Risk management is intimidating. At Quartzite Risk Management LLC (Quartzite), in addition to our role as an advisor, we want to go the extra mile by being an educational resource for our clients. We offer this note in that spirit. This piece aims to use an ordinary coin flip as examples to shed some light on a few of the terms we often use when advising our clients. We introduced several of these concepts in another article, "The Relationship Between Risk, Reward, Edge, and Risk Management" - feel free to check in there for a shorter and less-mathematical discussion.

We will start with a single coin flip. Assume that Ed and Jerry are bored and decide to bet a dollar on the flip of a fair coin to entertain themselves. We can use this simple bet to explain several concepts. Both Ed and Jerry could lose a dollar to the other - that is their potential risk. Each could also win a dollar from the other – that is their potential reward. We all understand risk and reward well enough to know this, but let's take the opportunity to dig a little deeper.

Before the coin flip, the wager's outcome is unknown, but we know it will fall within a certain range. We know that heads and tails are equally likely results. After the flip, we know that one of the participants will be a dollar richer and the other a dollar poorer. Our intuition tells us that this is a fair game, and the math supports that intuition. To see the math supporting that intuition, we need to calculate both Ed's and Jerry's expected values before the coin flip. In this case, we can use the same equation to calculate both of their expected values:

(probability of winning times reward) + (probability of losing times risk) = expected value

Or:

(0.5 x $1.00) + (0.5 x $-1.00) = $0.00

Ed and Jerry both have expected values of $0.00. Mathematically speaking, this is why the game is fair. Neither participant has a higher expected value than the other – so neither participant has an advantage. Now, we can take the story a bit further. Assume Ed knows both heads and tails are equally probable, so he lets Jerry call the flip. Jerry chooses heads. Ed flips the coin, and behold; it comes up heads. Jerry wins. Jerry is now a dollar richer, and Ed a dollar poorer. Being that he just won, and possibly prone to some very human decision-making errors, Jerry wants to play again. After all, in his mind, he has just demonstrated his skill at calling coin flips.

Ed, on the other hand, sees the game for what it is, unnecessary risk-taking, and is thus less inclined to play. Jerry, now a true believer in his coin-calling skill, insists. Jerry is so confident he offers to bet $1.50 against Ed's $1.00. Ed, a savvy probability student, agrees immediately. Here it makes sense to pause and revisit our expected value equation from above. This time we need to expand a bit because Ed and Jerry have different equations. The game is no longer fair:

Ed’s expected value = (0.5 x $1.50) + (0.5 x $-1.00) = $0.25

Jerry’s expected value = (0.5 x $1.00) + (0.5 x $-1.50) = $-0.25

The math (and probably our intuition) tells us that Ed has an advantage in this situation. That is what we, at Quartzite, call edge. However, Ed's edge is no guarantee in a single flip – he is still equally likely to win or lose the bet. Now, we can take our example through the actual coin flip. Again, Jerry chooses heads, Ed flips the coin, and the coin comes up heads a second time. Jerry adds another dollar to his growing fortune and some more unwarranted coin-calling-confidence to his brain. Ed, however, is another dollar poorer but not unhappy with the most recent bet. Ed knows if he consistently gets an edge, he should make money over the long run.

Not surprisingly, overconfident Jerry wants to play again, and even up the stakes. Ed likes the current game but senses the opportunity to get more edge. Ed suggests that he should win $2.00 for every $1.00 he risks. Jerry agrees, but he wants to wager more money on each flip. We should pause again to look at both Ed's and Jerry's expected values for each dollar wagered under these terms:

Ed’s expected value = (0.5 x $2.00) + (0.5 x $-1.00) = $0.50

Jerry’s expected value = (0.5 x $1.00) + (0.5 x $-2.00) = $-0.50

In this scenario, Ed has a massive edge – equaling half of what he has risked. The trouble is that advantage still doesn't guarantee Ed a profit. Ed still has risk, and overconfident Jerry wants to play for four times as much money. Ed knows betting four times as much money will net him four times as much edge (4 x $0.50 = $2.00) and wants to accept. However, he is worried about losing all of his lunch money. Ed needs some risk management. What are Ed's options?

The first is to accept the naked risk and put his $4.00 against Jerry's $8.00 on a single flip. The expected value math for that bet looks like this:

Ed’s expected value = (0.5 x $8.00) + (0.5 x $-4.00) = $2.00

That's a good bet, but it's hardly a sure thing, and Ed at least wants to have soup for lunch. Ed thinks about the problem and proposes to Jerry that they divide the $4.00 bet into four equal $1.00 wagers. Here the expected value is the same, but Ed's probability of winning overall is much higher. Four flips of the coin yield sixteen possible combinations. Table 1 shows each of those possible combinations and Ed's resulting profit or loss:

Table 1 demonstrates a few things for us. First, if we take the total dollar value of all of the combinations, we get $32.00 ($8.00 + $5.00 + $5.00 + $5.00 + $5.00 + $2.00 + $2.00 + $2.00 + $2.00 + $2.00 + $2.00 - $1.00 - $1.00 - $1.00 - $1.00 - $4.00 = $32.00). If we then divide that dollar amount by sixteen (the total number of combinations), we get $2.00 – Ed’s expected value is unchanged. We can also see that Ed wins in eleven of the sixteen combinations or 68.75% of the time, a big improvement over his initial 50% chance of winning. Splitting the larger bet into independent smaller bets demonstrates how diversification can be an excellent, though not perfect, risk management tool.

Jerry is in a hurry and only wants to bet on heads one more time. Ed is about to risk a hungry afternoon and make the one flip bet – after all, it is hard to turn down that much edge. Right before Ed agrees to the wager, he sees his friend Mark. Ed knows Mark. Mark is a good guy, has deep pockets, and is well diversified. Ed also knows that Mark bets on coin flips all the time. Ed waves to Mark and calls him over. Ed quickly explains to Mark that he is about to flip a coin with Jerry and asks if Mark would be willing to take tails on the same coin flip risking $5.95 to win $6.05. Mark agrees. You can see the potential outcomes for our three participants in Table 2:

In this scenario, Ed chose to give a small portion of his edge to Mark but has guaranteed himself a profit by doing so. Ed's wager with Mark is an excellent example of hedging. In this case, Ed gave up a portion of his profit potential to limit his potential loss. Ed found an opportunity to cancel out his risk and did some quick math to lock in most of his edge by giving Mark a little of that edge. At this point, Ed no longer cares if the coin comes up heads or tails – he profits equally in either scenario.

Many grain and soybean producers find themselves in the situation that Ed was in before Mark came along. Those producers are excellent at production and, as a result, have a considerable edge. However, to maximize their expected value, they find themselves taking uncomfortable amounts of risk that cannot be adequately diversified away. To manage this risk, those producers could employ a strategy like our Quartzite Precision Marketing program.

At Quartzite, we work with our clients to identify their edge, develop a robust risk management plan, and implement that plan. We take the job of protecting our clients seriously because we want to build long-term relationships and help our clients be more sustainable. We're committed to the long term so that you can be too. Contact us today if you're interested in learning more.