Present Value Functions and Recursions

Abstract

Using a Markov-induced decomposition of time in the labor force, we compute means, standard deviations, and other distributional characteristics of the present value of years of labor force activity. We also provide bootstrap estimates of the mean present value and the corresponding standard deviations of sample means. This paper combines years of future labor force activity, decomposed with a Markov process, with discounting that activity to the present, whereas previous literature has analyzed only undiscounted years of activity.A worker at exact age x, given activity status, sex and education, receives $1 for each future year in the labor force. The present value of this stream, which is assumed to grow at rate g and to be discounted at the interest rate r, is a random variable, since it depends on future labor market realizations and mortality experience. Letting NDR be the net discount rate, (r – g)/(1 + g), we define this random variable to be PVA (a, x, NDR) when the person begins active and PVA (i, x, NDR) when commencing inactive. We assume that future labor force activity status follows the usual Markov or increment-decrement model. In arranging these random variables into the row random vector PVA (x, NDR)′ ≡ PVA (a, x, NDR), PVA (i, x, NDR), we provide a recursion for its probability mass function (often abbreviated “pmf” below), which we show to be computationally intractable. Next, we indicate how we may nevertheless estimate the probability mass function. We then provide a computationally useful recursion for its expected value E [PVA (x, NDR)′]. These expected values have been tabulated in the United Kingdom, where they are associated with the Ogden Tables. From another point of view, E [PVA (x, NDR)′] is a generalization of the worklife expectancy since , the familiar worklife expectancies when NDR = 0. The same concepts apply to flows of $1 while in the inactive state, and our computationally feasible recursion covers it as well.It is well known that a person's worklife expectancy does not provide enough information to accurately compute the associated present value, since worklife will on average be allocated over future years in a particular pattern dictated by the underlying Markov model. Skoog and Ciecka (2006) took up a graphic way to perform this allocation. Other papers (Skoog and Ciecka, 2001a, 2001b and 2002) pointed out the intrinsic variability of the random variables YAx,a and YAx,i (years of activity starting active and inactive) around their respective means. The present paper brings these two ideas together, adds an extension of a recursion and decomposition found in Skoog (2002), to address what must surely be among the most natural and interesting questions for forensic economists: (1) What does the mean of the present value random variable, correctly calculated with the Markov-induced decomposition, look like at various NDR's? (2) What is the size of the standard error of the estimated mean of the present value random variable at various NDR's? (3) What does the present value distribution look like at different net discount rates? While the so-called Ogden Tables have asked about the first and second questions in the context of British data and a single NDR, it will be helpful to have these tables calculated for American data based on labor force participation and various NDR's. The third question has not even been previously asked. This is surprising; in the age of Daubert we feel that displaying variation as standard errors is important. Perhaps the lack of attention to this question has occurred because of the technical difficulties highlighted in this paper. We suggest another reason, however. Many forensic economists have implicitly understood or assumed that what we carefully define as the present value random variable instead meant expected present value. Questions about variability have not arisen because the discourse simply did not allow it. Our intuition about the variability comes from observing that, when the NDR is zero, present value is years of activity, whose pmf has been tabulated. We expect the same variability for the present value random variable when the NDR exceeds zero, but pulled leftward and shrunk, due to discounting.Beyond this introduction, the paper is organized as follows. Sections II and III contain notation to capture the probability structure and timing (of payments) convention used in the paper. Section IV deals with the intractable present value recursion for the present value function. On a first reading, this section may be skimmed to more quickly arrive at Sections V and VI containing mean recursions and tabular results, respectively. The paper concludes with a brief discussion of the Ogden Tables in Section VII and some final thoughts about the present value random variable and its expected value in Section VIII.We let Zx denote the state of our worker (referred to in the masculine) at exact age x, so that Zx = a or Zx = i. TA – 1 (where TA is “terminal age” or “truncation age”) is an age at which all transitions occurring ½ of a year later are to the death state; this is illustrated as age 110 below. As in our previous work, we assume that transitions occur at midpoints, so that for the next half-year at least, he continues in the same state he occupied at age x. At age x + ½, the first transition occurs, which we formalize by defining the “increment” random variables Zx,.5, depending on the state occupied at age x: In the same way, and, generally,Let t ≥ 0 refer to any (not necessarily integer or half integer) number of years beyond age x, and let Zx (t) be the random state occupied at this age, x+t, defined out of the increments in (1a)–(1d). We construct a random variable which is left continuous and constant between half integers, and it changes on the half integer whenever the labor force status changes. The stochastic structure of Zx (t) depends on the initial state of Zx and is induced by the probabilities . We can convert the function Zx(t), which takes on values of a, i and d, to a function which equals one for all time when in the active state, by use of the commonly used indicator function I[ Zx (t)=a ], defined as: for any event E, IE= 1 whenever the event E is true.Our earlier work focused on YAx, the years of additional labor force activity, starting at age x in state m. Evidently If we indicate both age x and the initial state, a or i, in our notation, we can express this last equality as Despite equation (3), this model is essentially discrete. A truly continuous model would posit instantaneous forces of increment, decrement and mortality, and payments would grow and be discounted with instantaneous forces. We take this model up elsewhere, but note here that in continuous time the “construction” of Zx (t) would not be required, simplifying the development above.We need to specify when earnings are paid, how they are discounted, and how they grow. We are motivated by the fact that our transitions will be observed once per year, but compensation is paid much more frequently, rarely daily or annually, but more often bi-weekly, semi-monthly or monthly. In the case of monthly payments, the average of the payment points of 1/12, 2/12,…, 12/12 of a year is 6.5 months or 13/24 of a year. Consequently assuming one payment is made at the midpoint of 12/24 introduces a very modest acceleration of 1/24 of a year.We allow for an age-earnings curve by including the sequence {aej} for appropriate indices j relative to the base earnings level for age x, although much forensic practice, and our tabulated tables, will set {aej} to unity. We follow standard forensic economic practice and assume that earnings grow at rate g and are discounted at rate r; these may be taken as either both nominal or both real. Since the effects of r and g separately enter the present value through the net discount rate, we use the relationsIn light of the timing issue between payments and transitions, and consistent with the convention adopted in our previous work, we continue to assume that transitions take place at mid-period. Our first inclination is to consider the possibility of assuming that payments are made when transitions take place, i.e. at mid-periods. There is one asymmetry, namely, that we begin at an exact age, x, and the first transition is after ½ of a period, followed thereafter at periods one year apart. To fix ideas, assume the worker is active at x. Then $.5 is earned between x and x+.5. Assume that the transition at x+.5 is active to active, so (i) $.5 is earned for the activity in [x+.5, x+1) and (ii) $.5 is earned in [x+1,x+1.5). We define two allocation conventions below: Convention A assumes that (i) $.5 is paid at x+.5, ½ of a period before the mid-point of the interval [x+.5, x+1.5) while (ii) another $.5 is paid at x+1.5, ½ of a period after the mid-point of the interval [x+.5, x+1.5). This convention splits payments for the interval into two pieces. Additionally, the transition which took place at age x –.5, and which is responsible for the activity in [x, x+.5), results in $.5 being paid at x+.5, so all future work at x and beyond is paid in the future.Convention B would put the entire payment at x+1, the mid-point of the [x+.5, x+1.5) interval. In this case, payments occur at exact ages, different points from transitions. Convention B requires that the payment for [x, x+.5) was made at x along with the payment for activity in [x–.5, x), and that these payments took place the instant before attaining age x, so that references to the present value of payments at exact age x start at age x+.5, with its associated one year interval [x+.5, x+1.5).The present value random variables associated with Convention A are: andOn Convention B, the expression on the right hand side does not depend on the initial state, although the distribution of I[Zx,j = a] does, so that we have:The equations (6) display the source of the randomness in the present value random variable, and provide the suggestion for how we will need to go about computing the probability mass function which summarizes its randomness. Recalling the connection of I[Zx,j = a|Zx = m] and Zx (t) emphasizes that any individual worker would have experienced a sample path of future labor market activity, which could be very short or very long, and so could have experienced a small or large present value of future compensation. We have a choice in studying the probability mass function of (6)—we could either attempt to discover its recursive structure, as we have done with the related random variables, or we could use (6) to generate a large number of realizations via simulation. We will start with the former and discern the necessity of doing the latter. We also have a choice of allocation methods. The theoretical and empirical work that follows utilizes Convention A.Equations (6) will be seen as new for most forensic economists and actuaries, who have slipped into the habit of thinking about the present value random variable as a number. By taking the mathematical expectations of the left hand sides in (6) and computing or estimating E{I[Zx.j= a|Zx= m]}on the right hand sides, they have been referring to the expected present value rather than the present value of the revenue stream. Nowhere has this emphasis been more pronounced than in actuarial science, where the notation for the random variable representing the payment of $1 per year for life is not typically distinguished from its expectation, ax. We will take up this point again in Section V.This section, as it describes the probability mass function of the present value function, should also help to fix ideas introduced in the notation of the previous section. Let us assume that TA, the truncation age, is 111. The last transition is at age 109.5, and everyone who survives this last transition, governed by either ap109a or ip109a, is dead at 110.5.1 The probability mass function for any age and initial condition is the function which assigns probabilities to each possible value of the present value random variable. We can do this from first principles. We will work with Convention A and starting active. There are two possibilities: at 109.5, the transition is to active, resulting in payments at 109.5 and 110.5, or the transition is to the inactive or dead states, resulting in no payment. In any case, for the first half of the interval, the beginning state of activity is continued (i.e., from age 109 to 109.5). There is a payoff for this half period of .5, which grows and is discounted over this interval, resulting in a present value contribution of .5β regardless of the transition. We generically define the probability mass function by the symbols px, m(pv), where the subscripts are the beginning age and state, m, the pv arguments are all possible values of the present value, and the value of the function (its range) at each of these pv values in the domain is the probability of that value of the present value occurring. Here, the pmf is Had we started inactive, there would be only one .5β term, and we would either realize a present value of .5β.5β3 or nothing. The pmf is We notice that the domain of p109,a (•) is, for the small NDR's encountered in practice (1% to 3%, say) “essentially” .5 and 1.5, which is exactly the domain encountered when β = 1 and the present value is counting years of activity. Similarly, the domain of p109,i (•) is, for these small NDR's encountered in practice “essentially” 0 and 1, again exactly the domain encountered when β = 1 and the present value is counting years of activity. These domains are exactly 0,.5β,.5β.5β3, β + .5β3 and do not overlap.2Moving back to age 108, we can begin to see the domains interleave and escalate. Starting active at x = 108, there will be .5β realized at least. If we go active, we will receive another .5β, and be active at 109. We can then use the two present values associated with p109,a(•), discounted by β2 since it is one year in the future, to complete this half of the pmf. Alternatively, we could go inactive at 108, contributing .5β and 0 at 108.5, and leaving us with β2 times the possibilities entailed in p109,i(•). Doing the accounting, showing how each of the two potential contributions for each transition contribute, and then grouping, gives us: The number of distinct points in the domain has doubled; let us define it as D109,a = {β+ β3+ .5β5, β + .5β3,. 5β + .5β3 + .5β5,.5β}. The 4 distinct points are essentially 2.5, 1.5 and .5, with there being now two distinct but slightly different ways to realize about 1.5. This problem will become worse when we get to 107, and will increase exponentially.For inactives at age 108 we have The number of distinct points in this domain has doubled, which we again define as D108,i = {.5β3 + .5β5,.5β + .5β3,.5β3 + .5β5,0}. The domain is essentially 2, 1 and 0, and there are now two distinct but slightly different ways to realize about 1. Again, this problem will become worse when we get to 107, and will increase exponentially thereafter. In fact, Figures 1a, 1b, 2a, and 2b show the general recursion, from which the age x – 1 = 107 domain elements and their corresponding probabilities evolve from their age x = 108 counterparts.The general pattern is now clear. Starting at 107, with three transitions left (at 107.5, 108.5, and 109.5), there are in each of the domains of D107,a and D107,j. Binomial coefficients capture the number of ways in which each of the essential values occurs, there being 3+1=4 such essential values for each domain of 8 points. The domains of D107,a and D107,i are non-overlapping, being essentially on half-integers and integers, respectively, although this distinction will blur as the number of periods less than the terminal age, three above, increases. The link between the domains and ranges one year apart is indicated in Figures 1a and 2a, holds in general, and expresses the recursion of the present value function.This recursion, although true, provides only insight into the mathematical structure and cannot be exploited for computational purposes, unless β = 1, in which case the number of distinct points in the domain does not increase exponentially, but equals (TA – x). For a person age x, the number of distinct points, 2(TA – x−1), becomes a number which is impossible to compute and store: at age 50, 2111–50–1= 260 = 250210, and 250 megabytes of information will never be computable. We state this as thePresent Value Function Properties and Recursion: The present value functions px,a (•) and px,i (•) for a person exact age x will contain 2(TA– x–1) distinct possible values (points in the domains Dx,a and Dx,i, respectively). These arise from 2(TA –x)− 1 sample paths, giving rise to (TA – x) essential values, which arise from grouping of the distinct values together (ignoring the powers of β). While computationally infeasible, the domain of D follows from the domains of Dx+1,a and Dx+1,i by use of the symbolic relation where the first half of Dx,a = .5β .5β + β2 Dx+1,a and the second half of Dx,i = .5β + 0β+ β2 Dx+1,i defined above, and the same is true for the first half of Dx,i = 0β + .5β + β2 Dx+1,a and the second half of Dx,i = 0β + 0β + β2 Dx+1,i. The ranges of px,a(•) (respectively, px,i(•)) follow from the ranges of px+1,a (•) and px+1,i (•) by applying to them the transition probabilities a pxa, a pxi and a pxd (respectively,i pxa, i pxi and i pxd) as indicated in Figures 1a and 1b (respectively, Figures 2a and 2b).In light of the analytic complexity and computational infeasibility of the present value function, it might appear unlikely that there would be a useful and elegant recursion to be found. The next section develops such a recursion for the means of the present value random variables.Let be the means or mathematical expectations of the probability mass functions defined above. We now define, in exactly the same way as we did for PVAx,a and PVAx,i, the present value functions paying in the active state, the new present value functions PVIx,a and PVIx,i which pay $1 for each year in the inactive state. These are not likely to be nearly as important, but their inclusion actually simplifies the theory which follows, which is enough justification. Additionally, the sum of these present values gives a standard annuity (which pays in both states, i.e., while alive); and annuities are of interest. Finally, the value of an income stream for all years while inactive might have independent interest, (e.g. in providing discretionary income or describing income requirements when there is no labor force income).We will determine the present value and expected present value of years of inactivity The present values are defined much like those for years of activity. We use only the Convention A below, and so only give equations corresponding to its timing convention: We allow for a different growth rate and a different age earnings profile, with the symbols gi and {aeij}. Expectations will be where we understand that, if we were to develop these probability mass functions, we should adopt superscripts to distinguish these functions and their domains from those given in (9).We now record the key equations for the mean value recursions, which will be combined into matrix form below. We consider (10a). The present value of all future activity, starting active, will have several components, the first of which is a payment ½ of a year from age x, hence the ½ year discount factor β on the $1 for ½ of a period, and therefore the .5, augmented by the age earnings factor, aex (which would typically be normalized to 1 for age x). Next, if there is a transition resulting in remaining active, which occurs at x + ½, then by our Convention A, .5 of this is paid then (this .5 must again be discounted back to x by β, and it occurs with probability apxa, so that its expected value is.5βaexapxa, while .5 is paid 1 year later, and so will be embedded in the present value aepvx+1a. The remaining payments will occur at x+1.5 and later, and their present value, as of x+1, aepvx+1a if the transition at x + ½ is to active, and is iepvx+1a if the x + ½ transition is to inactive. These happen with probabilities apxa and apxi, so that the expected present value at x+1 is {aepvx+1a apxa + iepvx+1a apxi}. However, we need to adjust these x+1 values back to age x, which requires discounting for two half periods, hence by β2.The derivation of (10b)–(10d) are similar, where in (10c)–(10d) we are counting inactivity rather than activity. Recursions (10a)–(10d) may be gathered into matrix equalities, where their structure becomes clearer. Gathering the four equations gives Giving each matrix its self-evident definition in (11a), and using the symbol * to denote matrix multiplication (since the juxtaposition of BAEx could be confused with B * AEx), results in which we record as the Expected Present Value Theorem. When labor force participation follows the Markov model, the expected present value of each base (age x) year's $1 of wages and fringe benefits is given by the (1,1) element of EPVx = B * AEx * .5{ I + Px } + B2 * EPV x+1 * Px when starting active, and by the (1,2) element when starting inactive. The expected present value of each base (age x) year's $1 in the inactive state is given by the (2,1) element when starting active, and by the (2,2) element when starting inactive.Several comments are in order.Comment 1. While derived by a forward looking argument, the great computational use of the result runs time (age) in the opposite direction. We need some value for EPVx+1 at some age x+1 on the right hand side to be able to use the equation to simply and quickly calculate all previous values EPVx, EPVx−1,. The natural age would equate x+1 to TA – 1, since at that age, only ½ year of life remains. When measuring activity, one receives payment for .5 aeTA–1 years discounted by β, if active at TA–1 and 0 otherwise with certainty. The same is true when measuring inactivity—5 aeiTA–1 is received and discounted at βi, if inactive at TA − 1 and nothing otherwise. Thus, to start the recursion we have the boundary conditionComment 2. If TA − 1 is taken to be 110, because of the values of AEx and Px between x = 80 and x = 109, the values in EPV80 are quite insensitive to either the choice of TA − 1 or the values chosen for EPVTA −1.Comment 3. Our present value or Scogden Tables, like the Ogden Tables, are calculated with no age earnings profile, i.e. with AEx = I for all ages x. Nevertheless, (11b) shows that it is quite straightforward to introduce age earnings sequences normalized with aex = aeix = 1 into the analysis.Comment 4. When β = βi = 1 so that B = I, and AEx = I, the recursion reduces to EPVx = 5{I + Px } + EPVx+1 Px. But this is the same recursion which Ex obeys: Ex =.5{ I + Px } + E x+1Px, (c.f. Skoog, 2002; Foster and Skoog. 2004). Hence with no discounting, since it follows that Ex = EPVx, where is the matrix of expected time in the upper right superscripted state, given that one has started at age x in the upper left superscripted state. Consequently, we have proved that in this case Ex= EPVx.Comment 5. Reflecting the dependence of the expected present values m epvxn on the growth rates and the interest rate, for any initial and final states m and n, EPVx (g, gi,r) is continuous (this is intuitive from the recursion, and is demonstrated below). Consequently the matrix function EPVx (g, gi,r) is continuous as g and gi approach r, and the limit is well known: Aside from the elegant theoretical interpretation of EPVx as an extension of a well known function off its boundary, the equality is useful in checking a computational algorithm for EPVx (g, gi, r).Comment 6. An exact expression for the means provides computational checks when the entire probability mass function is generated by simulation, as it must be when its general properties, aside from its mean, are being studied. In this case, the question will arise as to whether a simulated sample size is large enough to be assured that the strong law of large numbers has taken effect. By calculating the EPVx for the simulation and comparing the generated result with known answer via the recursion, one can check for the appropriateness of sample size—if the simulated mean is not accurate, a larger sample size is needed. Conversely, there is no need to run the computer for a week if the means converge with a sample size in the tens of thousands, which would be computed in seconds for one exact age.Comment 7. Since diagonal matrices B and AEx commute, the key recursion may be re-written as (not requiring * to resolve ambiguity).Comment 8. Setting AEx = I results in the equation:Comment 9. The information in recursion (14a) may be re-expressed in a form that makes the calculations more familiar to forensic economists, by repeated advancement of the age and substitution. Re-writing (14a) for ages x, x+1 and x+2, and noting that this may be continued to x +(TA – x − 1), we have:Substituting the second equation into the first results in while substituting the third expression yields Before expanding (15), we define, as in Skoog (2002), the jΠx matrices as: Elements of the matrices are probabilities that a worker in the state specified by the upper left superscripted state at age x will be in the state specified by the upper right superscript at age x + j. This is true by definition for 1Πx, and is evident if the terms in 2Πx = Px+1Px on the right hand side are written out.We return to (15), which needs expansion. Two groupings are possible, each with an interesting economic interpretation. The first expansion organizing the flows in age intervals (x, x+1], (x +1, x+2], (x +2, x+3], … is Thus, continuing the substitutions in (17a) we have where the second payment and the first payment of .5 from two adjacent transitions are combined.The second expansion, organizing the flows in transition intervals (x, x+.5], (x+.5, x+1.5], (x+1.5, x+2.5],…, resulting from re-arranging the grouping of terms in (15), is The first term in (18) is the second payment made at x+.5 from the previous, x–.5 transition. The terms in the summation capture the two payments of .5 generated from the x+.5, x+1.5,… transitions. In both representations, the half period discount factors B B3,... are centered on mid-points, and since multiplication and addition over a finite number of terms preserve continuity of the arguments, this establishes the continuity of EPV x in β, βi claimed in equation (13).Comment 10. We provide tables below which may be used with interpolation between ages for the injured or decedent when damages are computed as of the accident date. Support for calculation as of this date is found in the famous footnote 22 of Jones and Laughlin Steel Corp. v. Pfeifer, 462 U.S. 523. In such a case, under federal law, the amount so calculated would be advanced by a prejudgment interest factor. In most state cases, however, the damage period is split between pre-trial losses, on which no discounting or augmenting (prejudgment interest) is calculated, and future losses, which are discounted to present value. In such a case, where the damage flows are treated asymmetrically, either an approximation would be needed or an extension of these tables would need to be computed. For example, assuming that no more than 7 years lapse between the tort and the trial, for a personal injury case where the plaintiff is known to have survived to the trial date, the probabilities of being active and inactive in each of the pre-trial years could be computed in a separate or extended chart, zeroing out mortality. Similar calculations incorporating mortality would be pertinent in state wrongful death cases, where pre-trial survival would not be known. Such probabilities might be useful in refining calculations of pre-trial loss, and in determining the weights for a PVAx,a – PVAx,i weighted average post-trial calculation. In any event, the tables provided below, with or without such adjustments, will provide both a reality check on present value calculations as well as a clear statement of the underlying variation about the expected present value which forensic economists currently report.Comment 11. Since the present value calculation is nonlinear in the NDR, the following tables permit a quick approximate answer to the question about how present value calculations would change with a small increase or decrease in the NDR.Table 1 shows some characteristics of PVAx,a for x = 16,...,75 under Convention A using (6a) for initially active male high school graduates (assuming age-earnings factors aej of 1). This table contains simulation estimates4 for the expected present value, median present value, standard deviation, skewness, kurtosis and the 10th, 25th, 75th, and 90th percentile points of the present value pmf at NDR's of 0, .0050, .0100, .0125, .0150, .0175, .0200, .0250, .0300, .0350, .0400. Table 1 is based on simulating 30,000 sample paths at each age x. Certain obvious conclusions, present empirically in Table 1, follow from the foregoing mathematical treatment and discussion: (1) Expected present value varies inversely with NDR; (2) Expected present value varies inversely with age; and (3) As with life annuities and annuities certain, decrements (measured in absolute value) in expected present values become smaller for equal increases in the NDR.Beyond the magnitudes of the entries themselves, Table 1 reveals four less obvious insights about present value pmf's. (1) Given age, the standard deviation of the present value random variable declines with the NDR. Figures 3a–3c show present value pmf's at age 30 for NDR = 0, .02, and .04 – declining standard deviations with higher NDR's are apparent in these figures which have the same scaling of axes in order to facilitate comparability. In addition, at young ages the standard devia