The Value of Future Earnings in Perfect Foresight Equilibrium1
Scott Gilbert
Southern Illinois University Carbondale
Abstract
The present work considers the problem of valuing a future income stream in a perfect foresight
economy. In this setting, with competitive equilibrium in labor and asset markets, market valuation of
labor-generated income streams can be very simple. However, it can also be undone by moral hazard, in
which case valuation may be based instead on fair compensation. I show that perfect foresight valuation
emerges somewhat imperfectly in the forensic economics literature. To apply this type of valuation, the
economist must form an expectation E[ P ] about perfect foresight price P . I consider several models
of this expectation, some of which yield standard present value formulas. I find that, while standard
formulas ‘fit’ historical data well in some respects, they miss some dynamics that are better captured by
more advanced econometric methods.
1
For helpful input I thank three anonymous referees, executive editors Steven Shapiro and James Ciecka, William
Brandt, Lane Blume, and seminar participants at the 2010 Chicago meeting of the Illinois Economics Association
and the 2011 Denver meeting of the National Association of Forensic Economics. All remaining errors are my own.
1. Introduction
Forensic economists place a value on the future incomes that could have been earned by people who
were wrongfully injured or killed. They do this at the request of attorneys who represent the wronged
party or the alleged wrong-doer. There are many methods by which future incomes can be valued,2 but
most amount to a two-step process: first you pick an interest rate for discounting future earnings, a rate
of growth for future earnings, and an expected number of remaining work years; then you plug the
afore-mentioned values into a standard present value formula. This process does not require the
economist to write down a formal economic model. However, such models can potentially provide
additional economic insight and intuition into the valuation process. They can also suggest
new methods of valuation.
The present work considers the problem of valuing a future income stream, in the setting of
a perfect foresight economy. In such an economy, rational economic agents perfectly anticipate future
prices3 and the concept of present value is relatively simple and intuitive. The economist who wishes to
value an income stream may not have perfect foresight, and need not observe equilibrium prices, but
can nevertheless form sensible price expectations.
In a perfect foresight economy with equilibrium in highly stylized labor and asset markets, I show that
the equilibrium price P of income streams is synonymous with what forensic economists call the ”total
offset” formula, this being the simplest present value formula in use today. Hence explicit economic
modeling assumptions can endow known valuation methods with a new economic interpretation. At the
same time, joint modeling of labor and assets exposes a potential problem in valuing labor-generated
income streams: moral hazard may preclude a market for such assets, in which case no equilibrium price
may exist. This problem does not rule out valuation, however, as it remains possible to fairly
compensate an individual who has lost an income stream.
I next show that perfect foresight valuation, in various forms, emerges somewhat imperfectly in the
forensic economics literature on present value. For this I delineate various features of perfect foresight
price, and then show how these features emerge piece-meal in a series of present value studies by
2
3
see Martin (2010) for a survey of these
see for example Sargent and Wallace (1973) and Turnovsky (2000).
Schilling (1985), Dulaney (1987), Pelaez (1991), and Brush (2004). With these studies as precedent,
perfect foresight models provide an economic inroad to the valuation of future income streams.
To apply perfect foresight valuation, the economist must form an expectation E[ P ] about perfect
foresight price P . Under simplifying assumptions I show that empirical estimates of E[ P ] coincide with
a standard class of present value formulas used by forensic economists. When income growth and
interest rates are dynamic, the rational expectation E[ P ] is typically dynamic also. Approximating this
dynamic E[ P ] via regression (econometric) models, the resulting formulas are different than those
used by forensic economists.
I test several valuation formulas based on econometric specifications of E[ P ] . I find that standard
present value formulas ‘fit’ historical data well in some respects. However, they miss some dynamics in
P that are better captured by more advanced econometric methods.
The rest of this work is organized as follows. Section 2 characterizes the value of income streams in a
perfect foresight economy. Section 3 reconsiders the forensic economics literature on present value,
from the vantage point of perfect foresight equilibrium. Section 4 provides results on expected price
E[ P ] , Section 5 develops tests of expectations E[ P ] , Section 6 applies the methods to historical data,
and Section 7 concludes.
2. Perfect Foresight Economy
Consider an economy with markets in competitive equilibrium at each date. In the bond market,
suppose that at each date t there exists nominally riskless zero-coupon bonds of maturity m = 1, 2,..., M
and let R t ,m be the bond's gross return upon maturity. At date 0 the problem is to determine the
equilibrium value of a stream of non-negative future payments Y 1,...,Y N , all expressed in nominal terms,
withY i arriving at date i for each i = 1, 2,..., N .
Suppose that all economic agents have perfect foresight concerning future incomes.
Also, suppose that financial markets are perfectly competitive, in which case arbitrage is absent.
2.1. Equilibrium Price
I list here some well-known results about perfect foresight equilibrium, as they will be useful later.
At time 0 the competitive equilibrium price P0 of an income stream Y1 ,..., YN is:
N
P0  
i 1
Yi
R0,i
(1)
If all economic agents have perfect foresight concerning the bond market, and if the bond market is in
competitive equilibrium, then the gross return R0,m on a bond of maturity m  1 equals the product of
returns on successive 1-period bonds:
m 1
R0,m   Ri ,1
(2)
i 0
in which case:
N
P0  
i 1
Yi
(3)
i 1
R
j 0
j ,1
In terms of income growth rate w and rate of return r , defined as:
Yt  Yt 1
Yt 1
(4)
rt  Rt 1,1  1
(5)
wt 
for t  1, 2,..., N , the equilibrium price is:4
i
N
P0  Y0 
1 wj
i 1 j 1
(6)
1  rj
Equivalently:
i
N
P0  Y0  (1  g j )
(7)
i 1 j 1
where g j is the net growth rate:
gj 
1 wj
1  rj
1
(8)
*
Another way to simplify the appearance of pricing equation 6 is via geometric mean values w and r * :
 1  wi* 
P0  Y0  
* 
i 1  1  ri 
N
i
(9)
with:5
1/ i
 i

w    (1  w j )   1
 j 1

*
i
(10)
1/ i
 i

ri    (1  rj )   1
 j 1

*
(11)
for i  1, 2,..., N .
If income growth w and bond return r are constant over time, equilibrium price P0 is:
4
Explanation: With perfect foresight the income stream is riskless, and in competitive equilibrium there is no
arbitrage, in which case equations 1and 2 must hold. See Bailey (2005, equation 10.7).
1/ i
 i

5
Similarly, formula 7 can be written P0  Y0  (1  g ) with g    (1  g j )   1 .
i 1
 j 1

N
* i
i
*
i
N
P0  Y0 
i 1
(1  w)i
(1  r )i
(12)
which coincides with the basic textbook definition of present value.6 If income growth and/or returns
are not constant over time, the net growth rate g may nevertheless remain constant, in which case:
N
P0  Y0  (1  g )i
(13)
i 1
2.1 General Equilibrium
In computing equilibrium price P0 I have treated incomes Yi and bond returns ri as exogenous at each
date i , but in general equilibrium these are endogenous and depend on tastes and technology.
Restrictions on tastes and technology can narrow the possible range of incomes and bond returns, and
the range of prices P0 as well.
Consider a complete-markets representative agent economy with a single durable consumption good
which is consumed in each of the periods 0,1,..., N . The agent labors in each period to produce output:
Oi  Li
(14)
with Li the amount of labor and  a number in the range (0,1) . The agent's consumption in period i
is Ci , and leisure time is H i , with time constraint H i  Li  1. Taking consumption in period 0 as the
numeraire good, let qi be the competitive equilibrium price of the consumption good in period i , with
normalization q0  1 . The agent's intertemporal budget constraint is then:
N
N
i 0
i 0
 qiCi   qiOi
(15)
The agent chooses a consumption sequence C  (C0 ,..., CN ) and leisure sequence H  ( H 0 ,..., H N )
so as to maximize lifetime utility U (C, H ) subject to equations 14 and 15, with U (·) a function which I
suppose is log-linear and invariant with respect to each temporal reordering of consumption-leisure
pairs:
6
As in Cvitanic and Zapatero (2004, equation 2.3).
N
U (C, H )    log(Ci )  (1   ) log( H i )
(16)
i 0
*
with  a number in the range (0,1) . Denote by C and H * the utility-maximizing choices of
consumption and leisure, subject to equations 14 and 15.
N
*
Since U (·) is additively separable C maximizes the function
 log(C ) subject to:
i 0
N
q C
i
i 0
with “wealth” W 
 q (1  H
i
i
i
W
(17)
* 
i
) . In competitive equilibrium, the hyperplane in consumption space
i
defined by the consumer's budget constraint (17) is tangent to the indifference surface defined by
N
 log(C )  a for some scalar a , and since the indifference surface is fully symmetric in C ,..., C
i 0
0
i
N
,
equilibrium goods price qi equals 1 in each period i .
*
To solve for C and H * , the Lagrangian function is:
N
L (C, H,  )  U (C, H)    (Ci  (1  H i ) )
(18)
i 1
with  the Lagrangian multiplier. Differentiating L with respect to its arguments,
setting the derivatives to zero, and solving yields:




C 

 1   (1   ) 
*
i
H i* 
1
1   (1   )
(19)
(20)
for each period i .7 Hence the agent consumes the same amount ( C ) in each period, and the agent's
real income from labor is the same at each date, equal to C .
In this stylized general equilibrium model agents have perfect foresight, and hence the perfect foresight
asset pricing equation (6) holds. The real gross return Ri ,1 on a 1-period bond issued at time i must
equal the ratio
qi
of (durable) consumption prices, and with the above-described tastes and
qi 1
technology consumption prices equal 1 in each period, in which case Ri ,1  1 . If there is no inflation,
meaning that each nominal price equals the corresponding real price, then the growth rate wi of
nominal income equals 0 in each period, as does the net nominal return ri on a 1-period bond. Income
in each period is Yi  (1  H * ) , and if the agent can sell a contract at time 0 that delivers the future
income stream Y1 ,..., YN then, in the multiperiod market for consumption, the income stream has
equilibrium value:
P0  Y0 N
(21)
In forensic economics this is called the total offset pricing formula, as income growth just offsets
the bond return when discounting future income.
Nominal income and bond returns can be made-time varying, in the general equilibrium model,
by introducing non-distortionary inflation – with nominal goods price and nominal income undergoing
the same percentage change over time. Let  i the inflation rate between period i  1 and i . By
assumption wi   i . Also, with the 1-period bond's real net return equal to 0 in each period, the
nominal gross return ri equals the inflation rate  i . So ri   i  wi , and the net growth rate gi equals
0 in each period. In general equilibrium, this yields the total offset formula (21), as earlier, but now with
any sequence Y0 , Y1 ,..., YN of nominal incomes.
The foregoing does not imply that the total offset formula is always valid in general equilibrium. I made
simplifying assumptions about tastes and technology, and if these assumptions are relaxed then general
7
Here I use the fact that
q0 
 qN  1.
equilibrium can admit many possible values for P0 , as in Rowe (1991). Assuming perfect foresight, all
such equilibrium values must still conform to the pricing formula (6).
Stepping back further, general equilibrium does not even imply a market for all income streams.
I earlier assumed that the representative agent can sell a contract, a time 0, promising future delivery of
income stream Y1 ,..., YN . However, this income stream is the fruit of the agent's labor, and future
delivery is contingent on the agent's future work. A moral hazard then arises, as the agent may shirk the
future work. In this event the proposed contract may be unmarketable, and neither the total offset
formula (21) nor the more general perfect foresight formula (6) need offer a credible value for the
contract.
2.3. Fair Value
Even if a laborer's income stream can not be marketed or priced in equilibrium terms, it may still be
possible to attach a value to it. Consider a person who has been deprived of his claim to a future income
stream Y1 , Y2 ,..., YN , and wishes to be compensated fairly for this loss via a lump sum payment now. A
possible definition of “fair” is that the lump sum be just sufficient for the person to recoup his loss.
Definition: The fair value of a future income stream is that amount of money A0 which, at time 0 ,
just suffices to generate the income stream Y1 ,..., YN via a fund invested in bonds, withdrawing in each
period the specified income amount.
In a perfect foresight economy the fair value A0 is the amount P0 defined by equation 7, due to the
absence of arbitrage. To show this it suffices to split the lump sum P0 into N parts, the i -th part equal
to Yi / R0,i , and invest the i -th part in a bond that matures at time i , for i  1, 2,..., N . By construction,
this fund generates the income sequence exactly.
One can also view fair value as the solution to a dynamic programming problem. At each date suppose a
bond fund manager can buy or sell bonds of maturities m  1, 2,..., M for some fixed M . Over time
the bond fund earns interest and/or capital gains/losses, and pays out the amount Yi at date i .
Denoting by  i the vector of portfolio weights for the bond fund at date i , with weight distributed
across bonds of maturity 1,..., M , and let r i be its net return between dates i  1 and i . Fair value
then solves the following problem:
Minimize A0
subject to the conditions:
Ai  Ai 1 (1  r i )  Yi ,
i  1, 2,..., N
(22)
AN  0
(23)
with equation 23 stating that the fund is fully depleted at the final payout date. In perfect foresight
equilibrium the bond portfolio return must equal the interest rate on a 1-period bond:
r i  ri ,
i  1,..., N
(24)
in which case equation 22 reduces to:
Ai  Ai 1 (1  ri )  Yi ,
i  1, 2,..., N
(25)
To solve for A0 in terms of ri and wi , i  1, 2,..., N , we can use equation 4 to write Yi as:
Yi  (1  wi )Yi 1
(26)
Substituting for Yi in equation 25 via equation 26, and then substituting for AN 1 in equation 25 in
terms AN  2 , AN 2 in terms AN 3 , etc., and finally applying the depletion equation 23, one obtains:
N
N 1
N 2
A0 N
(1  ri )   (1  wi )  (1  rN ) (1  wi )  (1  rN )(1  rN 1 ) (1  wi ) 

Y0 i 1
i 1
i 1
i 1
 N

   (1  ri )  (1  w1 )
 i 2

Solving for A0 , it equals P0 as defined by equation 6.
3. Discussion
The literature on forensic economics, now over a quarter century old, has extensively studied the
present value of future income. Generally implicit in this work is the assumption that the economist
values future income in a manner consistent with market principles. Some authors come close to
explicitly stating an assumption of perfect foresight equilibrium. In particular, Schilling (1985) invokes
perfect foresight in computing a present value of an income stream via dynamic programming, similar
to the fair value concept A0 defined in Section 2.3. Subsequent works, including Dulaney (1987), Pelaez
(1991), and Brush (2004), also use language and methods consistent with the idea of a perfect foresight
economy, though they do not explicitly use the term perfect foresight.
The following passages discuss four works (Shilling 1985, Dulaney 1987, Pelaez 1991, Brush 2004) on
present value, from the vantage point of the perfect foresight equilibrium results in Section 2.
3.1. Schilling (1985)
In this work Shilling seeks a value, which I will denote S 0 , which is the “present value of an ex post
earnings series computed using ex post interest rates”. Using historical data:
“an ex post estimate of the present value cost of providing the lost income was performed for every
base year using the historical record of earnings and interest rates. The resulting estimate is equivalent
to what would be produced via perfect foresight. It represents a lower bound to the achievable
cost of providing income replacement...”
To provide the desired income replacement, Schilling uses bonds of various maturities. He does not
provide a formula for S 0 , and instead uses a computer to find “the time path of the maximum interest
yield portfolio” that just funds the income sequence.8
8
for this he assumes that any bond which is purchased is held to maturity.
To illustrate Schilling's present value S 0 , consider first a in a two-period economy with a single payment
Y1 at time 1. Here the maximum interest yield equals the return r1 on the one-period bond, and S 0
equals to the perfect foresight equilibrium price P0  Y1 / (1  r1 ) .
In a three-period economy with successive payments Y1 and Y2 , only the 1-period bond can fund
payment Y1 , whereas Y2 can be funded via back-to-back 1-period bonds or a two-period bond.
Choosing maximum yield, S 0 is then:
S0 
Y1
Y2

 P0
1  r1 max[( R0,2 ,(1  r1 )(1  r2 )]
(27)
Note that S 0 is similar to fair value as defined in Section 2.3, yet different in a crucial respect since in
computing S 0 Schilling does not enforce the perfect foresight equilibrium condition that the bond
portfolio return r 1 equal the interest rate r1 on a 1-period bond. By definition, r1  r1 , hence S0  P0 .
In perfect foresight equilibrium, r1  r1 and S0  P0 , just as in the 2-period case.
In a general multi-period setting, Schilling's present value S 0 solves the dynamic programming problem
stated in Section 2.3, but without the restriction that r i  ri for each i . The inequality r i  ri holds by
definition, for each i , in which case S0  P0 , with equality holding in perfect foresight equilibrium.
Applied to historical data, an exact equality r i  ri is possible but hardly likely at any date i , given that
Schilling is optimizing over bonds of numerous maturities (m = 1, 5, 10, 15, 25, 30 years). Hence
Schilling's calculated values for S 0 need never equal to the corresponding equilibrium value P0 , and it
is more likely that S0  P0 .
3.2. Dulaney (1987)
In this work Dulaney computes an “actual present value” of future earnings, which I will denote D0 , as
follows:
 1  wi 
D0   

i 1  1  ri 
N
i
(28)
Dulaney does not provide assumptions under which D0 is necessarily the market value of a future
income stream, but perfect foresight seems a natural vantage point. The present value equation 28
appears in subsequent literature, including Duncan and Pflaum (2010) where it is again presented with
language suggestive of perfect foresight, though not stating it outright.
To compare Dulaney's present value D0 to equilibrium P0 , first define:
Gi 
1  wi
, i  1, 2,..., N
1  ri
(29)
Since the income sequence Y1 ,..., YN is assumed non-negative, so are the terms 1  wi , and with gross
bond returns 1  ri also non-negative, so is each Gi .
In the case N  2 the difference between P0 and D0 is:
P0  D0  G1G2  G22
(30)
and if G1 and G2 are both positive then P0  D0  0 if and only if G1  G2 , whereas if G2  0 then the
value G1 is unrestricted.
For general N , if we assume that Gi is positive for i  1, 2,..., N then, by induction, in perfect foresight
equilibrium we obtain G1  G2 
 GN . Denoting by G the common value for each Gi , if D0  P0
then:
N
P0  Y0  G i
(31)
i 1
which coincides with the simplified equilibrium pricing equation 13 stated earlier.
Hence, Dulaney's present value D0 can be interpreted as equilibrium present value P0 in the situation
where net growth rates are constant over time, but otherwise differs from P0 .
3. 3 Pelaez (1991)
In this work Pelaez interprets present value as a “fair award”, being the “present value of an ex post
earnings series using ex post interest rates”, which I will denote by F0 . He does not further describe
those conditions under which F0 is fair, but his emphasis on ex post values is consistent with an
assumption of perfect foresight. To compute F0 Pelaez uses equation 9 in Section 2.1, which is one way
of writing the perfect foresight equilibrium price P0 .9 The idea of fair value, as defined in Section 2.3, is
then a natural way to reinterpret the sense in which F0 is a fair award.
Pelaez describes F0 as a “general present value model” in the sense that the values of wi* and ri* are
unrestricted at each date i . While F0 is indeed a generally valid model of equilibrium price in a perfect
foresight economy, it is not generally valid if perfect foresight fails. If, for example, agents have perfect
foresight about future incomes but not future bond returns, then the pricing equation 1 in Section 2.1
remains valid but equation 9 clearly does not. Hence, an explicit assumption of perfect foresight endows
Pelaez' fair value concept with economic meaning.
3.4 Brush (2004)
In this work Brush defines “actual present value”, which I will denote by B0 , this being ``the amount of
money needed to fund an income stream''. This is similar to the fair value concept from Section 2.3, as
Brush's fund invests only in bonds. However, Brush only allows 1-period bonds in his fund, whereas
longer-term bonds are allowed in the definition of fair value. Brush's ``actual present value'' is consistent
with what one would see with perfect foresight, and in perfect foresight equilibrium the discrepancy
between B0 and fair value vanishes because 1-period bonds span all investment opportunities, in which
case B0 equals the equilibrium price P0 .
9
Pelaez values a stream
Y0 , Y1 ,..., YN 1 , rather than Y1 ,..., YN , but his formula extends to the latter case.
To compute B0 , Brush solves the dynamic programming problem defined by equations 25 and 23 in
Section 2.3, via computer. As shown in Section 2.3, there is an explicit solution to these equations, which
is P0 as specified by equation 6. Brush appears to interpret the B0 as equal to Dulaney's value D0 , as
he reports that:
“An attempt was made to calculate ... actual present values in a manner consistent with the methods
described in Dulaney's paper. The Appendix contains a detailed description of the computational
methods used in the present paper. These methods ... do not in fact replicate Dulaney's results, for
reasons that are not entirely clear.”
Brush's value B0 is identical to equilibrium price P0 , but as discussed in Section 3.2 Dulaney's value D0
is equal to P0 only if net growth rates gi are the same at each date i .10 Hence to endow B0 with
Brush's intended meaning, in perfect foresight equilibrium it is necessary to assume a constant net
growth rate.
4. Price Expectations
Consider now an economist who is asked to value a stream of future income. If the economist has
perfect foresight, and the economy is in perfect foresight equilibrium, then the economist can compute
the equilibrium price P0 . If he does not have perfect foresight, but economic agents do, the economist
can at least hope to hold a rational expectation E0 [ P0 ] about P0 , given information available at time 0 .
As is usual in economic analysis, I define expectation as mean value:
E0 [ P0 ]   pdF0 ( p)
where F0 is the conditional (cumulative) distribution function for the random variable P0 .
In general, the conditional distribution F0 and expectation E0 for equilibrium price are of unknown
form. However, under simplifying assumptions it is possible for the economist to develop tractable
10
here I assume that the non-negative value 1  gi is positive for each i.
(32)
formulas for E0 [ P0 ] , and to estimate these formulas using historical data. Section 4.1 takes this
approach in the case of total offset, and Section 4.2 generalizes the situation to the case of serially
independent growth rates g1 ,..., g N . If simplifying assumptions fail then the posited form of E0 [ P0 ]
may be misspecified, and Section 4.3 considers this problem in the context of serially dependent growth
rates.
4.1. Total Offset
In the general equilibrium example presented in Section 2.2 a total offset of income growth and interest
rates implies that the net growth rate gi equals 0 in each period i , hence Gi  1  gi  1 for each i .
The economist therefore observes the perfect foresight price P0  Y0 N , so has expectation:
E0 [ P0 ]  Y0 N
(33)
4.2. IID Growth Rates
Generalizing the total offset case, suppose that net growth rates gi are independent and identically
distributed (IID) with finite variance. Equivalently, suppose that the series Gi  1  gi , i  1,..., N ,
is IID with finite variance and a common expectation E[G ] . Assume also that conditional expectation
E0 [ P0 ] uses only current and past G as conditioning information:
E0 [ P0 ]  E[ P0 | G0 , G1 ,...]
With independence the expected value of a product of G1G2
values E[G1 ]E[G2 ]
(34)
G j equals the product of expected
E[G j ] , in which case perfect foresight equation 7 yields a price expectation:
N
i
E0 [ P0 ]  Y0  E[G j ]
(35)
i 1 j 1
Also, with growth rates identically distributed over time, each expectation E[G j ] has the same value, in
which case:
N
E0 [ P0 ]  Y0  ( E[G ])i
(36)
i 1
In equation 36 the base income Y0 and number N of time periods are assumed given, and the only
unknown is E[G ] . This is still more complicated than total offset or the simple present value formula
(13), but using historical data we can hope to estimate E[G ] consistently, under the maintained
assumption of IID finite-variance net growth rates, yielding:
N
Eˆ [ P0 ]  Y0  (G )i
(37)
i 1
with G the average value of G computed over some historical period.
Equation 37, and variants thereof, is the method which most forensic economists use to value future
income streams.11 This method emerges naturally in the context of perfect foresight valuation so long as
growth rates G are IID.
4.3. Serially Dependent Growth Rates
If growth rates Gi are serially dependent then formula 36 is generally biased for E0 [ P0 ] . To illustrate,
denote by E0*[ P0 ] the value of E0 [ P0 ] obtained from (36), and let N  1. Here:
E0 [ P0 ]  Y0 E0 [G1 ]
(38)
E0*[ P0 ]  Y0 E[G1 ]
(39)
E0 [ P0 ]  E0*[ P0 ]  Y0 ( E0 [G1 ]  E[G1 ])
(40)
and the expectation bias is:
which is due to the difference between conditional expectation E0 and unconditional expectation E .
To counter the bias in expectation E0* , a relatively simple approach is to replace E with E0 on the
right-hand side of equation 35, yielding a valuation formula:
11
See Brush (2004) and references therein for discussion.
N
i
V0  Y0  E0 [G j ]
(41)
i 1 j 1
This formula takes as input forecasts E0 [G j ] of future growth rates G j . Variants of this forecasting
approach to present value are sometimes seen in forensic economic practice.12 It produces no bias for
E0 [ P0 ] when N  1, but if N  2 then:
E0 [ P0 ]  Y0 ( E0 [G1 ]  E0 [G1G2 ])
(42)
E0 [ P0 ]  V0  Y0 ( E0 [G1G2 ]  E0 [G1 ]E0 [G2 ])
(43)
The bias E0 [ P0 ]  V0 is then equal to Y0 times the conditional covariance Cov0 [G1 , G2 ] between G1
and G2 , given information available at time 0 . To evaluate this covariance consider the case where the
time series ..., G1 , G0 , G1 ,... is a stationary linear first-order autoregressive (AR(1)) process:
Gt     Gt 1   t
(44)
with  t an IID series having mean 0 and variance  2  0 . Some calculation yields:
Cov0 (G1 , G2 )  E0 [G1G2 ]  E0 [G1 ]E0 [G2 ]
 E0 [G1 (   G1 )]  E0 [G1]E0 [   G1 ]
  2
Noting that the autoregression slope  is equal to the correlation between Gt and Gt 1 ,
if this correlation is positive then V0 is a downward-based for E0 [ P0 ] , while if negative then it is
upward-biased.
12
Interest rate forecasts are sometimes used in present value calculations, as reported by Brookshire, Luthy and
Slesnick (2009). See Rosenberg (2010) for recent discussion of wage growth forecasts as in input to present value
calculation.
With bias in expectation formulas (35) and (41) when G is dynamic, let's consider more fully what the
expectation E0 [ P0 ] should look like. Generalizing the AR(1) case, suppose that Gt is a stationary
recurrent Markov process with a time-homogeneous transition density. As P0 is computable in terms of
G1 ,..., GN , if Gt is Markovian then the distribution of P0 conditional on G0 , G1 , G2 ,... is a function of
G0 alone, and if (34) also holds then the expectation E0 [ P0 ] takes the form:
E0 [ P0 ]  f (G0 )
(45)
for some function f (·) . One possibility is that f is linear, as in the regression model:
P    G  
(46)
with intercept  , slope  , and random error  . Here E0 [ P0 ]     G0 .
Since P0 is actually a nonlinear function of future values G1 ,..., GN , E0 [ P0 ] may be nonlinear in G0 , as
in the quadratic regression model:
P    1G   2G 2  
(47)
where E0 [ P0 ]    1G0   2G02 .
To my knowledge forensic economists have not used regression models like (46) or (47) as tools for
computing present value. A difficulty is that parameters  and  are unknown, but these can be
estimated from historical data via regression of P on G . The motivation for doing so is to address
possible bias in traditional formulas (35) and (41) for approximating the expected value E0 [ P0 ] of
perfect foresight value P0 . This motive is only compelling if perfect foresight is itself a tolerable
simplification of reality.
5. Tests
Having identified some tractable forms for price expectation E0 [ P0 ] , we can test the implications of
these forms via historical data. I first consider the static or unconditional valuation formula (36), whose
implementation is a present value method familiar to forensic economists. Comparing the fitted formula
to the average historical price P yields some tests of (36). I then briefly discuss tests of dynamic
valuation formulas based on regressions (46) and (47).
5.1. Total Offset
With total offset, if base income Y0 is set equal to 1 then the hypothesis to be tested is:
H 0 : E[ P]  N
and a corresponding z test statistic is:
z
PN
sP
(48)
with s P the standard error for P . The full implication of total offset is that Pt  N for all periods t ,
which would render the z statistic problematic as then sP  0 . However, I will test only the implication
H 0 , and will allow P to be stochastic and time-varying. Assuming that the time series Pt is stationary
and mean-reverting, a suitable standard error s P , robust to autocorrelation in Pt , is:
1
2
 K
ˆ (P , P ) 
sP     k 
t
t k 
 k 0

(49)
ˆ ( P , P ) the sample covariance between P and its lag P , and with lag weights  specified
with 
k
t
t k
t
t k
in the form of Bartlett (1946) with lag-truncation as in Newey and West (1987).
5.2. IID Growth Rate
When net growth rates are IID the conditional expectation E0 [ P0 ] equals the unconditional one
E[ P0 ]  E[ P1 ] 
N
, with a special form for the latter:
H 0 : E[ P]    E[G ]
i 1
i
I will test this “static” expectation hypothesis against a two-sided alternative, and for this purpose I
construct a z test statistic as follows:
U
sU
z
(50)
U  P   G 
N
I
(51)
i 1
sU  ˆVˆQ ˆ
(52)
Qt  ( Pt , Gt )
(53)
K
ˆ (Q , Q )
VˆQ    k 
t
t k
(54)
k 0
dU
dQ
(55)
ˆ1  1
(56)
ˆ 
ˆ2   i  G 
N
i 1
(57)
i 1
Here U is the discrepancy between the historical average P , of prices Pt over all dates at which it is
computable via equation 6 in Section 2.1, and the estimate Eˆ [ P ] defined by equation 37 in Section 4.2,
with G computed as the historical average of Gt over all available dates. The standard error of U is sU
, computed via the delta method -- which applies a first-order Taylor series expansion to U as a function
of P and G . To this end, VˆQ is an estimate of the variance/covariance matrix of the Q with
Q  ( P, G ) , and for robustness to serial dependence in P the formula for VˆQ incorporates
autocovariances with lag weights  k which I specify as the Newey-West autocorrelation-robust form,
analogous to equation 49. ̂ is the vector of partial derivatives of U with respect to P and
G.
5.3. Dynamic Growth Rate
The z tests defined by equations 48 and 50 examine how well price estimates P̂ fit a price average P ,
but if Pt is dynamic then it may be more useful to consider the fit of a time-varying estimate Pˆt to Pt
period-by-period. This is easy in the context of the regression models (46) and (47), via an R square
statistic. Non-zero slopes in these models imply a time-varying expectation E[ Pt | Gt ] , so long as the
growth rate Gt is time-varying. Hence if a test rejects the hypothesis of zero slope then it also rejects
the idea expressed in (36) that rational price expectations are static.
6. Application
I apply the methods in Section 5 to U.S. macroeconomic annual data for years 1953-2009. The measure
of income is U.S. hourly compensation for the business sector, obtained from the U.S. department of
Labor, Bureau of Labor Statistics, in the online Productivity and Costs tables, as an annual growth rate.
The measure of bond return is the market yield on U.S. Treasuries, 1-year constant maturity, quoted on
investment basis, and obtained as an annual average from the U.S. Federal Reserve Board, online table
H.15.13
For horizons N  1,5,10,15, 20 , I compute perfect foresight value Pt for each feasible base period t
via equation 6 in Section 2.1. For longer horizons the Pt series is shorter, due to the longer stretch of
leading values for G needed to compute P for higher N . I also compute the estimated price
Pˆ  Eˆ [ P ] based on the total offset formula 33) and the more general static or unconditional formula
(37). Table 1 reports the historical average of price and its estimates, for each horizon. In terms of `fit',
both estimates P̂ are fairly close to P on average, though the discrepancy grows with horizon N .
13
For the period 1953-1961 the Board does not report annual average yields, but does report a monthly series for
April 1953 through December 1961. I annualize the monthly data to get annual average yields for these years.
Table 2 reports z tests for the ”total offset” hypothesis and the more general “static” hypothesis
spelled out in Section 5. There is no significant evidence against either of them, at the 10 percent level,
and this is perhaps not surprising given the closeness of P and its estimate P̂ on average, as reported in
Table 1. On the whole, Tables 1 and 2 demonstrate a sense in which conventional present value
methods ‘fit’ U.S. macroeconomic data well, on average.
Table 3 reports statistics for the linear regression (46) of price Pt on growth rate Gt , at each horizon N .
At all horizons the R 2 statistic is non-negligible, though it is smaller at longer horizons. Tests based on t
statistics for this regression reject the hypothesis of zero slope, at the 1 percent significance level.
Standard errors, and t tests, are based on the Newey-West heteroskedasticity and autocorrelationrobust method.
Table 4 reports on the quadratic regression (47). The fit of this model is only marginally better than that
of the linear model, in terms of (adjusted) R 2 . Scatter plots (omitted, for brevity) of P on G show no
obvious nonlinear relationship. Wald tests of the zero slopes hypothesis 1   2  0 all reject the null
at the 1 percent significance level. Standard errors, and Wald tests, are based on the Newey-West
method.
The evidence in Tables 3 and 4 suggests that the conditional expectation E[ Pt | Gt ] does not reduce
to the unconditional or static expectation E[ Pt ] , when applied to U.S. macroeconomic data. Hence, if
present value is cast as the perfect foresight price of future income streams, an econometric model like
the linear regression (46) may serve as a useful tool for present value calculation, alongside calculations
based on the static formula (36). An econometric model adds complexity, but in applying the static
formula (36) some forensic economists already use an econometric model to select an appropriate value
for the income stream horizon N .
7. Conclusion
Perfect foresight about the economic future is an assumption which is trivially false, and while this fact
could render the present work irrelevant I argue instead that this strong assumption yields a tractable
economic model and relatively simple valuation methods, including a class of methods commonly used
by forensic economists to value future income streams. The model should not be relied upon
exclusively. One generalization of perfect foresight is risk neutrality, and future research might extend
the themes in the present work to a risk neutral economy.
Table 1: Perfect Foresight Value P and Estimates P̂
Horizon ( N )
1
5
time period
'53-'09
average P
0.997
average P̂ , total offset
average P̂ , static
15
20
'53-'04
'53-'99 '53-'94
'53-'89
4.959
9.776 14.405
18.964
1
5
0.997
4.957
10
10
15
20
9.842 14.656
19.402
Table 2: * Tests of Price Expectations
Horizon ( N )
hypothesis
1
5
10
15
20
E[ Pt ]  N
-0.497
-0.511
-0.840
-1.003
-1.004
0.000
-0.024
-0.10
-0.227
-0.411
N
E[ Pt ]    E[G ]
i
i 1
*
Note: All numbers are z statistics, asymptotically distributed standard normal under the null
hypothesis.
Table 3: * Linear Model of P on G
Horizon ( N )
statistic
R2
intercept
slope
*
1
5
10
15
20
0.582
0.460
0.316
0.259
0.185
0.233
-3.411
-11.883
-22.625
-29.854
(0.077)
(0.954)
(4.613)
(9.627)
(15.980)
0.766
8.388
21.733
37.091
48.774
(0.076)
(0.955)
(4.723)
(9.950)
(16.500)
Note: Standard errors are in parentheses.
Table 4: * Quadratic Model of P on G
Horizon ( N )
statistic
R2
intercept
Slope, G
2
Slope, G
Wald
1
5
10
15
20
0.577
0.455
0.318
0.261
0.184
-1.191
-32.129
-160.814
-328.054
-544.239
(2.205)
(36.094)
(119.750)
(271.136)
(562.210)
3.654
66.647
324.163
657.818
1095.514
(4.502)
(73.605)
(244.719)
-1.463
-29.523
-153.412
-315.099
-531.960
(2.296)
(37.458)
(124.832)
(282.674)
(588.109)
78.718
124.76
34.512
21.128
14.720
(554.072) (1150.914)
*Notes: Standard errors are in parentheses. Wald statistics have an asymptotic  2 distribution, with 2
degrees of freedom, under the null hypothesis of zero slopes.
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