St Peter Annals of Finance Sep 3 2015 - Semantic Scholar
The St. Petersburg Paradox and Capital Asset Pricing Assaf Eisdorfer*
Carmelo Giaccotto*
September 2015
ABSTRACT Durand (1957) shows that the classical St. Petersburg paradox can apply to the valuation of a firm whose dividends grow at a constant rate forever. To capture a more realistic pattern of dividends, we model the dividend growth rate as a mean reverting process, and then use the Capital Asset Pricing Model (CAPM) to derive the risk-adjusted present value. The model generates an equivalent St. Petersburg game. The long-run growth rate of the payoffs (dividends) is dominant in driving the value of the game (firm), and the condition under which the value is finite is less restrictive than that of the standard game. Keywords: Capital Asset Pricing Model (CAPM); Stochastic Dividends; St. Petersburg paradox JEL Classification: G0, G1
*University
of Connecticut. We thank Iskandar Arifin, John Harding, Po-Hsuan Hsu, Jose Martinez, Efdal Misirli, Tom O’Brien, Scott Roark, Jim Sfiridis, and participants at the finance seminar series at the University of Connecticut for valuable comments and suggestions. Please send all correspondence to Carmelo Giaccotto. Mailing address: University of Connecticut, School of Business, 2100 Hillside Road, Storrs, CT 06269-1041. Tel.: (860) 486-4360. E-mail: [email protected].
The St. Petersburg Paradox and Capital Asset Pricing 1. Introduction The St. Petersburg paradox is one of the most well-known and interesting problems in the history of financial economics. The paradox describes a situation where a simple game of chance offers an infinite expected payoff, and yet any reasonable investor will pay no more than a few dollars to participate in the game. Since the paradox was presented by Daniel Bernoulli in 1738, it has attracted a great deal of interest, mainly by theorists who provide solutions and derive its implications. One of the applications of the paradox is in the area of financial asset pricing. Durand (1957) shows that the St. Petersburg game can be transformed to describe a conventional stock pricing model for growth firms. The analogy is based on the assumption that the firm’s future dividends (as the game’s future payoffs) grow at a constant rate. Economic intuition and the historical evidence suggest, however, that the very high growth rates experienced by many young firms (e.g., firms in the high-tech industry) are expected to decline over time. Hence, the short-run growth rate is typically much greater than the expected long-run rate. In this study we model the dividend growth rate as a mean reverting process; we then find the risk-adjusted growth rate under the equilibrium setting of the classical Capital Asset Pricing Model (CAPM), and derive its equivalent modified St. Petersburg game.1 Our paper has several interesting results. First, we assume an autoregressive process for the dividend growth rate, and then use the CAPM to derive a closed form solution for the price of a growth stock. This result is a significant contribution to the asset pricing literature; it extends
1
Assuming a stochastic dividend growth is rather common in asset pricing studies; see, for example, Bansal and Yaron (2004), and Bhamra and Strebulaev (2010).
Rubenstein’s (1976) single factor arbitrage-free present value model by allowing mean reverting growth. Second, by distinguishing between the short-run and long-run growth rates, the model shows that the latter is the dominant factor in driving the value of a growth stock, and in turn, the properties of the St. Petersburg game, including its value. Third, we derive the condition under which the expected payoff of the game (or equivalently, the value of a growth firm) is finite; as expected, this condition is much less restrictive than the one required under constant growth. Fourth, and last, our solution to the paradox extends the work of Aase (2001) who framed the paradox as an arbitrage problem. His solution requires, in part, that economic agents discount future cash flows at a constant rate. By using the Capital Asset Pricing Model we are able to derive the required equilibrium discount factor, and show that the game price is finite. The rest of this paper is organized as follows. In the next section we describe the paradox and a number of solutions proposed over the years; in section 3 we describe the correspondence between the St. Petersburg paradox and the value of growth stocks – the original contribution of Durand (1957). Section 4 presents our stochastic dividend growth model and derive the valuation model implied by the CAPM. We find that the restriction to insure a finite asset value is by far less restrictive than the original one. In section 5 we present a modified St. Petersburg game for stochastic growth stocks, and in section 6 we consider two extensions of the basic valuation model along with the conditions necessary to preclude a St. Petersburg type paradox. The conclusion is in section 7.
2. The paradox and its common solutions The St. Petersburg paradox is based on the following simple game. A fair coin is tossed repeatedly until the first time it falls on ‘head’. The player’s payoff is 2n dollars (‘ducats’ in the
2
original Bernoulli’s paper), where n is the number of tosses. Since n is a geometric random variable with p=0.5, the expected payoff of the game is: E=
1 1 1 × 2 + × 4 + × 8 + ... = 1 + 1 + 1 + ... = ∞ 2 4 8
(1)
Yet, while this game offers an expected payoff of infinite dollars, a typical player will pay no more than a few dollars to participate in the game, reasoning that there is a very small probability to earn a significant amount of money. For example, the chance to earn at least 32 dollars is
2 −5 = 0.03125 , and at least 128 dollars is 2 −7 = 0.00713; hence, paying a game fee of even 1,000 dollars seems unreasonable. A number of solutions proposed to resolve the paradox rely on the concept of utility.2 The basic idea is that the relative satisfaction from an additional dollar decreases with the total amount of money received. Thus, the game fee should be based on the expected utility from the dollars earned, rather than the expected amount of dollars. Bernoulli himself suggested the log utility function; in this case, the expected value of the game is: ∞ 1 1 1 ln(2 n ) E (U ) = × ln(2) + × ln(4) + × ln(8) + ... = ∑ n = 2 ln(2) < ∞ 2 4 8 n=1 2
(2)
It turns out, however, that the utility-based solutions are incomplete: the game can be converted to a convex stream of payoffs, and this change reverses the benefit of a concave utility n
function. For example, if instead of 2 n , the game’s payoff is e 2 , then for the log utility just considered, the expected value again tends toward infinity. In general, for every unbounded utility function the payoffs can be changed such that the expected utility will be infinite. This generalization of the game often goes by the name “super St. Petersburg paradox.”
2
See Senetti (1976) on solving the paradox using expected utility in light of modern portfolio theory, and Sz´ekely and Richards (2004) for other suggested solutions to the paradox.
3
Proposed solutions to the super game rely on the concept of risk aversion (Friedman and Savage (1948), and Pratt (1964)); that is, holding everything else constant, a typical player prefers less risk, and therefore is willing to pay a lower fee for high-risk games. Weirich (1984) shows that, while the game may offer an expected infinite sum of money, it involves also an infinite amount of risk. This is because the dispersion of the possible payoffs results in an infinite standard deviation. Thus, the fair game fee resembles the difference between an infinite expected payoff and an infinite measure of risk. While the answer may actually be finite, it does not explain the low amounts typical players are willing to pay, which range between 2 to 25 dollars. Aase (2001) proposes an interesting solution by reframing the paradox as an arbitrage problem. He then shows that if (a) economic agents have finite credit at their disposal, and (b) agents discount cash flows to the present at a constant discount rate, then any arbitrage opportunity will disappear and the price of the game becomes finite. Our contribution is to frame the paradox as a growth stock – in the spirit of the Gordon dividend discount model, and then use the CAPM to derive the appropriate discount factor.
3. Application of the paradox to growth stocks One of the applications of the St. Petersburg paradox is in the area of asset pricing. Durand (1957) shows that with some modifications, the paradox can describe a conventional stock pricing model. A growth firm expects to generate an increasing stream of future earnings, and thus to pay an increasing stream of dividends. The value of such a firm is given by the present value of all future dividends:
P0 =
∞ Dt D1 D2 + + ... = ∑ t (1 + r ) (1 + r ) 2 t =1 (1 + r )
(3)
4
where Dt is the per-share dividend of year t and r is the discount rate. The constant growth model, also known as the Gordon (1962) model, assumes that future dividends grow at a constant rate (g); i.e., the dividend stream is D1 , D1 (1 + g ), D1 (1 + g ) 2 , ... for the years 1, 2, 3,… In that case, the present value of this growing perpetual future dividend stream equals:
D1 , if g < r D1 (1 + g ) t −1 r g − P0 = ∑ = (1 + r ) t t =1 ∞ , if g ≥ r ∞
(4)
Thus, the firm value is finite only if the growth rate is lower than the discount rate. Durand derives the St. Petersburg analogue of the constant growth model using the following analysis. Consider the St. Petersburg game, where instead of a fair coin, the probability that a ‘head’ appears is r (1 + r ) , where r > 0 . Assume further that instead of earning a single payment when the game ends (i.e., when ‘head’ appears in the first time), the player earns a specific amount of dollars as long as the game continues. Specifically, the player will earn D1 if the first toss is ‘tail’, D1 (1 + g ) if the second toss is ‘tail’, D1 (1 + g ) 2 if the third toss is ‘tail’, and so on. That is, if the game lasts for n tosses, instead of earning 2n , the player will earn n −1
∑ D1 (1 + g ) j −1 = j =1
D1[(1 + g ) n−1 − 1] . Therefore, the expected payoff of the game is: g
D1 , if g < r D1[(1 + g ) n−1 − 1] ∞ D1 (1 + g ) n −1 r ( r − g ) E=∑ = = ∑ n g (1 + r ) n n =1 (1 + r ) n =1 ∞, if g ≥ r ∞
(5)
which is identical to the value of a constant dividend growth firm (as appears in Equation 4). Note that the analogy is based on the translation of the discount rate r to the coin probability
r (1 + r ) . That is, while in the original St. Petersburg game any future payment will be paid with some probability, in the modified game, which is equivalent to the dividend stream generated by 5
growth firms, any future payment will be paid for sure, but will be evaluated with a discount factor. This analogy helps explaining the condition under which the expected payoff of the game is infinite. The value of the stock is infinite only when the dividend grows at an equal or higher rate than the discount rate; and in the same way, the expected payoff of the game is infinite only when the payoffs are increasing at an equal or higher rate than the rate at which the correspondent probabilities are decreasing. An important characteristic of Durand’s constructed game is that its representation is not unique, as different forms of the payoff series and the coin probability can yield the same expected value. To illustrate, consider the same setup of Durand’s game outlined above where the probability that a ‘head’ appears is now (r − g ) (1 + r ) instead of r (1 + r ) , for g < r . The payoff series also changes, by a factor of r (r − g ) : the player will earn D1 r ( r − g ) if the first toss is ‘tail’, D1 (1 + g ) r (r − g ) if the second toss is ‘tail’, D1 (1 + g ) 2 r ( r − g ) if the third toss is ‘tail’, and so on. That is, if the game lasts for n tosses, the player will earn n −1
∑ D1 (1 + g ) j −1 r (r − g ) = j =1
D1[(1 + g ) n −1 − 1] r ( r − g ) . Therefore, the expected payoff of the game g
is:
D1 (r − g ) D1[(1 + g ) n−1 − 1] r (r − g ) ∞ D1 (1 + g ) n−1 =∑ = n n g (r − g ) (1 + r ) n=1 (1 + r ) n=1 ∞
E=∑
(6)
which is the value of a constant dividend growth firm (as appears in Equation 5). While these two representations of Durand’s game yield the same expected value, they generate different probability distribution of the payoffs; particularly, the first representation offers lower payments than the second representation but with higher probability to stay in the game. This difference in the payoff probability distribution may have implications with respect to the investor preferences. For example, consider a constant growth stock with D1 = $2 , r = 15% 6
, and g = 10%. The value of the stock is
D1 $2 = = $40 , which is also the expected ( r − g ) 0.15 − 0.1
value of the payoffs under the two representations of Durand’s game (Equations 5 and 6). However, the balance between the periodic payments and the corresponding probabilities is different under the two representations; for instance, under the first representation of the game the probability that the game will last at least 5 tosses is 49.7% at which the player will earn at least $12.21, whereas under the second representation of the game the probability that the game will last at least 5 tosses is 16.6% at which the player will earn at least $36.63. Assuming that a typical investor has a decreasing utility function (as discussed in Section 2), the first representation of the game might be more appealing to the investor, and therefore she will be willing to pay a higher fee to participate in the game. We believe the paradox arises for two reasons. First, the assumption that the dividend stream will grow at a constant rate permanently is unrealistic. For example, economic intuition, as well as the historical evidence, suggests that high growth tech firms (such as IBM, Microsoft and now Google or Facebook) may grow very rapidly in the short run. But nothing attracts competition like market success; therefore, in the long run new market entrants will force the earnings growth rate to slow down (almost surely) to a level consistent with the growth rate of the overall economy. The second reason is that the degree of risk implicit in the dividend stream may cause investors to change the risk-adjusted discount rate, i.e., the probability of actually receiving the expected dividends. In the next section we formalize these ideas within the context of the Capital Asset Pricing Model of Sharpe (1964) and Lintner (1965). We model the dividend growth rate as a mean reverting process so that the current rate can be very large but the long run rate is expected to be much lower. We then use the CAPM to derive the appropriate risk-adjusted present value. We
7
find that the equity price can be finite without the unreasonable condition required by the constant growth model.
4. Stochastic dividend valuation in the CAPM world Let Dt be the time-t value of dividends or earnings (for simplicity we will use these two g terms interchangeably), and assume these values grow at rate g t +1 : Dt +1 = (e t +1 )Dt . We use a
first order autoregressive process ( AR(1) ) to model mean reversion:
gt +1 = (1 − φ ) g + φ gt + ε t +1
(7)
where g is the long run (unconditional) mean growth rate and φ is the autoregressive coefficient (we assume 0 ≤ φ < 1 to be consistent with the smooth behavior of dividends). We make the usual assumptions to insure the process is stationary and the growth rate is mean reverting. The innovation terms ε t +1 are normally distributed random variables with mean zero, variance σ ε2 , no serial correlation, and constant covariance with the market portfolio return. The major implication of the mean-reverting dividend model is that while the current growth rate can be abnormally large, in the long run earnings growth should slow down to a lower rate. Intuitively, we expect g to be close to the growth rate for the overall economy because of competitive pressures brought about by new startup companies. To obtain the risk-adjusted present value of each future dividend we use the CAPM of Sharpe (1964) and Lintner (1965): ER = R f + [ ERm − R f ]β ROR
(8)
where ER is the single period expected rate of return on the asset, R f is the risk-free rate of interest, [ ERm - R f ] is the expected excess market portfolio return, and market risk is measured 8
by the rate of return beta ( β ROR ) . Then, the CAPM price for an asset that pays a stochastic dividend stream {Dt +τ }τ∞=1 is given by the sum of expected future dividends adjusted for market risk, and discounted to the present at the riskless rate of interest (Equations 9 and 10 are derived in the Appendix): τ
∞
(Et Dt +τ )∏ [1 − ( ERm − R f ) z j β g ]
τ =1
(1 + R f ) τ
Pt = ∑
j =1
(9)
where the deterministic variable zj captures the effect of mean reversion on cash flows and the market risk. It may be computed recursively as: z j = φ z j −1 + 1 for j=1, 2, … and starting value
z0 = 0 . The growth rate beta, β g , is defined as the covariance between the growth rate innovation (εt+1) and the market return, divided by the variance of the market return. The expected dividend series is an exponential affine function of the current dividend growth rate and long run growth: Et Dt +τ = Dt e A(τ ) + B (τ ) g + C (τ ) g t
and
(10)
τ
τ
j =1
j =1
A(τ ) = (σ ε2 / 2) ∑ z 2j , B(τ ) = (1 − φ ) ∑ z j , and C (τ ) = φ zτ , Equations (9) and (10) show that the current price depends on the current growth gt, and the
long run rate g ; however, the impact of the latter is much stronger because it is multiplied by
B(τ ) (the sum of zj). We note that z j converges to z = 1 /(1 − φ ) for large τ, thus B(τ ) increases with the time horizon. On the other hand, C (τ ) converges to a constant finite value φ /(1 − φ ) . Intuitively, stronger mean reversion implies faster reversal to the long run mean; therefore, a currently high growth rate has only a transitory impact on the equity price. The permanent component of price is driven by long run dividend growth.
9
Today’s price depends also on the appropriate risk adjustment. Since the market risk, represented by z j β g , increases with j -- up to β g /(1 − φ ) , the adjustment for risk becomes increasingly larger for distant expected dividends, and this helps obtain a finite present value. Two special cases of the general model are worth mentioning. The first is Gordon’s deterministic growth model which is obtained by setting φ = 0 and σ ε2 = 0 .
In this case,
dividends are expected to grow in a deterministic fashion at a constant rate g . The beta factor ( β g ) equals zero and the discount rate r equals the riskless rate R f . The second, originally developed by Rubenstein (1976) within the context of a single factor arbitrage-free model, allows stochastic growth but no serial correlation: gt +1 = g + ε t +1 . In this case, (log) earnings or dividends follow a random walk with drift, and the stock price has a closed-form solution similar to Durand’s formula: Pt =
given by
(1 + R f ) 1 − ( ERm − R f ) β g
Dt (e g* ) r − e g*
, where the discount rate r is
2 , and the adjusted growth rate is g * = g +σε / 2 . The major
drawback of these two models is that unlike our autoregressive model, they do not allow a distinction between current growth – which can be abnormally high, and long-run growth. We can now derive the condition under which the asset price is finite. Observe that for large j, z j converges to a constant value of 1 (1 − φ ) ; therefore, the expected future dividend will
evolve along the
2 g + 1 σ ε T 2 (1−φ )2 path e
. Also, for dividends far into the future, the risk-adjusted T
1 − ( ERm − R f ) β g (1 − φ ) present value factor may be approximated by . Therefore, the 1+ R f present value of a single dividend DT , for large T, may be approximated as follows: 10
V t ≈ Dt e
β g 1 σ ε2 T − R f + ( ERm − R f ) g + 2 2 ( 1 −φ ) (1−φ )
(11)
This value will converge to zero, and thereby the sum of the present values of all future dividends (i.e., the firm value) will be finite, provided the expression inside the square brackets is negative. Thus, the restriction on long term growth is:
g ≤ R f + ( ERm − R f )
βg 1 σ ε2 − (1 − φ ) 2 (1 − φ )2
(12)
The right hand side of this expression is similar to the CAPM risk-adjusted return with two modifications. First, risk is measured by the growth rate beta adjusted by the degree of predictability in the dividend stream, and second, since the growth rate is continuously compounded, we need to subtract one half the long-run variance of dividend shocks. Clearly this condition is less restrictive than for constant growth (i.e., g < r ); and, more importantly, it imposes no restrictions on the short term dividend growth rate. Figure 1 illustrates the upper bound condition described by Equation (12). We vary the level of mean reversion, φ , on the horizontal axis from 0.0 (strong) to 0.6 (weak), and include three levels of β g : 0.5, 1.0, and 1.5. To set the riskless rate and the market risk premium, we use data from Professor Ken French’s website. We have R f = 0 .037 and E ( Rm − R f ) = 0.0804 , which 2 correspond to the sample averages from 1927 thru 2013. Last, σ ε is set arbitrarily at 0.01
because its exact value has a marginal effect on the upper bound.3 The horizontal line at 3.7% corresponds to the constant growth Gordon model; clearly this is a low bar for the long run growth rate. The more realistic cases reflect varying degrees of mean reversion. In the strongest case, where φ = 0.0, the upper bound increases with beta. When β g = 3
We consider both a 50 percent decrease and a 50 percent increase in the variance of shocks to the dividend growth rate. As expected the new upper bound on long term dividend growth is not substantially affected by these changes.
11
0.5, the risk-adjusted upper bound is 7.2%, and increases to 11.2% for β g = 1.0, and to 15.3% for β g = 1.5. Figure 1 shows that these upper bounds increase even faster as mean reversion weakens. No firm can be expected to grow permanently at such high rates.
5. A modified St. Petersburg game for stochastic growth stocks As Durand (1957) shows, the classical St. Petersburg game (with some modifications) can describe an investment in a stock with a constant dividend growth rate. To capture the more realistic pattern of growth firms (as outlined in the previous section), we present a modified St. Petersburg game that is analogous to a stock with a mean reverting dividend growth rate in the equilibrium setting of the CAPM. Consider the St. Petersburg game, where instead of a fair coin, the probability that a ‘head’ appears is R f ( 1 + R f ) , where R f > 0 (this constant probability is consistent with that in Durand’s game using the risk-free rate, R f , instead of r). Assume further that the player earns an j
Dj∏aj i =1
amount of
Rf
dollars in the jth toss (including the last toss of the game), where D j
denotes the dividend series as defined in Section 4, and a j = 1 − ( ER m − R f ) z j β g . That is, if the j
game lasts for n tosses, the player will earn
n
Dj ∏a j
j =1
Rf
∑
i =1
dollars. The expected payoff of the
game is therefore:
∞
(E 0 Dn )∏ [1 − ( ERm − R f ) z j β g ]
n =1
(1 + R f ) n
n
∞
E=∑
Dn ∏ a j
Rf
n =1 (1 +
Rf )
j =1
n
Rf
n
=
∑
j =1
(13)
12
which matches exactly the stock value in the CAPM world (Equation 9). Note that, as for Durand’s game, the representation of the modified game is also not unique where different forms of payoffs and corresponding probabilities can achieve the same expected value. The condition under which the value of the game is finite, therefore, is identical to the one that makes the stock price finite, as given in Equation (12). Since this condition is less restrictive than the classical St. Petersburg game (as discussed above), it provides an indirect solution to the paradox. That is, under the more realistic setup of a stochastic mean reverting growth rate, it is more likely that the value of the game is finite, and therefore the game fee the players are willing to pay is finite.
6. Extensions In this section we consider two extensions of the basic valuation model (Equations 9 and 10) and derive the conditions necessary to preclude a St. Petersburg paradox. The first extension relaxes the assumption of a constant long-run mean dividend growth rate; while the second deals with firms that may or may not pay dividends.4
6.1 Valuation when long-run growth varies with the business cycle In section 4, for simplicity of exposition, we modeled dividend growth as a first order mean reverting process with constant long run dividend growth. But as the economy moves across the business cycle, g may be time-varying. Let g~t be the long run dividend growth rate, then we rewrite Equation (7) as gt = (1 − φ ) g~t + φ gt −1 + ε t . It can be easily shown that if g~t is itself mean
4
It may seem like a natural extension to consider also time variation in the rate of return beta. However, Fama (1977) shows that the single period CAPM precludes uncertainty about the riskless rate, the market risk premium as well as beta.
13
reverting (e.g., AR(1) ), then the reduced form model for dividend growth becomes ARMA(2, 1), that is second order autoregressive process with a first order moving average component:
gt = (1 − φ1 − φ2 ) g + φ1 gt −1 + φ2 gt −2 + ε t − θ ε t −1
(14)
where ( φ1, φ2 , θ ) are, respectively, the autoregressive and moving-average coefficients.5 Again, we assume the process is stationary and shocks to the growth rate have constant covariance with the market return. To compute asset prices we need to define two sets of auxiliary equations to account for the autoregressive and moving average components of serial correlation in the cumulative growth rate. The two sequences are: w j = z j − θ z j −1 , and z j = φ1 z j −1 + φ2 z j − 2 + 1 , with a starting value of z0 = 0 . Then, the time-t price is given by (proofs of Equations (15) and (16) are provided in the appendix): τ
∞
(E0 Dt +τ )∏ [1 − ( ERm − R f ) w j β g ]
τ =1
(1 + R f ) τ
Pt = ∑
j =1
(15)
and the expected future dividend is
Et Dt +τ = Dt e [ A(τ ) + θ The
other
parameters
zτ ]σ ε2 / 2 + B (τ ) g + C (τ ) g t + D (τ ) g t −1
2 2
are
defined
(16) τ
as: A(τ ) = ∑ w2j , j =1
τ
B(τ ) = (1 − φ1 − φ2 ) ∑ z j , j =1
C (τ ) = φ1 zτ + φ2 zτ −1 , and D(τ ) = φ2 zτ . Using numerical derivatives, one may show that this asset pricing model has some interesting characteristics. First, the current price increases with both the current and long run dividend growth rates. However, as was the case for the AR(1) model, price is much more sensitive to 5
Fama and French (2000) provide empirical evidence for mean reversion in profitability. They find that the strength of mean reversion varies depending on whether profitability is above or below its mean. To the extent that dividends follow earnings, the ARMA(2,1) model captures these dynamics.
14
long run than short run dividend growth. Second, price decreases with growth rate beta and the market risk premium. And third, growth rate volatility has a positive impact on price because of the convexity of compounded cash flows. The condition under which the asset price is finite is very similar to that of the constant long run dividend growth (Equation 12). We note that for long horizon cash flows, z j converges to a constant value of 1 (1 − φ1 − φ2 ) , and w j converges to (1 − θ ) /(1 − φ1 − φ2 ) . Thus, for large τ, both C (τ ) and D(τ ) play a negligible role. B(τ) increases linearly with τ ; and A(τ ) converges
(σ ε2 / 2)(1 − θ ) τ. to (1 − φ1 − φ2 )
The expected dividend series will evolve along the
2 g + 1 σ ε (1−θ ) T 2 (1−φ1 −φ2 ) 2 path e
, while the risk-
T
1 − ( ERm − R f ) β g (1 − θ ) (1 − φ1 − φ2 ) adjusted discount factor converges to . Therefore, for 1+ Rf large T, the present value of a single dividend DT may be approximated as:
V t ≈ Dt
β g (1−θ ) 1 σ ε2 (1−θ )2 T − R f + ( ERm − R f ) g + 2 2 (1−φ1 −φ2 ) (1−φ1 −φ2 ) e
(17)
Finite pricing of assets requires that the following restriction holds for long term growth:
β g (1 − θ ) 1 σ ε2 (1 − θ )2 g ≤ R f + ( ERm − R f ) − (1 − φ1 − φ2 ) 2 (1 − φ1 − φ2 )2
(18)
The right hand side of Equation (18) again consists of three terms: the first two represent the CAPM risk-adjusted return with a modified beta to account for predictability in the dividend stream. The third term is needed because the analysis is in terms of continuously compounded dividend growth. Once again we find that the condition is less restrictive than a simple g < r ; furthermore, there is no restriction on short term dividend growth. 15
6.2 Valuation of Non-Dividend Paying Firms In this section we generalize the model to the universe of firms that may or may not pay dividends. We show that the main results hold provided we take into account mean reversion in profitability. The foundation for this analysis is the “Clean Surplus” identity for a firm that is financed by equity only and expects no new equity issues. This accounting relationship then states that the current book value of equity equals last period’s book value plus current income minus current dividends: Bt = Bt −1 + I t − Dt . The assumption of equity financing is not critical to the analysis that follows. The main result in this section holds in the presence of debt financing provided that the accounting rate of return on book equity (ROE) is independent of the level of debt. For firms under financial distress this independence hypothesis may be inappropriate; but it is likely to hold for healthy firms that maintain a fairly constant debt to equity ratio. There are many ways to model dividend payments. For example, one may choose a dividend rate set at a constant fraction of earnings; but earnings may be negative from time to time and negative dividends would be inappropriate. Also earnings are quite volatile over the business cycle, and a constant relationship would make dividends quite erratic. Because dividends are typically smooth, we assume that dividends are paid as a percent of book value, and examine the implications for valuation by the CAPM. Let c be the constant proportion of book equity paid out as a periodic dividend: Dt+1 = cBt. Setting c = 0 allows us to model firms that pay no dividends. We define the accounting rate of return on book equity (ROE) ρt+1 as firm’s earnings – at end of period t+1, divided by book value of equity as of period t. Then, the clean surplus relation implies that book equity grows as:6
(
)
Bt +1 = e ρ t +1 − c Bt
(19)
(
The clean surplus relationship implies that the rate of growth in book value is Bt +1 = e The approximation is exact only in continuous time. 6
ln(1+ ρ t +1 − c )
)B ≈ (e t
ρ t +1 − c
)B . t
16
To model the profitability rate, we assume a first order autoregressive process:
ρt +1 = (1 − φ )ρ + φ ρt + ζ t +1 , where ρ represents long run mean profitability. Shocks to the accounting profitability rate ζ t +1 are modeled as a white noise process with zero mean, variance
σζ2 , and constant covariance with the market portfolio. This covariance -- divided by the variance of the market return, defines the profitability rate beta βρ. To complete the model, we assume that at a future date T competition will force abnormal returns down to the point where market value and book value equal one another: MT = BT .7 It is rather surprising that the traditional CAPM leads to a straightforward relationship between the market to book ratio and the accounting measure of profitability. To show this result, suppose the CAPM holds, and the time series behavior of the profitability rate follows an AR(1) model with a long run profitability rate ρ . Define the autocovariance variable z j = 1 + φ z j −1 for j=1, … , T and with starting values z0 = 0. Then, the market to book ratio is
given by (the derivation of Equation (20) is analogous to that of (8) and (9) in the appendix): τ
(e A(τ )+ B (τ ) ρ +C (τ ) ρt −τ c )∏ (1 − ( ERm − R f ) z j β ρ )
T −1 c Mt = +c ∑ Bt 1 + R f τ =1
j =1
(1 + R f )τ T
+
where
(e A(T )+ B (T ) ρ +C (T ) ρt − cT )∏ (1 − ( ERm − R f ) z j β ρ ) j =1
(20)
(1 + R f )T τ
τ
j =1
j =1
A(τ ) = (σ ζ2 / 2) ∑ z 2j , B(τ ) = (1 − φ ) ∑ z j , and C (τ ) = φ zτ .
Consistent with intuition, we find that the ratio of market to book value is positively related to the current ROE rate ρt , the long run mean rate ρ , and the volatility of accounting profits. An 7
Pastor and Veronesi (2003) present this model in continuous time, and discuss the assumption of a fixed time horizon T at length.
17
increase in the risk-free rate, the market risk premium, or profitability rate beta lead to a lower market to book ratio. Interestingly, Pastor and Veronesi (2003) obtain similar results with a continuous time model and a stochastic discount factor, whereas ours are based on the CAPM. For non-dividend paying firms we set c=0, and obtain the market to book ratio as: T
Mt = Bt
(e A(T )+ B (T ) ρ +C (T ) ρt )∏ (1 − ( ERm − R f ) z j β ρ ) j =1
(1 + R f )T
(21)
The condition for a finite market to book ratio is roughly identical to the stochastic dividend growth model. Provided T is large, the restriction on long term profitability is:
ρ ≤ R f + ( ERm − R f )
2 βρ 1 σζ − (1 − φ ) 2 (1 − φ )2
(22)
The right hand side of this expression is similar to the condition (11); however, the profitability beta replaces the growth rate beta adjusted by the degree of predictability in the book equity and the long-run variance of ROE shocks. Once again there is no restrictions on the short run ROE.
7. Conclusions The St. Petersburg paradox describes a simple game of chance with infinite expected payoff, and yet any reasonable investor will pay no more than a few dollars to participate in the game. Researchers throughout history have provided a number of solutions as well as variations of the original paradox. One of these, developed by Durand (1957), shows that the standard St. Petersburg game can describe an investment in a firm with a constant growth rate of dividends. To capture a more realistic growth pattern, we present a model that allows mean reversion in dividends. We then derive the risk adjustment required in a CAPM environment, and propose an equivalent St. Petersburg game. We show that the expected payoff of the modified game (or equivalently, the value of growth firms) is driven mainly by the long-run growth rate of the 18
payoffs (dividends), while the short-term growth rate has a minor effect on the properties of the game or the firm. The model further shows that the condition under which the value of the game or the firm is finite is much less restrictive than that of the classical St. Petersburg game, and this might provide an indirect solution to the paradox.
19
References Aase, K., 2001, “On the St. Petersburg Paradox,” Scandinavian Actuarial Journal 1, 69-78. Ali, M., 1977, “Analysis of Autoregressive-Moving Average Models: Estimation and Prediction,” Biometrika 64, 535-545. Bansal, R., and A. Yaron, 2004, “Risks for the Long Run: A Potential Resolution of Asset Pricing Puzzles,” Journal of Finance 59, 1481-1509. Bernoulli, D., 1738, “Specimen Theoriae Novae de Mensura Sortis,” Commentarii Academiae Scientiarum Imperialis Petropolitanea V, 175-192. Translated and republished as “Exposition of a New Theory on the Measurement of Risk,” 1954, Econometrica 22, 23-36. Bhamra, H., Kuehn, L., and I. Strebulaev, 2010, “The Levered Equity Risk Premium and Credit Spreads: A Unified Framework,” Review of Financial Studies 23, 645-703. Durand, D., 1957, “Growth Stocks and the Petersburg Paradox,” Journal of Finance 12, 348363. Fama, E., 1977. “Risk-Adjusted Discount Rates and Capital Budgeting Under Uncertainty,” Journal of Financial Economics, 5 (August): 3-24. Fama, E., and K. French, 2000, “Forecasting Profitability And Earnings,” Journal of Business 73, 161-175. Friedman, M. and L. J. Savage, 1948, “The Utility Analysis of Choices Involving Risk,” Journal of Political Economy 56, 279-304. Gordon, M. (1962), The Investment, Financing, and Valuation of the Corporation. Homewood, Ill.: Irwin. Lintner, J., 1965, “The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets,” Review of Economics and Statistics 47, 1337-1355. Pastor, L., and P. Veronesi, 2003, “Stock valuation and learning about profitability,” Journal of Finance 58, 1749–1789. Pratt, J., 1964, “Risk Aversion in the Small and in the Large,” Econometrica 32, 122-136. Rubinstein M., 1976, Valuation of Uncertain Income Streams and the Pricing of Options. Bell Journal of Economics (Autumn), 407-425. Senetti J. T., 1976, “On Bernoulli, Sharpe, Financial Risk, and the St. Petersburg Paradox.” Journal of Finance 31, 960-962.
20
Sz´ekely, G. J., and Richards, D. St. P. (2004), “The St. Petersburg Paradox and the Crash of High-Tech Stocks in 2000,” The American Statistician, 58, 225-231. Sharpe, W. F., 1964, “Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk,” Journal of Finance 19, 425-442. Weirich, P., 1984, “The St. Petersburg Gamble and Risk,” Theory and Decision 17, 193-202.
21
Figure 1. Upper bound on long run dividend growth rate The three sloping lines represent the upper bound on the long run dividend growth rate, computed from Equation (12), as a function of the degree of mean reversion ( φ ), for three levels of β g . The model parameters are: R f = 0.037 , market risk premium E ( Rm − R f ) = 0.0804 , 2 and σ ε = 0.01. The horizontal line, set at R f = 0.037 , represents the upper bound on the constant (deterministic) growth rate in the Gordon model.
0.35
0.3
Upper bound
0.25
0.2 Rf Beta g = 0.5 0.15
Beta g = 1.0 Beta g = 1.5
0.1
0.05
0 0.00
0.10
0.20
0.30
0.40
0.50
0.60
Phi
22
Appendix: Proof of Equations (9) and (10) Without loss of generality, we set time t at 0, and define the sequence of future growth rates as a row vector G ' ≡ ( g1, K, gτ ) . We then use the following system of equations to describe potential
sample paths from time periods 1 thru τ:
0 0 . . 1 − φ 1 0 . . 0 −φ 1 0 . . . . 0 . . 0 −φ
0 0 0 1
g1 (1 − φ ) g φ gt ε1 g (1 − φ ) g 0 ε 2 2 . = . + . + . . . . . gτ (1 − φ ) g 0 ετ
(1)
A compact representation for this system is ΦG = (1 − φ ) g i + G0 + Ε , where i is a column vector of 1s, G0 is a column vector with φ g0 in the first row and 0 in the remaining rows, and E ' ≡ (ε1 , ε 2 , K , ετ ) is the vector of random innovation terms. This set implies that the
cumulative growth rate
τ
∑ j =1 g j
' −1 ' −1 has conditional mean (1 − φ ) g i Φ i + i Φ G0 and
conditional variance σ ε2 i ' Φ −1 (Φ −1 ) ' i . Using a result from time series analysis (Ali, 1977), we show next that these moments may be computed without inverting the Φ matrix. Define the vector Z ' ≡ ( zT , zT −1 , K , z1 ) = i ' Φ −1 and note that each element may be computed recursively from the previous one: z j = φ z j −1 + 1 for j=1, 2, … , τ, and starting value z0=0. Hence, each τ τ future expected dividend is given by: E 0 Dτ = D0 exp (1 − φ ) g ∑ z j + zτ φ g0 + (σ ε2 / 2) ∑ z 2j , j =1 j =1
and Equation (10) follows immediately. Next, we derive the present value of each future dividend Dτ starting from τ=1, 2, and so on. Let R =
D1 − 1 be the rate of return on a claim that pays off a single cash flow $ D1 at τ=1, V0
23
and sells for V0 .
Plug this return into the security market line (Equation 8) to show that the
present value is given by the expected dividend multiplied by a discount factor
1 − ( ERm − R f )Cov( D0 /( Et D1 ), Rm 1 ) / σ m2 . Using Stein’s lemma it follows that: V 0 = (E0 D1 ) 1+ Rf
(
)
(
)
D Cov 1 , Rm1 = Cov ε1, Rm1 . Define the growth rate beta β g = Cov ε1, Rm1 / σ ε2 . E0 D1 Therefore, the present value of the first dividend is given by Equation (9) with τ=1. Next, let V1 be the time-1 value of a single cash flow $D2 expected one period later. V Again, let R = 1 − 1 be the rate of return (from 0 to 1) from holding the claim on $D2. The V0
security
market
line
(Equation
8)
implies
that
2 1 − ( ERm − R f )Cov(V1 /( E0 V1), Rm1) / σ m . Using a similar argument as in Fama 1+ R f
V 0 = (E0 V1 )
1 − ( ERm − R f ) z1β g (1977), we can show that E0V1 = (E0 D2 ) . Moreover, the ratio of V1 to its 1+ Rf conditional expectation one period prior, E0V1, equals the ratio of cash flow expectations:
V V1 = E1 D 2 . Then, from Stein’s lemma we have Cov 1 , Rm,1 = Cov z 2ε 1 + z1ε 2 , Rm ,1 = E 0V 1 E 0 D 2 E0 V1
(
(
)
)
z 2 Cov ε1 , Rm ,1 . Therefore, the present value of the second dividend is given by Equation (9)
with τ=2. Proceeding in this fashion one may show that for any Dτ , the present value is:
24
τ
V0=
(E0 Dτ )∏ [1 − ( ERm − R f ) z j β g ] j =1
(1 + R f )τ
. Thus, Equation (9) holds by the principle of value
additivity. ■
Proof of Equations (15 and 16) Consider the sequence of future growth rates G ' ≡ ( g t +1 , K , g t +T ) may be described as a multivariate system ΦG = (1 − φ1 − φ2 ) g i + G0 + ΘΕ , where
1 − φ 1 Φ = − φ2 0
0
0
.
.
1
0
.
.
− φ1 1 . .
0 .
.
− φ2
− φ1
.
.
0 1 − θ 0 0 , Θ = 0 0 1
0
0 .
.
1 −θ .
0 . 1 0 . .
. .
.
.
0 −θ
0 0 0 I
G, g , i, and Ε were defined in the previous proof, while the initial conditions vector G0 consists of φ1 gt + φ2 gt −1 − θ ε t in the first row, φ2 gt in the second row, and 0s in the τ-3 remaining rows. The first two moments of cumulative growth are:
(
)(
Et i'G = (1 − φ ) gi'Φ−1i + i'Φ−1G0 and
)
' Vt i 'G = σ ε2 i 'Φ −1Θ i 'Φ −1Θ . Again, define Z ' ≡ ( zT , zT −1 , K, z1 ) = i 'Φ −1 so that each element
may be computed recursively from the previous one: z j = φ1 z j −1 + φ2 z j − 2 + 1 , and starting value of z0 = 0. Define also the vector W ' ≡ ( wT , wT −1 , K, w1 ) = Z 'Θ to aggregate serial correlation induced by the moving average component of growth. Each element may be computed recursively as: w j = z j − θ1 z j −1 for j=1, 2, … , τ. Given these transformation, the conditional expected
future
cash
flow
has
a
closed
form
solution
given
by
T T 2 2 E t Dt +T = Dt exp(1 − φ1 − φ2 ) g ∑ z j + zT φ1 gt + zT φ2 gt −1 − zT θ ε t + (σ ε / 2) ∑ z j . We note that j =1 j =1 25
this expectation is conditional on the time t shock to the current growth rate. This shock is unobservable, therefore we use a property of normal random variables to obtain the unconditional value. To obtain this result, observe that the expectation of Et [ Dt +T | ε t ] is analogous to the moment generating function of εt evaluated at the point zTθ . Then, Equation (16) follows immediately. The rest of the proof is by induction on t. From the proof of (8) above, we know that at τ-1 1 − ( ERm − R f ) β g the dividend discounted value is given by: V τ −1 = Eτ −1 Dτ . Thus, (15) holds 1+ R f as of τ-1 because the first value of w is 1. Assume the result holds for time period t = τ+1. From Stein’s
lemma
(
we
have
T Covτ ∑ g s , Rm,τ +1 = s =τ +1
(
Covτ W ' E , Rm,τ +1
)
=
)
wT −τ Covτ ετ +1, Rm,τ +1 . Using the same logic as in Proposition 1, as we move back one time
1 − ( ERm − R f ) wτ βε period from τ+1 to τ, the discount factor is . Thus, the time t=T-τ price is 1 + R f
given by: τ
(ET −τ DT )∏ [1 − ( ERm − R f )w j β g ] V t,T =
j =1
(1 + R f )τ
This last step shows that the proposition holds for time period t = T-τ, and all other times t. ■
26
Carmelo Giaccotto*
September 2015
ABSTRACT Durand (1957) shows that the classical St. Petersburg paradox can apply to the valuation of a firm whose dividends grow at a constant rate forever. To capture a more realistic pattern of dividends, we model the dividend growth rate as a mean reverting process, and then use the Capital Asset Pricing Model (CAPM) to derive the risk-adjusted present value. The model generates an equivalent St. Petersburg game. The long-run growth rate of the payoffs (dividends) is dominant in driving the value of the game (firm), and the condition under which the value is finite is less restrictive than that of the standard game. Keywords: Capital Asset Pricing Model (CAPM); Stochastic Dividends; St. Petersburg paradox JEL Classification: G0, G1
*University
of Connecticut. We thank Iskandar Arifin, John Harding, Po-Hsuan Hsu, Jose Martinez, Efdal Misirli, Tom O’Brien, Scott Roark, Jim Sfiridis, and participants at the finance seminar series at the University of Connecticut for valuable comments and suggestions. Please send all correspondence to Carmelo Giaccotto. Mailing address: University of Connecticut, School of Business, 2100 Hillside Road, Storrs, CT 06269-1041. Tel.: (860) 486-4360. E-mail: [email protected].
The St. Petersburg Paradox and Capital Asset Pricing 1. Introduction The St. Petersburg paradox is one of the most well-known and interesting problems in the history of financial economics. The paradox describes a situation where a simple game of chance offers an infinite expected payoff, and yet any reasonable investor will pay no more than a few dollars to participate in the game. Since the paradox was presented by Daniel Bernoulli in 1738, it has attracted a great deal of interest, mainly by theorists who provide solutions and derive its implications. One of the applications of the paradox is in the area of financial asset pricing. Durand (1957) shows that the St. Petersburg game can be transformed to describe a conventional stock pricing model for growth firms. The analogy is based on the assumption that the firm’s future dividends (as the game’s future payoffs) grow at a constant rate. Economic intuition and the historical evidence suggest, however, that the very high growth rates experienced by many young firms (e.g., firms in the high-tech industry) are expected to decline over time. Hence, the short-run growth rate is typically much greater than the expected long-run rate. In this study we model the dividend growth rate as a mean reverting process; we then find the risk-adjusted growth rate under the equilibrium setting of the classical Capital Asset Pricing Model (CAPM), and derive its equivalent modified St. Petersburg game.1 Our paper has several interesting results. First, we assume an autoregressive process for the dividend growth rate, and then use the CAPM to derive a closed form solution for the price of a growth stock. This result is a significant contribution to the asset pricing literature; it extends
1
Assuming a stochastic dividend growth is rather common in asset pricing studies; see, for example, Bansal and Yaron (2004), and Bhamra and Strebulaev (2010).
Rubenstein’s (1976) single factor arbitrage-free present value model by allowing mean reverting growth. Second, by distinguishing between the short-run and long-run growth rates, the model shows that the latter is the dominant factor in driving the value of a growth stock, and in turn, the properties of the St. Petersburg game, including its value. Third, we derive the condition under which the expected payoff of the game (or equivalently, the value of a growth firm) is finite; as expected, this condition is much less restrictive than the one required under constant growth. Fourth, and last, our solution to the paradox extends the work of Aase (2001) who framed the paradox as an arbitrage problem. His solution requires, in part, that economic agents discount future cash flows at a constant rate. By using the Capital Asset Pricing Model we are able to derive the required equilibrium discount factor, and show that the game price is finite. The rest of this paper is organized as follows. In the next section we describe the paradox and a number of solutions proposed over the years; in section 3 we describe the correspondence between the St. Petersburg paradox and the value of growth stocks – the original contribution of Durand (1957). Section 4 presents our stochastic dividend growth model and derive the valuation model implied by the CAPM. We find that the restriction to insure a finite asset value is by far less restrictive than the original one. In section 5 we present a modified St. Petersburg game for stochastic growth stocks, and in section 6 we consider two extensions of the basic valuation model along with the conditions necessary to preclude a St. Petersburg type paradox. The conclusion is in section 7.
2. The paradox and its common solutions The St. Petersburg paradox is based on the following simple game. A fair coin is tossed repeatedly until the first time it falls on ‘head’. The player’s payoff is 2n dollars (‘ducats’ in the
2
original Bernoulli’s paper), where n is the number of tosses. Since n is a geometric random variable with p=0.5, the expected payoff of the game is: E=
1 1 1 × 2 + × 4 + × 8 + ... = 1 + 1 + 1 + ... = ∞ 2 4 8
(1)
Yet, while this game offers an expected payoff of infinite dollars, a typical player will pay no more than a few dollars to participate in the game, reasoning that there is a very small probability to earn a significant amount of money. For example, the chance to earn at least 32 dollars is
2 −5 = 0.03125 , and at least 128 dollars is 2 −7 = 0.00713; hence, paying a game fee of even 1,000 dollars seems unreasonable. A number of solutions proposed to resolve the paradox rely on the concept of utility.2 The basic idea is that the relative satisfaction from an additional dollar decreases with the total amount of money received. Thus, the game fee should be based on the expected utility from the dollars earned, rather than the expected amount of dollars. Bernoulli himself suggested the log utility function; in this case, the expected value of the game is: ∞ 1 1 1 ln(2 n ) E (U ) = × ln(2) + × ln(4) + × ln(8) + ... = ∑ n = 2 ln(2) < ∞ 2 4 8 n=1 2
(2)
It turns out, however, that the utility-based solutions are incomplete: the game can be converted to a convex stream of payoffs, and this change reverses the benefit of a concave utility n
function. For example, if instead of 2 n , the game’s payoff is e 2 , then for the log utility just considered, the expected value again tends toward infinity. In general, for every unbounded utility function the payoffs can be changed such that the expected utility will be infinite. This generalization of the game often goes by the name “super St. Petersburg paradox.”
2
See Senetti (1976) on solving the paradox using expected utility in light of modern portfolio theory, and Sz´ekely and Richards (2004) for other suggested solutions to the paradox.
3
Proposed solutions to the super game rely on the concept of risk aversion (Friedman and Savage (1948), and Pratt (1964)); that is, holding everything else constant, a typical player prefers less risk, and therefore is willing to pay a lower fee for high-risk games. Weirich (1984) shows that, while the game may offer an expected infinite sum of money, it involves also an infinite amount of risk. This is because the dispersion of the possible payoffs results in an infinite standard deviation. Thus, the fair game fee resembles the difference between an infinite expected payoff and an infinite measure of risk. While the answer may actually be finite, it does not explain the low amounts typical players are willing to pay, which range between 2 to 25 dollars. Aase (2001) proposes an interesting solution by reframing the paradox as an arbitrage problem. He then shows that if (a) economic agents have finite credit at their disposal, and (b) agents discount cash flows to the present at a constant discount rate, then any arbitrage opportunity will disappear and the price of the game becomes finite. Our contribution is to frame the paradox as a growth stock – in the spirit of the Gordon dividend discount model, and then use the CAPM to derive the appropriate discount factor.
3. Application of the paradox to growth stocks One of the applications of the St. Petersburg paradox is in the area of asset pricing. Durand (1957) shows that with some modifications, the paradox can describe a conventional stock pricing model. A growth firm expects to generate an increasing stream of future earnings, and thus to pay an increasing stream of dividends. The value of such a firm is given by the present value of all future dividends:
P0 =
∞ Dt D1 D2 + + ... = ∑ t (1 + r ) (1 + r ) 2 t =1 (1 + r )
(3)
4
where Dt is the per-share dividend of year t and r is the discount rate. The constant growth model, also known as the Gordon (1962) model, assumes that future dividends grow at a constant rate (g); i.e., the dividend stream is D1 , D1 (1 + g ), D1 (1 + g ) 2 , ... for the years 1, 2, 3,… In that case, the present value of this growing perpetual future dividend stream equals:
D1 , if g < r D1 (1 + g ) t −1 r g − P0 = ∑ = (1 + r ) t t =1 ∞ , if g ≥ r ∞
(4)
Thus, the firm value is finite only if the growth rate is lower than the discount rate. Durand derives the St. Petersburg analogue of the constant growth model using the following analysis. Consider the St. Petersburg game, where instead of a fair coin, the probability that a ‘head’ appears is r (1 + r ) , where r > 0 . Assume further that instead of earning a single payment when the game ends (i.e., when ‘head’ appears in the first time), the player earns a specific amount of dollars as long as the game continues. Specifically, the player will earn D1 if the first toss is ‘tail’, D1 (1 + g ) if the second toss is ‘tail’, D1 (1 + g ) 2 if the third toss is ‘tail’, and so on. That is, if the game lasts for n tosses, instead of earning 2n , the player will earn n −1
∑ D1 (1 + g ) j −1 = j =1
D1[(1 + g ) n−1 − 1] . Therefore, the expected payoff of the game is: g
D1 , if g < r D1[(1 + g ) n−1 − 1] ∞ D1 (1 + g ) n −1 r ( r − g ) E=∑ = = ∑ n g (1 + r ) n n =1 (1 + r ) n =1 ∞, if g ≥ r ∞
(5)
which is identical to the value of a constant dividend growth firm (as appears in Equation 4). Note that the analogy is based on the translation of the discount rate r to the coin probability
r (1 + r ) . That is, while in the original St. Petersburg game any future payment will be paid with some probability, in the modified game, which is equivalent to the dividend stream generated by 5
growth firms, any future payment will be paid for sure, but will be evaluated with a discount factor. This analogy helps explaining the condition under which the expected payoff of the game is infinite. The value of the stock is infinite only when the dividend grows at an equal or higher rate than the discount rate; and in the same way, the expected payoff of the game is infinite only when the payoffs are increasing at an equal or higher rate than the rate at which the correspondent probabilities are decreasing. An important characteristic of Durand’s constructed game is that its representation is not unique, as different forms of the payoff series and the coin probability can yield the same expected value. To illustrate, consider the same setup of Durand’s game outlined above where the probability that a ‘head’ appears is now (r − g ) (1 + r ) instead of r (1 + r ) , for g < r . The payoff series also changes, by a factor of r (r − g ) : the player will earn D1 r ( r − g ) if the first toss is ‘tail’, D1 (1 + g ) r (r − g ) if the second toss is ‘tail’, D1 (1 + g ) 2 r ( r − g ) if the third toss is ‘tail’, and so on. That is, if the game lasts for n tosses, the player will earn n −1
∑ D1 (1 + g ) j −1 r (r − g ) = j =1
D1[(1 + g ) n −1 − 1] r ( r − g ) . Therefore, the expected payoff of the game g
is:
D1 (r − g ) D1[(1 + g ) n−1 − 1] r (r − g ) ∞ D1 (1 + g ) n−1 =∑ = n n g (r − g ) (1 + r ) n=1 (1 + r ) n=1 ∞
E=∑
(6)
which is the value of a constant dividend growth firm (as appears in Equation 5). While these two representations of Durand’s game yield the same expected value, they generate different probability distribution of the payoffs; particularly, the first representation offers lower payments than the second representation but with higher probability to stay in the game. This difference in the payoff probability distribution may have implications with respect to the investor preferences. For example, consider a constant growth stock with D1 = $2 , r = 15% 6
, and g = 10%. The value of the stock is
D1 $2 = = $40 , which is also the expected ( r − g ) 0.15 − 0.1
value of the payoffs under the two representations of Durand’s game (Equations 5 and 6). However, the balance between the periodic payments and the corresponding probabilities is different under the two representations; for instance, under the first representation of the game the probability that the game will last at least 5 tosses is 49.7% at which the player will earn at least $12.21, whereas under the second representation of the game the probability that the game will last at least 5 tosses is 16.6% at which the player will earn at least $36.63. Assuming that a typical investor has a decreasing utility function (as discussed in Section 2), the first representation of the game might be more appealing to the investor, and therefore she will be willing to pay a higher fee to participate in the game. We believe the paradox arises for two reasons. First, the assumption that the dividend stream will grow at a constant rate permanently is unrealistic. For example, economic intuition, as well as the historical evidence, suggests that high growth tech firms (such as IBM, Microsoft and now Google or Facebook) may grow very rapidly in the short run. But nothing attracts competition like market success; therefore, in the long run new market entrants will force the earnings growth rate to slow down (almost surely) to a level consistent with the growth rate of the overall economy. The second reason is that the degree of risk implicit in the dividend stream may cause investors to change the risk-adjusted discount rate, i.e., the probability of actually receiving the expected dividends. In the next section we formalize these ideas within the context of the Capital Asset Pricing Model of Sharpe (1964) and Lintner (1965). We model the dividend growth rate as a mean reverting process so that the current rate can be very large but the long run rate is expected to be much lower. We then use the CAPM to derive the appropriate risk-adjusted present value. We
7
find that the equity price can be finite without the unreasonable condition required by the constant growth model.
4. Stochastic dividend valuation in the CAPM world Let Dt be the time-t value of dividends or earnings (for simplicity we will use these two g terms interchangeably), and assume these values grow at rate g t +1 : Dt +1 = (e t +1 )Dt . We use a
first order autoregressive process ( AR(1) ) to model mean reversion:
gt +1 = (1 − φ ) g + φ gt + ε t +1
(7)
where g is the long run (unconditional) mean growth rate and φ is the autoregressive coefficient (we assume 0 ≤ φ < 1 to be consistent with the smooth behavior of dividends). We make the usual assumptions to insure the process is stationary and the growth rate is mean reverting. The innovation terms ε t +1 are normally distributed random variables with mean zero, variance σ ε2 , no serial correlation, and constant covariance with the market portfolio return. The major implication of the mean-reverting dividend model is that while the current growth rate can be abnormally large, in the long run earnings growth should slow down to a lower rate. Intuitively, we expect g to be close to the growth rate for the overall economy because of competitive pressures brought about by new startup companies. To obtain the risk-adjusted present value of each future dividend we use the CAPM of Sharpe (1964) and Lintner (1965): ER = R f + [ ERm − R f ]β ROR
(8)
where ER is the single period expected rate of return on the asset, R f is the risk-free rate of interest, [ ERm - R f ] is the expected excess market portfolio return, and market risk is measured 8
by the rate of return beta ( β ROR ) . Then, the CAPM price for an asset that pays a stochastic dividend stream {Dt +τ }τ∞=1 is given by the sum of expected future dividends adjusted for market risk, and discounted to the present at the riskless rate of interest (Equations 9 and 10 are derived in the Appendix): τ
∞
(Et Dt +τ )∏ [1 − ( ERm − R f ) z j β g ]
τ =1
(1 + R f ) τ
Pt = ∑
j =1
(9)
where the deterministic variable zj captures the effect of mean reversion on cash flows and the market risk. It may be computed recursively as: z j = φ z j −1 + 1 for j=1, 2, … and starting value
z0 = 0 . The growth rate beta, β g , is defined as the covariance between the growth rate innovation (εt+1) and the market return, divided by the variance of the market return. The expected dividend series is an exponential affine function of the current dividend growth rate and long run growth: Et Dt +τ = Dt e A(τ ) + B (τ ) g + C (τ ) g t
and
(10)
τ
τ
j =1
j =1
A(τ ) = (σ ε2 / 2) ∑ z 2j , B(τ ) = (1 − φ ) ∑ z j , and C (τ ) = φ zτ , Equations (9) and (10) show that the current price depends on the current growth gt, and the
long run rate g ; however, the impact of the latter is much stronger because it is multiplied by
B(τ ) (the sum of zj). We note that z j converges to z = 1 /(1 − φ ) for large τ, thus B(τ ) increases with the time horizon. On the other hand, C (τ ) converges to a constant finite value φ /(1 − φ ) . Intuitively, stronger mean reversion implies faster reversal to the long run mean; therefore, a currently high growth rate has only a transitory impact on the equity price. The permanent component of price is driven by long run dividend growth.
9
Today’s price depends also on the appropriate risk adjustment. Since the market risk, represented by z j β g , increases with j -- up to β g /(1 − φ ) , the adjustment for risk becomes increasingly larger for distant expected dividends, and this helps obtain a finite present value. Two special cases of the general model are worth mentioning. The first is Gordon’s deterministic growth model which is obtained by setting φ = 0 and σ ε2 = 0 .
In this case,
dividends are expected to grow in a deterministic fashion at a constant rate g . The beta factor ( β g ) equals zero and the discount rate r equals the riskless rate R f . The second, originally developed by Rubenstein (1976) within the context of a single factor arbitrage-free model, allows stochastic growth but no serial correlation: gt +1 = g + ε t +1 . In this case, (log) earnings or dividends follow a random walk with drift, and the stock price has a closed-form solution similar to Durand’s formula: Pt =
given by
(1 + R f ) 1 − ( ERm − R f ) β g
Dt (e g* ) r − e g*
, where the discount rate r is
2 , and the adjusted growth rate is g * = g +σε / 2 . The major
drawback of these two models is that unlike our autoregressive model, they do not allow a distinction between current growth – which can be abnormally high, and long-run growth. We can now derive the condition under which the asset price is finite. Observe that for large j, z j converges to a constant value of 1 (1 − φ ) ; therefore, the expected future dividend will
evolve along the
2 g + 1 σ ε T 2 (1−φ )2 path e
. Also, for dividends far into the future, the risk-adjusted T
1 − ( ERm − R f ) β g (1 − φ ) present value factor may be approximated by . Therefore, the 1+ R f present value of a single dividend DT , for large T, may be approximated as follows: 10
V t ≈ Dt e
β g 1 σ ε2 T − R f + ( ERm − R f ) g + 2 2 ( 1 −φ ) (1−φ )
(11)
This value will converge to zero, and thereby the sum of the present values of all future dividends (i.e., the firm value) will be finite, provided the expression inside the square brackets is negative. Thus, the restriction on long term growth is:
g ≤ R f + ( ERm − R f )
βg 1 σ ε2 − (1 − φ ) 2 (1 − φ )2
(12)
The right hand side of this expression is similar to the CAPM risk-adjusted return with two modifications. First, risk is measured by the growth rate beta adjusted by the degree of predictability in the dividend stream, and second, since the growth rate is continuously compounded, we need to subtract one half the long-run variance of dividend shocks. Clearly this condition is less restrictive than for constant growth (i.e., g < r ); and, more importantly, it imposes no restrictions on the short term dividend growth rate. Figure 1 illustrates the upper bound condition described by Equation (12). We vary the level of mean reversion, φ , on the horizontal axis from 0.0 (strong) to 0.6 (weak), and include three levels of β g : 0.5, 1.0, and 1.5. To set the riskless rate and the market risk premium, we use data from Professor Ken French’s website. We have R f = 0 .037 and E ( Rm − R f ) = 0.0804 , which 2 correspond to the sample averages from 1927 thru 2013. Last, σ ε is set arbitrarily at 0.01
because its exact value has a marginal effect on the upper bound.3 The horizontal line at 3.7% corresponds to the constant growth Gordon model; clearly this is a low bar for the long run growth rate. The more realistic cases reflect varying degrees of mean reversion. In the strongest case, where φ = 0.0, the upper bound increases with beta. When β g = 3
We consider both a 50 percent decrease and a 50 percent increase in the variance of shocks to the dividend growth rate. As expected the new upper bound on long term dividend growth is not substantially affected by these changes.
11
0.5, the risk-adjusted upper bound is 7.2%, and increases to 11.2% for β g = 1.0, and to 15.3% for β g = 1.5. Figure 1 shows that these upper bounds increase even faster as mean reversion weakens. No firm can be expected to grow permanently at such high rates.
5. A modified St. Petersburg game for stochastic growth stocks As Durand (1957) shows, the classical St. Petersburg game (with some modifications) can describe an investment in a stock with a constant dividend growth rate. To capture the more realistic pattern of growth firms (as outlined in the previous section), we present a modified St. Petersburg game that is analogous to a stock with a mean reverting dividend growth rate in the equilibrium setting of the CAPM. Consider the St. Petersburg game, where instead of a fair coin, the probability that a ‘head’ appears is R f ( 1 + R f ) , where R f > 0 (this constant probability is consistent with that in Durand’s game using the risk-free rate, R f , instead of r). Assume further that the player earns an j
Dj∏aj i =1
amount of
Rf
dollars in the jth toss (including the last toss of the game), where D j
denotes the dividend series as defined in Section 4, and a j = 1 − ( ER m − R f ) z j β g . That is, if the j
game lasts for n tosses, the player will earn
n
Dj ∏a j
j =1
Rf
∑
i =1
dollars. The expected payoff of the
game is therefore:
∞
(E 0 Dn )∏ [1 − ( ERm − R f ) z j β g ]
n =1
(1 + R f ) n
n
∞
E=∑
Dn ∏ a j
Rf
n =1 (1 +
Rf )
j =1
n
Rf
n
=
∑
j =1
(13)
12
which matches exactly the stock value in the CAPM world (Equation 9). Note that, as for Durand’s game, the representation of the modified game is also not unique where different forms of payoffs and corresponding probabilities can achieve the same expected value. The condition under which the value of the game is finite, therefore, is identical to the one that makes the stock price finite, as given in Equation (12). Since this condition is less restrictive than the classical St. Petersburg game (as discussed above), it provides an indirect solution to the paradox. That is, under the more realistic setup of a stochastic mean reverting growth rate, it is more likely that the value of the game is finite, and therefore the game fee the players are willing to pay is finite.
6. Extensions In this section we consider two extensions of the basic valuation model (Equations 9 and 10) and derive the conditions necessary to preclude a St. Petersburg paradox. The first extension relaxes the assumption of a constant long-run mean dividend growth rate; while the second deals with firms that may or may not pay dividends.4
6.1 Valuation when long-run growth varies with the business cycle In section 4, for simplicity of exposition, we modeled dividend growth as a first order mean reverting process with constant long run dividend growth. But as the economy moves across the business cycle, g may be time-varying. Let g~t be the long run dividend growth rate, then we rewrite Equation (7) as gt = (1 − φ ) g~t + φ gt −1 + ε t . It can be easily shown that if g~t is itself mean
4
It may seem like a natural extension to consider also time variation in the rate of return beta. However, Fama (1977) shows that the single period CAPM precludes uncertainty about the riskless rate, the market risk premium as well as beta.
13
reverting (e.g., AR(1) ), then the reduced form model for dividend growth becomes ARMA(2, 1), that is second order autoregressive process with a first order moving average component:
gt = (1 − φ1 − φ2 ) g + φ1 gt −1 + φ2 gt −2 + ε t − θ ε t −1
(14)
where ( φ1, φ2 , θ ) are, respectively, the autoregressive and moving-average coefficients.5 Again, we assume the process is stationary and shocks to the growth rate have constant covariance with the market return. To compute asset prices we need to define two sets of auxiliary equations to account for the autoregressive and moving average components of serial correlation in the cumulative growth rate. The two sequences are: w j = z j − θ z j −1 , and z j = φ1 z j −1 + φ2 z j − 2 + 1 , with a starting value of z0 = 0 . Then, the time-t price is given by (proofs of Equations (15) and (16) are provided in the appendix): τ
∞
(E0 Dt +τ )∏ [1 − ( ERm − R f ) w j β g ]
τ =1
(1 + R f ) τ
Pt = ∑
j =1
(15)
and the expected future dividend is
Et Dt +τ = Dt e [ A(τ ) + θ The
other
parameters
zτ ]σ ε2 / 2 + B (τ ) g + C (τ ) g t + D (τ ) g t −1
2 2
are
defined
(16) τ
as: A(τ ) = ∑ w2j , j =1
τ
B(τ ) = (1 − φ1 − φ2 ) ∑ z j , j =1
C (τ ) = φ1 zτ + φ2 zτ −1 , and D(τ ) = φ2 zτ . Using numerical derivatives, one may show that this asset pricing model has some interesting characteristics. First, the current price increases with both the current and long run dividend growth rates. However, as was the case for the AR(1) model, price is much more sensitive to 5
Fama and French (2000) provide empirical evidence for mean reversion in profitability. They find that the strength of mean reversion varies depending on whether profitability is above or below its mean. To the extent that dividends follow earnings, the ARMA(2,1) model captures these dynamics.
14
long run than short run dividend growth. Second, price decreases with growth rate beta and the market risk premium. And third, growth rate volatility has a positive impact on price because of the convexity of compounded cash flows. The condition under which the asset price is finite is very similar to that of the constant long run dividend growth (Equation 12). We note that for long horizon cash flows, z j converges to a constant value of 1 (1 − φ1 − φ2 ) , and w j converges to (1 − θ ) /(1 − φ1 − φ2 ) . Thus, for large τ, both C (τ ) and D(τ ) play a negligible role. B(τ) increases linearly with τ ; and A(τ ) converges
(σ ε2 / 2)(1 − θ ) τ. to (1 − φ1 − φ2 )
The expected dividend series will evolve along the
2 g + 1 σ ε (1−θ ) T 2 (1−φ1 −φ2 ) 2 path e
, while the risk-
T
1 − ( ERm − R f ) β g (1 − θ ) (1 − φ1 − φ2 ) adjusted discount factor converges to . Therefore, for 1+ Rf large T, the present value of a single dividend DT may be approximated as:
V t ≈ Dt
β g (1−θ ) 1 σ ε2 (1−θ )2 T − R f + ( ERm − R f ) g + 2 2 (1−φ1 −φ2 ) (1−φ1 −φ2 ) e
(17)
Finite pricing of assets requires that the following restriction holds for long term growth:
β g (1 − θ ) 1 σ ε2 (1 − θ )2 g ≤ R f + ( ERm − R f ) − (1 − φ1 − φ2 ) 2 (1 − φ1 − φ2 )2
(18)
The right hand side of Equation (18) again consists of three terms: the first two represent the CAPM risk-adjusted return with a modified beta to account for predictability in the dividend stream. The third term is needed because the analysis is in terms of continuously compounded dividend growth. Once again we find that the condition is less restrictive than a simple g < r ; furthermore, there is no restriction on short term dividend growth. 15
6.2 Valuation of Non-Dividend Paying Firms In this section we generalize the model to the universe of firms that may or may not pay dividends. We show that the main results hold provided we take into account mean reversion in profitability. The foundation for this analysis is the “Clean Surplus” identity for a firm that is financed by equity only and expects no new equity issues. This accounting relationship then states that the current book value of equity equals last period’s book value plus current income minus current dividends: Bt = Bt −1 + I t − Dt . The assumption of equity financing is not critical to the analysis that follows. The main result in this section holds in the presence of debt financing provided that the accounting rate of return on book equity (ROE) is independent of the level of debt. For firms under financial distress this independence hypothesis may be inappropriate; but it is likely to hold for healthy firms that maintain a fairly constant debt to equity ratio. There are many ways to model dividend payments. For example, one may choose a dividend rate set at a constant fraction of earnings; but earnings may be negative from time to time and negative dividends would be inappropriate. Also earnings are quite volatile over the business cycle, and a constant relationship would make dividends quite erratic. Because dividends are typically smooth, we assume that dividends are paid as a percent of book value, and examine the implications for valuation by the CAPM. Let c be the constant proportion of book equity paid out as a periodic dividend: Dt+1 = cBt. Setting c = 0 allows us to model firms that pay no dividends. We define the accounting rate of return on book equity (ROE) ρt+1 as firm’s earnings – at end of period t+1, divided by book value of equity as of period t. Then, the clean surplus relation implies that book equity grows as:6
(
)
Bt +1 = e ρ t +1 − c Bt
(19)
(
The clean surplus relationship implies that the rate of growth in book value is Bt +1 = e The approximation is exact only in continuous time. 6
ln(1+ ρ t +1 − c )
)B ≈ (e t
ρ t +1 − c
)B . t
16
To model the profitability rate, we assume a first order autoregressive process:
ρt +1 = (1 − φ )ρ + φ ρt + ζ t +1 , where ρ represents long run mean profitability. Shocks to the accounting profitability rate ζ t +1 are modeled as a white noise process with zero mean, variance
σζ2 , and constant covariance with the market portfolio. This covariance -- divided by the variance of the market return, defines the profitability rate beta βρ. To complete the model, we assume that at a future date T competition will force abnormal returns down to the point where market value and book value equal one another: MT = BT .7 It is rather surprising that the traditional CAPM leads to a straightforward relationship between the market to book ratio and the accounting measure of profitability. To show this result, suppose the CAPM holds, and the time series behavior of the profitability rate follows an AR(1) model with a long run profitability rate ρ . Define the autocovariance variable z j = 1 + φ z j −1 for j=1, … , T and with starting values z0 = 0. Then, the market to book ratio is
given by (the derivation of Equation (20) is analogous to that of (8) and (9) in the appendix): τ
(e A(τ )+ B (τ ) ρ +C (τ ) ρt −τ c )∏ (1 − ( ERm − R f ) z j β ρ )
T −1 c Mt = +c ∑ Bt 1 + R f τ =1
j =1
(1 + R f )τ T
+
where
(e A(T )+ B (T ) ρ +C (T ) ρt − cT )∏ (1 − ( ERm − R f ) z j β ρ ) j =1
(20)
(1 + R f )T τ
τ
j =1
j =1
A(τ ) = (σ ζ2 / 2) ∑ z 2j , B(τ ) = (1 − φ ) ∑ z j , and C (τ ) = φ zτ .
Consistent with intuition, we find that the ratio of market to book value is positively related to the current ROE rate ρt , the long run mean rate ρ , and the volatility of accounting profits. An 7
Pastor and Veronesi (2003) present this model in continuous time, and discuss the assumption of a fixed time horizon T at length.
17
increase in the risk-free rate, the market risk premium, or profitability rate beta lead to a lower market to book ratio. Interestingly, Pastor and Veronesi (2003) obtain similar results with a continuous time model and a stochastic discount factor, whereas ours are based on the CAPM. For non-dividend paying firms we set c=0, and obtain the market to book ratio as: T
Mt = Bt
(e A(T )+ B (T ) ρ +C (T ) ρt )∏ (1 − ( ERm − R f ) z j β ρ ) j =1
(1 + R f )T
(21)
The condition for a finite market to book ratio is roughly identical to the stochastic dividend growth model. Provided T is large, the restriction on long term profitability is:
ρ ≤ R f + ( ERm − R f )
2 βρ 1 σζ − (1 − φ ) 2 (1 − φ )2
(22)
The right hand side of this expression is similar to the condition (11); however, the profitability beta replaces the growth rate beta adjusted by the degree of predictability in the book equity and the long-run variance of ROE shocks. Once again there is no restrictions on the short run ROE.
7. Conclusions The St. Petersburg paradox describes a simple game of chance with infinite expected payoff, and yet any reasonable investor will pay no more than a few dollars to participate in the game. Researchers throughout history have provided a number of solutions as well as variations of the original paradox. One of these, developed by Durand (1957), shows that the standard St. Petersburg game can describe an investment in a firm with a constant growth rate of dividends. To capture a more realistic growth pattern, we present a model that allows mean reversion in dividends. We then derive the risk adjustment required in a CAPM environment, and propose an equivalent St. Petersburg game. We show that the expected payoff of the modified game (or equivalently, the value of growth firms) is driven mainly by the long-run growth rate of the 18
payoffs (dividends), while the short-term growth rate has a minor effect on the properties of the game or the firm. The model further shows that the condition under which the value of the game or the firm is finite is much less restrictive than that of the classical St. Petersburg game, and this might provide an indirect solution to the paradox.
19
References Aase, K., 2001, “On the St. Petersburg Paradox,” Scandinavian Actuarial Journal 1, 69-78. Ali, M., 1977, “Analysis of Autoregressive-Moving Average Models: Estimation and Prediction,” Biometrika 64, 535-545. Bansal, R., and A. Yaron, 2004, “Risks for the Long Run: A Potential Resolution of Asset Pricing Puzzles,” Journal of Finance 59, 1481-1509. Bernoulli, D., 1738, “Specimen Theoriae Novae de Mensura Sortis,” Commentarii Academiae Scientiarum Imperialis Petropolitanea V, 175-192. Translated and republished as “Exposition of a New Theory on the Measurement of Risk,” 1954, Econometrica 22, 23-36. Bhamra, H., Kuehn, L., and I. Strebulaev, 2010, “The Levered Equity Risk Premium and Credit Spreads: A Unified Framework,” Review of Financial Studies 23, 645-703. Durand, D., 1957, “Growth Stocks and the Petersburg Paradox,” Journal of Finance 12, 348363. Fama, E., 1977. “Risk-Adjusted Discount Rates and Capital Budgeting Under Uncertainty,” Journal of Financial Economics, 5 (August): 3-24. Fama, E., and K. French, 2000, “Forecasting Profitability And Earnings,” Journal of Business 73, 161-175. Friedman, M. and L. J. Savage, 1948, “The Utility Analysis of Choices Involving Risk,” Journal of Political Economy 56, 279-304. Gordon, M. (1962), The Investment, Financing, and Valuation of the Corporation. Homewood, Ill.: Irwin. Lintner, J., 1965, “The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets,” Review of Economics and Statistics 47, 1337-1355. Pastor, L., and P. Veronesi, 2003, “Stock valuation and learning about profitability,” Journal of Finance 58, 1749–1789. Pratt, J., 1964, “Risk Aversion in the Small and in the Large,” Econometrica 32, 122-136. Rubinstein M., 1976, Valuation of Uncertain Income Streams and the Pricing of Options. Bell Journal of Economics (Autumn), 407-425. Senetti J. T., 1976, “On Bernoulli, Sharpe, Financial Risk, and the St. Petersburg Paradox.” Journal of Finance 31, 960-962.
20
Sz´ekely, G. J., and Richards, D. St. P. (2004), “The St. Petersburg Paradox and the Crash of High-Tech Stocks in 2000,” The American Statistician, 58, 225-231. Sharpe, W. F., 1964, “Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk,” Journal of Finance 19, 425-442. Weirich, P., 1984, “The St. Petersburg Gamble and Risk,” Theory and Decision 17, 193-202.
21
Figure 1. Upper bound on long run dividend growth rate The three sloping lines represent the upper bound on the long run dividend growth rate, computed from Equation (12), as a function of the degree of mean reversion ( φ ), for three levels of β g . The model parameters are: R f = 0.037 , market risk premium E ( Rm − R f ) = 0.0804 , 2 and σ ε = 0.01. The horizontal line, set at R f = 0.037 , represents the upper bound on the constant (deterministic) growth rate in the Gordon model.
0.35
0.3
Upper bound
0.25
0.2 Rf Beta g = 0.5 0.15
Beta g = 1.0 Beta g = 1.5
0.1
0.05
0 0.00
0.10
0.20
0.30
0.40
0.50
0.60
Phi
22
Appendix: Proof of Equations (9) and (10) Without loss of generality, we set time t at 0, and define the sequence of future growth rates as a row vector G ' ≡ ( g1, K, gτ ) . We then use the following system of equations to describe potential
sample paths from time periods 1 thru τ:
0 0 . . 1 − φ 1 0 . . 0 −φ 1 0 . . . . 0 . . 0 −φ
0 0 0 1
g1 (1 − φ ) g φ gt ε1 g (1 − φ ) g 0 ε 2 2 . = . + . + . . . . . gτ (1 − φ ) g 0 ετ
(1)
A compact representation for this system is ΦG = (1 − φ ) g i + G0 + Ε , where i is a column vector of 1s, G0 is a column vector with φ g0 in the first row and 0 in the remaining rows, and E ' ≡ (ε1 , ε 2 , K , ετ ) is the vector of random innovation terms. This set implies that the
cumulative growth rate
τ
∑ j =1 g j
' −1 ' −1 has conditional mean (1 − φ ) g i Φ i + i Φ G0 and
conditional variance σ ε2 i ' Φ −1 (Φ −1 ) ' i . Using a result from time series analysis (Ali, 1977), we show next that these moments may be computed without inverting the Φ matrix. Define the vector Z ' ≡ ( zT , zT −1 , K , z1 ) = i ' Φ −1 and note that each element may be computed recursively from the previous one: z j = φ z j −1 + 1 for j=1, 2, … , τ, and starting value z0=0. Hence, each τ τ future expected dividend is given by: E 0 Dτ = D0 exp (1 − φ ) g ∑ z j + zτ φ g0 + (σ ε2 / 2) ∑ z 2j , j =1 j =1
and Equation (10) follows immediately. Next, we derive the present value of each future dividend Dτ starting from τ=1, 2, and so on. Let R =
D1 − 1 be the rate of return on a claim that pays off a single cash flow $ D1 at τ=1, V0
23
and sells for V0 .
Plug this return into the security market line (Equation 8) to show that the
present value is given by the expected dividend multiplied by a discount factor
1 − ( ERm − R f )Cov( D0 /( Et D1 ), Rm 1 ) / σ m2 . Using Stein’s lemma it follows that: V 0 = (E0 D1 ) 1+ Rf
(
)
(
)
D Cov 1 , Rm1 = Cov ε1, Rm1 . Define the growth rate beta β g = Cov ε1, Rm1 / σ ε2 . E0 D1 Therefore, the present value of the first dividend is given by Equation (9) with τ=1. Next, let V1 be the time-1 value of a single cash flow $D2 expected one period later. V Again, let R = 1 − 1 be the rate of return (from 0 to 1) from holding the claim on $D2. The V0
security
market
line
(Equation
8)
implies
that
2 1 − ( ERm − R f )Cov(V1 /( E0 V1), Rm1) / σ m . Using a similar argument as in Fama 1+ R f
V 0 = (E0 V1 )
1 − ( ERm − R f ) z1β g (1977), we can show that E0V1 = (E0 D2 ) . Moreover, the ratio of V1 to its 1+ Rf conditional expectation one period prior, E0V1, equals the ratio of cash flow expectations:
V V1 = E1 D 2 . Then, from Stein’s lemma we have Cov 1 , Rm,1 = Cov z 2ε 1 + z1ε 2 , Rm ,1 = E 0V 1 E 0 D 2 E0 V1
(
(
)
)
z 2 Cov ε1 , Rm ,1 . Therefore, the present value of the second dividend is given by Equation (9)
with τ=2. Proceeding in this fashion one may show that for any Dτ , the present value is:
24
τ
V0=
(E0 Dτ )∏ [1 − ( ERm − R f ) z j β g ] j =1
(1 + R f )τ
. Thus, Equation (9) holds by the principle of value
additivity. ■
Proof of Equations (15 and 16) Consider the sequence of future growth rates G ' ≡ ( g t +1 , K , g t +T ) may be described as a multivariate system ΦG = (1 − φ1 − φ2 ) g i + G0 + ΘΕ , where
1 − φ 1 Φ = − φ2 0
0
0
.
.
1
0
.
.
− φ1 1 . .
0 .
.
− φ2
− φ1
.
.
0 1 − θ 0 0 , Θ = 0 0 1
0
0 .
.
1 −θ .
0 . 1 0 . .
. .
.
.
0 −θ
0 0 0 I
G, g , i, and Ε were defined in the previous proof, while the initial conditions vector G0 consists of φ1 gt + φ2 gt −1 − θ ε t in the first row, φ2 gt in the second row, and 0s in the τ-3 remaining rows. The first two moments of cumulative growth are:
(
)(
Et i'G = (1 − φ ) gi'Φ−1i + i'Φ−1G0 and
)
' Vt i 'G = σ ε2 i 'Φ −1Θ i 'Φ −1Θ . Again, define Z ' ≡ ( zT , zT −1 , K, z1 ) = i 'Φ −1 so that each element
may be computed recursively from the previous one: z j = φ1 z j −1 + φ2 z j − 2 + 1 , and starting value of z0 = 0. Define also the vector W ' ≡ ( wT , wT −1 , K, w1 ) = Z 'Θ to aggregate serial correlation induced by the moving average component of growth. Each element may be computed recursively as: w j = z j − θ1 z j −1 for j=1, 2, … , τ. Given these transformation, the conditional expected
future
cash
flow
has
a
closed
form
solution
given
by
T T 2 2 E t Dt +T = Dt exp(1 − φ1 − φ2 ) g ∑ z j + zT φ1 gt + zT φ2 gt −1 − zT θ ε t + (σ ε / 2) ∑ z j . We note that j =1 j =1 25
this expectation is conditional on the time t shock to the current growth rate. This shock is unobservable, therefore we use a property of normal random variables to obtain the unconditional value. To obtain this result, observe that the expectation of Et [ Dt +T | ε t ] is analogous to the moment generating function of εt evaluated at the point zTθ . Then, Equation (16) follows immediately. The rest of the proof is by induction on t. From the proof of (8) above, we know that at τ-1 1 − ( ERm − R f ) β g the dividend discounted value is given by: V τ −1 = Eτ −1 Dτ . Thus, (15) holds 1+ R f as of τ-1 because the first value of w is 1. Assume the result holds for time period t = τ+1. From Stein’s
lemma
(
we
have
T Covτ ∑ g s , Rm,τ +1 = s =τ +1
(
Covτ W ' E , Rm,τ +1
)
=
)
wT −τ Covτ ετ +1, Rm,τ +1 . Using the same logic as in Proposition 1, as we move back one time
1 − ( ERm − R f ) wτ βε period from τ+1 to τ, the discount factor is . Thus, the time t=T-τ price is 1 + R f
given by: τ
(ET −τ DT )∏ [1 − ( ERm − R f )w j β g ] V t,T =
j =1
(1 + R f )τ
This last step shows that the proposition holds for time period t = T-τ, and all other times t. ■
26
