Fair Value - How Much is a Bet Worth?

Fair Value - How Much is a Bet Worth?

Let’s say you are offered the chance to play the following game indefinitely: A coin is flipped and you win $1 if it lands “heads” but lose your bet amount if it lands “tails.” How much should you be willing to wager on this game?

In order to solve this, you must understand that the value of any bet is determined by the reward and the probability of winning that reward. As the reward or probability of winning increases, so does the value of the bet. In this coin-tossing problem, we know that the reward is one dollar so all we need to determine is the probability of winning.

We can probably guess that the coin will land heads and tails about equally often so we might decide to use 50% as the probability of winning (which implies there is a 50% chance of losing since the total probabilities must add up to one). Now that we know the reward is $1 and the probability of winning the reward is 50%, we can figure out how much to pay for this bet. We will show you a mathematical way shortly but first let’s see if we can figure it out intu­itively. What would happen if you paid $1.50 for the bet? Since we reasoned the probability is 50%, we can guess that you would win $1.00 half the time and lose $1.50 half the time. This results in a loss of 50 cents every two flips, or 25 cents on average. Although you might win an occasional bet here and there, we can be sure that you would end up on the losing side after hundreds of flips. So $1.50 is too high of a price to pay.

What would happen if you paid 50 cents? We would now expect that you would win $1.00 half the time and lose 50 cents half the time, which means you should make about 50 cents every 2 flips or 25 cents on average. You could certainly lose some bets but, over the long run, you would be sure to end up on the winning side.

Well, if $1.50 is too high of a price to pay and 50 cents is too low, then there must be a price in between that is neither too high nor too low. That price is called the fair value of the bet. If a bet is priced at fair value then neither player will sustain a long-run advantage. Both players are expected to just break even over the long run.

You may have guessed that the fair value of this bet is SI.00. If you wager $1.00, then you would win $1.00 half the time and lose $1.00 half the time thus making you no richer or poorer over the long run. At a price of one dollar, the bet is fair to both gamblers.