In accounting, accounts receivable and accounts payable refer to revenues and expenses that have not yet been paid. An account receivable is money owed to a company that has not yet been paid. An account payable is money owed by a company that has not yet been paid. On a balance sheet, accounts receivable are listed as assets and accounts payable are listed as liabilities.
In the cash method of accounting, transactions are recorded only when payment is received or made. The accrual method of accounting, by contrast, records revenues and expenses at the moment of a transaction, regardless of when payment occurs. In particular, the accrual method treats accounts receivable as accounts received and accounts payable as accounts paid. The cash method only cares about money that is actually received and paid.
The key difference between accrual and cash methods is the difference between expected payments and realized payments. In a perfect world, all expected revenues (accounts receivable) are eventually realized (received), and both accrual and cash work out the same in the long run. In practice, there’s a big difference between receivable and received, payable and paid, expected and realized.
The same principle applies when accounting for expected value and realized value in betting and other risky or uncertain contexts.
Expected value is the average profit one can expect if the same (or similar) bet can be repeated indefinitely over a large number of trials. Much like accountants regard accounts receivable and accounts payable as assets and liabilities, respectively, gamblers and risk analysts often regard expected value as an accruing asset. In the accrual method, accountants treat accounts receivable as assets, on the expectation that the accounts will actually be received. In a similar way, professional gamblers and investors ignore short-term fluctuations and treat expected value as an asset, on the expectation that the value will eventually be realized. Call this the accrual method of risk accounting.
It is a core underlying assumption of the accrual method that expected gains can actually be realized. A coffee shop that accepts credit card payment for a cup of coffee does not receive $3 at the point of sale. To realize the $3 of revenue, the shop must later collect it from the payment processor. The shop owner agrees to this arrangement on the assumption that there will be no problems collecting the payment.
Applied to risk accounting, the accrual method for expected value only applies if the expected value can actually be realized. The obvious consideration of whether the counterparty is trustworthy and has enough money to pay is a relevant preliminary step, but not the most important for determining whether expected value can rightly be treated as an asset. Many professional risk takers erroneously believe that expected value is “banked” on each bet, a common misunderstanding of what expected value is.
Expected Value is a Long-Run Average (under optimal conditions)
By definition, expected value is a long-run average. So while expected value is accrued on individual transactions, it is only realized by repetition for an indefinite sequence of trials. If the bettor cannot transact a positive expected value bet in a way that could be sustained for as long as necessary, then the expected value will never be fully realized.
For a concrete example, consider an even money bet on a biased coin toss for which there is a 55% probability to land heads. The expected value of this bet is 10% of the amount risked: $1 for every $10. Now, consider a gambler with a bankroll of $100,000 who risks all of it on heads on one toss of the coin. Then the gambler has accrued $10,000 of expected value, but either -$100,000 or +$100,000 of realized value. Importantly, the gambler who loses on the first toss has given up the opportunity to ever realize the $10,000 of value because he is broke. In this extreme example, the +$10,000 of accrued expected value is merely a theoretical number, as the bettor is not executing a strategy that can ever expect to realize this value.
Value Expected, Value Realized, Expected Value Payable and Expected Value Receivable
To bring the accounting analogy full circle, let’s define the terms expected value payable (EVP) and expected value receivable (EVR) from the terms value expected (VE) and value realized (VR).
Mathematically, value expected is defined the same way was expected value (EV). Practically, VE differs from EV in that it can only be accrued by a strategy that can reasonably expect to realize the value with a high degree of certainty. Value realized is actual profit and loss.
Assuming the probability of winning each round is known*, VE and VR can be calculated after any given round in a sequence of bets. (*This is a big assumption that rarely holds in practice.)
After computing these two quantities, we can compute Expected Value Payable (if VR exceeds VE) or Expected Value Realized (if VE exceeds VR). Let’s discuss EVP first.
In the above example, we have a 55% chance of winning an even money bet. For simplicity, suppose we bet $100 on each round, for a VE of $10 per round. After ten rounds of play, suppose the results are: 7 wins, 3 losses.
In this case, we assumed the same bet of $100 on each round, for a value expected of $10 per round. With 7 wins and 3 losses, we have a realized value of $400 over ten rounds for VR of $40 per round, indicating that we have realized $30 more value per round than expected. We call this $30 the expected value payable and mark it as a liability on our risk balance sheet. If we continue betting the same strategy for an indefinite period, we should expect that this $30 will be reduced to $0 per round.
On the other side, suppose we’ve made the same $100 bets per round, but with 3 wins and 7 losses in the first 10 rounds. In the case, we have the same value expected of $10 per round, but we have lost $400 for a value realized of -$40 per round. The difference of $50 ($10 - (-$40)) is called the expected value receivable, which we mark as an asset on the risk balance sheet.
In both cases, the difference between VE and VR is only truly payable or receivable if the strategy can be sustainable indefinitely.
Gambling as an accounting problem
In highlighting the difference between value expected and value realized, the above discussion emphasizes the importance of accounting and bankroll management in risky propositions. Accounting is important not because we want to know how much money we’ve won or lost, but instead to make sure that our positive expected value opportunities also have positive value expected. Most importantly, to ensure that we are positioned to turn our Expected Value Receivable into Value Realized.