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How Much Would You Wager on The Flip of a Coin?

Updated: May 14


close up photo of a coin

How much would you bet on a fair coin flip that where you double your money on heads and get back only half of your bet on tails?  

 

If you were to play this game over and over, you would find that your short-term outcomes would vary significantly. Sometimes you win. Sometimes you lose. If you were to play this game for long enough, you would eventually find your results converging to an average win of 25% per game. This average outcome over many games is what we call the expected value or expected return. 

 

So how much of your wealth should you bet on this game? 


Intuitively

Since this game has a positive expected value, you might think that maximizing your expected wealth requires going all-in. This would be true if you had an infinite number of chances with infinitely deep pockets. Reality, though, has its limits. An all-in strategy for this game ignores the very real risk of ruin. 

 

We have a name for investors who ignore risk while chasing returns: “risk neutral.” Risk neutral investors eventually go bust. Their luck will one day run out. Everyone else is risk averse to some degree. Being risk averse isn’t something to be ashamed of. It’s necessary for survival and long-term success. 

 

Intuitively, how much you should invest in a venture with uncertain outcomes should scale up with your expectations for returns and scale downwards with the degree of dispersion of results, or volatility. For example, if you only lost a small percentage of your investment on each losing coin flip, you’d be much more confident betting larger sums.  



Mathematically

Two researchers independently discovered a formula that tells us how much to invest in uncertain ventures: J. L. Kelly Jr., a researcher at Bell Labs in 1956, and the Nobel Prize winner in economics, Robert C. Merton in 1969. 

 

Kelly realized that a gambler could grow their wealth exponentially, and beat all other strategies, by investing and reinvesting a fixed proportion of their wealth. Instead of maximizing their expected wealth, requiring that they go all-in on every bet, they would maximize the logarithm of their expected wealth. This is done by scaling their bet proportionately with the expected return of their gamble and inversely with the square of the volatility. 

 

Merton took a different approach. He sought to directly solve the question of how to construct a portfolio under uncertainty and was able to incorporate individual differences in risk aversion. Merton’s solution is identical to Kelly’s when an investor exhibits a Constant Relative Risk Aversion (CRRA) of 1.  

 


Constant Relative Risk Aversion is a popular economic model that reasonably approximates our behavior as investors. The utility we obtain from wealth follows a logarithmic path. We feel losses more than gains. Further gains offer ever-diminishing utility; thus, we are risk averse. Risk aversion is logarithmic in nature, hence the connection to the Kelly Criterion. 

 


 

At CRRA factor 0 is the risk neutral investor who eventually goes bust. Someone with infinite CRRA has extreme aversion to risk and would never invest in anything with uncertainty. CRRA factor 1 describes the Kelly bettor attempting optimal geometric growth. Someone investing of their wealth in US stocks who is also expecting a repeat of 1972 - 2023 might have a CRRA factor of slightly above 2. 




The Unusual Power of Compounding Links Everything Together

Compound interest is the eighth wonder of the world. He who understands it, earns it; he who doesn't, pays it.  - Supposedly Albert Einstein

Investors earn compound returns when they can reinvest their earnings. Effectively, they earn interest on their interest, causing exponential growth. It starts slowly at first, later accelerating to rates that eventually blow your mind. 


diagram comparing simple vs compound growth

Both Kelly and Merton agree that maximizing the expected log return is the way to go because they ultimately derive their power from the same fundamental source: compounding. Compound returns are calculated by taking the average of the log of returns.

 

An important fact about compound returns; it hates volatility.


Back to the coin toss with the 25% expected return. Let's say you lose the first game. Half your money is instantly zapped. Then you win the next round, doubling your now diminished account. The average return of your two flips is 25%, yet you've only broken even. Volatility hurts compound returns because you have less available for investing the next time around, slowing down your time to recover.


Achieving the best results requires investing just the right amount. Invest too little; you're leaving money on the table. Invest too much; the effect of volatility becomes overwhelming.



Making This Work For You

All this boils down to a few important investment principals that we hold very closely. Investing is inherently uncertain and risky. Managing risk is our #1 job. All other things being equal, we prefer investments and portfolios with lower volatility and tend to advise our clients toward that direction. But we are also not afraid to advise our clients to take appropriate risk and get the money.  

 

While Kelly tells us that a CRRA factor of 1 gets the most money (in theory), it requires that people are machines with minimal risk aversion and perfect estimates of market conditions. Unrealistic. These graphs and formulas cannot be used in isolation to determine what or when to buy. Properly applying Kelly or Merton requires careful analysis of forward-looking market expectations, which we do for our clients.

 

We work with clients to understand their personal CRRA level, tailor portfolio recommendations to that level, continuously monitor and update market expectations, and work with clients on implementation strategies that meet them where they are. 

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