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: Carmelo.Giaccotto@business.uconn.edu.

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. ■