Realidad problemática

An alternative procedure is to recognize that the victim might have died either much earlier or later than the average time for a person having the same starting age. The probability of dying at each age can be found in standard mortality tables in the form of a “mortality rate” per thousand. These mortality rates have been entered as column (9) of Exhibit 3.6. Remember that, since the rates are in thousands, their further use requires that the decimal points be moved leftward three places, i.e., 19.9/1000 = .0199,

Starting Age 60 Real Income Growth 1.00% Starting Salary $20,000 Inflation Factor 7.00%

(1) (2) (7) (8) (9) (10) (11) (12)

Yr. Age PV Cumul Mort Alive ExVal Cumul

1 60 19,547.51 19,547.51 19.1 0.9809 19,174.15 19,174.15 2 61 18,899.46 38,446.97 21.1 0.9602 18,147.32 37,321.47 3 62 17,756.20 56,203.17 22.2 0.9389 16,671.06 53,992.53 4 63 16,413.48 72,616.66 25.0 0.9154 15,025.14 69,017.67 5 64 15,262.46 87,879.12 26.9 0.8908 13,595.64 82,613.31 6 65 14,459.17 102,338.29 28.8 0.8651 12,509.13 95,122.45 7 66 13,698.16 116,036.45 30.8 0.8385 11,485.76 106,608.20 8 67 12,977.21 129,013.66 33.2 0.8107 10,519.99 117,128.19 9 68 12,294.20 141,307.86 36.0 0.7815 9,607.52 126,735.70 10 69 11,647.13 152,954.99 39.2 0.7508 8,745.06 135,480.77 11 70 11,034.13 163,989.12 42.6 0.7188 7,931.86 143,412.63 12 71 10,453.38 174,442.50 46.3 0.6856 7,166.48 150,579.11 13 72 9,903.20 184,345.70 50.3 0.6511 6,447.80 157,026.91 14 73 9,381.98 193,727.69 54.8 0.6154 5,773.70 162,800.61 15 74 8,888.19 202,615.88 59.8 0.5786 5,142.72 167,943.33 16 75 8,420.39 211,036.28 65.3 0.5408 4,553.91 172,497.24 17 76 7,977.22 219,013.49 71.1 0.5024 4,007.49 176,504.72 18 77 7,557.36 226,570.85 77.4 0.4635 3,502.71 180,007.44 19 78 7,159.61 233,730.46 84.4 0.4244 3,038.29 183,045.73 20 79 6,782.79 240,513.25 91.0 0.3857 2,616.45 185,662.17 21 80 6,425.80 246,939.04 98.3 0.3478 2,235.08 187,897.25 22 81 6,087.60 253,026.64 105.9 0.3110 1,893.21 189,790.46 23 82 5,767.20 258,793.84 113.7 0.2756 1,589.64 191,380.09 24 83 5,463.66 264,257.50 121.3 0.2422 1,323.30 192,703.39 25 84 5,176.10 269,433.60 128.4 0.2111 1,092.68 193,796.07 26 85 4,903.67 274,337.27 136.0 0.1824 894.39 194,690.46 27 86 4,645.58 278,982.86 144.1 0.1561 725.26 195,415.72 28 87 4,401.08 283,383.94 152.6 0.1323 582.25 195,997.97 29 88 4,169.44 287,553.38 161.6 0.1109 462.47 196,460.44 30 89 3,950.00 291,503.38 171.2 0.0919 363.12 196,823.56 31 90 3,742.11 295,245.49 181.3 0.0753 281.64 197,105.20 32 91 3,545.15 298,790.64 192.0 0.0608 215.59 197,320.78 33 92 3,358.57 302,149.20 203.4 0.0484 162.70 197,483.48 34 93 3,181.80 305,331.00 215.4 0.0380 120.93 197,604.42 35 94 3,014.34 308,345.34 228.2 0.0293 88.42 197,692.84

21.1/100 = .0211, etc. At each year, the cohort of current 60 year olds will be reduced by the proportion that dies. Hence, at age 61 the fraction (1.00 - .0199) = .98090 will still be alive, and that is the probability that a sixty year old will live to earn the full income from his 60th year. During the next year, the fraction .0211 of the .98090 still alive will die, yielding the .98090 (1- .0211) =.9602 of the original cohort of 60 year-olds alive to reap the second year’s income. These fractions left alive, as listed in column 10, are interpretable as the probabilities that the 60 year old victim would actually be alive to receive income in any particular year.

The next step is to use the probability for each year to “weight” its corresponding income figure from column 7. In effect, this method “credits” a person with a proportion of potential income that appropriately reflects the probability that the income will indeed be received. Of course, this probability-weighted figure is really what Chapter II identified as the “expected value” of the income in that year. We are interested in the expected value of the income for the victim’s lifetime, rather than for any single year. Therefore the expected values for each year, as listed in column 11, are cumulated in column 12. In principle, we should keep summing up until the column 11 figure becomes zero, but the reader can see that, because both the present value discounting and death factor operate so strongly in later years, little change in the “bottom line” number would occur if the table were carried out for additional periods. Hence, we can see that the expected value of the victim’s income stream is really less than $200,000. Contrast this result with the much higher figure achieved by using the average life expectancy.

In a real-world application, the income for each year should be weighted by realization probabilities much lower than those shown in the same. Obviously, being alive is a necessary but not a sufficient condition that income will be earned. A further adjustment should be made for the probability that a live individual is still an active participant in the workforce (i.e., not retired) and, if active, is not unemployed. Unemployment rates and labor force participation rates by age and sex are available to analysts, so that the appropriate joint probability of being alive, participating, and employed can be computed. Inclusion of these factors would make Exhibit 3.6 become somewhat more complicated, but the principle is the same. There is no doubt that the expected value method is conceptually superior to the average life expectancy method and, as illustrated, the methods give rather different results. The specific numbers in Exhibit 3.6 and some of the queries in the last question block above should help you be able to explain in commonsense terms what the fallacy is in the average life expectancy method.

3. RISK SELECTION AND RISK COMPENSATION

Kay Glehn owns and manages a number of apartment units in a city which is the site of a large university. Most of her tenants are students and she has been plagued by a midyear “skipping” problem at around the end of first semester. Typically, about six percent of her tenants default after making only five monthly payments toward an annual lease. Of those remaining, four percent default after one additional payment. The rest stay through the year and fulfill their entire contractual obligation. Because of the seasonal nature of the rental market, a midyear default has the practical implication that an apartment will remain empty for the entire defaulted period. Al- though it is in theory possible to obtain a judgment for the lost rent, that alternative is impractical because of a combination of legal expenses and the low probability of recovery against persons who have left the jurisdiction, are impecunious, etc. Hence, although she attempts to create an impression otherwise, her actual policy is merely to absorb the loss from any default.

Since rental housing is essentially a competitive market, Kay’s rental schedule of $500 per month presumably suffices to give her a reasonable rate of return on invested capital and operating costs. Hence, she (and other landlords in the market) are “charging” renters for the default risk that landlords must bear. This charge, of course, manifests itself in the form of a contract rent that is slightly higher than it would otherwise be.

Experience suggests to Kay that the tenants are composed of three types: Alphas, who are sure that they will perform on the entire lease; Betas who regard themselves as sure “skippers” at mid-year; and Gammas, who regard the probability of performance as a fifty-fifty proposition. Unfortunately, she cannot tell in advance which prospective renter belongs to which category. A worksheet similar to Exhibit 3.7 can be found in Kay’s files.