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).
2
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 n2 dollars (‘ducats’ in the
3
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:
∞=+++=+×+×+×= ...111...88
14
4
12
2
1E (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
03125.02 5 =− , and at least 128 dollars is 00713.02 7 =− ; 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:
∞
4
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:
∑∞
= +=+
++
+=
12
210
)1(...
)1()1( tt
t
r
D
r
D
r
DP (3)
5
where tD 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 ...,)1(),1(, 2111 gDgDD ++ for the years 1, 2, 3,…
In that case, the present value of this growing perpetual future dividend stream equals:
≥∞
<−=
++=∑
∞
=
−
rgif
rgifgr
D
r
gDP
tt
t
,
,
)1(
)1( 1
1
11
0 (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 )1( rr + , where 0>r . 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 1D if the first
toss is ‘tail’, )1(1 gD + if the second toss is ‘tail’, 2
1 )1( gD + if the third toss is ‘tail’, and so on.
That is, if the game lasts for n tosses, instead of earningn2 , the player will earn
g
gDgD
nn
j
j ]1)1[()1(1
11
1
11
−+=+−−
=
−∑ . Therefore, the expected payoff of the game is:
≥∞
<−=
++=−+
+= ∑∑
∞
=
−∞
=
−
rgif
rgifgr
D
r
gD
g
gD
r
rE
nn
n
n
n
n
,
,)(
)1(
)1(]1)1[(
)1(
1
1
11
1
11 (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
)1( rr + . 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
6
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 )1()( rgr +− instead of )1( rr + , for rg < . The
payoff series also changes, by a factor of )( grr − : the player will earn )(1 grrD − if the first
toss is ‘tail’, )()1(1 grrgD −+ if the second toss is ‘tail’, )()1(2
1 grrgD −+ if the third toss is
‘tail’, and so on. That is, if the game lasts for n tosses, the player will earn
g
grrgDgrrgD
nn
j
j )(]1)1[()()1(1
11
1
11
−−+=−+−−
=
−∑ . Therefore, the expected payoff of the game
is:
)()1(
)1()(]1)1[(
)1(
)( 1
1
11
1
11
gr
D
r
gD
g
grrgD
r
grE
nn
n
n
n
n −=
++
=−−+
+−= ∑∑
∞
=
−∞
=
−
(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 2$1 =D , %15=r
7
, and %.10=g The value of the stock is 40$1.015.0
2$
)(1 =
−=
− grD
, which is also the expected
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
8
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 tD be the time-t value of dividends or earnings (for simplicity we will use these two
terms interchangeably), and assume these values grow at rate 1+tg : ttg
t De D )( 11 ++ = . We use a
first order autoregressive process ( AR(1) ) to model mean reversion:
11 )1( ++ ++−= ttt gg g εφφ (7)
where g is the long run (unconditional) mean growth rate and φ is the autoregressive
coefficient (we assume 10
9
by the rate of return beta )( RORβ . Then, the CAPM price for an asset that pays a stochastic
dividend stream ∞=+ 1}{ ττtD 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):
( ) [ ]( )∑
∏∞=
=+
+
−−
=1
1
1
)(1
ττ
τ
τ β
f
jgjfmtt
tR
zRERDE
P (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: 11+= −jj z z φ for j= 1, 2, … and starting value
00 =z . The growth rate beta,gβ , 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:
tgCgBAttt eDDE
)()()( ττττ
+++ = (10)
and ∑==
τεστ
1
22 )2/()(j
jzA , ∑−==
τφτ
1)1()(
jjzB , and τφτ zC =)( ,
Equations (9) and (10) show that the current price depends on the current growth gt, and the
long run rateg ; however, the impact of the latter is much stronger because it is multiplied by
)(τB (the sum of zj). We note that jz converges to )1/(1 φ−=z for large τ, thus )(τB increases
with the time horizon. On the other hand, )(τC converges to a constant finite value )1/( φφ − .
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.
10
Today’s price depends also on the appropriate risk adjustment. Since the market risk,
represented by gjz β , increases with j -- up to )1/( φβ −g , 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 0=φ and 02 =εσ . In this case,
dividends are expected to grow in a deterministic fashion at a constant rateg . The beta factor
( gβ ) equals zero and the discount rate r equals the riskless rate fR .
The second, originally developed by Rubenstein (1976) within the context of a single factor
arbitrage-free model, allows stochastic growth but no serial correlation: 11 ++ += tt g g ε . 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: *
* )(g
gt
ter
eD P
−= , where the discount rate r is
given by gfm
f
RER
R
β)(1)1(
−−+
, and the adjusted growth rate is 2/* 2εσ+= g g . 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, jz converges to a constant value of )1(1 φ− ; therefore, the expected future dividend will
evolve along the pathTg
e
−+
2
2
)1(21
φσε
. Also, for dividends far into the future, the risk-adjusted
present value factor may be approximated by
T
f
gfm
R
RER
+−−−
1
)1()(1 φβ. Therefore, the
present value of a single dividend TD , for large T, may be approximated as follows:
11
TRERRg
tt
gfmf
eD V
−−+−
−+
≈)1(
)()1(2
12
2
φβ
φσε
(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:
2
2
)1(2
1
)1()(
φσ
φβ ε
−−
−−+≤ gfmf RERRg (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., rg < ); 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 gβ : 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 037.0=fR and 0804.0)( =− fm RRE , which
correspond to the sample averages from 1927 thru 2013. Last, 2εσ 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 gβ = 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.
12
0.5, the risk-adjusted upper bound is 7.2%, and increases to 11.2% for gβ = 1.0, and to 15.3%
for gβ = 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(R ff +1 , where 0>fR (this constant probability is consistent with that in
Durand’s game using the risk-free rate, fR , instead of r). Assume further that the player earns an
amount of f
j
ijj
R
aD ∏=1 dollars in the jth toss (including the last toss of the game), where jD
denotes the dividend series as defined in Section 4, and gjfmj zRERa β)(1 −−= . That is, if the
game lasts for n tosses, the player will earn ∑∏
=
=n
j f
j
ijj
R
aD
1
1 dollars. The expected payoff of the
game is therefore:
=+
=∑∏∞
=
=
1
1
)1(n f
n
jjn
nf
f
R
aD
R
RE
( ) [ ]∑
∏∞=
=
+
−−
1
10
)1(
)(1
nn
f
n
jgjfmn
R
zRERDE β (13)
13
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 tg~ be the long run dividend growth rate, then we
rewrite Equation (7) as tttt gg g εφφ ++−= −1~)1( . It can be easily shown that if tg~ 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.
14
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:
1221121 )1( −−− −+++−−= ttttt ggg g εθεφφφφ (14)
where ),,( 21 θφφ 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: 1−−= jjj zz w θ , and 12211 ++= −− jjj zz z φφ , with a starting value
of 00 =z . Then, the time-t price is given by (proofs of Equations (15) and (16) are provided in
the appendix):
( ) [ ]( )∑
∏∞=
=+
+
−−
=1
10
1
)(1
ττ
τ
τ β
f
jgjfmt
tR
wRERDE
P (15)
and the expected future dividend is
1222 )()()(2/])([ −++++
+ = ttgDgCgBzA
ttt eDDEτττσθτ
τετ (16)
The other parameters are defined as: ∑==
ττ
1
2)(j
jwA , ∑−−==
τφφτ
121 )1()(
jjzB ,
121)( −+= ττ φφτ zzC , and τφτ zD 2)( = .
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.
15
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, jz converges to a
constant value of )1(1 21 φφ −− , and jw converges to )1/()1( 21 φφθ −−− . Thus, for large τ,
both )(τC and )(τD play a negligible role. B(τ) increases linearly with τ ; and )(τA converges
to τφφ
θσε)1()1)(2/(
21
2
−−−
.
The expected dividend series will evolve along the pathTg
e
−−−+
221
2
)1(
)1(
21
φφθσε
, while the risk-
adjusted discount factor converges to
T
f
gfm
R
RER
+−−−−−
1
)1()1()(1 21 φφθβ . Therefore, for
large T, the present value of a single dividend TD may be approximated as:
TRERRg
tt
gfmf
eD V
−−−
−+−−−−+
≈)1(
)1()(
)1(
)1(21
21221
22
φφθβ
φφθσε
(17)
Finite pricing of assets requires that the following restriction holds for long term growth:
221
22
21 )1(
)1(21
)1(
)1()(
φφθσ
φφθβ ε
−−−−
−−−
−+≤ gfmf RERRg (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 rg < ;
furthermore, there is no restriction on short term dividend growth.
16
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: tttt DIBB −+= −1 . 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
( ) tct Be B t −+ += 11 ρ (19) 6 The clean surplus relationship implies that the rate of growth in book value is ( ) ( ) tctct Be Be B tt −−++ ++ ≈= 11 )1ln(1 ρρ . The approximation is exact only in continuous time.
17
To model the profitability rate, we assume a first order autoregressive process:
11 )1( ++ ++−= ttt ζρφρφρ , where ρ represents long run mean profitability. Shocks to the
accounting profitability rate 1+tζ are modeled as a white noise process with zero mean, variance
2ζσ , 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
11 −+= jj z z φ 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):
∑∏−
=
=
−++
+
−−
++
=1
1
1
)()()(
)1(
))(1()(
1
T
f
jjfm
cCBA
ft
t
R
zRERe
cR
c
B
M
t
ττ
τ
ρτρτρττ β
T
f
T
jjfm
cTTCTBTA
R
zRERe
t
)1(
))(1()(1
)()()(
+
−−
+∏
=
−++ρ
ρρ β (20)
where ∑==
τζστ
1
22 )2/()(j
jzA , ∑−==
τφτ
1)1()(
jjzB , and τφτ zC =)( .
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.
18
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
f
T
jjfm
TCTBTA
t
t
R
zRERe
B
Mt
)1(
))(1()(1
)()()(
+
−−
=∏
=
++ρ
ρρ β (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:
2
2
)1(2
1
)1()(
φσ
φβ
ρ ζρ−
−−
−+≤ fmf RERR (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
19
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.
20
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, 348-
363. 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.
21
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.
22
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 gβ . The model parameters are: 037.0=fR , market risk premium 0804.0)( =− fm RRE ,
and 2εσ = 0.01. The horizontal line, set at 037.0=fR , represents the upper bound on the constant (deterministic) growth rate in the Gordon model.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.00 0.10 0.20 0.30 0.40 0.50 0.60
Up
pe
r b
ou
nd
Phi
Rf
Beta g = 0.5
Beta g = 1.0
Beta g = 1.5
23
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 ),,( 1'
τggG K≡ . We then use the following system of equations to describe potential
sample paths from time periods 1 thru τ:
+
+
−
−−
=
−
−−
ττ ε
εεφ
φ
φφ
φ
φφ
.
.
0
.
.
0
)1(
.
.
)1(
)1(
.
.
10..0
...
0.010
0..01
0..001
2
1
2
1 tg
g
g
g
g
g
g
(1)
A compact representation for this system is Ε++−=Φ 0)1( Gig G φ , where i is a column vector
of 1s, 0G is a column vector with 0gφ in the first row and 0 in the remaining rows, and
),,,( 21'
τεεε K E ≡ is the vector of random innovation terms. This set implies that the
cumulative growth rate ∑ =τ
1j jg has conditional mean 0
1'1')1( Giiig −− Φ+Φ−φ and
conditional variance ii '11'2 )( −− ΦΦεσ . 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 1'11' ),,,( −− Φ=≡ izz zZ TT K and note that each element may be computed recursively
from the previous one: 11 += −jj z z φ for j=1, 2, … , τ, and starting value z0=0. Hence, each
future expected dividend is given by:
∑++∑−===
τετ
ττ σφφ
1
220
100 )2/()1(exp
jj
jj zgzzgD DE ,
and Equation (10) follows immediately.
Next, we derive the present value of each future dividend τD starting from τ=1, 2, and so
on. Let 10
1 −=V
D R be the rate of return on a claim that pays off a single cash flow $ 1D at τ=1,
24
and sells for 0V . 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
( )
+−−
=f
mmtfm
R
RDEDCovRERDE V
1
/)),/(()(1 2110100
σ. Using Stein’s lemma it follows that:
( )11110
1 ,, mm RCovRDE
DCov ε=
. Define the growth rate beta ( ) 211 /, εσεβ mg RCov= .
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.
Again, let 10
1 −=V
V R be the rate of return (from 0 to 1) from holding the claim on $D2. The
security market line (Equation 8) implies that
( )
+−−
=f
mmfm
R
RVEVCovRERVE V
1
/)),/(()(1 21101100
σ. Using a similar argument as in Fama
(1977), we can show that ( ) .1
)(1 12010
+−−
=f
gfm
R
zRERDE VE
β Moreover, the ratio of V1 to its
conditional expectation one period prior, E0V1, equals the ratio of cash flow expectations:
DE
DE VE
V
20
21
10
1 = . Then, from Stein’s lemma we have
1,
10
1 , mRVE
VCov ( )1,2112 , mRzzCov εε += =
( )1,12 , mRCovz ε . 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:
25
( ) [ ]( )τ
τ
τ β
f
jgjfm
R
zRERDE
V+
−−
=∏
=
1
)(11
0
0 . Thus, Equation (9) holds by the principle of value
additivity. ■
Proof of Equations (15 and 16)
Consider the sequence of future growth rates ),,( 1'
Ttt ggG ++≡ K may be described as a
multivariate system ΘΕ++−−=Φ 021 )1( Gig G φφ , where
−−
−−−
=Φ
1..0
...
0.01
0..01
0..001
12
12
1
φφ
φφφ
,
−
−−
=Θ
Iθ
θθ
0..0
...
0.010
0..01
0..001
,,, igG and Ε were defined in the previous proof, while the initial conditions vector 0G consists
of ttt gg εθφφ −+ −121 in the first row, tg2φ in the second row, and 0s in the τ-3 remaining
rows.
The first two moments of cumulative growth are: 01'1'' )1( Giiig G iEt
−− Φ+Φ−= φ and
( )( )'1'1'2' ΘΦΘΦ= −− ii G iVt εσ . Again, define 1'11' ),,,( −− Φ=≡ izz zZ TT K so that each element may be computed recursively from the previous one: 12211 ++= −− jjj zz z φφ , and starting value
of z0 = 0. Define also the vector Θ=≡ −'
11' ),,,( Zw wwW TT K to aggregate serial correlation
induced by the moving average component of growth. Each element may be computed
recursively as: 11 −−= jjj zz w θ for j=1, 2, … , τ. Given these transformation, the conditional
expected future cash flow has a closed form solution given by
∑+−++∑−−==
−=
+T
jjtTtTtT
T
jjtTtt zzgzgzzgD DE
1
22121
121 )2/()1(exp εσεθφφφφ . We note that
26
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 ]|[ tTtt DE ε+ is
analogous to the moment generating function of tε evaluated at the point θTz . 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
the dividend discounted value is given by:
+−−
= −−f
gfm
R
RERDE V
1
)(111
βτττ . Thus, (15) holds
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
+
+=∑ 1,
1
, ττ
τ mT
ss RgCov = ( )1,' , +ττ mREWCov =
( )1,1, ++− ττττ ε mT RCovw . Using the same logic as in Proposition 1, as we move back one time
period from τ+1 to τ, the discount factor is
+−−
f
fm
R
wRER
1
)(1 ετ β . Thus, the time t=T-τ price is
given by:
( ) [ ]( )τ
ττ β
f
jgjfmTT
TtR
wRERDE
V+
−−
=∏
=−
1
)(11
,
This last step shows that the proposition holds for time period t = T-τ, and all other times t. ■