Generalizing Kelly, we consider the following simple strategy of the investor: he regularly checks the asset’s current price p(t), and sells or buys some asset shares in order to keep the current market value of his asset holdings a pre-selected fraction r of his total capital. These readjustments are made periodically at a fixed interval, which we refer to as readjustment interval, and select it as the discrete unit of time. In this work the readjustment time interval is selected once and for all, and we do not attempt optimization of its length.
i.e. at each time step the random number η(t) is drawn from some probability distribution π(η), and is independent of it’s value at previous time steps. This exponential notation is particularly convenient for working with multiplicative noise, keeping the necessary algebra at minimum. Under these rules of dynamics the logarithm of the asset’s price, ln p(t), performs a random walk with an average drift v = ⟨η⟩ and a dispersion D = ⟨η2⟩ − ⟨η⟩2.
Let us assume that the value of the investor’s capital at t = 0 is W(0) = 1. The evolution of the expectation value of the expectation value of the total capital ⟨W (t)⟩ after t time steps is obviously given by the recursion ⟨W (t + 1)⟩ = (1 − r + r⟨eη⟩)⟨W (t)⟩. When ⟨eη⟩ > 1, at first thought the investor should invest all his money in the risky asset. Then the expectation value of his capital would enjoy an exponential growth with the fastest growth rate. However, it would be totally unreasonable to expect that in a typical realization of price fluctuations, the investor would be able to attain the average growth rate determined as vavg = d⟨W(t)⟩/dt. This is because the main contribution to the expectation value ⟨W(t)⟩ comes from exponentially unlikely outcomes, when the price of the asset after a long series of favorable events with η > ⟨η⟩ becomes exponentially big. Such outcomes lie well beyond reasonable fluctuations of W (t), determined by the standard deviation √Dt of ln W (t) around its average value ⟨ln W (t)⟩ = ⟨η⟩t. For the investor who deals with just one realization of the multiplicative process it is better not to rely on such unlikely events, and maximize his gain in a typical outcome of a process. To quantify the intuitively clear concept of a typical value of a random variable x, we define xtyp as a median of its distribution, i.e xtyp has the property that Prob(x > xtyp) = Prob(x < xtyp) = 1/2. In a multiplicative process (2) with r = 1, W (t + 1) = eη(t)W (t), one can show that Wtyp(t) – the typical value of W(t) – grows exponentially in time: Wtyp(t) = e⟨η⟩t at a rate vtyp = ⟨η⟩, while the expectation value ⟨W(t)⟩ also grows exponentially as ⟨W(t)⟩ = ⟨eη⟩t, but at a faster rate given by vavg = ln⟨eη⟩. Notice that ⟨lnW(t)⟩ always grows with the typical growth rate, since those very rare outcomes when W (t) is exponentially big, do not make significant contribution to this average.
The question we are going to address is: which investment fraction r provides the investor with the best typical growth rate vtyp of his capital. Kelly has answered this question for a particular realization of multiplicative stochastic process, where the capital is multiplied by 2 with probability q > 1/2, and by 0 with probability p = 1 − q. This case is realized in a gambling game, where betting on the right outcome pays 2:1, while you know the right outcome with probability q > 1/2. In our notation this case corresponds to η being equal to ln 2 with probability q and −∞ otherwise. The player’s capital in Kelly’s model with r = 1 enjoys the growth of expectation value ⟨W(t)⟩ at a rate vavg = ln2q > 0. In this case it is however particularly clear that one should not use maximization of the expectation value of the capital as the optimum criterion. If the player indeed bets all of his capital at every time step, sooner or later he will loose everything and would not be able to continue to play. In other words, r = 1 corresponds to the worst typical growth of the capital: asymptotically the player will be bankrupt with probability 1. In this example it is also very transparent, where the positive average growth rate comes from: after T rounds of the game, in a very unlikely (Prob = qT) event that the capital was multiplied by 2 at all times (the gambler guessed right all the time!), the capital is equal to 2T. This exponentially large value of the capital outweighs exponentially small probability of this event, and gives rise to an exponentially growing average. This would offer condolence to a gambler who lost everything.
We generalize Kelly’s arguments for arbitrary distribution π(η). As we will see this generalization reveals some hidden results, not realized in Kelly’s “betting” game. As we learned above, the growth of the typical value of W(t), is given by the drift of ⟨lnW(t)⟩ = vtypt, which in our case can be written as
One can check that vtyp(0) = 0, since in this case the whole capital is in the form of cash and does not change in time. In another limit one has vtyp(1) = ⟨η⟩, since in this case the whole capital is invested in the asset and enjoys it’s typical growth rate (⟨η⟩ = −∞ for Kelly’s case). Can one do better by selecting 0 < r < 1? To find the maximum of vtyp(r) one differentiates (3) with respect to r and looks for a solution of the resulting equation: 0 = v’typ(r) = ∫ dη π(η) (eη −1)/(1+r(eη −1)) in the interval 0 ≤ r ≤ 1. If such a solution exists, it is unique since v′′typ(r) = − ∫ dη π(η) (eη − 1)2 / (1 + r(eη − 1))2 < 0 everywhere. The values of the v’typ(r) at 0 and 1 are given by v’typ(0) = ⟨eη⟩ − 1, and v’typ(1) = 1−⟨e−η⟩. One has to consider three possibilities:
(2) ⟨eη⟩ > 1 , and ⟨e−η⟩ > 1. In this case v’typ(0) > 0, but v’typ(1) < 0 and the maximum of v(r) is realized at some 0 < r < 1, which is a unique solution to v’typ(r) = 0. The typical growth rate in this case is always positive (because you could have always selected r = 0 to make it zero), but not as big as the average rate ln⟨eη⟩, which serves as an unattainable ideal limit. An intuitive understanding of why one should select r < 1 in this case comes from the following observation: the condition ⟨e−η⟩ > 1 makes ⟨1/p(t)⟩ to grow exponentially in time. Such an exponential growth indicates that the outcomes with very small p(t) are feasible and give dominant contribution to ⟨1/p(t)⟩. This is an indicator that the asset price is unstable and one should not trust his whole capital to such a risky investment.
(3) ⟨eη⟩ > 1 , and ⟨e−η⟩ < 1. This is a safe asset and one can invest his whole capital in it. The maximum vtyp(r) is achieved at r = 1 and is equal to vtyp(1) = ln⟨η⟩. A simple example of this type of asset is one in which the price p(t) with equal probabilities is multiplied by 2 or by a = 2/3. As one can see this is a marginal case in which ⟨1/p(t)⟩ = const. For a < 2/3 one should invest only a fraction r < 1 of his capital in the asset, while for a ≥ 2/3 the whole sum could be trusted to it. The specialty of the case with a = 2/3 cannot not be guessed by just looking at the typical and average growth rates of the asset! One has to go and calculate ⟨e−η⟩ to check if ⟨1/p(t)⟩ diverges. This “reliable” type of asset is a new feature of the model with a general π(η). It is never realized in Kelly’s original model, which always has ⟨η⟩ = −∞, so that it never makes sense to gamble the whole capital every time.
An interesting and somewhat counterintuitive consequence of the above results is that under certain conditions one can make his capital grow by investing in asset with a negative typical growth rate ⟨η⟩ < 0. Such asset certainly loses value, and its typical price experiences an exponential decay. Any investor bold enough to trust his whole capital in such an asset is losing money with the same rate. But as long as the fluctuations are strong enough to maintain a positive average return per capital ⟨eη⟩ − 1 > 0) one can maintain a certain fraction of his total capital invested in this asset and almost certainly make money! A simple example of such mind-boggling situation is given by a random multiplicative process in which the price of the asset with equal probabilities is doubled (goes up by 100%) or divided by 3 (goes down by 66.7%). The typical price of this asset drifts down by 18% each time step. Indeed, after T time steps one could reasonably expect the price of this asset to be ptyp(T) = 2T/2 3−T/2 = (√2/3)T ≃ 0.82T. On the other hand, the average ⟨p(t)⟩ enjoys a 17% growth ⟨p(t + 1)⟩ = 7/6 ⟨p(t)⟩ ≃ 1.17⟨W (t)⟩. As one can easily see, the optimum of the typical growth rate is achieved by maintaining a fraction r = 1/4 of the capital invested in this asset. The typical rate in this case is a meager √(25/24) ≃ 1.02, meaning that in a long run one almost certainly gets a 2% return per time step, but it is certainly better than losing 18% by investing the whole capital in this asset.
Of course the properties of a typical realization of a random multiplicative process are not fully characterized by the drift vtyp(r)t in the position of the center of mass of P(h,t), where h(t) = lnW(t) is a logarithm of the wealth of the investor. Indeed, asymptotically P (h, t) has a Gaussian shape P (h, t) =1/ (√2π D(r)t) (exp(−(h−vtyp(r)t)2)/(2D(r)t), where vtyp(r) is given by eq. (3). One needs to know the dispersion D(r) to estimate √D(r)t, which is the magnitude of characteristic deviations of h(t) away from its typical value htyp(t) = vtypt. At the infinite time horizon t → ∞, the process with the biggest vtyp(r) will certainly be preferable over any other process. This is because the separation between typical values of h(t) for two different investment fractions r grows linearly in time, while the span of typical fluctuations grows only as a √t. However, at a finite time horizon the investor should take into account both vtyp(r) and D(r) and decide what he prefers: moderate growth with small fluctuations or faster growth with still bigger fluctuations. To quantify this decision one needs to introduce an investor’s “utility function” which we will not attempt in this work. The most conservative players are advised to always keep their capital in cash, since with any other arrangement the fluctuations will certainly be bigger. As a rule one can show that the dispersion D(r) = ∫ π(η) ln2[1 + r(eη − 1)]dη − v2typ monotonically increases with r. Therefore, among two solutions with equal vtyp(r) one should always select the one with a smaller r, since it would guarantee smaller fluctuations. Here it is more convenient to switch to the standard notation. It is customary to use the random variable
which is referred to as return per unit capital of the asset. The properties of a random multiplicative process are expressed in terms of the average return per capital α = ⟨Λ⟩ = ⟨eη⟩ − 1, and the volatility (standard deviation) of the return per capital σ = √(⟨Λ2⟩ – ⟨Λ⟩2. In our notation, α = ⟨eη⟩ – 1, is determined by the average and not typical growth rate of the process. For η ≪ 1 , α ≃ v + D/2 + v2/2, while the volatility σ is related to D ( the dispersion of η) through σ ≃ √D.
If the above formula prescribes ropt > 1, the investor is advised to trust his whole capital to this asset. We remind you that in this paper the risk-free return per capital is set to zero (investor keeps the rest of his capital in cash). In a more realistic case, when a risk-free bank deposit brings a return p during a single readjustment interval, the formula for the optimal investment ratio should be generalized to:
In a hypothetical case discussed by Merton, when asset’s price follows a continuous multiplicative random walk (i.e. price fluctuations are uncorrelated at the smallest time scale) and the investor is committed to adjust his investment ratio on a continuous basis, one should use infinitesimal quantities α → αdt and σ2 → σ2dt. Under these circumstances the term α2dt2, being second order in infinitesimal time increment dt, should be dropped from the denominator. Then one recovers an optimal investment fraction for “logarithmic utility” derived by Merton.