each method has strengths and weaknesses, it is important to use more than one to confirm estimates or identify the range of probable estimates.

Casualty actuaries often use the incurred development method because it is easily explainable and uses the loss data of a company to the fullest extent possible. Known incurred losses-payments plus outstanding reserves are developed to estimate ultimate losses using incurred loss development factors. This method can be used in most situations, except with exposure having low frequency and high severity claim potential. It is most effective when historical data from a particular exposure is used to calculate loss development factors, since the method assumes that patterns of loss development will be repeated.

The accuracy of incurred development estimates suffers when significant changes have occurred in the company's claim handling procedures or loss reserving philosophies. If current case reserves set by claims examiners are significantly higher than in past years, applying historical loss development factors could result in redundant estimates of ultimate losses. If case reserves are lower than in the past, applying historical development factors could lead to deficient estimates of ultimate losses.

The paid development method also uses historical loss development factors but, unlike incurred development, includes only paid losses. The advantage of this method is that it does not depend on case reserves set by claims examiners. Since only payments are used, altering reserving procedures has no effect on estimates. The drawback to this method is that for early ages of development, the factors required are large. Because of this high leverage effect, the presence or absence of one or two large loss payments may significantly change estimates of ultimate losses generated.

The frequency and severity method may be useful if experience analysis is needed for an acquisition or divestiture. Probable adjustments to estimated ultimate losses should be quantified by looking separately at frequency and severity elements and then recombining the two to arrive at estimated ultimate losses. Analyzing frequency and severity separately makes the analysis more detailed and may uncover important information hidden within aggregate incurred or paid amounts. But unless the volume is great, this method may have limited value because the data could lack stable patterns.

The protection method may be best for estimating losses for the current policy period. This method relies on estimates of ultimate losses for prior historical policy periods, then adjusts those estimates to current year cost levels by considering changes in exposures and average loss costs. The estimated ultimate losses for recent prior years are adjusted to current year cost levels by using trend factors.

Trend factors can be generated from the Consumer Price Index, but should probably come from a more specific index. For example, the medical care component of the Consumer Price Index may be a better choice if most of the losses are attributable to medical expenses. In this method an exposure adjustment should adjust inflation-sensitive exposure bases, such as payroll, to the current level. Indexes compiled by the Bureau of Labor Statistics can provide an appropriate measure of such trends for payroll.

The projection method has the advantage of not being influenced by changes in claim handling and reserve philosophies or the presence or absence of large claim payments in immature current experience. However, this may also be a disadvantage, because important information contained in current loss data could go unused.

The completion method is a hybrid of two or more of the other methods. In the first step expected ultimate losses are estimated based on permissible loss ratios or a loss projection method. A completion factor is then calculated equal to (1-1/LDF), where LDF is the account's incurred loss development factor based on data maturity. Multiplying this completion factor by the expected losses yields the amount of expected loss development.

The completion amount, when added to known losses, yields a new estimate of expected ultimate losses, which is quite viable because it uses known losses. A completion-type method such as the Bornhuetter-Ferguson method can be helpful when dealing with loss frequencies and severities that are unstable from year to year. The completion part of the estimate has the disadvantage of being based on expected losses without regard to known losses.

Applying Professional Judgment

Experienced judgment regarding the strengths and weaknesses of the various methods is often the most important element of actuarial analysis. It is especially critical in the incurred development method when selecting individual age-to-age development factors; selections based on simple mechanical averages may be inappropriate. The example on page 82 explains the basics of loss development methodology and illustrates the importance of informed judgment allied with basic mathematical analysis. It represents the incurred loss development history for XYZ Corp. and the calculated development factors based on that history.

The ultimate cost of claims incurred for a specific time period is usually not known until several years after the close of that period. Loss development factors quantify the late developing aspects of certain losses, such as claims involving medical complications unrecognized in early treatment stages or verdicts for litigated claims that differ from the amount reserved to pay the claims. They also account for losses incurred during the policy period but not reported until a later date, commonly referred to as incurred but not reported losses (IBNRs).

XYZ Corp.'s losses incurred during the 1986 accident year are shown in yearly increments from 12 to 48 months after the beginning of that year. Generally, the value of reported losses increases from one evaluation to the next as a result of developing reported claims and substantiated IBNR losses. Age-to-age factors are the rates by which losses developed from year to year. For example, the 12 to 24 months age-to-age factor for the 1986 accident year is 2.148. This number represents the 24-month incurred amount ($562,996) divided by the 12-month incurred amount ($262,135). The 2.148 age-to-age factor means that the value of losses incurred during the 1986 accident year increased by 114.8 percent during the 12-to-24-months interval. An age-to-age development factor less than 1.0 indicates that the value of reported losses declined, possibly due to claims being settled for less than what was reserved.

The chart also shows three averages of age-to-age factors from one evaluation to the next, with the "average" being the mean of all age-to-age factors in the column; "midaverage," the point reached after excluding the highest and lowest age-to-age factor in each column; and "three-year average," the average of the three most current factors. In the line labeled selected, the average, midaverage and three-year average factors are evaluated, and the factors that best represent XYZ Corp.'s expected loss development are shown. The loss development factors are based on the selected age-to-age factors. The 48-month to ultimate loss development factor, for instance, is found by multiplying the 48-to-60-months age-to-age selected factor by the 60 month to ultimate loss development factor. The loss development factors shown on the chart are used to develop the current amounts of reported losses to their expected ultimate value.

For XYZ Corp., the goal was to select age-to-age factors and calculate loss development factors appropriate to develop losses for 1986, 1987, 1988 and 1989 to their expected ultimate levels. For example, the 1.21 loss development factor for 48 months to ultimate multiplied by 1986 losses of $1,267,484 would yield estimated ultimate losses of $1,534,000. Such estimates of ultimate losses would be input to reserve calculations for those years and provide a basis for a projection method to estimate losses for the 1990 accident year.

Important judgments are made in the age-to-age factors selection for each column. For example, the last factor shown in each column of historical factors is based on the loss development that took place during calendar year 1989. The 2.052 factor for 1988 indicates the increase in incurred losses between the Dec. 31, 1988, and Dec. 31, 1989, evaluation dates. Likewise, each number up the last diagonal, 2.052, 1.721, 1.393, 1.132 and so on, indicates the change in insured losses that took place during 1989. The first prior diagonal, 2.188, 1.616, 1.645 and so on, indicates changes in incurred losses during 1988.

When selecting factors, it is common to emphasize the most recent historical factors in each column. However, the three-year average indicated factors are significantly higher than those indicated by other averages for the yearly periods between 36 and 84 months. The risk manager and claims administrator indicated that significant reserve strengthening had occurred during 1988 and 1989 and cautioned against applying factors based on the last two diagonals.

If the level of reserves had been significantly increased, applying factors generated by those increases would compound them and generate excessive estimates of ultimate losses. This seemed a real possibility. Note that the factors cited, including those in the 24 to 36 months column, appear higher than factors for previous years in the same columns. It was suggested that selected development factors might have to be lower than even longer term historical averages to adjust for reserve strengthening.

High and Low Factors

Although part of this reasoning was solid, further analysis leads to different conclusions. The chart indicates those factors that are too low or too high. It is not unusual to see isolated reversals for a single accident year. For example, in 1978 the 48 to 60 months factor is high and the 60 to 72 months factor is low Conversely, for 1980 the 36 to 48 months factor is low and the 48 to 60 months factor is high. In each case a low factor is immediately offset by a high factor, caused by slight differences in development timing. However, for 1981 to 1985 accident years, something very different and more systematic occurred.

By adding lines to the chart that enclose the groups of low and high factors, a different scenario is defined than the theory of much stronger overall reserves originally presented. High factors in the last two diagonals for 1981 to 1985 appear to be a necessary follow-up to the low factors for the same years found in the previous two diagonals. During 1986 and 1987, which have diagonals marked low, incurred losses did not rise as fast as they had historically or would again in subsequent years. Reserve levels must have been held down by a change in reserve philosophy, claims management or some other action.

The action in 1986 and 1987 was apparently temporary. Note that the age-to-age development factors for 1986 and 1987 (horizontal) are similar to those in 1982. After comparing the factors for 1986, 1987 and 1988 to those for 1979 to 1982, it becomes clear that factors for 1986 and subsequent occurrence years are not high but instead represent a return to normal patterns.

Two diagonals of historical factors have been distorted low, followed by many factors in the latest two diagonals that were distorted high to compensate. As a result, the goal becomes selecting factors that represent more normal development patterns for XYZ Corp. Note that the factors selected are significantly lower than the three-year average for the four periods between 36 and 84 months. However, they are significantly higher than any average for the 24 to 36 months column. In no case is it appropriate to select factors lower than long-term historical averages; doing so may result in underestimating ultimate losses. The indicated loss development factors multiplied by known losses yield XYZ Corp.'s estimates of ultimate incurred losses as shown on page 84.

The large factor applied to 1989 causes any large claim amounts to be highly leveraged in estimating ultimate losses, indicating that the potential range around this estimate is wide and that the results of other methods should be considered. Most loss development patterns are not this difficult to analyze. However, this example of an actual development history shows where thorough analysis by an experienced professional may avoid erroneous conclusions.

Other areas of actuarial analysis where experienced judgment plays an important role includes choosing methods to calculate estimated ultimate losses; limiting the effect of large claims when applying loss development and trend factors; and segregating large insureds' data to account for differences in loss development patterns and/or differing frequency and severity characteristics. Other analytical techniques include adjusting for the effect of any acquisitions or divestitures that have different loss severity or frequency characteristics than the total historical operations, and selecting trend factors based on the entity's own data or from reliable government or industry sources.

Confidence Level

After using two or more methods to independently estimate ultimate expected losses, results will be weighted by the actuary. Those weights will depend on the credence assigned to each estimate after evaluating its strengths and weaknesses. Given the variation in results produced by the individual methods, it is clear that there is a wide range of potential estimates. Indeed, the definition of range may be as important as the estimate itself. The selected estimate is approximately equal to the expected or mean estimate at the 50 percent confidence level, or center of the range. That means there is a 50 percent chance that the ultimate aggregate losses will be less than the selected number and a 50 percent chance they will be more.

A range of results in terms of confidence levels can become an important tool. For mature experience periods, considering the potential for additional development of the few individual claims remaining open will help define a probable range. The experienced actuary will evaluate the IBNR potential in estimating the range around the selected estimated ultimate losses. The actuary may also indicate that he or she is highly confident that the losses will not exceed 1.30 times the expected losses.

This is a reasonable approach. However, defining a range on any individual year is problematical when a single jury verdict could radically change the value of any significant claims remaining open. Therefore, the estimated range for a given year should not be considered individually, but rather as part of an overall estimate of the range around the incurred losses and/or reserves for all experience periods combined.

A mathematical modeling approach can be used to estimate amounts for confidence levels around the estimated ultimate amounts. Because modeling the combined frequency and severity effects is difficult, it is common to examine separately the frequency distribution of claim counts and the severity distribution of amounts of individual losses. These distributions may be estimated based on historical empirical distributions from a corporation's own data. The empirical distributions might then be input to a computer model designed to fit a statistically defined distribution to the empirical frequency distribution and then to the empirical severity distribution. The definition of approximate statistical distributions subsequently allows computer modeling or simulation of the distribution of losses that may ultimately occur.

By random sampling from the fitted statistical distributions, a series of potential frequency and severity outcomes can be simulated or modeled, an approach often referred to as Monte Carlo simulation. If the individual outcomes are then ranked by magnitude, a range of aggregate loss estimates can be defined. By splitting the range into percentiles, confidence levels can be established. One significant problem is the definition of empirical claim distributions with claims developed to ultimate levels. Accurate confidence levels will be established if statistical distributions fit well to the empirical distributions and an appropriate number of simulated random trials are run. The output from such a simulation for XYZ Corp. is shown below.

XYZ Corp.'s aggregate loss probability distribution was created by rank ordering aggregate losses over many simulated years. Its 1988 expected losses of $2,133,000 have a 51.6 percent probability of containing the actual losses. This probability means that approximately 52 times out of 100, losses will fall below $2,133,000. Conversely, 48 percent of the time losses will exceed this figure. As the level of aggregate losses is increased, the probability that losses will be less than the aggregate amount also increases, thus the probability that the losses will exceed that level declines. By moving down the scale, aggregate limits can be selected that have a higher probability of not being exceeded.

With the distribution of expected ultimate losses defined by confidence level, the risk manager and corporate officers can select the level of estimated ultimate losses appropriate to their independent objectives and philosophy. Although this decision-making process has taken place in previous years, it may not have been done with such structured input.

Reserves and Discounting

After selecting the confidence level of expected ultimate losses, loss reserve liabilities proceeds can be calculated by subtracting the amount of prior loss payments from the selected estimated ultimate losses to yield the estimated reserves necessary to pay remaining claims. The reserve number from this equation is a dollar amount that does not recognize the time value of money. Since all unpaid claims will not be paid immediately, loss reserves might be invested to generate income until the claims are paid. The investment income from such a fund might be used initially to provide a buffer fund which, if added to loss reserves, would raise the confidence level that reserves would not eventually be exceeded by payments.

A more aggressive approach is to discount loss reserves to a level where the initial reserve fund plus expected investment income on those funds are sufficient to pay claims. The initial reserve established in this funding approach is known as the discounted or present value loss reserve. The term "discounted" refers to reserves discounted by an interest rate from the time of expected payment to the present. The term "present value" means the amount that at the evaluation date would be sufficient if invested at interest to pay all loss amounts as they become due. To make these discounting or present value calculations, the probable timing of future loss payments and the interest rate are the data required in addition to the estimated loss reserves.

A conservative approach should be used to determine the discount rate. There is variability in the estimate of ultimate losses, in the potential timing of payment and in potential reinvestment rates. Because of the risk created by this variability, a discount rate should be selected at, or slightly less then, current short-term interest rates.

The decision to include only discounted or present value reserves in a corporate financial statement should be carefully weighed and discussed with external auditors before being implemented. Without external auditors' prior approval, a conditional or qualified statement might be given by the auditors. If a corporation retains any primary risk, potential balance sheet liability should be seriously considered. Large retained risks and long time periods between accident and claim payment dates make professional actuarial input important.

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Author: | Alff, Gregory N. |
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Publication: | Risk Management |

Date: | Apr 1, 1991 |

Words: | 3366 |

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