# Estimating the present value of future income losses: an historical simulation 1900-1982.

Estimating the Present Value of Future Income Losses: An Historical
Simulation 1900-1982: Comment

A number of methods for estimating the present value of lost future earnings to injured parties in tort litigation have been presented in this Journal during the past decade. Professor Schilling (1985) uses historical simulation for all possible five, 12 and 30 year loss periods from 1900 through 1982 to compare the performance of three valuation methods (1) to each other and to a benchmark estimate. He concluded that "the 'Alaska' method is found to be the best relative to the benchmark estimate although none of the [three] methods performs very well in an absolute sense relative to the benchmark estimate" [p. 100].

We do not question Schilling's conclusion regarding the superiority of the Alaska method. There is good economic logic to explain this finding. However, we question whether his benchmark estimate provides an appropriate standard for evaluating the performance of alternative valuation methods.

Estimating the Present Value of Future Earnings

The present value of a worker's future earnings can be represented as

[Mathematical Expression Omitted]

where [PV.sub.0] = present value of future earnings at t=0; WL = expected working life in years at t=0; [E.sub.0] = annual earnings rate at t=0; [g.sub.[alpha]] = rate of growth in earnings in period [alpha] ([alpha] = 1, 2, ..., WL); [d.sub.[alpha]] = discount rate in period [alpha] ([alpha] = 1, 2, ..., WL); and [pi] = multiplication operator.

Using continuous compounding, an average growth rate (g) and discount rate (d), equation (1) converts into

[PV.sub.0] = [E.sub.0][1 - [e.sub.-(d-g)(WL)]]/[[e.sub.(d-g)-1]]. (2)

From equation (2) it is apparent that the size of the (d-g) differential, not the expected levels of d and g, determines the present value of a future stream of earnings.

A strong covariance between nominal rates of growth in the annual earnings of workers (g) and annual interest rates (d) is to be expected because of their common determinant, inflation, and because the hire price for any factor of production embodies the real interest rate. Historical earnings growth and interest rates data provide support for the expected covariance between [d.sub.[alpha]] and [g.sub.[alpha]]; (2) moreover, over extended periods of time these same data show that the (d-g) differential was not markedly different from zero. Thus, it is not surprising that Schilling found that the Alaska Method outperformed the two valuation methods employing averages of historical wage growth and discount rates. Nor it is surprising that he supports the recent Model Periodic Payments of Judgment Act (National Conference of Commissioners on Uniform State Laws, 1982). This Act essentially adopts the Alaska Method for its recommendation that tort awards be paid in installments over the loss period with the annual growth rate ([g.sub.[alpha]]) of installments set equal to the yield on one-year U.S. Treasury securities (Henderson, 1980 and 1981, National Conference of Commissioners on Uniform State Laws, 1982).

What is surprising as that Schilling concludes on the basis of his simulation study that the Alaska Method performs tolerably well for short (five-year) loss periods, but that it does not perform very well for longer (12 and 30 year) loss periods (National Conference of Commissioners on Uniform State Laws, 1982, p. 114). (3) It is our view that the support for this conclusion relates to Schilling's benchmark estimation procedure.

Schilling's Benchmark Valuation Estimate

Schilling develops his benchmark present value estimates for all possible five, 12 and 30 year loss periods from 1900 through 1983 using perfect foresight. First, he estimated interest rates that would actually be available each year during an n-year loss period for zero coupon, high-grade corporate bonds with maturities from one to n (five, 12 or 30) years. He then used dynamic programming to identify the time configuration of investment decisions that would create the minimum cost (maximum interest yield) portfolio, and would provide exactly the year-by-year earnings (also known with certainty). Schilling's maximum interest yield portfolio was determined using an investment strategy that permitted portfolio adjustments to occur in each year of the loss period.

We have several reservations regarding the use of Schilling's benchmark valuation estimates as the standard for assessing the performance of alternative methodologies for valuing future income streams. One reservation relates to the procedures he followed in determining the minimum cost portfolio required to replicate the future earnings stream. The logic of our reservation can best be conveyed by an example. Suppose Schilling had selected common stocks rather than zero coupon bonds as the investment vehicle. Using his perfect foresight investment strategy, the time path of the maximum yield (minimum cost) stock portfolio would have been obtained simply by investing each year in the one common stock that provided the largest one-year holding period return. We doubt Schilling would argue that the stock benchmark valuation estimate provides a useful standard for assessing the performance of other valuation methods. Yet it is clear that the difference between the stock benchmark estimate and his bond benchmark estimate is one of degree, not kind.

A second reservation we have regarding Schilling's conclusions relates to the systematic understatement of the wage growth measure ([g.sub.alpha]). As observed earlier, the logic of the alaska Method reolves around the covariance between [d.sub.alpha] and [g.sub.alpha]. The [g.sub.alpha] measure used by Schilling captures the growth in money earnings only. However, workers' total earnings, money earnings and non-money (employee benefit) earnings, grew more rapidly than money earnings from 1900 through 1982. The average U.S. worker received little or no employer provided benefits in 1900; by 1982, workers received in-kind payments from employers that were equal to over 20 percent of their money earnings (Employee Benefits 1985, 1986, p. 31). Thus, the [g.sub.alpha] values were systematically biased downward.

Finally, Schilling's use of overlapping data in examining all five, 12 and 30 year loss periods from 1900 through 1982 is likely to have biased his results. The use of overlapping data results in the progressive underweighting of experience as the loss period moves toward the beginning and end of the data set, and a corresponding over-weighting of experience as the loss period moves toward the middle of the set. For example, in the 30 year loss period analysis the g and d values associated with the year 1929 through 1953 enter 30 times, but the g and d values for other years (1900 through 1928 and 1954 through 1982) enter the analysis less frequently. (4) To the extent that there are trend dependent variables operating on the data, the use of overlapping data can cause results t be systematically biased.

Concluding Observations

The research design employed by Schilling suffers from his decision to evaluate alternative valuation methodologies in terms of the extent to which they differ from results of an approach based on perfect foresight. Any evaluation of alternative valuation methods should, in our opinion, compare the performance of each valuation technique according to an objective and a set of constraints. We agree with Schilling that a sensible objective would be to minimize the cost of an award. But an important constraint would limit the acceptable profitability that an award might prove to be inadequate in replacing the lost earnings. (5) Historical simulation could be used to evaluate each valuation method according to the specified objective and constraint. Obviously, no valuation method will match the perfect foresight benchmark method. But this is not the issue of interest.

(*1) University of Illinois (Champaign-Urbana).

(1) Three methods were examined. One was the so-called Alaska Method, the valuation approach that assumes the annual growth rate in a worker's earnings will match or offset the appropriate discount rate. This approach allows the present value of the future earnings to be derived by multiplying the current earnings rate by the worklife expectancy. A second approach may be referred to as the "compound and discount method." This approach involves projecting future earnings from a least-squares trend line fitted to the n (five, 12, or 30) years of earning data preceding each analysis base year. Discount rates consist of the yields available at the analysis date for issues maturing in each year of the n year loss periods. The third approach may be referred to as the "simplified compound-discount method." According to this approach, the earning growth rate and the discount rate were based upon the n (five, 12 and 30) year geometric average of the wage growth rates and interest rates during the n (five, 12, and 30) years preceding each analysis base year.

(2) From 1953 to 1980, for example, the variance in the yield on one-year U.S. securities (constant maturity) was 6.75. During the same period the variance in the rate of growth in wages was 3.92. But the variance in the differential between that measure of d and g was only 1.75. Recall that Var (d-g) = Var d + g - 2 Cov (d,g). The fact that the variance of the differential is smaller than the sum of the variances of d and g implies a positive covariance between d and g. From 1953 to 1980 (d-g) = -.39; later, from 1981 to 1985 (d-g) = 6.50; for the 1953 to 1985 period as a whole (d-g) = .65. For evidence that the effective or realized (d-g) differential is zero, see Bryan and Linke (1988).

(3) In a subsequent article, Schilling expressed a more favorable evaluation of the Alaska Method.

It is somewhat ironic that I find myself arguing for the Alaska approach. My research was begun with the intention of helping to refute the heretical idea that simply multiplying current income by the number of future years on which it is to be received could give a result superior to that achieved by the arcane method of econometrics. The data, however, refused to agree (Schilling, 1986, p. 496).

(4) g and d values for years in the 1900 through 1928 time span enter the analysis (19XX-1899) times. The g and d values for years in the 1954 through 1982 time span enter the analysis (1983-19XX) times.

(5) For a discusssion of such an approach see Bryan and Linke (1, pp. 282-283).

References

[1.] Bryan, William R., and Charles M. Linke, 1988, "Estimating Present Value of Future Earnings: Experience With Dedicated Portfolios," The Journal of Risk and Insurance, 55: 273-86.

[2.] Employee Benefits 1985, 1986, (Washington, D.C.: U.S. Chamber of Commerce).

[3.] Henderson, Roger C., 1980, "Periodic Payments of Bodily Injury Awards," The American Bar Association Journal, 66: 736-37.

[4.] Henderson, 1981, "Restoring the Tort Victim to Pre-Injury Position," The American Bar Association Journal, 67: 301-02.

[5.] National Conference of Commissioners on Uniform State Laws, 1982, Handbook and Proceedings of the Eighty-Ninth Year (New York: William Shein & Co.).

[6.] Schilling, Don, 1985, "Estimating the Present Value of Future Income Losses: A Historical Simulation 1900-1982," The Journal of Risk and Insurance, 52: 100-16.

[7.] Schilling, 1986, "Rejoinder," The Journal of Risk and Insurance, 53: 496-97.

A number of methods for estimating the present value of lost future earnings to injured parties in tort litigation have been presented in this Journal during the past decade. Professor Schilling (1985) uses historical simulation for all possible five, 12 and 30 year loss periods from 1900 through 1982 to compare the performance of three valuation methods (1) to each other and to a benchmark estimate. He concluded that "the 'Alaska' method is found to be the best relative to the benchmark estimate although none of the [three] methods performs very well in an absolute sense relative to the benchmark estimate" [p. 100].

We do not question Schilling's conclusion regarding the superiority of the Alaska method. There is good economic logic to explain this finding. However, we question whether his benchmark estimate provides an appropriate standard for evaluating the performance of alternative valuation methods.

Estimating the Present Value of Future Earnings

The present value of a worker's future earnings can be represented as

[Mathematical Expression Omitted]

where [PV.sub.0] = present value of future earnings at t=0; WL = expected working life in years at t=0; [E.sub.0] = annual earnings rate at t=0; [g.sub.[alpha]] = rate of growth in earnings in period [alpha] ([alpha] = 1, 2, ..., WL); [d.sub.[alpha]] = discount rate in period [alpha] ([alpha] = 1, 2, ..., WL); and [pi] = multiplication operator.

Using continuous compounding, an average growth rate (g) and discount rate (d), equation (1) converts into

[PV.sub.0] = [E.sub.0][1 - [e.sub.-(d-g)(WL)]]/[[e.sub.(d-g)-1]]. (2)

From equation (2) it is apparent that the size of the (d-g) differential, not the expected levels of d and g, determines the present value of a future stream of earnings.

A strong covariance between nominal rates of growth in the annual earnings of workers (g) and annual interest rates (d) is to be expected because of their common determinant, inflation, and because the hire price for any factor of production embodies the real interest rate. Historical earnings growth and interest rates data provide support for the expected covariance between [d.sub.[alpha]] and [g.sub.[alpha]]; (2) moreover, over extended periods of time these same data show that the (d-g) differential was not markedly different from zero. Thus, it is not surprising that Schilling found that the Alaska Method outperformed the two valuation methods employing averages of historical wage growth and discount rates. Nor it is surprising that he supports the recent Model Periodic Payments of Judgment Act (National Conference of Commissioners on Uniform State Laws, 1982). This Act essentially adopts the Alaska Method for its recommendation that tort awards be paid in installments over the loss period with the annual growth rate ([g.sub.[alpha]]) of installments set equal to the yield on one-year U.S. Treasury securities (Henderson, 1980 and 1981, National Conference of Commissioners on Uniform State Laws, 1982).

What is surprising as that Schilling concludes on the basis of his simulation study that the Alaska Method performs tolerably well for short (five-year) loss periods, but that it does not perform very well for longer (12 and 30 year) loss periods (National Conference of Commissioners on Uniform State Laws, 1982, p. 114). (3) It is our view that the support for this conclusion relates to Schilling's benchmark estimation procedure.

Schilling's Benchmark Valuation Estimate

Schilling develops his benchmark present value estimates for all possible five, 12 and 30 year loss periods from 1900 through 1983 using perfect foresight. First, he estimated interest rates that would actually be available each year during an n-year loss period for zero coupon, high-grade corporate bonds with maturities from one to n (five, 12 or 30) years. He then used dynamic programming to identify the time configuration of investment decisions that would create the minimum cost (maximum interest yield) portfolio, and would provide exactly the year-by-year earnings (also known with certainty). Schilling's maximum interest yield portfolio was determined using an investment strategy that permitted portfolio adjustments to occur in each year of the loss period.

We have several reservations regarding the use of Schilling's benchmark valuation estimates as the standard for assessing the performance of alternative methodologies for valuing future income streams. One reservation relates to the procedures he followed in determining the minimum cost portfolio required to replicate the future earnings stream. The logic of our reservation can best be conveyed by an example. Suppose Schilling had selected common stocks rather than zero coupon bonds as the investment vehicle. Using his perfect foresight investment strategy, the time path of the maximum yield (minimum cost) stock portfolio would have been obtained simply by investing each year in the one common stock that provided the largest one-year holding period return. We doubt Schilling would argue that the stock benchmark valuation estimate provides a useful standard for assessing the performance of other valuation methods. Yet it is clear that the difference between the stock benchmark estimate and his bond benchmark estimate is one of degree, not kind.

A second reservation we have regarding Schilling's conclusions relates to the systematic understatement of the wage growth measure ([g.sub.alpha]). As observed earlier, the logic of the alaska Method reolves around the covariance between [d.sub.alpha] and [g.sub.alpha]. The [g.sub.alpha] measure used by Schilling captures the growth in money earnings only. However, workers' total earnings, money earnings and non-money (employee benefit) earnings, grew more rapidly than money earnings from 1900 through 1982. The average U.S. worker received little or no employer provided benefits in 1900; by 1982, workers received in-kind payments from employers that were equal to over 20 percent of their money earnings (Employee Benefits 1985, 1986, p. 31). Thus, the [g.sub.alpha] values were systematically biased downward.

Finally, Schilling's use of overlapping data in examining all five, 12 and 30 year loss periods from 1900 through 1982 is likely to have biased his results. The use of overlapping data results in the progressive underweighting of experience as the loss period moves toward the beginning and end of the data set, and a corresponding over-weighting of experience as the loss period moves toward the middle of the set. For example, in the 30 year loss period analysis the g and d values associated with the year 1929 through 1953 enter 30 times, but the g and d values for other years (1900 through 1928 and 1954 through 1982) enter the analysis less frequently. (4) To the extent that there are trend dependent variables operating on the data, the use of overlapping data can cause results t be systematically biased.

Concluding Observations

The research design employed by Schilling suffers from his decision to evaluate alternative valuation methodologies in terms of the extent to which they differ from results of an approach based on perfect foresight. Any evaluation of alternative valuation methods should, in our opinion, compare the performance of each valuation technique according to an objective and a set of constraints. We agree with Schilling that a sensible objective would be to minimize the cost of an award. But an important constraint would limit the acceptable profitability that an award might prove to be inadequate in replacing the lost earnings. (5) Historical simulation could be used to evaluate each valuation method according to the specified objective and constraint. Obviously, no valuation method will match the perfect foresight benchmark method. But this is not the issue of interest.

(*1) University of Illinois (Champaign-Urbana).

(1) Three methods were examined. One was the so-called Alaska Method, the valuation approach that assumes the annual growth rate in a worker's earnings will match or offset the appropriate discount rate. This approach allows the present value of the future earnings to be derived by multiplying the current earnings rate by the worklife expectancy. A second approach may be referred to as the "compound and discount method." This approach involves projecting future earnings from a least-squares trend line fitted to the n (five, 12, or 30) years of earning data preceding each analysis base year. Discount rates consist of the yields available at the analysis date for issues maturing in each year of the n year loss periods. The third approach may be referred to as the "simplified compound-discount method." According to this approach, the earning growth rate and the discount rate were based upon the n (five, 12 and 30) year geometric average of the wage growth rates and interest rates during the n (five, 12, and 30) years preceding each analysis base year.

(2) From 1953 to 1980, for example, the variance in the yield on one-year U.S. securities (constant maturity) was 6.75. During the same period the variance in the rate of growth in wages was 3.92. But the variance in the differential between that measure of d and g was only 1.75. Recall that Var (d-g) = Var d + g - 2 Cov (d,g). The fact that the variance of the differential is smaller than the sum of the variances of d and g implies a positive covariance between d and g. From 1953 to 1980 (d-g) = -.39; later, from 1981 to 1985 (d-g) = 6.50; for the 1953 to 1985 period as a whole (d-g) = .65. For evidence that the effective or realized (d-g) differential is zero, see Bryan and Linke (1988).

(3) In a subsequent article, Schilling expressed a more favorable evaluation of the Alaska Method.

It is somewhat ironic that I find myself arguing for the Alaska approach. My research was begun with the intention of helping to refute the heretical idea that simply multiplying current income by the number of future years on which it is to be received could give a result superior to that achieved by the arcane method of econometrics. The data, however, refused to agree (Schilling, 1986, p. 496).

(4) g and d values for years in the 1900 through 1928 time span enter the analysis (19XX-1899) times. The g and d values for years in the 1954 through 1982 time span enter the analysis (1983-19XX) times.

(5) For a discusssion of such an approach see Bryan and Linke (1, pp. 282-283).

References

[1.] Bryan, William R., and Charles M. Linke, 1988, "Estimating Present Value of Future Earnings: Experience With Dedicated Portfolios," The Journal of Risk and Insurance, 55: 273-86.

[2.] Employee Benefits 1985, 1986, (Washington, D.C.: U.S. Chamber of Commerce).

[3.] Henderson, Roger C., 1980, "Periodic Payments of Bodily Injury Awards," The American Bar Association Journal, 66: 736-37.

[4.] Henderson, 1981, "Restoring the Tort Victim to Pre-Injury Position," The American Bar Association Journal, 67: 301-02.

[5.] National Conference of Commissioners on Uniform State Laws, 1982, Handbook and Proceedings of the Eighty-Ninth Year (New York: William Shein & Co.).

[6.] Schilling, Don, 1985, "Estimating the Present Value of Future Income Losses: A Historical Simulation 1900-1982," The Journal of Risk and Insurance, 52: 100-16.

[7.] Schilling, 1986, "Rejoinder," The Journal of Risk and Insurance, 53: 496-97.

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Author: | Bryan, William R.; Linke, Charles M.; Schilling, Don |

Publication: | Journal of Risk and Insurance |

Date: | Sep 1, 1989 |

Words: | 1837 |

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