LEVEL: INTERMEDIATE
Risk management is generally considered a defensive strategy as the techniques that are used are focused on minimizing loses and avoiding the risk of ruin. A basic concept of risk is that it is highly correlated to reward. As an investor increases the amount of capital they are willing to risk, their potential reward increases. The key to a successful investment strategy is to determine the optimal risk to assume to achieve a specific return profile.
Risk
Risk can be defined as the possibility of loss. For example if an investor owns a stock, and there is a possibility of a price decline, the investor is at risk. The loss itself is not the risk; instead the possibility of loss is the risk. There are a number of techniques to control the risk, which includes selling a stock using a stop loss, as well as, hedging with other financial instruments. To achieve returns above a risk free rate (purchasing US treasury bills), investors need to assume the potential for loss, and the key is to manage that potential loss with risk management.
Bet Size
To achieve the optimal portfolio, investors need to determine most efficient amount of capital to use when making an investment. There are a number of strategies that can be used to determine the bet size of an investment, which include a fixed bet or a fixed-fraction bet. In a fixed betting system, the amount of capital remains the same no matter how large the portfolio grows. In this instance, an investor would place trades of $50 dollars regardless of whether the bet becomes proportionately too large or too small. A $50 dollar bet on an account size of $1,000 dollars seems reasonable, but it would be considered large on a $100 dollar portfolio and small on a $100,000 portfolio.
To remedy this problem of the equity within the portfolio drifting out of proportion to the fixed bet, an investor can define a bet size as a fixed fraction of the equity within a portfolio. A 5% fixed-fraction bet would, on our original $10,000, also lead to a $500 bet. If the equity rises or falls, the fixed-fraction bet stays in proportion to the equity.
Generating the optimal bet size is a process similar to probability. This process of finding this optimal number has been best described by the Kelly criterion, developed by John Kelly. Kelly created a simple formula that describes the optimal strategy for non-correlated bets.
The formula can best be described by observing the probability of a coin toss. For example let’s say that you’re going to make a series of 10 bets on a coin flip. The coin is weighted and thus lands on heads 70% of the time. You have $100. The question is how much to bet on each coin toss?
Many might believe that betting your entire portfolio on this outcome is the optimal bet size, but this is the incorrect answer. While risking your portfolio does maximize your expected value for any individual bet, it ignores the fact that you have a limited amount of capital, and once you lose it, you also lose the opportunity to make future profitable bets is gone. If you bet your entire bankroll each time, you will likely go bankrupt, so you’ll miss out on future advantageous bets.
One way to think about Kelly’s criterion is to consider the outcome if all the capital is risked on each bet. For example, let’s assume the entire portfolio is bet on that 70% weighted coin. If there are 10 bets, we can expect to double our portfolio 7 times, but then we will go bankrupt. If instead we only bet 10% of our portfolio, the equity will grow by a factor of 1.1 when on winning tosses, and shrink by a factor of 0.9 on losing tosses. The growth looks like this: 1.1*1.1*1.1*1.1*1.1*1.1*1.1*0.9*0.9*0.9 for a total return of about 42%.
Kelly Criterion formula:
- f is the fraction of the current portfolio
- b is the net odds received on the wager
- p is the probability of winning;
- q is the probability of losing, which is 1 – p
Risk vs. Reward
There are a number of ways investors can modify the Kelly Criterion to find a process that is more in tune with their trading style. One of the most important concepts in determining bet size is risk relative to reward. If the reward of a trade is compensated beyond the relative risk it is considered a robust bet.
For example, if a coin toss paid an investor 10% for every 5% wagered on a $100 dollar portfolio, than over the course of time, this scenario would pay off handsomely. The payout on an equal weighted coin would look at follows:
1.1 * 1.1 * 1.1 * 1.1 * 1.1 * .95* .95 * .95 * .95 *.95
This payout would generate an overall return of approximately 24%. This concept not only incorporates bet sizes but other techniques in risk management which will help an investor find the best trades to incorporate into their portfolios.
Summary
Bet size is an important concept that should be analyzed prior to creating a portfolio of stocks in an effort to achieve the most efficient risk adjusted returns. Bet sizes that are too large will potential create the risk of losing an investor’s entire portfolio at some point, while bet sizes that are too small will never meet and investors expected returns.