Maximizing relative wealth using leverage: The role of risk aversion

By applying leverage, investors can increase the returns and profit of an investment, but also increase the relative performance of a portfolio against a benchmark.
Christian Lundström Tjurhufvud, PhD

Head of Fund Selection and Compliance Unit

Swedish Pensions Agency

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In investing it is sometimes that the relative wealth is more important than absolute wealth, fund managers trying to outperform a benchmark for example. By applying leverage, investors can increase the returns and profit of an investment, but also increase the relative performance of a portfolio against a benchmark. For example, fund managers may use leverage to outperform their benchmark by taking larger risk than the benchmark and, implicitly, higher beta against the market. In this paper coauthored with Jarkko Peltomäki from Stockholm Business School, we study leverage for maximizing relative wealth and the role of risk aversion.

The so-called Kelly criterion is traditionally used to determine the optimal leverage factor for maximizing an investor’s absolute wealth (Kelly 1956; Thorp 1969) and the Kelly criterion has many valuable long-run properties for investors. In the long run, Breiman (1961) showed that the Kelly criterion maximizes long-term growth and asymptotically outperforms any other essentially different leverage rule. Breiman (1961) also showed that the Kelly criterion minimizes the expected time it takes to reach a certain level of capital, which can be valuable to investors that experienced a severe drawdown and would like to use leverage to minimize the time to get back to the previous high. However, investors are risk averse to some degree and therefore not comfortable to use the level of leverage suggested by the Kelly criterion.

In fact, Breiman (1961) also show that rational investors only with a log utility function maximize both utility and expected profit when applying the Kelly criterion in investing. This suggests that investors with other utility functions would find the Kelly-optimal-leverage too aggressive to use.

How much leverage should risk-averse investors then use when maximizing their relative wealth? The importance of this question is increasing due to the recent democratization of investment leverage. Investors have gained easier access to leverage via popular trading platforms such as Robinhood, and the margin debt levels are continuing to hit record highs as of February, 2021.

In this paper, we expand the Kelly criterion to situations where investors seek to maximize wealth relative to a generic reference, applicable also for risk averse investors. Using a generic reference, our model is applicable to many situations, for example, when the reference may be an investment benchmark. Our model is also applicable when the reference may consist of behavioural features in line with Orr (2017) and Lo et al. (2018), or when investors compare their return relative to their peers or other possible references of importance. Besides expanding the Kelly criterion to investors seeking to maximize relative wealth, our ambition is to link the Kelly criterion to a broader utility class than log utility, thereby enabling investor risk aversion in the investment decision.

To relate our risk-adjusted Kelly criterion to expected utility theory, we utilize the quadratic utility function (e.g., Levy and Markowitz 1979), which is widely used to study the behaviour of risk-averse investors in financial markets. The quadratic utility function is, in contrast to the utility functions that underlie the leverage criteria of other authors, independent of the returns-density-function assumption. This assumption is an advantage of our methodology and it is useful in investment applications as returns of financial assets are typically unknown with both kurtosis and skewness (e.g. Cont 2001).

Our findings show that it is theoretically possible to determine optimal leverage given investor risk aversion. We show that the standard Kelly criterion results holds also in relative wealth maximization. That is, too much leverage (too large leverage factors) will actually lead to lower expected profit than the reference portfolio, even if the unleveraged profit is larger. The intuition is that a too large leverage factor leads to severe drawdowns in capital relative to the reference portfolio that takes too long time to recover from even if the strategy has a positive expected return per trade and the number of trades are very large.

Assuming quadratic utility, we find that it is optimal for risk-averse investors to use leverage. Interestingly, optimal leverage decreases below 1 only for very high degree of risk aversion. These theoretical findings suggest that the availability of leverage is important in relative wealth accumulation and they show why it may be rational for some investors to use high levels of leverage even though they are risk averse. Thus, our risk-adjusted Kelly criterion can be a valuable tool in decision making also in practical applications.

The article is based on the the paper “Maximizing relative wealth using leverage: The role of risk aversion“ by Christian Lundström Tjurhufvud, PhD, Head of Fund Selection and Compliance Unit, Swedish Pensions Agency and Jarkko Peltomäki, PhD Associate Professor, Stockholm Business School. The full paper is now available for reading on SSRN.

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References Cont, R.: Empirical properties of asset returns: stylized facts and statistical issues. Quantitative Finance 1, 223-236 (2001), Breiman, L.: Optimal gambling systems for favorable games. Fourth Berkeley Symposium on Probability and Statistics 1, 65-78 (1961), Kelly, J. L.: A new interpretation of information rate. Bell System Technical Journal 35, 917–926 (1956), Levy, H., Markowitz, H.: Approximating expected utility by a function of mean and variance. The American Economic Review 69, 308-317 (1979), Lo, A.W., Orr, A., Zhang, R.: The Growth of Relative Wealth and the Kelly Criterion. Journal of Bioeconomics 20, 49-67 (2017), Thorp, E. O.: Optimal gambling systems for favorable games. Review of the International Statistical Institute 37, 273–293 (1969), Orr, H. A.: Evolution, finance, and the population genetics of welfare wealth. Journal of Bioeconomics 20, 29-48 (2017).

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Maximizing relative wealth using leverage: The role of risk aversion