1. Problem: Reframe the Kelly Approach as a Convective-Diffusion Equation
The purpose of this article is to analyze a sequence of independent bets by modeling it with a convective-diffusion equation (CDE). The CDE approach approximates innovation investing for the case where the projects are mostly independent (i.e., low correlation between successive events). “Innovation investments” differ from other investments like stocks, bonds, or commodities since there is no market for options or short selling with innovation [
1].
Various models have been used to model sequential investments at both the portfolio and individual investment levels. At the portfolio level, early strategies focused on gaining high returns while avoiding risk. Modern portfolio theory (MPT) [
2] introduced the concept of diversifying assets, and employs historical variance and covariance of the assets to assess risk. The Capital Asset Pricing Model (CAPM) followed from MPT, using a simple model with a parameter beta (
) that captures the risk of an asset relative to historical market outcomes. In 1973, the Black–Scholes model [
1] was developed to analyze dynamic investment. The authors emphasized the role of options and shifted the focus from “risk aversion” to “risk neutrality”. They derived the Black–Scholes equation (BSE), which they called the “heat-transfer equation of physics”, and solved it for the price of an option that is risk neutral. Since the BSE has both drift and random walk terms, the equation they derived is actually more like the “convective-diffusion equation” than the “heat equation” or the “diffusion equation”.
Another model used to analyze sequential bets is the Kelly Criterion [
3,
4,
5]. In 1956, Kelly examined the gambling or betting problem—or here, we would say the investing problem—with the goal of
maximizing the growth of a bankroll rather than seeking risk neutrality. He considered a sequence of independent Bernoulli trials that accounts explicitly for the wins and losses of a bet, which have a given probability (
p) of winning. The output from the Kelly model is the fraction (
f) of current resources that should be bet in order to maximize future resources. The Kelly model has been applied to innovation investing [
6]. The risk of the bet is specified by the value of
p — which usually must be estimated for innovation investments — rather than from using the historical variance of the asset value as the BSE does. A champion of the Kelly approach to betting is Edward O. Thorp, a mathematician who had significant winnings both in the casino [
7] and in the stock market [
8]. Interestingly, based on Warren Buffet’s focus on long-term growth maximization instead of the avoidance of short-term losses, Davis et al. surmise that he is a Kelly Criterion bettor [
9].
The Kelly Criterion has a number of variations. An important one is the use of “fractional Kelly”, in which one reduces risk by betting perhaps a half or a fourth of what the Kelly Criterion recommends [
9]. Another is a Kelly Criterion with learning [
10], which accounts for a change in parameters (e.g., a change in probability of winning) as expenditures are made. For example, as work is conducted on a project, the probability of success might rise. A third variation is Kelly with draw down control, determining how much to pull out as profit after each bet [
11]. Other variations account for constraints in the betting, such as having to keep the total number of bets below some number. The focus of the Kelly Criterion and its variations has been on finding the betting fraction
f that maximizes the median growth (0.50 quantile) for a given scenario.
Sometimes, however, we might seek a best “poor case” (similar to maximin) in order to avoid ruin, and sometimes we might seek a best “good case” to see what is possible. In this article, we remove the limitation of optimizing for the best median (i.e., 50th percentile or 0.50 quantile) return. Instead, we optimize our proportional betting parameter f so that the desired quantile is optimized. For example, we might conservatively want to optimize a poor-case scenario, say the 10th percentile (i.e., ).
To do this, we reframe the Kelly approach as a CDE and provide solutions to the equation. The use of the CDE clarifies the role of steady growth (characterized by a velocity U) and random fluctuations (characterized by a diffusion coefficient D) to predict a probability distribution of the bankroll at any time. We analyze three main sub-problems:
Optimal allocation f for the gamma () quantile case, using our CDE;
Optimal allocation f for a given “ruin” tolerance, using an absorptive boundary condition;
Analysis of the return–risk trade-off for “fractional Kelly” betting.
2. Analysis Using the Convective-Diffusion Equation (CDE)
To analyze the questions above, we start by looking at a single investment, and we will proceed in several steps: (1) We will show that a binary win–loss game with proportional betting can be represented as a biased random walk. (2) We will model this random walk using a convective-diffusion equation (CDE). (3) We will use known CDE solutions to draw conclusions about the present problem.
This approach based on the CDE was used in the famous Black–Scholes model for pricing options [
12]. They called their partial differential equation “the heat-transfer equation of physics”, although it also describes geometric Brownian motion including both deterministic drift (convection) and random movements (diffusion), which is why we call this equation the CDE. The Black–Scholes formula specifies conditions that determine the fair price of a risk-neutral option. Here, we take a different direction and provide solutions for the investment fraction that maximizes different objective functions, which depend on the predicted distribution of returns.
First, let us show that a binary win–loss game with proportional betting can be represented as a biased random walk. Starting with an initial bankroll
for innovation, we want to predict a future bankroll
B if we know the probability
p of winning the bet, the winning odds
b (such that if you bet
$1 and win, you win
b dollars plus your original
$1), and the losing odds
a (such that if you bet
$1 and lose, you lose a fraction
a of your original
$1). We wager a fraction
f of our bankroll on each such bet, holding the remainder in cash or allocating it for other bets. After
W wins and
L losses in
t betting events (where
), the final bankroll
is related to the initial bankroll
by the following equation
Taking the logarithm of both sides gives
We identify the distance
between the log bankroll at time
t and the log bankroll at
as
. Wins and losses correspond to steps along the
y axis to the right or left starting from
(
Figure 1a). When we win a bet, we move to the right by a step size
; when we lose a bet, we move to the left with a step size
. Equation (
2) now becomes
We denote the probability density of being at location
y at (discrete) time
t as
. Initially,
, where
is the Dirac delta function since we start at
by definition. The probability density is analogous to the concentration of a conserved quantity which is transported to the right or left by a random walk with drift. During a single betting event, the concentration at position
y is updated due to steps to the right from position
(i.e., a win with probability
p) and steps to the left from position
(i.e., a loss with probability
). Mathematically, this updating is expressed as
Starting from the initial concentration
, the repeated application of this updating rule leads to the following distributions after one and two betting events
Following this pattern, the general solution can be expressed as
where the sum is evaluated over the possible number of wins
W from 0 to
t. The binomial coefficient describes the number of different ways of obtaining
W wins in
t bets. As illustrated in
Figure 1b, this distribution is well summarized by its mean
and variance
Integrating the delta functions from Equation (
6) in Equations (7) and (8) gives rise to summations, and the analytical summations give the final results of Equations (7) and (8). According to the central limit theorem, higher moments are increasingly negligible after many betting events
t. For example, the third standardized moment known as skewness
decays with time
t as
For large
t, the exact solution of Equation (
6) is therefore well approximated by a normal distribution with mean
, standard deviation
, and zero skewness. To accurately describe the discrete distribution of Equation (
6) by a continuous normal distribution, we further require that the width of the distribution
be much larger than the step size
. Substituting Equation (8) for the standard deviation, this condition is valid when
.
Because the result from Equations (
7) and (8) give a normal distribution for large
t, the continuum approximation
to the exact solution of Equation (
6) satisfies the
convective-diffusion equation (CDE) with velocity (sometimes called “drift”)
U and diffusion coefficient
D (to be determined)
For the initial distribution
on an unbounded domain (
), the solution to Equation (
10) is well known from diffusion theory [
13]
The result is a normal distribution with mean
and standard deviation
. By comparison to Equations (
7) and (8) for the moments of the discrete solution, we identify the velocity and the diffusion coefficient to be
Equations (
11) and (
12) give the probability density
for being at any value of
y (i.e., the log bankroll) at some dimensionless time
t. This result is for a sequence of binary win–lose outcomes. We have added a result in
Appendix A for the case when there are more than two possible outcomes to consider. It is important to remember that the given solution depends implicitly on our choice of
f since the velocity
U and diffusivity
D both depend on
f by way of the step sizes
and
. In the next section, we will explore how we should choose an optimal betting fraction
f for different candidate objective functions.
4. Avoiding Ruin
If our bankroll becomes too small during the course of repeated betting events, it may become impossible to make further gambles. We have “gone bust”. We define a ruin tolerance
such that we can make no further bets when the bankroll falls below
. The possibility of ruin can be described by modifying the random walk to include an absorbing boundary at
, where
is the distance from the initial condition to ruin. When the random walker steps below this ruin length, betting ceases and the walk ends. (We could also place an “absorbing boundary” at a position
, in which case we stop when we have won enough to be satisfied rather than having lost enough.) In our continuum CDE description of Equation (
10), the possibility of ruin is described by the following boundary condition
Starting from the initial distribution
, we can solve the convection–diffusion equation subject to Equation (
23) to obtain
on the domain
(
Figure 3a; see
Supplementary Materials for derivation).
Using this solution, we can evaluate the ruin rate
J, which characterizes the flux of probability through the absorbing boundary at
Substituting Equation (
24), the ruin rate can be expressed as
The quantity
describes the probability of ruin within the differential time interval
t to
. We can integrate the ruin rate with respect to time from 0 some later time
t to obtain the accumulated probability of ruin as
At long times, the ruin probability approaches a constant value that depends on the dimensionless Péclet number
, which characterizes the relative rates of convection and diffusion over the ruin length
ℓ. When convection transports the distribution
away from the absorbing ruin boundary (i.e., when
), the asymptotic ruin probability is
For large Péclet numbers (
), this asymptotic limit is achieved for times
, which is the time required to convect the distance
ℓ. For small Péclet numbers (
), the asymptotic behavior of Equation (
28) is realized when
beyond which convection is dominant compared to diffusion. By contrast, when convection transports the distribution
towards the absorbing boundary (i.e., when
), ruin becomes inevitable at long times.
Figure 3 plots the ruin probability as a function of time for “winning” investments with
. Ruin can be avoided when the Péclet number is large
—for example, by increasing the ratio
or by lowering our ruin tolerance
r (thereby increasing
ℓ). If, however, we seek to
minimize the ruin probability by varying the betting fraction
f, we are led to the conservative strategy in which
. That is, we can avoid ruin by risking nothing. Alternatively, we can use the ruin probability as a constraint on the optimization of other objective functions such as the
-quantile described above.
Figure 4 shows how the ruin probability can be used to constrain the optimization of the
-quantile. Here, we specify a maximum ruin probability of
and seek investment strategies for which
. When the chance of ruin is small (
), the absorbing boundary solution of Equation (
24) is well approximated by the unbounded solution of Equation (
11). The maximum ruin length
and the associated ruin tolerance
can then be determined from Equation (
28) as
where the velocity
U and diffusivity
D are evaluated at the Kelly betting fraction
. This assignment ensures that the optimal
f for a specified quantile
will avoid ruin at long times
. For the present example with
, this condition implies that
and
. At intermediate times
t, the betting fraction
f must be set below some time-dependent value to ensure the desired ruin probability
(
Figure 4, black curve). This ruin constraint prohibits some of the riskier strategies for quantiles
.
When the ruin probability is not negligibly small, it is no longer possible to decouple the optimization of the betting fraction
f from the enforcing of the ruin constraint. The probability density
of Equation (
11) which neglects ruin becomes qualitatively different from that of Equation (
24), which accounts for it. In this case, we must search for the betting fraction
f that optimizes the
-quantile of Equation (
24) subject to the ruin constraint that
. This problem can be solved numerically but is not considered further here.
5. Fractional Kelly
Betting a fraction of the Kelly Criterion
—or “Fractional Kelly”—is a common heuristic for reducing risk and avoiding ruin while maintaining a high return [
9]. Although fractional Kelly is sometimes viewed as an ad hoc correction, we can use our analysis to relate this betting policy to that associated with (1) a particular quantile or (2) a particular ruin rate or probability.
Figure 2a can be used to relate a fractional Kelly policy to the corresponding quantile policy for a given time horizon
t (for better precision, solve Equation (
20) for
specifying
f equal to fractional KC). For example, in our game from this paper,
, which corresponds to
. If we choose “half Kelly”, that is
, this gives added safety at small times, corresponding to
at
or
at
.
Alternatively, and perhaps of more value, we might want to relate some fraction of the KC to the median growth rate and ruin probability. We can use the parameters of a game (in this article,
,
,
) to find
and
, and thus
U and
D. We use Equation (
1) to evaluate the median growth per flip, and subtract 1 = 100% to get the growth rate. We can then use the
U and
D in Equation (
28) to obtain the asymptotic ruin probability
or Equation (
27) to obtain
at any earlier time. Here we’ll use a ruin tolerance of
.
For the full Kelly Criterion, , giving and . The resulting Péclet number is . This gives the median return of 1.0206/flip, or a growth rate of 2.06% per flip. The asymptotic ruin probability is . For “half Kelly”, , giving and , which gives . The median return per flip is 1.0155, or a growth of 1.55% per flip (75% as much as full Kelly). drops considerably, to . For “quarter Kelly”, , giving . The median return per flip is 1.0091, or a growth of 0.91% per flip (44% as much as full Kelly), while drops to . For this game, the quarter Kelly policy would almost never reach ruin, and yet give almost half the growth rate.
6. Conclusions
The purpose of this article is to analyze a sequence of independent bets by modeling it with the convective-diffusion equation (CDE). This provides a model for innovation investing when the projects are independent. We have combined the CDE formulation with known results from the physico-chemical literature, to produce solutions for the growth of a bankroll due to sequential innovation bets, as well as the entropic fluctuations. By applying an absorptive boundary condition, we have derived a readily usable result for the ruin rate in terms of the system Peclet number. Finally, we have shown how the previously ad hoc fractional Kelly heuristic affects returns and risk.
The fluctuations resulting from the diffusive effects of innovation investments increase the variance of the final outcomes. Thus, while previous Kelly Criterion modeling emphasized the best median result—which is independent of diffusion—our work shows how diffusion can lead to very favorable outcomes as well as unfavorable outcomes for the same underlying scenario. The quantitative intuition gained from our results can guide professionals away from situations where the possibility of ruin is greater than desired.
This work can, in the future, be extended in several directions by pursuing the following questions: (1) Can we use the analytical machinery we have derived to create a practical tool for guiding innovation investments and portfolio management? (2) Can the present technical analysis impact actual companies, which have internal psychological and political forces? And perhaps most immediately, (3) what other CDE solutions might be used to provide new insights into innovation investing? By framing our problem in terms of the CDE, we aim to inspire others to extend insights from the chemico-physical literature to gain an edge in sequential betting, perhaps including innovation investing, as well as other aspects of life.