Working Paper 149

This paper was rejected by two major journals and has not been revised since 1984. I considered this one of my better papers over the years, so now I am providing it for the world to judge. Referee comments that the paper is not well written are justified criticisms. However, i think that the analysis was timely in 1984 when criticism of efficient market theories was tantamount to heresy. I apologize for equations that do not read as well in HTML as they did in the original DOC file.

DOES A ROSS ECONOMY LUNCH REALLY COST AS MUCH AS
HIRSHLEIFER CUISINE COMPLETE WITH s_m² DESSERT?

Suppose s depicts intensity of states of the economy on the real line where s < 0 are "risky" depressions, s = 0 is "risk" neutral, and s > 0 are "risky" booms. A "world" w will be termed specified at time t if p ( t, s_w ) is specified for a joint combination (point) of s = s_w and p ( t, s ) = p ( t, s_w ) in the Cartesian space of state s and p ( t, s ) probabilities. By "no assumptions in the underlying stochastic processes in the economy," Ross is referring to an arbitrage pricing theory (APT) computation of net present value. APT does, however, implicitly assume that, without specifying s_w and p ( t, s_w ), analysts can somehow forecast both C ( t, s_w ) future prices of a marketed comparison asset and X ( t, s_w ) cash flows of a nonmarketed asset whose m_m net present value is to be estimated from only C ( t, s_w ) price forecasts.

APT literature to date is vague about how cash flows of a capital budeting project can be forecasted when the underlying s_w state of the world economy is unspecified. This paper shows how cash flows may be forecast for a s = s_x state even though comparison asset prices are given for an s = s_c unknown state of the economy. The APT valuation approach is appropriate if s_x meets necessary but not sufficient s_c compatibility tests given in this paper.

This paper also shows how APT has some hidden assumptions that either limit or render meaningless the APT m_mnet present value (NPV):

Suppose s < 0 depicts the intensity of a depression and s > 0 depicts the intensity of a boom in the market-wide economy. For the time being assume s lies on the real line, and p ( t, s ) depicts the probability that the economy is in state s at time t. Under a Market Risk Proposition, equal cash flows do not have equal marginal utilities in the following "risk" conceptualization in Hirshleifer [1965] :

This proposition is stated concisely by Banz and Miller [1978, p. 656], who publish S ( t, s ) risk adjustment present value factors (that are intended for practical application in adjusting a cash flow stream for market risk differentials) as conceptualized below:

Consistent with the Market Risk Proposition, suppose there exist multiplicative m ( t, s ) market risk preference modifiers (for each $1.00 of cash flow in state s at time t) that monotonically decrease on s as follows under the Market Risk Proposition:

1 < m ( t, s ) < ¥	if s < 0 signifies depression states where current claim value of a depression dollar is higher because "funds are hard to come by."
m ( t, s ) = 1	if s = 0 signifies a market risk neutral state.
0 < m ( t, s ) < 1	if s > 0 signifies boom states where claim value of a boom dollar "will have a lower value per dollar of expected return."

The m ( t, s ) is also a measure of "intensity" of a boom or depression. For example, m ( t, s ) = .95 implies s is a mild boom state in which $1.00 cash flow is only worth $ .95 under the Market Risk Proposition. It is "mild" relative to a m ( t, s ) = .75 boom. Conversely, m ( t, s ) = 1.05 implies s is a mild depression in which each $1.00 return is worth $1.05. This is mild relative to a m ( t, s ) = 1.25 depression. The m ( t, s ) market risk modifiers adjust for market (systematic) risk only in the well known context of Markowitz [1959], Sharpe [1963], and Lintner [1965]. Hence, a $1.00 return is adjusted by the same m ( t, s ) whether it flows in at time t in state s from ownership of a solid government bond asset or of a temperamental race horse asset. Individual asset (unsystematic) risk is considered diversifiable in portfolios and is not the type of risk being discussed in this paper. For example, under the capital asset pricing model (CAPM), the bracketed terms in Equation (10) in Bierman and Smidt [1980, p. 326] constitute m ( t, s ) estimates that conform to the Market Risk Propositon.

Without loss of generality, the s meaure of depression / boom intensity of the economy will be somewhat arbitrarily defined as:

It is important to distinguish a risk neutral economy (where m ( t, s ) = 1 for all s on the real line) and a risk neutral state in a risk preferential economy (where m ( t, s ) = 1 only when s = 0). The Market Risk Proposition assumes the economy is not risk neutral. We note the following with a conceptualization of a ( t, 0 ) as an arbitrage probability in a risk preferential economy:

m ( t, s ) =	1 for only s = 0 at time t such that the Market Risk Proposition holds for all s on the real line.	(5)
p ( t, s ) m ( t, s ) =	a ( t, 0 ) by assumptions based upon some general arbitrage free market conditions to be discussed later on, where s = 0 depicts a risk neutral state.	(6)

Equation (6) holds under some very general arbitrage assumptions in APT about the market for all assets. The a ( t, 0 ) values correspond to risk neutral state "arbitrage probabilities" in the context of Gehr [1981] and Cox, Ross, and Rubinstein [1979].

Suppose d ( t, s ) denotes a discount factor in any risk neutral state where buyers and sellers are content with risk free returns. The expected net present values of an asset's X ( t, s ) expected cash flow is m without market risk adjustment and m_m with market risk adjustment as shown below for Y ( t, s ) discounted cash flow of the asset and a ( t, 0 ) = p ( t, s ) m ( t, s ) :

Formulas for discrete time and discrete states are slightly more complicated and need not be discussed here.

In theory, under the Market Risk Proposition, an asset's net present values without and with market risk adjustment are such that:

m_m > m	for an asset whose investment performance excels more in depression states, thereby increasing its expected value under the Market Risk Proposition.
m_m < m	for an asset whose investment performance excels more in boom states, thereby decreasing its expected value under the Market Risk Proposition.

If a p ( t, s ) m ( t, s ) = a ( t, 0 ) condition in Equation (6) holds for ( o < a ( t, 0 ) < 1 ) , it follows from Theorem I and its corollary that p ( t, s ) probability curves (on either side of the s = 0 risk neutral state) are (i) asymmetric, (ii) convex across depression ( s < 0 ) states, and (iii) disjointed across boom ( s > 0 ) states of the world. Probability bounds also vary for depressions versus booms.

Theorem I:	If p ( t, s ) m ( t, s ) - a ( t, 0 ) under Market Risk Proposition and ( 0 < a ( t, 0 ) < 1 ) in APT, this implies:
	IA. ( 0 < p ( t, s ) < a ( t, 0 ) < 1 ) for s < 0 Depressions.
	IB. [ 0 < p ( t, s ) < 1 ] for s > 0 Booms.
Proof:	See Appendix.
Corollary:	Under Theorem I, the m ( t, s ) risk preferences are bounded as follows:
CA:	[ 1 < m ( t, s ) < + ¥ ] for s < 0 Depressions.
CB:	[ a ( t, 0 ) < m ( t, s ) < 1 ) for s > 0 Booms.
Proof:	See Appendix

Proofs of Theorem I and its corollary are rather trivial and are shown in the Appendix of this paper. The implications are that:

Any p ( t, s_w ) curve for a world (point) w specified in Cartesian space by s = s_w and
p ( t, s ) = p ( t, s_w ) will be defined to be "a ( t, 0 ) compatible" and "a ( t, 0 ) incompatible" by the following tests:

From Theorem I and its corollary, a ( t, 0 ) compatability curves plotted over all s on the real line must be asymmetric and disjointed as pictured in Figure 1 for a ( t, 0 ) = .30, .60, and .90 illustrative a ( t, 0 ) compatibility curves. Note the nonlinear continuity across s < 0 depression states versus linearity and disjointedness across s > 0 boom states. Figure 1 illustrates that it is impossible for depressions to have probabilities in excess of a ( t, 0 ) and feasible booms to have probabilities less than a ( t, 0 ).

The a ( t, 0 ) compatibility curves may be plotted on m ( t, s ) instead of s. In that case
p ( t, 0 ) = 0 for ( 0 < m ( t, s ) < a ( t, 0 ) ) and is a nonlinear convex function for
m ( t, s ) > a ( t, 0 ). Any world w defined as a point p ( t, s ) = p ( t, s_w ) and s = s_w on a curve in Figure 2 is "a ( t, 0 ) compatible" provided p ( t, s_w ) > 0. An illustration of
a ( t, 0 ) = .30, .60, and .90 compatibility is shown in Figure 2. Note the nonlinear continuity across m ( t, s ) > 1 depressions versus nonlinear disjointedness across m ( t, s ) < 1 boom states of the economy.

Both Theorem I and the definition of a ( t, 0 ) compatibility of any world w assumed
p ( t, s_w ) m ( t, s_w ) = a ( t, 0 ). Examples of compatible worlds (1 through 12) and incompatible worlds (13 through 18) for a ( t, 0 ) = .60 compatibility curves are illustrated below:

Compatible?	World	State of the World			p ( t, s_w )	m ( t, s_w )	a ( t, 0 )
Yes	w = 1	s_w =	+ 0.65	Boom (severe)	.99	0.61	.60
Yes	w = 2	s_w =	+ 0.50	Boom	.90	0.67	.60
Yes	w = 3	s_w =	+ 0.33	Boom	.80	0.75	.60
Yes	w = 4	s_w =	+ 0.18	Boom	.70	0.85	.60
Yes	w = 5	s_w =	0.00	Neutral	.60	1.00	.60
Yes	w = 6	s_w =	- 0.20	Depression	.50	1.20	.60
Yes	w = 7	s_w =	- 0.50	Depression	.40	1.50	.60
Yes	w = 8	s_w =	- 1.00	Depression	.30	2.00	.60
Yes	w = 9	s_w =	- 2.00	Depression	.20	3.00	.60
Yes	w = 10	s_w =	- 5.00	Depression	.10	6.00	.60
Yes	w = 11	s_w =	- 11.00	Depression	.05	12.00	.60
Yes	w = 12	s_w =	- 59.00	Depression (severe)	.01	60.00	.60

No	w = 13	s_w =	- 59.00	Depression (severe)	.01	61.00	.61
No	w = 14	s_w =	+ 0.20	Depression (mild)	.51	1.20	.61
No	w = 15	s_w =	+ 0.65	Boom (severe)	.98	0.61	.59
No	w = 16	s_w =	0.64	Boom (severe)	.99	0.62	.61
No	w = 17	s_w =	0.18	Boom (mild)	.50	0.85	.43
No	w = 18	s_w =	- 0.69	Boom (severe)	.01	61.00	.61

A market is comprised of many buyers and sellers having different subjective probabilities and risk preferences. The p ( t, s_w ) and m ( t, s_w ) may be viewed as composite (unknown) risk indices that "explain" price movements in an asset market complete in an Arrow-Debreu sense. The p ( t, s ) probabilities are analogous to odds cranked out at a race track to explain positions held by all persons placing bets on a race. Possible outcomes of the race comprise possible "states" of the world.

In a formal proof of the "General Theory of Arbitrage Valuation," Ross [1978] shows how the m_m market risk adjusted net present value of any X ( t, s_c ) stream of cash flows (over t and s_c) from a nonmarketed "comparison" asset may be derived by arbitrage pricing theory (APT) as a linear aggregation of the prices of any comparison asset or bundle of assets in the set of all marketed assets. Most importantly the minimal input requirements for APT derivation of m_m (market) risk adjusted NPV of a cash flow stream are:

The APT method requires no estimates of p ( t, s_c ) stochastic process parameters and the m ( t, s_c ) market risk preferences in the s_c state of the world that generated prices and cash flows, i.e., it derives m_m by leapfrogging the need to compute p ( t, s_c ) and m ( t, s_c ) parameters. It makes few assumptions about individuals' utility functions other than investors prefer more wealth to less wealth. It is not necessary to examine a market portfolio, i.e., only one comparison asset is needed to find an equivalent m_m net present value of any stream of cash flows. The APT conceives of "risk" consistent with the Market Risk Proposition and assumes prices of a given asset do not confound differing individual asset (unsystematic) risks. This assumption is implicit in the "arbitrage free" equilibrium market and justifies using risk free rate of return discounting for valuation.

The APT approach to valuing a market risk adjutsed m_m NPV, in an Arrow-Debreau market, generates a vector of state (contingent) claims prices comprised of estimated S ( t, s_c ) prices such that

The S ( t, s_c ) prices are also termed (market) risk adjustment present value factors (RAPVFs). Since we are free to asume "any" world, there is a tremendous advantage in estimating S ( t, s_c ) state prices by assuming (for computational purposes) that the world is risk neutral i.e. m ( t, s ) = 1 for all t and s even though a risk neutral world is never likely to be the "real" state (s_c) of world ( c ) that generated a C ( t, s_c ) comparison asset price. The S ( t, s_c ) estimates based upon a risk neutrality state assumption ( where s = 0 ) will be denoted S ( t, 0 ). Risk neutrality computational approaches to computing S ( t, 0 ) state prices are described in Cox, Ross, and Rubinstein [1979] and Gehr [1981]. Gehr [1981, p. 15] states (emphasis added):

In order to simplify notation, single period cases will be focused upon initially without loss of generality. In single time period horizon

because m (1, 0 ) = 1.00 under the Market Risk Proposition for a risk neutral state. The a ( t, 0 ) scalers are usually termed the arbitrage probabilities that would be "fair odds" state transition probabilities in a risk neutral world. They are not, however, the "fair odds" p ( t, s_c ) probabilities that generate C ( t, s_c ) comparison asset prices in the s_c states of a risk preferential world. The general equivalency of m_m ( t, s_c ) computed for risk neutral worlds and m_m ( t, s ) for risk preferential worlds is given in Ross [1978, pp. 462-463]. This formed the basis for the APT p ( t, s ) m ( t, s ) = a ( t, 0 ) condition in Theorem I and Equation (6).

Contingency claims theory proposed in Hirshleifer [1965] had its early roots in the "state" conceptualizations in a time state preference model (TSPM) originating in a seminal paper by Arrow [1952], translated into English in Arrow [1964] and Debreu [1959]. In the early Arrow, Debreu, and Hirshleifer theory, it was assumed that s_w "states" in TSPM were conceptualized as to some specified attribute of interest. Hirshleifer [1965, p. 524], for example, focuses on a two-state world generally described as follows: "The two states might be thought of as war versus peace, or prosperity versus depression." The prosperity versus depression conceptualization is further narrowed down in this paper to the m ( t, s ) risk preference definitions under the Market Risk Proposition.

In the early Hirshleifer contingency claims theory, it was necessary to know or assume conceptualized s_w states of the "real" world at each node of a TSPM network. In this early theory, no m_m market risk adjusted present value could be generated without specifying a known underlying stochastic process of each conceptualized s_w state of the "real" world. Hirshleifer [1965] uses p_a and p_b to depict p ( t, a ) and p ( t, b ) transition probabilities. Thus, if "lunch" is described as the process of deriving a m_m expected value, a Hirshleifer lunch is very expensive, because it consists of paying for required types of recipe ingredients difficult to obtain by economic chefs who are mere mortals:

The Hirshleifer lunch would, in theory, generate both a m_m market risk adjusted "entree" and a s_m² "dessert." Hirshleifer [1965, p. 518] states:

The seminal paper originating arbitrage theory appeared in Ross [1976] and was later extended to contingency claims m_m valuation theory in various papers (e.g., Ross [1978] and Cox, Ross, and Rubinstein [1979]. Arbitrage valuation in essence provides a Ross fast food lunch version of the Hirshleifer lunch. Some might be tempted to also call it a relatively free lunch since it derives m_m without requiring the two most expensive recipe ingredients in a Hirshleifer lunch, i.e., the m ( t , s_w ) risk preferences and p ( t, s_w ) probabilities.

The Ross and Hirshleifer lunches have identical m_m entrees. However, the Ross lunch has no s_m² dessert, because s_m² cannot be prepared without knowing p ( t, s_w ) probabilities for s_w states in which m ( t, s_w ) risk preferences are also known.

But is the Ross lunch a free lunch? If it were an almost free lunch, managers in General Motors, Exxon, and every other business firm would dine on it regularly whenever a capital budgeting proposal arose. There are, problematically, two interacting expenses entailed in a Ross lunch that will be termed "forecasting expense" and "CX coupling expense."

The forecasting expense in a Ross lunch is the obvious expense of forecasting risk free discount factors, comparison asset prices, and project cash flows for future time periods. Probably even more important are the hidden costs of assuming the forecasted C ( t, s_c ) comparison asset prices are arbitrage free prices among hypothetically cleared prices of all marketed assets in a future equilibrium state.

However, since both the Hirshleifer and Ross lunches (and any other lunches that value future cash flow streams) entail expensive forecast ingredients, it would be tacky to criticize a Ross lunch for forecasting expense. We may, however, be somewhat critical of the Ross lunch for its obscure and rather confusing CX coupling expense that interacts with forecasting expense. The CX coupling expense in a Ross lunch encompasses all costs or problems of mapping each forecasted C ( t, s_c ) comparison asset price (or its derivative estimate of S ( t, 0 ) ) with a forecasted X ( t, s_c ) cash flow without knowing the s_c state of the world in which they are both being forecasted.

It is this subtle difference between the Hirshleifer and Ross lunches that makes subsequent notation of this paper somewhat awkward. In a Hirshleifer lunch, each s_w state of the world is defined explicitly in terms of Equations (1), (2), and (3) of this paper. These equations utilize the "expensive" recipe ingredient m ( t, s_w ) risk preference for each conceptualized s_w in a TSPM network.

In contrast, a Ross lunch with minimal ingredients does not entail specification of any s_w state or s_c state that generates a C ( t, s_c ) comparison price forecast. It, therefore, has underlying unknown m ( t, s_c ) risk preferences and p ( t, s_c ) stochastic process parameters. Hence, we can only conceptualize a "state" in a Ross lunch as a comparison asset price, e.g., in an illustration to be taken up shortly with a C ( 1, s_c ) = $30 comparison asset price, nothing else is known about the s_c state. We don't even know if the s_c economic state is a boom or a depression! Forecasting a X ( 1, s_c ) cash flow, of a proposed capital budgeting project, to couple with a mere "$30" would be difficult or impossible without at least knowing something about the s_c state of the economy that generated "$30" as a comparison asset price forecast.

There is a way around this dilemma, at least in theory. This is the focus of the next two sections of this paper. It will be shown how cash flow forecasts may be based upon a known economic state s_x that is compatible with the unknown s_c comparison asset economic state.

The APT literature to date generally assumes that each C ( t, s_c ) forecasted comparison asset price is coupled (mapped) to a X ( t, s_c ) cash flow without detailing how this coupling takes place with s_c not fully specified. Much ingenuity is thereby required to relate each price and cash flow in unknown s_c state s. In practice, one has to assume that the minimal Ross lunch ingredients are not sufficient for meaningful CX coupling of forecasts if s_c is to be the basis of a X ( t, s_c ) cash flow forecast, i.e., then s_c would have to be conceptualized "somewhat" even if it is not fully specified in terms of p ( t, s_c ) and m ( t, s_c ).

It will be shown in this paper that the s_c comparison asset state and a s_x state of the world (conceptualized by an analyst who forecasts X ( t, s_x ) ) need not be identical as long as s_x is compatible with s_c. Theorem I and its corollary can be utilized to generate a "s_c compatibility curve" such as the heavy curve in Figures 1 or 2, although "compatibility" is restricted to only non-zero portions of the curve. The terms "c compatibility," "s_c compatibility" "a ( t, 0 ) compatibility," and "S ( t, 0 ) compatibility" are synonymous in this paper and all refer to ( t, 0 ) compatibility curves such as those illustrated in Figures 1 and 2.

What is interesting from a theoretical standpoint is that the Ross fast food lunch, although not free, is seemingly far less expensive than a Hirshleifer lunch. Hirshleifer would require that risk preferences and stochastic processes be specified. For a Ross lunch, we may allow the C ( t, s_c ) and X ( t, s_x ) to have differing s_c and s_x states of the world so long as they are compatible, e.g., so long as world x lies on the same compatibility curve as world c. This gives forecasters more flexibility, because they do not even have to have a weak conceptualization of s_c as long as they can show, via Theorem I and its corollary, that s_x is compatible with s_c rather than having to show s_c and s_x are identical states of the world underlying both the comparison asset price forecast and the cash flow forecast.

What is even more important is that, consistent with APT, a s_c compatiblity curve can be generated from a ( t, 0 ) risk neutral arbitrage probabilities without knowing anything about the s_c state of the world. Thus, even though s_c is an unknown world state ( s_c ) that generated the comparison asset price C ( t, s_c ) in mystery state s_c, an equivalent m_m net present value may be computed from a ( t, 0 ) arbitrage probabilities (in a risk neutral state) that spans other s_x states of alternative worlds, including the s_c world specified by unknown m ( t, s_c ) risk preference and p ( t, s_c ) stochastic process parameters.

Theorem I might be termed a "World Compatiblity Partitioning Theorem" in that it enables us to partition the set all s_w states of hypothetical worlds (i.e., the set of all combinations of m ( t, s_w ) risk preferences and p ( t, s_w ) probabilities) into:

Suppose an unknown world c has been the basis for a Ross lunch valuation with minimal ( .02, $21, $30, $10 ) input ingredients, where .02 is the risk free rate and $21 is the current ( t = 0 ) price of a comparison asset. The single period ( t = 1 ) forecasted comparison asset prices in two ( binomial ) states of the world are $30 and $10. For instance, let s_c ( 30 ) depict the s_c state that generated C (1, s_c (30) ) = $30, the forecasted higher price. Assume absolutely nothing is known about the s_c (30) state of the world that generated the $30 price forecast, e.g., this forecast may have been generated from an "external" forecasting service that disclosed nothing about forecast assumptions. Notationally, let s_c ( 30 ) and s_c ( 10 ) depict the s_c unknown states of c ( 30 ) and c ( 10 ) worlds that generated the comparison asset price forecasts of $30 and $10.

Without knowing anything about s_c ( 30 ) and s_c ( 10 ) states of the world, we may use APT methods described in Cox, Ross, and Rubinstein [1979] and Gehr [1981] to derive the following Ross lunch from only ( .02, $21, $30, $10 ) ingredients where subscripts are utilized in an obvious manner for transition from t = 0 to t = 1:

In the customary APT valuation approach, such as that illustrated in Gehr [1981], the analyst is required to forecast X₃₀ ( 1, s_c ( 30 ) ) and X₁₀ ( 1, s_c ( 10 ) ) cash flows to be coupled (mapped, related) to the $30, and $10 comparison asset prices. Most forecasters, however, would find this impossible when virtually nothing is known about the s_c ( 30 ) and s_c ( 10 ) states of the future c ( 30 ) and c ( 10 ) worlds that generated those prices.

If we obtained added information that this particular comparison asset's prices covary positively with returns of a market portfolio, we still wouldn't even know whether s_c ( 30 ) and s_c ( 10 ) are both boom states, both depression states, or unlike states. For example, s_c ( 30 ) could be a moderate depression and s_c ( 10 ) could be a severe depression, but we don't know this or anything else about those states.

However, the a₃₀ ( 1, 0 ) = .60 arbitrage probability (in a risk neutral world) that corresponds to C ( 1, s_c ( 30 ) ) = $30 provides a basis for generating a ".60 compatibility curve" or the equivalent "s_c ( 30 ) compatibility curve" shown in Figure 3.

For example, worlds w that are s_c ( 30 ) compatible (lie on the curve where p ( t, s_w ) > 0 in FIgure 3) or incompatible (lie off the curve or lie on the curve where
p ( t, s_w ) = 0 in Figure 3) are listed in Table 1.

We do not know what world c with state s_c ( 30 ) = ? generated the C ( 1, s_c ( 30 ) ) = $30 comparison asset price. It could have been any of the twelve s_c ( 30 ) compatible states listed in Table 1 or any among all possible s_c ( 10 ) compatible worlds on the Figure 3 curve where p ( t, s_w ) > 0. In choosing a world x for purposes of forecasting X₃₀ ( 1, s_x ) cash flow, the analyst may specify any world x with state s_x that is compatible with the unknown world c ( 30 ) with unknown state s_c ( 30 ).

For example, suppose the analyst specifies world x = w = 4 in Table 1. This is a moderate boom specified by p ( 1, .18 ) = .70 probability of a boom with a market modifier m ( 1, .18 ) = 0.85. Suppose that in such a world, the analyst forecasts a X₃₀ ( 1, .18 ) = $100 cash flow from a proposed capital budgeting project. Since in Table 1, world x = 4 with state sx = + 0.18 is compatible with the unknown world c ( 30 ) = ? with unknown s_c ( 30 ) = ?, the market risk adjusted value of this $100 at time zero is

Similarly, if X₁₀ ( 1, - .20 ) = $80 was forecasted for some specified w = 6 world compatible with the unknown world c ( 10 ) = ? having unknown states s_c ( 10 ) = ?, the market risk adjusted net present value of both cash flows becomes

This m_m - $90 is an appropriate valuation of contingency $100 and $80 forecasted cash flows provided the analyst's chosen states (for the cash flow forecasts) are compatible with the unknown s_c ( 30 ) and s_c ( 10 ) states that generated the $30 and $10 comparison asset price forecasts. The m_m = $90 is not an appropriate valuation if the analyst selected one or more incompatible states. For example if the analyst forecasted a X₃₀ ( 1, .18 ) = $100 corresponding to the x = w = 17 incompatible world (see Table 1) having a lower probability p ( 1, .18 ) = .50 than the x = w = 4 compatible world having p ( 1, .18 ) = .70, then m_m ( 1, s_c ( 30 ) ) = $58.82 is not an appropriate valuation of this $100 cash flow and m_m = $90 would not be an appropriate NPV across both states.

Recall that both Hirshleifer and Ross lunches offered m_m entrees, but they differed in the ability to provide s_m² desserts, because for such a dessert in the Ross lunch one would have to additionally specify the world c having a known state s_c for which both p ( t, s_c ) and m ( t, s_c ) are known in each s_c state. SInce world c and its s_c state are assumed unknown in APT valuation, calculation of a s_m² variance to accompany a m_m NPV is not possible.

For example, suppose s_x ( 30 ) and s_c ( 30 ) plus s_x ( 10 ) and s_c ( 10 ) are all compatible in our preceeding illustration and that forecasted cash flows for four assets are as follows:

According to the Ross luncheon menu, all four Assets A, B, C, and D are identical lunches, i.e., have the same APT net present value of $90 (ignoring slight rounding differences).

But a true connoisseur may not have equivalent preferences for Assets A, B, C, and D in a Hirshleifer luncheon, because there are differing s_m² ( c ) desserts that differentiate Assets A, B, C, and D. The Ross luncheon cannot deal with these differences, i.e., the c worlds (that generated the comparison asset price forecasts) are unknown. Note also, however, that if s_x = s_c, the X ( t, s_c ) = ? cash flows in s_c states have also not been forecasted for compatible s_x states. Hence, a s_m² ( c ) could not be measured even if the s_c states were specified in a Ross luncheon.

If world x is subjectively specified for each X ( t, s_x ) subjective cash flow forecast in a Ross luncheon, then a s_m² ( x ) dessert of sorts becomes feasible. It is not s_m² ( c ) that corresponds to s_c comparison asset worlds, but it is a s_m² ( x ) that corresponds to X ( t, s_x ) cash flow forecasts. For example, suppose world s = w = 4 in Table 1 was specified as a s_c compatible basis for forecasting all X₃₀ ( 1, .18 ) cash flows for Assets A, B, C, and D above. This world has p ( 1, .18 ) = .70 and m ( 1, .18 ) = .85 that can be utilized in a s_m² ( x ) variance calculation. For illustrative purposes, assume the s_x ( 30 ) and s_x ( 10 ) states were such that:

Asset A:	s_m² ( x )	=	.70 [ ( .85 ) ( 1 / 1.02 ) ( $100 ) ] ² + .30 [ ( 1.33 ) ( 1 / 1.02 ) ( $80 ) ] ² - [ 90 ] ²	=	26
Asset B:	s_m² ( x )	=	.70 [ ( .85 ) ( 1 / 1.02 ) ( $92 ) ] ² + .30 [ ( 1.33 ) ( 1 / 1.02 ) ( $92 ) ] ² - [ 90 ] ²	=	332
Asset C:	s_m² ( x )	=	.70 [ ( .85 ) ( 1 / 1.02 ) ( $220 ) ] ² + .30 [ ( 1.33 ) ( 1 / 1.02 ) ( - $100 ) ] ² - [ 90 ] ²	=	20528
Asset D:	s_m² ( x )	=	.70 [ ( .85 ) ( 1 / 1.02 ) ( - $100 ) ] ² + .30 [ ( 1.33 ) ( 1 / 1.03 ) ( $ 380 ) ] ² - [ 90 ] ²	=	68991

These APT value equivalent assets do indeed have greatly differing variances of market risk adjusted cash flows. Purists might argue that these s_m² ( x ) variances rely upon subjective world x specifications and, accordingly, are not s_m² ( c ) variances one would get if the world c (that generated C ( 1, s_c ( 30 ) ) = $30 and C ( 1, s_c ( $10 ) ) = $10 comparison asset prices) was specified. In this sense s_m² ( x ) variances are not "market driven," but since the $30 and $10 prices are also forecasted market prices in future markets that have not cleared, the $30 and $10 prices can hardly be considered "market driven." The analysis does, however, make m_m valuations based upon external C ( t, s_c ) forecasts and compatible internal X ( t, s_x ) forecasts. The s_m² ( x ) is merely a little "sweetener" added to the m_m lunch at the APT restaurant.

The Cartesian space of all subjective worlds w (specified where s = s_w and p ( t, s ) = p ( t, s_w ) ) is reduced to a single curve connecting a relatively small subset of worlds that are s_c compatible. Although m_m ( t, s_c ) becomes an appropriate APT valuation for any X ( t, s_x ) cash flow (generated from an s_x state that is compatible with the unknown s_c comparison asset state), other s_c compatible worlds on the curve do not yield the same m_m ( t, s_c ) value unless they have the same cash flow. Suppose m_m ( t, s_c ) compatiblity is defined as follows:

Although some assets (e.g., government bonds) generate identical X ( t, s ) cash flows across all states s of the economy, most capital budgeting projects are going to have cash flows that vary considerably with s. Indeed, the entire underpinning of the Market Risk Proposition is an assumption that most assets in the marketwide economy return more in boom states than in depression states.

Since it is common in APT literature to read that "APT valuation makes no assumption about subjective risk preferences in the market and their subjective probabilities," the reader must be extremely cautious in interpreting the generality of APT valuation outcomes. Recall m_m ( 1, s_c ( 30 ) ) = $58.82 was the market risk adjusted APT valuation of the X₃₀ ( 1, s_x ) = $100 cash flow where s_x = .18 was s_c ( 30 ) compatible. It is tempting to assume that this $58.82 is the value of the capital budgeting asset in other economic states s_y ( 30 ) that are s_c ( 30 ) compatible. This is not the case, however, if the asset has a different forecasted cash flow in s_y ( 30 ).

For example, s_x ( 30 ) = .18 depicts a "moderate" boom state. Consider s_y ( 30 ) = .33 for world y = w = 3 in Table 1 rather than x = w = 4. Both s_x ( 30 ) and
s_y ( 30 ) are s_c ( 30 ) compatible. However, s_y ( 30 ) depicts a much more ecstatic boom than the s_x ( 30 ) boom. It is highly likely that the capital budgeting project will do even better in a more ecstatic boom, say X ( 1, .33 ) = $120. Its market risk adjusted APT value is, thereby, increased to

Even though s_y ( 30 ) and s_x ( 30 ) are s_c ( 30 ) compatible on the same s_c curve, they have differing m_m ( 1, s_c ) APT values.

Whenever X ( 1, s_w ) cash flow covaries across all s_w states of the world, as is highly likely in capital budgeting assets, the APT market risk adjusted value m_m ( t, s_c ) applies to only a single world w = x specified where s = s_x and p ( t, s ) = p ( t, s_x ). In this context an APT valuation applies only to a subjective world (point) x specified subjectively by an analyst for purposes of forecasting the X ( t, s_x ) cash flow. It is not the value in the s_c state that generated the C ( t, s_c ) comparison asset price if X ( t, s_x ) ¹ X ( t, s_c ).

Illustrations up to now have been for single period transitions. Consider the multiperiod APT illustration repeated below from Gehr [1981, p. 16] with a slight error correction:

t	C ( t, s_c )	X ( t, s_c )	d ( t, 0 )	a ( t, o )	S ( t, 0 )

1	$ 12	$ 100	1 / 1.08	a₁₂ ( 1, 0 ) = .60	S₀₂ ( 1, 0 ) = $ .5556
1	$ 09	$ 75	1 / 1.08	a₀₉ ( 1, 0 ) = .40	S₀₉ ( 1, 0 ) = $ .3704
2	$ 14	$ 125	1 / 1.08	a₁₄ ( 1, 0 ) = .73	S₁₄ ( 1, 0 ) = $ .3704
2	$ 11	$ 75	1 / 1.08	a₁₁ ( 2, 0 ) = .27	S₁₁ ( 2, 0 ) = $ .1347
2	$ 10	$ 90	1 / 1.03	a₁₀ ( 2, 0 ) = .77	S₁₀ ( 2, 0 ) = $ .2690
2	$ 08	$ 65	1 / 1.03	a₀₈ ( 2, 0 ) = .23	S₀₈ ( 2, 0 ) = $ .0804

1	m_m ( t, s_c ( 12 ) )	=	( $ .5556 ) ( $100 )	=	$ 56
1	m_m ( t, s_c ( 09 ) )	=	( $ .3704 ) ( $ 75 )	=	28
2	m_m ( t, s_c ( 14 ) )	=	( $ .3704 ) ( $125 )	=	46
2	m_m ( t, s_c ( 11 ) )	=	( $ .1347 ) ( $ 75 )	=	10
2	m_m ( t, s_c ( 10 ) )	=	( $ .2960 ) ( $ 90 )	=	24
2	m_m ( t, s_c ( 08 ) )	=	( $ .0804 ) ( $ 65 )	=	5
			m_m	=	$ 169

Gehr's answer of $159.17 results from a slight arithmetic error. The corrected answer is $169.17.

Gehr [1981, p. 15], as quoted earlier, contends: "We are, therefore, free to make any assumptions about preferences and probabilities that are consistent with the observed prices" of the comparison asset. Gehr implicitly assumes all such C ( t, s_c ) prices were somehow coupled with the X ( t, s_c ) cash flows for all unknown s_c states of the economy.

The question we might raise here is whether the m_m = $169 NPV computed by APT methods applies to more than just the one combination of unknown s_c ( 12 ), s_c ( 09 ), s_c ( 14 ), s_c ( 11 ), s_c ( 10 ), and s_c ( 08 ) states. Does m_m = $169 really apply to other subjective worlds having different m ( t, s_w ) risk preferences and p ( t, s_w ) probabilities? The answer is no if the capital budgeting asset's cash flows are unique to the unknown s_c states. The m_m = $169 is not an appropriate NPV in alternate worlds having different subjective X ( t, s_w ) cash flow forecasts from X ( t, s_c ), different subjective m ( t, s_w ) risk preferences from unknown m ( t, s_c ), and different subjective p ( t, s_w ) probabilities from unknown p ( t, s_c ). Even if the cash flows are identical in alternative subjective worlds, the m_m = $169 NPV is not an appropriate valuation unless the s_w states are also compatible in the context of s_c compatibility curves. It is more than likely, however, that the a capital budgeting asset's cash flows covary (positively or negatively) with the economy and are, therefore, not identical in states other than the unknown s_c states. It would seem, therefore, that the m_m = $169 is of limited utility since we cannot identify any worlds in which it is appropriate. This NPV is appropriate for the c worlds that generated the C ( t, s_c ) comparison asset prices, but we don't know what those c worlds are or what are the underlying s_c economic states.

In reality, analysts are probably unable to make X ( t, s_c ) subjective cash flow forecasts corresponding to C ( t, s_c ) comparison asset prices without specifying the s_c intensities on a depression / boom spectrum. If s_c states are specified in terms of m ( t, s_c ) risk preferences under the Market Risk Proposition, this also specifies p ( t, s_c ) = a ( t, 0 ) / m ( t, s_c ) under APT theory. In this context, the claim that APT m_m values can be computed without specifying the worlds of comparison asset prices is dubious. The claim depends upon how well each X ( t, s_c ) cash flow can be forecasted without specifying the s_c state of the economy underlying each C ( t, s_c ) price. Even then, the value may not apply to any other state of the economy.

In the early years of contingency claims theory, market risk adjusted ( m_m ) NPV valuations required full specification of "real" world states (s_w ) and p ( t, s_w ) state transition probabilities. This precluded most real world applications of such theory. More recently, APT valuation offered more hope by freeing analysts from the need to fully specify the world w at known s_w states with p ( t, s_w ) known probabilities.

In theory, all that is needed for APT m_m valuation of NPV of a nonmarketed asset are forecasted C ( t, s_c ) comparison asset prices (of a marketed asset), forecasted X ( t, s_c ) nonmarketable asset cash flows, and forecasted risk free rates.

In practice, the requisite coupling of C ( t, s_c ) and X ( t, s_c ) inputs for APT computations of m_m is virtually impossible without specifying the s_c intensity on a depression / boom spectrum for the s_c states of the world that generated the C ( t, s_c ) comparison asset price. In this paper it was shown how such couplings can be avoided. An X ( t, s_x ) forecasted cash flow from any state s_x that is s_c compatible may be utilized in computing m_m. This can be accomplished even when s_c remains totally unknown.

In the context of (i) and (ii) above, APT is a comparison asset around the barn way of taking us back to the old days where m_m NPV requires specification of "real" world states ( s_w ). Specification of s_c in terms of m ( t, s_c ) intensity on a depression / boom spectrum also indirectly yields p ( t, s_c ) = a ( t, 0 ) / m ( t, s_c ) in APT valuation. Hence, after going around the barn with APT, we are back to where s_w and p ( t, s_w ) are specified for w = c, the only difference being that c is a subjective estimate of the "real" world w that explains comparison asset prices. A Ross economy lunch providing m_m as an entree may end up costing as much as Hirshleifer cuisine sweetened afterwards with s_m² dessert.

m ( t, s ) =	p ( t, s ) [ d ( t, s ) X ( t, s ) ]	(7)
=	p ( t, s ) Y ( t, s ).
m_m ( t, s ) =	p ( t, s ) m ( t, s ) Y ( t, s )	(8)
=	a ( t, 0 ) Y ( t, s ).

m =	_ò_ò	m ( t, s ) dtds.	(9)
m_m =	_ò_ò	m_m ( t, s ) dtds.	(10)

m_m ( 1, s_c )	= a ( 1, 0 ) m ( 1, 0 ) Y ( 1, s_c )
	= a ( 1, 0 ) Y ( 1, s_c ) ,	(13)

m_m( 1, s_c ( 30 ) )	=	S₃₀ ( 1, 0 ) X₃₀ ( 1, .18 )
	=	( $ .5882 ) ( $100 )
	=	$ 58.82.

m_m	=	S₃₀ ( 1, 0 ) X₃₀ ( 1, .18 ) + S₁₀ ( 1, 0 ) X ( 1, - .20 )
	=	( $ .5882 ) ( $ 100 ) + ( $ .3992 ) ( $ 80 )
	=	$ 90.

s =	1 - m ( t, s ) for s < 0 values of values of depression states.	(1)
s =	- 1 + 1 / m ( t, s ) for s > 0 values of boom states.	(2)
s =	0 for a market risk neutral state.	(3)

			m_m
Asset	X₃₀ ( 1, s_x )	X₁₀ ( 1, s_x )	NPV

Asset A	+ $ 100	+ $ 80	$90
Asset B	+ $ 92	+ $ 92	$90
Asset C	+ $ 220	- $ 100	$90
Asset D	- $ 100	+ $380	$90

s_x ( 30 )			s_x ( 10 )
p ( 1, s_x )	=	.70	p ( 1, s_x )	=	.30
m ( 1, s_x )	=	.85	m ( 1, s_x )	=	1.33
d ( 1, s_x )	=	1 / 1.02	d ( 1, s_x )	=	1 / 1.02
s_x	=	+ .18	s_x	=	- .33

Asset A	m_m = $ 90	s_m ( x ) »	$ 5
Asset B	m_m = $ 90	s_m ( x ) »	$ 18
Asset C	m_m = $ 90	s_m ( x ) »	$ 143
Asset D	m_m = $ 90	s_m ( x ) »	$ 263

m_m ( 1, s_c )	=	S₃₀ ( 1, 0 ) X₃₀ ( 1, .33 )
	=	( $ .5882 ) ( $ 120 )
	=	$ 70.58.