# The Present Value of a Bet's Payoff

Future income flows should always be expressed in present value terms. This practice is as important in the prediction market setting as it is elsewhere. The payoffs on winning prediction market bets frequently do occur many months from the present. Indeed, it is not uncommon to see prediction market bets concerning events that might occur two or more years in the future. If the wager was made using a currency which experiences inflation, then the payoff should be discounted to reflect the change in the value of the currency. When we calculate the value of the future payoff in terms of the value of the currency today, we are calculating the present value or the present discounted value of the future income flow.

The present value of a future income flow can be illustrated with an example. Suppose the date is January 1st and it is possible to earn 2% interest on a 1-year time deposit (aka certificate of deposit) with a major bank. How much should one be willing to pay on January 1st for a future income flow of 100 in one year? Clearly it must be less than 100 because it is possible to deposit 100 with the major bank and receive 102 in one year’s time. Accepting a price of 100 would “lock in” a loss of 2. The commonly accepted answer is $$\frac{1}{1 + r} 100 \approx 98.04$$, where $$r$$ is equal to the “risk-free” rate (2%). We say that the present value of 100 in a year’s time is 98.04. It would be wasteful to pay more than 98.04 for a future income flow of 100 because you would be better off depositing your money in the bank and withdrawing it, with interest, in a year’s time.

(Energetic students willing to engage in light algebra may derive the present value formula using principles mentioned above.)

Using a present-value adjustment in the prediction market setting is relatively straightforward. Consider a contract that costs 0.98 today and pays out with certainty in one year. (It is not difficult to find contracts fitting this description on PredictIt.) Assuming a 2% risk-free rate, what is the return on this contract, in present value terms? The present value of 1 in one year is $$\frac{1}{1+r} 1 \approx 0.9804$$, making the return a measly 0.04%. In fact, the return is negative if we account for PredictIt’s fees. What appeared on the surface to be a sure bet is, upon reflection, a losing wager.

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