Leonard C. MacLean

The Kelly Capital Growth Investment Criterion: Theory and Practice

This volume provides the definitive treatment of fortunes formula or the Kelly capital growth criterion as it is often called. The strategy is to maximize long run wealth of the investor by maximizing the period by period expected utility of wealth with a logarithmic utility function. Mathematical theorems show that only the log utility function maximizes asymptotic long run wealth and minimizes the expected time to arbitrary large goals. In general, the strategy is risky in the short term but as the number of bets increase, the Kelly bettors wealth tends to be much larger than those with essentially different strategies. So most of the time, the Kelly bettor will have much more wealth than these other bettors but the Kelly strategy can lead to considerable losses a small percent of the time. There are ways to reduce this risk at the cost of lower expected final wealth using fractional Kelly strategies that blend the Kelly suggested wager with cash. The various classic reprinted papers and the new ones written specifically for this volume cover various aspects of the theory and practice of dynamic investing. Good and bad properties are discussed, as are fixed-mix and volatility induced growth strategies. The relationships with utility theory and the use of these ideas by great investors are featured.

Capital growth with security

Abstract This paper discusses the allocation of capital over time with several risky assets. The capital growth log utility approach is used with conditions requiring that specific goals are achieved with high probability. The stochastic optimization model uses a disjunctive form for the probabilistic constraints, which identifies an outer problem of choosing an optimal set of scenarios, and an inner (conditional) problem of finding the optimal investment decisions for a given scenarios set. The multiperiod inner problem is composed of a sequence of conditional one period problems. The theory is illustrated for the dynamic allocation of wealth in stocks, bonds and cash equivalents.

Growth versus security tradeoffs indynamic investment analysis

This paper presents an approach to the problem of optimal dynamic choice in discrete orcontinuous time where there is a direct tradeoff of growth versus security. In each period,the investor must allocate the available resources among various risky assets. The maximizationof the expected logarithm of the period‐by‐period wealth, called the capital growthor the Kelly criterion, has many desirable properties such as maximizing the asymptoticrate of asset growth. However, this strategy has low risk aversion and typically has verylarge wagers which yield high variance of wealth. With uncertain parameters, this can leadto overbetting and loss of wealth. Using fractional Kelly strategies leads to a less volatileand safer sequence of wealth levels with less growth. The investor can choose a desirabletradeoff of growth and security appropriate for the problem under consideration. Thisapproach yields simple two‐dimensional graphs analogous to static mean variance analysisthat capture the essence of the dynamic problem in a form useful for sound investmentanalysis. Use of the approach in practice is illustrated on favorable investments in blackjack,horse racing, lotto games, index and commodity futures and options trading.

Time to wealth goals in capital accumulation

This paper considers the problem of investment of capital in risky assets in a dynamic capital market in continuous time. The model controls risk, and in particular the risk associated with errors in the estimation of asset returns. The framework for investment risk is a geometric Brownian motion model for asset prices, with random rates of return. The information filtration process and the capital allocation decisions are considered separately. The filtration is based on a Bayesian model for asset prices, and an (empirical) Bayes estimator for current price dynamics is developed from the price history. Given the conditional price dynamics, investors allocate wealth to achieve their financial goals efficiently over time. The price updating and wealth reallocations occur when control limits on the wealth process are attained. A Bayesian fractional Kelly strategy is optimal at each rebalancing, assuming that the risky assets are jointly lognormal distributed. The strategy minimizes the expected time to the upper wealth limit while maintaining a high probability of reaching that goal before falling to a lower wealth limit. The fractional Kelly strategy is a blend of the log-optimal portfolio and cash and is equivalently represented by a negative power utility function, under the multivariate lognormal distribution assumption. By rebalancing when control limits are reached, the wealth goals approach provides greater control over downside risk and upside growth. The wealth goals approach with random rebalancing times is compared to the expected utility approach with fixed rebalancing times in an asset allocation problem involving stocks, bonds, and cash.

Capital growth: Theory and practice*

Publisher Summary This chapter is a survey of the theoretical results and practical uses of the capital growth approach. It presents the alternative formulations for capital growth models in discrete and continuous time. Various criteria for performance and requirements for feasibility are related in an expected utility framework. Typically, there is a trade-off between growth and security with a fraction invested in an optimal growth portfolio determined by the risk aversion criteria. Models for calculating the optimal fractional Kelly investment with alternative performance criteria are formulated. In capital accumulation under uncertainty, a decision maker must determine how much capital to invest in riskless and risky investment opportunities over time. The investment strategy yields a steam of capital, with investment decisions made so that the dynamic distribution of wealth has desirable properties. The distribution of accumulated capital to a fixed point in time and the distribution of the first passage time to a fixed level of accumulated capital are variables controlled by the investment decisions. An investment strategy which has many attractive and some not attractive properties is the optimal strategy, where the expected logarithm of wealth is maximized. This strategy is also referred to as the Kelly strategy. It maximizes the rate of growth of accumulated capital asymptotically. With the Kelly strategy, the first passage time to arbitrary large wealth targets is minimized, and the probability of reaching those targets is maximized. However, the strategy is very aggressive since the Arrow-Prat risk aversion index is essentially zero. Hence, the changes of losing a substantial portion of wealth are very high, particularly if the estimates of the returns distribution are in error. In the time domain, the chances are high that the first passage to subsistence wealth occurs before achieving the established wealth goals

Mean-variance versus expected utility in dynamic investment analysis

Given the existence of a Markovian state price density process, this paper extends Merton’s continuous time (instantaneous) mean-variance analysis and the mutual fund separation theory in which the risky fund can be chosen to be the growth optimal portfolio. The CAPM obtains without the assumption of log-normality for prices. The optimal investment policies for the case of a hyperbolic absolute risk aversion (HARA) utility function are derived analytically. It is proved that only the quadratic utility exhibits the global mean-variance efficiency among the family of HARA utility functions. A numerical comparison is made between the growth optimal portfolio and the mean-variance analysis for the case of log-normal prices. The optimal choice of target return which maximizes the probability that the mean-variance analysis outperforms the expected utility portfolio is discussed. Mean variance analysis is better near the mean and the expected utility maximization is better in the tails.

Long-term capital growth: the good and bad properties of the Kelly and fractional Kelly capital growth criteria

The main advantage of the Kelly criterion, which maximizes the expected value of the logarithm of wealth period by period, is that it maximizes the limiting exponential growth rate of wealth. The main disadvantage of the Kelly criterion is that its suggested wagers may be very large because the Arrow–Pratt risk aversion, the reciprocal of current wealth, is small compared with other commonly chosen utility functions. Hence, the Kelly criterion is relatively risky in the short term. And although most of the time Kelly bettors will have a lot of final wealth after a long sequence of favorable bets, it is possible, through bad scenarios, to lose most of one’s wealth. Hence, care in the use of Kelly and fractional Kelly strategies is crucial. In the one asset two valued payoff case, the optimal Kelly wager is the edge (expected return) divided by the odds. If there are multiple assets in a continuous time model with a lognormal distribution for returns, the .

Growth-security profiles in capital accumulation under risk

This paper considers the tradeoff between growth and security in the problem of capital accumulation under risk. It is shown how growth can be continuously traded for security with simple deterministic strategies generated from the optimal growth and optimal security problems. A lower bound is derived for the error resulting from the use of such strategies.

How does the Fortune's Formula-Kelly capital growth model perform?

William Poundstone’s book, Fortune’s Formula, brought the Kelly capital growth criterion to the attention of investors. But how do full and fractional Kelly strategies perform in practice? The authors study three simple investment situations and simulate the behavior of these strategies over medium-term horizons using a large number of scenarios. The results show 1) the great superiority of full Kelly and close-to-full Kelly strategies over longer horizons in that they earn very large gains a large fraction of the time, 2) the very risky short-term performance of Kelly and high-fractional Kelly strategies, 3) a consistent trade-off of growth versus security as a function of the bet size determined by the various strategies, and 4) no matter how favorable the investment opportunities are, or how long the finite horizon is, a sequence of bad scenarios can lead to very poor final wealth outcomes with a loss of most of the investor’s initial capital. Hence, in practice, financial engineering is important for dealing with the short-term volatility and long-run situations in a sequence of bad scenarios. Properly used, however, the strategy has much to commend it, especially in trading with many repeated investments.

The bond--stock yield differential as a risk indicator in financial markets

The trading prices for securities in financial markets can exhibit sudden shifts or reversals in direction. In this paper a methodology for asset price dynamics is presented where the diffusive component is combined with a risk process. The risk process accommodates deviations from an equilibrium process and reversions. The bond‐stock yield differential is considered as a risk factor affecting the risk process. An approach using a “peaks over threshold” technique and conditional maximum likelihood is used to estimate parameters in the model. Numerical results for the period 1985‐2004 in the US market validate the effectiveness of the model.