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Taxation and compensation to dependents of accident victims
Znternutional Review of Law and Economics (1988), 8(85-95)
TAXATION AND COMPENSATION TO DEPENDENTS OF ACCIDENT VICTIMS JOHN CREEDY Department
of Economics,
University of Melbourne, Australia
Parkville,
Victoria
3052,
I. INTRODUCTION This article examines the problems raised by the taxation of interest income arising from an award made to dependents of accident victims, and the treatment of taxation in calculating the appropriate value of the award. These problems are discussed in Section II, where they are considered in the context of the British system, under the United Kingdom’s Fatal Accidents Acts 1848-1959. The British system has been examined by Kidner and Richards (1974) and Richards (1975), both of whom concentrated on its inability to allow for inflation and the growth of real earnings, and Parkman (1983, who criticized the method used in England to deal with the delay between the accident and the settlement. Analyses are usually carried out in terms of a “net of tax interest rate,” although it will be seen below that this concept is not entirely straightforward. Section III then briefly discusses the allowance for mortality and inflation. Furthermore, the role of savings seems to have been given very little attention in the literature on compensation; thus, the circumstances under which savings are irrelevant are also considered in Section III. Brief conclusions are given in Section IV. First, however, the nature of the British system is summarized. The British procedure represents a considerable simplification of what most economists have in mind when considering the calculation of “expected future earnings.” The calculation of an award involves three basic stages. First, the period of dependency is obtained; this is normally based on the expected working life of the deceased. Second, the value of the “dependency” is calculated; this is expressed as a constant value for each year of the dependency period, with no allowance for the general growth of real earnings or inflation but allowing for deduction of taxation and the probable amount that would be spent on the deceased. The final, third stage then requires the choice of an interest rate for the calculation of the present value of the dependency over the specific period. The third stage, in fact, involves the calculation of a “multiplier,” so that the award is equal to the annual dependency multiplied by the multiplier. It is necessary to introduce a number of relevant actuarial terms at this stage since they are a basic part of the following analysis. The multiplier is the present value of an “annuity certain due” whereby fl is paid at the beginning of each year for the specified length of the dependency. The present value of such an annuity certain due of IZpayments of fl , with the first payment being made at the beginning of the first year, is denoted L&If i is the rate of interest and u = l/(1 + i), then: ii;;l = 1 + u + 02 + . . . + u”-’ I am very grateful to a referee whose comments led to substantial improvements in the exposition of this paper. My interest in this topic was revived as a result of conversations with Jim Rodgers of Pennsylvania State University. 0 1988 Butterworth
Publishers
86
Taxation of Compensation
and using the sum of a geometric
in Accident Awards
progression
a;;l
it is easily seen that
(1 - U”)liV.
=
(1)
This contrasts with the present value of an “annuity certain immediate,” a;;l in which n payments of fl are made, with the first received at the end of the first year. In the latter case: u + d + . . . + u”
a;;l =
(1 -
=
u”)li.
(2)
Hence it can be seen that (1) and (2) are related by iia=1+aq
(3)
It must be stressed at this stage that this article is concerned only with some of the more technical aspects of measuring the loss of a dependency in the British system. The more fundamental question of the appropriate value of compensation is outside the scope of the present discussion, although it may be noted that the present value of an expected future dependency is not necessarily the measure that further economic analysis might suggest. For example, Shave11 (1978) suggests that in perfect markets compensation should reflect the insurance coverage against the accident that a rational individual, with full information, would have purchased (as long as his income is regarded as “socially acceptable”). This view of “optimal” compensation is discussed further in Friedman (1982) and Fraser (1984). Furthermore, the many issues concerning the precise estimation of expected future earnings cannot be examined here (the relevant literature is much too extensive to cite). II. TAXATION
AND COMPENSATION
An example discussed by Kidner and Richards (1974) is the case of Howitt v. Heads (1972), where the period of dependency was judged to be forty years and the dependency was assessed at f936 per year. They state that “the judge awarded a capital sum of &16,848 by applying to the multiplicand a multiplier of 18. The figure 18 is the present value of fl per annum received at the beginning of the year for 40 years at a rate of discount of about 5% net of tax. In other words, if from the capital sum of f16,848, f936 were to be drawn out of the fund annually, the fund would be exhausted after 40 years so long as 5% net of tax were gained on the capital still available each year (including capital appreciation as well as interest, both taken net of tax)” (1974, pp. 132-133). Although the rate of interest is stated as being 5 percent “net of tax,” there is no discussion of the way in which its value was calculated. The judge in Howitt v. Heads did not provide an explanation, and the issue was not pursued by Kidner and Richards, perhaps understandably in view of their main objectives. The problem is essentially to make some adjustment to a gross (before tax) rate of interest in order to allow for the subsequent taxation of interest income received by the widow over the dependency period. The present article shows that there is not necessarily a simple relationship between the gross rate and the net rate; the question is complicated by the assumption made about the way in which the capital sum is invested, the tax structure, and the other circumstances of the widow that affect her tax position.
J.
87
CREEDY
A proportional
tax system
The issue can be resolved easily only in the special case where the tax system involves a proportional rate applied to all earned income and interest income, where the widow simply invests the award in a form that yields a fixed gross interest rate over the period, there are no capital gains, and the interest and capital is consumed steadily over the specified period of the dependency. If the tax rate is 7, then the interest rate net of tax, i’, is given by: ” = i( 1 - 7)
(4)
2
The required approach in this case would involve the simple use of Equation 4, substituting appropriate values for i and r. For example, a net of tax rate of 5 percent is consistent with the use of a gross rate of interest of 7.5 percent and a tax rate of 33 percent. There are, however, no grounds for assuming that this type of calculation was actually made by the judge in Howitt v. Heads. It is most unfortunate that the assumptions and procedures used by judges are not set out in much more detail. A rather different approach is suggested by the case of Taylor v. O’Connor (1970). The judge in this case, Lord Reid, took the dependency of f3,OOOper year and then added f500 per year to allow for the tax; see Kidner and Richards (1974, p. 132, n. 1). Hence Lord Reid was attempting to produce an adjusted value of the dependency, so that after discounting at a gross rate of interest, the capital sum would provide the required amount, net of tax, for the assumed dependency period. This procedure is equivalent to adding an appropriate lump sum to the award to allow for the subsequent taxation of the interest income. Again, it is unfortunate that the judge did not explain the way in which the additional figure of f500 per year was calculated. Indeed, it seems that the value was arrived at by guesswork; see Kidner and Richards (1974, p. 132, n. 1). The problem can be considered explicitly as follows. Suppose that the dependency is denoted d, where d is calculated net of tax. For the widow to receive d net of tax for each of n years, the award must be increased by A, where:
where iii indicates that the present value of the annuity certain due is calculated using the net rate i’ = i(1 - 7). Equation 5 simply states that the award, if calculated using a net rate of interest, i’, must equal the value that would be obtained using a gross rate of interest, i, plus the adjustment. Alternatively, an amount 6 could be added to the dependency for each period, and it can be seen that the two systems are equivalent if A = &A. Thus from Equation 5 it can be seen that: A = d(ii$/-ii;;l)
and
6 = d(&lii;;l -
1)
These results can be used to obtain precise values for the type of adjustment required by Lord Reid. If a proportional tax system is assumed, there is no need for any guesswork in allowing for taxation of interest income. For example, assume that IZ = 40 and i = 0.05, with 7 = 0.30, so that the net of tax rate i’ = 0.035. Using the general result in Equation 1 it can be found that iii = 22, hence: A = d(22 -
18) = 4d
and
Using the figures relating to Taylor v. O’Connor,
6 = 0.222d the judge should therefore
88
Taxation
of Compensation
in Accident Awrds
add a capital value to the award of four times the annual dependency; thus the award should be increased by f12,OOO. This is equivalent to adding f666 = (0.222)f3000 to the dependency. Unfortunately, this latter figure cannot be compared directly with the X500 added by Lord Reid, since the values of n, i and T actually used are not stated. Notice, however, that the interest rate and tax rate used in the above example are quite low, yet the addition to the dependency exceeds that given by Lord Reid. It might perhaps be suggested that Lord Reid’s addition of f500 was based on an entirely different and simpler type of reasoning. If instead of a proportional tax there were a tax free allowance of f1500, then a tax rate of 33 percent applied to the remaining f(3,OOO - 1,500) would yield approximately f500. It must be stressed, however, that such an argument would be entirely inappropriate since only the interest component of the annuity of f3000 would anyway be liable to tax and this component decreases steadily over the period of dependency. Indeed, Lord Reid explicitly recognized that only the interest income would be taxable. This aspect will be examined in more detail later in this section. So far it has been assumed that the widow simply takes the award and invests it in a form yielding a fixed gross rate of interest over the dependency period. An alternative strategy is to use the award to purchase a whole-life annuity. In this case the widow would receive, in each period, a specified amount for the remainder of her life. This is in fact an attractive alternative if the widow reasonably expects to live longer than the dependency period used in the calculation of the award. But it has different tax implications from the above system in which the capital and interest are consumed at a constant rate until it is exhausted after n years. This can be seen as follows. The term a, represents the present value of an annuity immediate of&l paid each year, as long as the recipient survives, where the first payment occurs at the end of the first year. If 1, is, using standard notation, the number of people alive at age X, then: a, = {u lx+1 + v2 lx+2 + . * . + U’VPXl,,,}/l,
(7)
where lH.+, = 0. The present value of a whole-life annuity due of fl per year, ii,, is given using Equation 7 combined with the fact that ii, = 1 + a,, where of course the values of Ii would be taken from tables relating to women. As before, let d denote the value of the annual dependency and i the gross rate of interest, then with the award the widow could purchase an annuity giving her a gross annual payment of (&/d&f for the rest of her life. A nonproportional
tax system
If the tax system is such that no tax is paid on income below a threshold level (which may vary depending on the circumstances of the household), then the convenient results used in the previous subsection do not hold. Suppose for convenience that with income, y, the tax schedule involves tax payments of ~(y - m) for y > m, with no tax paid for y 9 m. In this case there is no longer a simple proportional relationship between the gross and net of tax interest rates. If an amount A is invested at a gross rate i for one year and if the interest, Ai, exceeds the threshold, m, the net interest obtained is rn7 + Ai(1 - 7); that is, A{i(l - T) + (m/A)T}. Hence when Ai > m the net rate of interest, i’, is given by: 1” = i(1 - 7) + (m/A)7
(8)
J.
89
CREEDY
This situation is complicated not only by the awkward addition of (m/A)7 in Equation 8. As the interest and capital are consumed over the dependency period, in the later years no tax will be paid after the interest component falls to such an extent that Ai < m (since the relevant value of A falls). It does not therefore seem possible to produce a simple net of tax interest rate that can be applied directly to all years. This can be seen as follows. If interest and capital are consumed at a steady rate, as in the main discussion of the previous subsection, it is possible to trace the precise time pattern of interest payments and, therefore (in principle), calculate the amount of tax paid each year for any type of tax schedule. Even in the simpler case of a proportional tax, the implied flow of net income will not be constant since more tax will be paid in earlier years, but when a threshold operates, the pattern will be more awkward. As in the previous case, it is nevertheless more fruitful to consider the implications of withdrawing a constant gross amount each period. Suppose that the widow begins with a capital sum equal to the present value of an annuity certain due for n years, ii;;l = A. This is consumed at the rate of fl per year, from the beginning of the first year. Hence the capital value of A - 1 provides interest of (A - 1)i at the beginning of the second year, leaving a capital sum of (A - 1)(1 + i) - 1. At the beginning of the third year interest of {(A l)( 1 - i) - l}i will be obtained, and after withdrawing fl, the capital sum remaining will be {(A - I)(1 + i) - I}(1 + i) - 1. It can therefore be seen that the capital remaining at the beginning of period t, K,, is given by: K, = (ii4 -
where s;;l = 1 + (1 resents the value that amount to in n years, Using the result given can be rewritten as:
1)(1 + i)r-’
s,_ l
(9)
+ i) + . . . + (1 + i)“- I = ((1 + i)” -
l}li, which reppayments of &l, accumulating at compound interest, will with the first payment made at the end of the first year. earlier that ii;;l = 1 + a-, the expression in Equation 9
K, = a-(1
The interest received
-
+ i)r-’
-
sq.
(10)
at the beginning of period t, I,, for t 2 2, is equal to: I, = [a-(1
+ i)‘-*
-
sali.
(11)
Equation 11 could then be used to find the tax payments made, for any given award and tax schedule, until the interest obtained falls below the tax threshold, assuming that there is no other income. The question of how long tax would be paid can be answered directly using Equation 11. If, as before, d represents the value of the dependency (where d > m), then tax will be paid as long as dZf > m. Substituting for s;;l = {(I + i)” - 1)/i into Equation 11, it can be seen that tax will continue to be paid for 1 such that: (1 + i)‘p2{ia;;=?1 -
l} > (m/d) -
1
hence (1 + i)r-2 < (1 -
m/d)/{1 -
ia;;=T}
(12)
90
Taxation of Compensation
in Accident Awards
Using the fact that a;;l = (1 - u”)li, it can be seen that 1 - ia, Equation 12 becomes: (1 + i)‘-* < (1 - mld)(l After taking logarithms and rearranging, paid as long as
Equation
+ i)nP’
= uflP I, so that
(13)
13 gives the result that tax is
t < 1 + n + log(1 - mld)llog(l
+ i)
(14)
The final term in Equation 14 is of course always negative since 1 - m/d < 1. Some numerical examples may be useful at this point. Suppose, for example, that the annual dependency is twice the tax threshold (so that d = 2m), i = 0.05 and 12= 40, then substitution into Equation 14 shows that tax is paid for only the first 27 years. If however d = 4m it can be seen that tax is paid for the first 35 years: when d = 8m tax is paid for 38 years. The dependency would have to be very large for the interest always to exceed the tax threshold because by setting t = n, this would require d/m > (1 + Q/i. But even in this situation, it is not possible to produce a convenient interest rate net of tax that can apply to all years. With the nonproportional tax system the net rate of interest during any period when tax is paid is, from suitable adaptation of Equation 8, given by: 1” = i(l
-
7) + {ml(dK,_
,)}7
where K, is given in Equation 10 above. When i = 0.05, it is required that m/d > 21 for tax to be paid in each year. If r = 0.3, then rmld = 0.014. Obviously K is relatively large in the early years, so that it is quite reasonable to neglect the final term in the expression for i’; but this will be inappropriate towards the later years as K falls. It may of course be suggested that a widow might have additional sources of income that would exhaust her tax allowance in each year, thereby making an assumption of proportional taxation of interest income for all years more reasonable. The existence of such sources, however, would have to be established in each individual case. As argued earlier, it is most unfortunate that British judges do not even bother to say whether or not they are using the convenient assumption of a proportional tax system. The results derived in the present paper could of course be used directly in helping ajudge to work out the time stream of net payments that would be available (corresponding to the constant annuity) from any given award, for assumed values of the relevant variables. The calculations could easily be programmed for a personal computer. The use of a gross dependency
It might perhaps be thought that the complexities arising from the taxation of interest income could be avoided by the simple expedient of calculating the value of the constant dependency without first deducting taxation. The resulting increase in the dependency would, it might be argued, compensate for the subsequent payment of tax by the widow. But it can be shown that such a system would provide an overcompensation rather than a simple way of avoiding double taxation. This result is demonstrated by comparing the present value of the tax stream that would have been paid had the dependency been received as a flow of earnings,
J.
CREEDY
91
with the present value of the tax stream arising from the taxation of interest income on the (declining) capital sum. The latter stream will have a lower present value than the first because the stream of earnings has been converted into a capital value whose consumption is not taxable. For convenience of presentation only, the following analysis uses a proportional tax system. Suppose that the gross dependency (that is, income after deduction of the husband’s consumption but without deducting taxation) is calculated at fl per year. Assume also that taxation is a constant proportion, T, of income. If income is received and taxation is paid at the beginning of the year, then the present value of the tax that would be paid on the constant income stream is simply 7ii;;l.The present value, P,, of the taxation paid on interest arising from the capital value of the award, if it is exhausted after IZyears, is given by:
(13 where I, is the interest received in year t, as in Equation 11. The substitution of (11) into (15) and its simplification is quite tedious, although fairly straightforward. It is eased by the use of the precise form for s, = ((1 + i)” - l}li and the fact that iid = 1 + aa. After some manipulation, it can be found that: P, = ~[iia{(n - 1)iv + 1} - n] Hence the present value of the taxation paid on the interest arising from the award is less than that arising from a constant income stream, having the same value as the dependency, by the amount: 7{n - (n - 1)ivii;;O
(17)
For example, if i = 0.05 and n = 40, it is found that iin = 18, and the difference in present values is equal to 6.57, so that whatever the tax rate, the award gives rise to a tax stream having a present value that is 36.5 percent lower than the tax stream arising from a constant flow of income. The calculation of awards based on gross earnings would therefore produce excessive values, even allowing for the subsequent taxation of interest income. The algebra is significantly more awkward using a nonproportional tax system; but it is clear that such a system would produce larger differences than shown in Equation 17 because no tax would be paid on the interest after it falls below the tax threshold.
III. FURTHER
ISSUES
The effect of mortality It is clear that the British method does not allow for the effects of mortality because the dependency is assumed to represent a certuin stream of payments for a specified period of time. This was pointed out by Kidner and Richards (1974, pp. 133-134), who also noted that it does not allow for the possibility of sickness or unemployment. The allowance for mortality involves the concept of the wholelife annuity, discussed earlier. Kidner and Richards (1974, p. 134, n. 2) suggest that for a man obtaining a stream of earnings from age n to retirement at age m,
92
Taxation of Compensation
in Accident Awards
the multiplier should be: -g ti-”
(lx/l,)
(18)
n+l
It can be seen, using Equation 7, that Equation 18 is equivalent to a, - a,, ,. It can be argued, however, that for consistency the multiplier should be ii, - ii, + , since payments have been assumed to begin at the beginning of each period. Hence, the relevant concept is that of the “whole-life annuity due.” Nevertheless, there will be little difference between the measures because ii, = 1 + a,. As Kidner and Richards argue, however, allowing for mortality makes little difference except for older workers. This is of course because the future earnings of the younger workers are so heavily discounted in the years when mortality operates most strongly. Inflation
and real earnings
Within the framework of using a multiplier based on payments received with certainty over a specified period, Kidner and Richards (1974) produce a neat method of calculating a multiplier when allowance is made for a constant percentage growth of earnings. They add the assumed inflation rate over the period to the assumed general growth in real earnings (assumed to be shared equally among all individuals) and then add the result to an assumed constant growth rate of earnings over the life cycle of the individual. Kidner and Richards simply use a “net of tax” interest rate, although it has been shown above that this requires the implicit assumption that a proportional tax system is used. (It might also be added that the average percentage increase in earnings over the life cycle is not generally constant but declines steadily with age. But this is not the main issue here.) The three effects produce a growth in the dependency of, say, f percent per year. Kidner and Richards then use a net of tax rate of return of r percent per year. If the initial dependency is fl, the present value of the income stream is thus (see Kidner and Richards, 1974, p. 138): 1 + (1 + f)ll
+ r) + . . . + ((1 + f)l(l
+ r)>“-*
(19)
It is then suggested, though without presenting the formulae, that for f < r, the series in Equation 19 becomes ii, evaluated for a rate of interest of (1 + r)l(l + f) - 1; similarly for S > r it can be seen that Equation 19 is equivalent to s, for a rate of interest of (1 + f)l(l + r) - 1. Since the multiplier in the Howitt v. Heads case was stated to be Equation 18, it is then possible, using the formulae given earlier for ii;;l and s, to work out how long the award would actually last ifthe widow withdrew a net amount each year equal to the dependency she would have received had her husband been alive (and experienced the growth of nominal earnings assumed). For f < r the award would have lasted for n = {log(l - 18iv))llog u years, while for f > r it would have lasted for n = {log( 18i + l)}/log( 1 + i). Kidner and Richards (1974, p. 139) give several examples, ranging from the situation in which f = 0.11 and Y = 0.13, to one in which f = 0.11 with r = 0.07 (implying a very low real rate of return). In the first example, the multiplier is 29 and the award actually made would last 22 years (rather than the 40 intended); in the second example, the multiplier needs to be 89, so that the actual award lasts only 14 years.
J.CREEDY
93
Kidner and Richards suggest that, based on the first example, the award should have been f936 multiplied by 29; that is, f27,144. But it is possible to argue that the judge had already allowed for possible promotion between the death and the time of trial by increasing the net earnings from f20.46 to 526 per week, giving after appropriate deductions a dependency of El8 per week (that is f936 per year). A small part of the expected growth of earnings with age, up to the trial, had therefore been allowed by the judge. An alternative calculation would thus involve taking the value of weekly earnings at the time of death, before applying the growth rate over the whole of the dependency period (since the dependency period is measured from the date of death). Taking the lower initial value of net-of-tax earnings, and a dependency ratio of 0.69 (that is, f18/f26), the initial dependency in this new context of allowing for growth over the life cycle would be f14 per week, or f734 per year. Hence the award would be 29 multiplied by f734, or f2 1,286. Despite this minor criticism, and the associated problems of taxation and the need for more realistic assumptions about earnings profiles, a major implication of these examples is that they clearly demonstrate the sensitivity of the award to changes in the basic assumptions. It would be very helpful if judges were to carry out, and report the results of, a number of sensitivity analyses. This would involve very little additional work but would help to emphasize those assumptions about an uncertain future that require particular consideration. Kidner and Richards’s discussion itself exemplifies the problems of attempting to predict inflation because they suggested that the “best expectation” of a rate of inflation for the “near future” would be 6 percent. Within a very short time of the publication of their paper inflation rates were considerably higher and real rates of return were negative. The irrelevance of savings The method of calculating the award, summarized in Section II, makes no reference to savings. But in their valuable early attempt to measure compensation using information about expected future earnings (allowing for mortality, sickness and unemployment over the life cycle) Dublin and Lotka (1946) went to considerable trouble to allow for variations in saving rates at different ages and for different income groups. The British method simply ignores these very awkward problems. This subsection therefore concentrates on the question of whether or not the award would be affected by allowing for savings over life. There is a variety of possible approaches to this problem. For example, it may be assumed that accumulated savings are used to purchase a whole-life annuity for the retirement period. In order to simplify the discussion, however, the following approach retains the main framework used earlier, whereby a flow of income is assumed to arise with certainty over a specified period. The effect of saving is of course to shift the flow of consumption to a later date than the receipt of earnings. The savings accumulate until retirement, at which time the interest and capital begin to be consumed. Suppose that 8 represents the proportion of the dependency, d, that would be saved by the widow at the beginning of each year. (The deduction from earnings of the amount that the husband would have been expected to consume is now taken to include the savings made on behalf of the husband.) An amount 8d is therefore saved each year by the widow, giving rise to a capital sum after n years of I%‘~. In view of the complexities arising from taxation, it is most convenient to assume first that the tax system is directly proportional so that (from the earlier argument) net of tax interest rates can be used throughout. If the widow is expected to live for m years after the accident, the capital sum from the savings would then be consumed at a steady
94
Taxation of Compensation
in Accident Awards
rate of q per year, for a further m - n years, after the award has been exhausted. The value of q is given by the requirement that:
hence
The complete consumption stream must then be discounted back to the time of death of the husband; that is, back to period 1. If, then, the award is required to finance the consumption stream of the widow during the full m years (m > n), the appropriate award is the present value, V, of that stream, given by: v = d(1 - 0)ti;;l + 6M($J/;llii,;;l)(u” + un+l + . . . + urn-‘) The final term in Equation
v =
(21)
19 is equal to ii;;;l - ii;;l, so that:
d&J + Bd{(ii;;;l - iiJ)(fJ/liim-;;l)
- &I}
(22)
The first term in Equation 22 is of course the award that would be given with no allowance for saving. The substitution of explicit expressions for ii, and 3, in terms of the rate of interest shows, after some manipulation, that the term in curly brackets in Equation 22 is equal to zero. The British system is therefore vindicated and there is no need to make any attempt to allow for savings. This is very useful indeed since a vast amount of detailed information about saving propensities would be required. This result, however, would need to be qualified if an allowance were made for mortality, if savings were exempt from taxation, or if a nonproportional tax system were used. Similarly, if a whole-life annuity were purchased at retirement, the tax treatment would be different and so savings would have an effect on the present value of consumption. IV. CONCLUSIONS This article has examined a number of technical aspects relating to the calculation of compensation to dependents of accident victims in Britain. The main problems relate to the effects of taxation, which will also depend on the way in which the lump sum is invested. It was seen that only in the case of proportional taxation is it possible to apply a simple net-of-tax interest rate. The appropriate adjustment of either the award or the assumed dependency in order to allow for taxation was also examined. A nonproportional tax system with a threshold was then considered, and the length of time during which the widow would pay tax on the interest obtained from the compensation was obtained. It was also shown that a failure to adjust for the taxation of expected earnings, on the grounds that the interest would subsequently be taxed, would lead to a significant reduction in the present value of the tax paid compared with the tax that would have been paid on the earnings stream (and hence overcompensation). The effects of inflation after the granting of the award, when nominal interest rates are slow to adjust, are seen to have dramatic effects on the ability of the award to compensate for the loss of the dependency. One method of overcoming this problem would be to make periodic payments (which could be indexed in some way) rather than using a lump sum award. The many issues arising from the use of such payments provide a subject for future research.
J. CREEDY
95
REFERENCES Dublin, L.I., and A.J. Lotka, The Money Value of a Man, 2d ed., New York: The Ronald Press, 1946). Fraser, C.D., What Is Fair Compensation for Death or Injury?, 4 Intf. Rev. L. & Econ. 83-88 (1984). for Death or Injury? 2 Intl. Rev. L. & Econ. Friedman, D., What Is “Fair Compensation” 81-93 (1982). Kidner, R., and K. Richards, Compensation to Dependants of Accident Victims, 84 Econ. J. 130-142 (1974). Parkman, A.M., The Multiplier in English Fatal Accident Cases: What Happens When Judges Teach Judges Economics, 5 Zntl. Rev. L. & Econ. 187-197 (1985). Richards, K., Actuarial Aids for Judges, 119 Solicitors’ J. 70-73 (1975). Shavell, S., Theoretical Issues in Medical Malpractice, in S. Rottenberg, The Economics of Medical Malpractice 35-64 (Washington: American Enterprise Institute, 1978).
TAXATION AND COMPENSATION TO DEPENDENTS OF ACCIDENT VICTIMS JOHN CREEDY Department
of Economics,
University of Melbourne, Australia
Parkville,
Victoria
3052,
I. INTRODUCTION This article examines the problems raised by the taxation of interest income arising from an award made to dependents of accident victims, and the treatment of taxation in calculating the appropriate value of the award. These problems are discussed in Section II, where they are considered in the context of the British system, under the United Kingdom’s Fatal Accidents Acts 1848-1959. The British system has been examined by Kidner and Richards (1974) and Richards (1975), both of whom concentrated on its inability to allow for inflation and the growth of real earnings, and Parkman (1983, who criticized the method used in England to deal with the delay between the accident and the settlement. Analyses are usually carried out in terms of a “net of tax interest rate,” although it will be seen below that this concept is not entirely straightforward. Section III then briefly discusses the allowance for mortality and inflation. Furthermore, the role of savings seems to have been given very little attention in the literature on compensation; thus, the circumstances under which savings are irrelevant are also considered in Section III. Brief conclusions are given in Section IV. First, however, the nature of the British system is summarized. The British procedure represents a considerable simplification of what most economists have in mind when considering the calculation of “expected future earnings.” The calculation of an award involves three basic stages. First, the period of dependency is obtained; this is normally based on the expected working life of the deceased. Second, the value of the “dependency” is calculated; this is expressed as a constant value for each year of the dependency period, with no allowance for the general growth of real earnings or inflation but allowing for deduction of taxation and the probable amount that would be spent on the deceased. The final, third stage then requires the choice of an interest rate for the calculation of the present value of the dependency over the specific period. The third stage, in fact, involves the calculation of a “multiplier,” so that the award is equal to the annual dependency multiplied by the multiplier. It is necessary to introduce a number of relevant actuarial terms at this stage since they are a basic part of the following analysis. The multiplier is the present value of an “annuity certain due” whereby fl is paid at the beginning of each year for the specified length of the dependency. The present value of such an annuity certain due of IZpayments of fl , with the first payment being made at the beginning of the first year, is denoted L&If i is the rate of interest and u = l/(1 + i), then: ii;;l = 1 + u + 02 + . . . + u”-’ I am very grateful to a referee whose comments led to substantial improvements in the exposition of this paper. My interest in this topic was revived as a result of conversations with Jim Rodgers of Pennsylvania State University. 0 1988 Butterworth
Publishers
86
Taxation of Compensation
and using the sum of a geometric
in Accident Awards
progression
a;;l
it is easily seen that
(1 - U”)liV.
=
(1)
This contrasts with the present value of an “annuity certain immediate,” a;;l in which n payments of fl are made, with the first received at the end of the first year. In the latter case: u + d + . . . + u”
a;;l =
(1 -
=
u”)li.
(2)
Hence it can be seen that (1) and (2) are related by iia=1+aq
(3)
It must be stressed at this stage that this article is concerned only with some of the more technical aspects of measuring the loss of a dependency in the British system. The more fundamental question of the appropriate value of compensation is outside the scope of the present discussion, although it may be noted that the present value of an expected future dependency is not necessarily the measure that further economic analysis might suggest. For example, Shave11 (1978) suggests that in perfect markets compensation should reflect the insurance coverage against the accident that a rational individual, with full information, would have purchased (as long as his income is regarded as “socially acceptable”). This view of “optimal” compensation is discussed further in Friedman (1982) and Fraser (1984). Furthermore, the many issues concerning the precise estimation of expected future earnings cannot be examined here (the relevant literature is much too extensive to cite). II. TAXATION
AND COMPENSATION
An example discussed by Kidner and Richards (1974) is the case of Howitt v. Heads (1972), where the period of dependency was judged to be forty years and the dependency was assessed at f936 per year. They state that “the judge awarded a capital sum of &16,848 by applying to the multiplicand a multiplier of 18. The figure 18 is the present value of fl per annum received at the beginning of the year for 40 years at a rate of discount of about 5% net of tax. In other words, if from the capital sum of f16,848, f936 were to be drawn out of the fund annually, the fund would be exhausted after 40 years so long as 5% net of tax were gained on the capital still available each year (including capital appreciation as well as interest, both taken net of tax)” (1974, pp. 132-133). Although the rate of interest is stated as being 5 percent “net of tax,” there is no discussion of the way in which its value was calculated. The judge in Howitt v. Heads did not provide an explanation, and the issue was not pursued by Kidner and Richards, perhaps understandably in view of their main objectives. The problem is essentially to make some adjustment to a gross (before tax) rate of interest in order to allow for the subsequent taxation of interest income received by the widow over the dependency period. The present article shows that there is not necessarily a simple relationship between the gross rate and the net rate; the question is complicated by the assumption made about the way in which the capital sum is invested, the tax structure, and the other circumstances of the widow that affect her tax position.
J.
87
CREEDY
A proportional
tax system
The issue can be resolved easily only in the special case where the tax system involves a proportional rate applied to all earned income and interest income, where the widow simply invests the award in a form that yields a fixed gross interest rate over the period, there are no capital gains, and the interest and capital is consumed steadily over the specified period of the dependency. If the tax rate is 7, then the interest rate net of tax, i’, is given by: ” = i( 1 - 7)
(4)
2
The required approach in this case would involve the simple use of Equation 4, substituting appropriate values for i and r. For example, a net of tax rate of 5 percent is consistent with the use of a gross rate of interest of 7.5 percent and a tax rate of 33 percent. There are, however, no grounds for assuming that this type of calculation was actually made by the judge in Howitt v. Heads. It is most unfortunate that the assumptions and procedures used by judges are not set out in much more detail. A rather different approach is suggested by the case of Taylor v. O’Connor (1970). The judge in this case, Lord Reid, took the dependency of f3,OOOper year and then added f500 per year to allow for the tax; see Kidner and Richards (1974, p. 132, n. 1). Hence Lord Reid was attempting to produce an adjusted value of the dependency, so that after discounting at a gross rate of interest, the capital sum would provide the required amount, net of tax, for the assumed dependency period. This procedure is equivalent to adding an appropriate lump sum to the award to allow for the subsequent taxation of the interest income. Again, it is unfortunate that the judge did not explain the way in which the additional figure of f500 per year was calculated. Indeed, it seems that the value was arrived at by guesswork; see Kidner and Richards (1974, p. 132, n. 1). The problem can be considered explicitly as follows. Suppose that the dependency is denoted d, where d is calculated net of tax. For the widow to receive d net of tax for each of n years, the award must be increased by A, where:
where iii indicates that the present value of the annuity certain due is calculated using the net rate i’ = i(1 - 7). Equation 5 simply states that the award, if calculated using a net rate of interest, i’, must equal the value that would be obtained using a gross rate of interest, i, plus the adjustment. Alternatively, an amount 6 could be added to the dependency for each period, and it can be seen that the two systems are equivalent if A = &A. Thus from Equation 5 it can be seen that: A = d(ii$/-ii;;l)
and
6 = d(&lii;;l -
1)
These results can be used to obtain precise values for the type of adjustment required by Lord Reid. If a proportional tax system is assumed, there is no need for any guesswork in allowing for taxation of interest income. For example, assume that IZ = 40 and i = 0.05, with 7 = 0.30, so that the net of tax rate i’ = 0.035. Using the general result in Equation 1 it can be found that iii = 22, hence: A = d(22 -
18) = 4d
and
Using the figures relating to Taylor v. O’Connor,
6 = 0.222d the judge should therefore
88
Taxation
of Compensation
in Accident Awrds
add a capital value to the award of four times the annual dependency; thus the award should be increased by f12,OOO. This is equivalent to adding f666 = (0.222)f3000 to the dependency. Unfortunately, this latter figure cannot be compared directly with the X500 added by Lord Reid, since the values of n, i and T actually used are not stated. Notice, however, that the interest rate and tax rate used in the above example are quite low, yet the addition to the dependency exceeds that given by Lord Reid. It might perhaps be suggested that Lord Reid’s addition of f500 was based on an entirely different and simpler type of reasoning. If instead of a proportional tax there were a tax free allowance of f1500, then a tax rate of 33 percent applied to the remaining f(3,OOO - 1,500) would yield approximately f500. It must be stressed, however, that such an argument would be entirely inappropriate since only the interest component of the annuity of f3000 would anyway be liable to tax and this component decreases steadily over the period of dependency. Indeed, Lord Reid explicitly recognized that only the interest income would be taxable. This aspect will be examined in more detail later in this section. So far it has been assumed that the widow simply takes the award and invests it in a form yielding a fixed gross rate of interest over the dependency period. An alternative strategy is to use the award to purchase a whole-life annuity. In this case the widow would receive, in each period, a specified amount for the remainder of her life. This is in fact an attractive alternative if the widow reasonably expects to live longer than the dependency period used in the calculation of the award. But it has different tax implications from the above system in which the capital and interest are consumed at a constant rate until it is exhausted after n years. This can be seen as follows. The term a, represents the present value of an annuity immediate of&l paid each year, as long as the recipient survives, where the first payment occurs at the end of the first year. If 1, is, using standard notation, the number of people alive at age X, then: a, = {u lx+1 + v2 lx+2 + . * . + U’VPXl,,,}/l,
(7)
where lH.+, = 0. The present value of a whole-life annuity due of fl per year, ii,, is given using Equation 7 combined with the fact that ii, = 1 + a,, where of course the values of Ii would be taken from tables relating to women. As before, let d denote the value of the annual dependency and i the gross rate of interest, then with the award the widow could purchase an annuity giving her a gross annual payment of (&/d&f for the rest of her life. A nonproportional
tax system
If the tax system is such that no tax is paid on income below a threshold level (which may vary depending on the circumstances of the household), then the convenient results used in the previous subsection do not hold. Suppose for convenience that with income, y, the tax schedule involves tax payments of ~(y - m) for y > m, with no tax paid for y 9 m. In this case there is no longer a simple proportional relationship between the gross and net of tax interest rates. If an amount A is invested at a gross rate i for one year and if the interest, Ai, exceeds the threshold, m, the net interest obtained is rn7 + Ai(1 - 7); that is, A{i(l - T) + (m/A)T}. Hence when Ai > m the net rate of interest, i’, is given by: 1” = i(1 - 7) + (m/A)7
(8)
J.
89
CREEDY
This situation is complicated not only by the awkward addition of (m/A)7 in Equation 8. As the interest and capital are consumed over the dependency period, in the later years no tax will be paid after the interest component falls to such an extent that Ai < m (since the relevant value of A falls). It does not therefore seem possible to produce a simple net of tax interest rate that can be applied directly to all years. This can be seen as follows. If interest and capital are consumed at a steady rate, as in the main discussion of the previous subsection, it is possible to trace the precise time pattern of interest payments and, therefore (in principle), calculate the amount of tax paid each year for any type of tax schedule. Even in the simpler case of a proportional tax, the implied flow of net income will not be constant since more tax will be paid in earlier years, but when a threshold operates, the pattern will be more awkward. As in the previous case, it is nevertheless more fruitful to consider the implications of withdrawing a constant gross amount each period. Suppose that the widow begins with a capital sum equal to the present value of an annuity certain due for n years, ii;;l = A. This is consumed at the rate of fl per year, from the beginning of the first year. Hence the capital value of A - 1 provides interest of (A - 1)i at the beginning of the second year, leaving a capital sum of (A - 1)(1 + i) - 1. At the beginning of the third year interest of {(A l)( 1 - i) - l}i will be obtained, and after withdrawing fl, the capital sum remaining will be {(A - I)(1 + i) - I}(1 + i) - 1. It can therefore be seen that the capital remaining at the beginning of period t, K,, is given by: K, = (ii4 -
where s;;l = 1 + (1 resents the value that amount to in n years, Using the result given can be rewritten as:
1)(1 + i)r-’
s,_ l
(9)
+ i) + . . . + (1 + i)“- I = ((1 + i)” -
l}li, which reppayments of &l, accumulating at compound interest, will with the first payment made at the end of the first year. earlier that ii;;l = 1 + a-, the expression in Equation 9
K, = a-(1
The interest received
-
+ i)r-’
-
sq.
(10)
at the beginning of period t, I,, for t 2 2, is equal to: I, = [a-(1
+ i)‘-*
-
sali.
(11)
Equation 11 could then be used to find the tax payments made, for any given award and tax schedule, until the interest obtained falls below the tax threshold, assuming that there is no other income. The question of how long tax would be paid can be answered directly using Equation 11. If, as before, d represents the value of the dependency (where d > m), then tax will be paid as long as dZf > m. Substituting for s;;l = {(I + i)” - 1)/i into Equation 11, it can be seen that tax will continue to be paid for 1 such that: (1 + i)‘p2{ia;;=?1 -
l} > (m/d) -
1
hence (1 + i)r-2 < (1 -
m/d)/{1 -
ia;;=T}
(12)
90
Taxation of Compensation
in Accident Awards
Using the fact that a;;l = (1 - u”)li, it can be seen that 1 - ia, Equation 12 becomes: (1 + i)‘-* < (1 - mld)(l After taking logarithms and rearranging, paid as long as
Equation
+ i)nP’
= uflP I, so that
(13)
13 gives the result that tax is
t < 1 + n + log(1 - mld)llog(l
+ i)
(14)
The final term in Equation 14 is of course always negative since 1 - m/d < 1. Some numerical examples may be useful at this point. Suppose, for example, that the annual dependency is twice the tax threshold (so that d = 2m), i = 0.05 and 12= 40, then substitution into Equation 14 shows that tax is paid for only the first 27 years. If however d = 4m it can be seen that tax is paid for the first 35 years: when d = 8m tax is paid for 38 years. The dependency would have to be very large for the interest always to exceed the tax threshold because by setting t = n, this would require d/m > (1 + Q/i. But even in this situation, it is not possible to produce a convenient interest rate net of tax that can apply to all years. With the nonproportional tax system the net rate of interest during any period when tax is paid is, from suitable adaptation of Equation 8, given by: 1” = i(l
-
7) + {ml(dK,_
,)}7
where K, is given in Equation 10 above. When i = 0.05, it is required that m/d > 21 for tax to be paid in each year. If r = 0.3, then rmld = 0.014. Obviously K is relatively large in the early years, so that it is quite reasonable to neglect the final term in the expression for i’; but this will be inappropriate towards the later years as K falls. It may of course be suggested that a widow might have additional sources of income that would exhaust her tax allowance in each year, thereby making an assumption of proportional taxation of interest income for all years more reasonable. The existence of such sources, however, would have to be established in each individual case. As argued earlier, it is most unfortunate that British judges do not even bother to say whether or not they are using the convenient assumption of a proportional tax system. The results derived in the present paper could of course be used directly in helping ajudge to work out the time stream of net payments that would be available (corresponding to the constant annuity) from any given award, for assumed values of the relevant variables. The calculations could easily be programmed for a personal computer. The use of a gross dependency
It might perhaps be thought that the complexities arising from the taxation of interest income could be avoided by the simple expedient of calculating the value of the constant dependency without first deducting taxation. The resulting increase in the dependency would, it might be argued, compensate for the subsequent payment of tax by the widow. But it can be shown that such a system would provide an overcompensation rather than a simple way of avoiding double taxation. This result is demonstrated by comparing the present value of the tax stream that would have been paid had the dependency been received as a flow of earnings,
J.
CREEDY
91
with the present value of the tax stream arising from the taxation of interest income on the (declining) capital sum. The latter stream will have a lower present value than the first because the stream of earnings has been converted into a capital value whose consumption is not taxable. For convenience of presentation only, the following analysis uses a proportional tax system. Suppose that the gross dependency (that is, income after deduction of the husband’s consumption but without deducting taxation) is calculated at fl per year. Assume also that taxation is a constant proportion, T, of income. If income is received and taxation is paid at the beginning of the year, then the present value of the tax that would be paid on the constant income stream is simply 7ii;;l.The present value, P,, of the taxation paid on interest arising from the capital value of the award, if it is exhausted after IZyears, is given by:
(13 where I, is the interest received in year t, as in Equation 11. The substitution of (11) into (15) and its simplification is quite tedious, although fairly straightforward. It is eased by the use of the precise form for s, = ((1 + i)” - l}li and the fact that iid = 1 + aa. After some manipulation, it can be found that: P, = ~[iia{(n - 1)iv + 1} - n] Hence the present value of the taxation paid on the interest arising from the award is less than that arising from a constant income stream, having the same value as the dependency, by the amount: 7{n - (n - 1)ivii;;O
(17)
For example, if i = 0.05 and n = 40, it is found that iin = 18, and the difference in present values is equal to 6.57, so that whatever the tax rate, the award gives rise to a tax stream having a present value that is 36.5 percent lower than the tax stream arising from a constant flow of income. The calculation of awards based on gross earnings would therefore produce excessive values, even allowing for the subsequent taxation of interest income. The algebra is significantly more awkward using a nonproportional tax system; but it is clear that such a system would produce larger differences than shown in Equation 17 because no tax would be paid on the interest after it falls below the tax threshold.
III. FURTHER
ISSUES
The effect of mortality It is clear that the British method does not allow for the effects of mortality because the dependency is assumed to represent a certuin stream of payments for a specified period of time. This was pointed out by Kidner and Richards (1974, pp. 133-134), who also noted that it does not allow for the possibility of sickness or unemployment. The allowance for mortality involves the concept of the wholelife annuity, discussed earlier. Kidner and Richards (1974, p. 134, n. 2) suggest that for a man obtaining a stream of earnings from age n to retirement at age m,
92
Taxation of Compensation
in Accident Awards
the multiplier should be: -g ti-”
(lx/l,)
(18)
n+l
It can be seen, using Equation 7, that Equation 18 is equivalent to a, - a,, ,. It can be argued, however, that for consistency the multiplier should be ii, - ii, + , since payments have been assumed to begin at the beginning of each period. Hence, the relevant concept is that of the “whole-life annuity due.” Nevertheless, there will be little difference between the measures because ii, = 1 + a,. As Kidner and Richards argue, however, allowing for mortality makes little difference except for older workers. This is of course because the future earnings of the younger workers are so heavily discounted in the years when mortality operates most strongly. Inflation
and real earnings
Within the framework of using a multiplier based on payments received with certainty over a specified period, Kidner and Richards (1974) produce a neat method of calculating a multiplier when allowance is made for a constant percentage growth of earnings. They add the assumed inflation rate over the period to the assumed general growth in real earnings (assumed to be shared equally among all individuals) and then add the result to an assumed constant growth rate of earnings over the life cycle of the individual. Kidner and Richards simply use a “net of tax” interest rate, although it has been shown above that this requires the implicit assumption that a proportional tax system is used. (It might also be added that the average percentage increase in earnings over the life cycle is not generally constant but declines steadily with age. But this is not the main issue here.) The three effects produce a growth in the dependency of, say, f percent per year. Kidner and Richards then use a net of tax rate of return of r percent per year. If the initial dependency is fl, the present value of the income stream is thus (see Kidner and Richards, 1974, p. 138): 1 + (1 + f)ll
+ r) + . . . + ((1 + f)l(l
+ r)>“-*
(19)
It is then suggested, though without presenting the formulae, that for f < r, the series in Equation 19 becomes ii, evaluated for a rate of interest of (1 + r)l(l + f) - 1; similarly for S > r it can be seen that Equation 19 is equivalent to s, for a rate of interest of (1 + f)l(l + r) - 1. Since the multiplier in the Howitt v. Heads case was stated to be Equation 18, it is then possible, using the formulae given earlier for ii;;l and s, to work out how long the award would actually last ifthe widow withdrew a net amount each year equal to the dependency she would have received had her husband been alive (and experienced the growth of nominal earnings assumed). For f < r the award would have lasted for n = {log(l - 18iv))llog u years, while for f > r it would have lasted for n = {log( 18i + l)}/log( 1 + i). Kidner and Richards (1974, p. 139) give several examples, ranging from the situation in which f = 0.11 and Y = 0.13, to one in which f = 0.11 with r = 0.07 (implying a very low real rate of return). In the first example, the multiplier is 29 and the award actually made would last 22 years (rather than the 40 intended); in the second example, the multiplier needs to be 89, so that the actual award lasts only 14 years.
J.CREEDY
93
Kidner and Richards suggest that, based on the first example, the award should have been f936 multiplied by 29; that is, f27,144. But it is possible to argue that the judge had already allowed for possible promotion between the death and the time of trial by increasing the net earnings from f20.46 to 526 per week, giving after appropriate deductions a dependency of El8 per week (that is f936 per year). A small part of the expected growth of earnings with age, up to the trial, had therefore been allowed by the judge. An alternative calculation would thus involve taking the value of weekly earnings at the time of death, before applying the growth rate over the whole of the dependency period (since the dependency period is measured from the date of death). Taking the lower initial value of net-of-tax earnings, and a dependency ratio of 0.69 (that is, f18/f26), the initial dependency in this new context of allowing for growth over the life cycle would be f14 per week, or f734 per year. Hence the award would be 29 multiplied by f734, or f2 1,286. Despite this minor criticism, and the associated problems of taxation and the need for more realistic assumptions about earnings profiles, a major implication of these examples is that they clearly demonstrate the sensitivity of the award to changes in the basic assumptions. It would be very helpful if judges were to carry out, and report the results of, a number of sensitivity analyses. This would involve very little additional work but would help to emphasize those assumptions about an uncertain future that require particular consideration. Kidner and Richards’s discussion itself exemplifies the problems of attempting to predict inflation because they suggested that the “best expectation” of a rate of inflation for the “near future” would be 6 percent. Within a very short time of the publication of their paper inflation rates were considerably higher and real rates of return were negative. The irrelevance of savings The method of calculating the award, summarized in Section II, makes no reference to savings. But in their valuable early attempt to measure compensation using information about expected future earnings (allowing for mortality, sickness and unemployment over the life cycle) Dublin and Lotka (1946) went to considerable trouble to allow for variations in saving rates at different ages and for different income groups. The British method simply ignores these very awkward problems. This subsection therefore concentrates on the question of whether or not the award would be affected by allowing for savings over life. There is a variety of possible approaches to this problem. For example, it may be assumed that accumulated savings are used to purchase a whole-life annuity for the retirement period. In order to simplify the discussion, however, the following approach retains the main framework used earlier, whereby a flow of income is assumed to arise with certainty over a specified period. The effect of saving is of course to shift the flow of consumption to a later date than the receipt of earnings. The savings accumulate until retirement, at which time the interest and capital begin to be consumed. Suppose that 8 represents the proportion of the dependency, d, that would be saved by the widow at the beginning of each year. (The deduction from earnings of the amount that the husband would have been expected to consume is now taken to include the savings made on behalf of the husband.) An amount 8d is therefore saved each year by the widow, giving rise to a capital sum after n years of I%‘~. In view of the complexities arising from taxation, it is most convenient to assume first that the tax system is directly proportional so that (from the earlier argument) net of tax interest rates can be used throughout. If the widow is expected to live for m years after the accident, the capital sum from the savings would then be consumed at a steady
94
Taxation of Compensation
in Accident Awards
rate of q per year, for a further m - n years, after the award has been exhausted. The value of q is given by the requirement that:
hence
The complete consumption stream must then be discounted back to the time of death of the husband; that is, back to period 1. If, then, the award is required to finance the consumption stream of the widow during the full m years (m > n), the appropriate award is the present value, V, of that stream, given by: v = d(1 - 0)ti;;l + 6M($J/;llii,;;l)(u” + un+l + . . . + urn-‘) The final term in Equation
v =
(21)
19 is equal to ii;;;l - ii;;l, so that:
d&J + Bd{(ii;;;l - iiJ)(fJ/liim-;;l)
- &I}
(22)
The first term in Equation 22 is of course the award that would be given with no allowance for saving. The substitution of explicit expressions for ii, and 3, in terms of the rate of interest shows, after some manipulation, that the term in curly brackets in Equation 22 is equal to zero. The British system is therefore vindicated and there is no need to make any attempt to allow for savings. This is very useful indeed since a vast amount of detailed information about saving propensities would be required. This result, however, would need to be qualified if an allowance were made for mortality, if savings were exempt from taxation, or if a nonproportional tax system were used. Similarly, if a whole-life annuity were purchased at retirement, the tax treatment would be different and so savings would have an effect on the present value of consumption. IV. CONCLUSIONS This article has examined a number of technical aspects relating to the calculation of compensation to dependents of accident victims in Britain. The main problems relate to the effects of taxation, which will also depend on the way in which the lump sum is invested. It was seen that only in the case of proportional taxation is it possible to apply a simple net-of-tax interest rate. The appropriate adjustment of either the award or the assumed dependency in order to allow for taxation was also examined. A nonproportional tax system with a threshold was then considered, and the length of time during which the widow would pay tax on the interest obtained from the compensation was obtained. It was also shown that a failure to adjust for the taxation of expected earnings, on the grounds that the interest would subsequently be taxed, would lead to a significant reduction in the present value of the tax paid compared with the tax that would have been paid on the earnings stream (and hence overcompensation). The effects of inflation after the granting of the award, when nominal interest rates are slow to adjust, are seen to have dramatic effects on the ability of the award to compensate for the loss of the dependency. One method of overcoming this problem would be to make periodic payments (which could be indexed in some way) rather than using a lump sum award. The many issues arising from the use of such payments provide a subject for future research.
J. CREEDY
95
REFERENCES Dublin, L.I., and A.J. Lotka, The Money Value of a Man, 2d ed., New York: The Ronald Press, 1946). Fraser, C.D., What Is Fair Compensation for Death or Injury?, 4 Intf. Rev. L. & Econ. 83-88 (1984). for Death or Injury? 2 Intl. Rev. L. & Econ. Friedman, D., What Is “Fair Compensation” 81-93 (1982). Kidner, R., and K. Richards, Compensation to Dependants of Accident Victims, 84 Econ. J. 130-142 (1974). Parkman, A.M., The Multiplier in English Fatal Accident Cases: What Happens When Judges Teach Judges Economics, 5 Zntl. Rev. L. & Econ. 187-197 (1985). Richards, K., Actuarial Aids for Judges, 119 Solicitors’ J. 70-73 (1975). Shavell, S., Theoretical Issues in Medical Malpractice, in S. Rottenberg, The Economics of Medical Malpractice 35-64 (Washington: American Enterprise Institute, 1978).