Expected Value

Expected value is the average result one would expect from an uncertain situation if it could be repeated many times. It is calculated by multiplying each possible outcome by its probability and adding the products together. The result is a single number that summarizes a range of possible outcomes, weighted by how likely each one is.

Expected value is a basic tool of probability theory and decision analysis. In business and accounting it is used to compare investment options, set prices and budgets, value insurance and guarantees, and measure some amounts in the financial statements, such as variable consideration in revenue contracts and expected credit losses. It is usually considered together with a measure of risk, such as the standard deviation, because two choices with the same expected value can carry very different risks.

Imagine a game where you flip a coin: heads you win $10, tails you win nothing. On any single flip you get either $10 or $0, never $5. But if you played a thousand times, you would win about half the time and average $5 a flip. That $5 is the expected value.

Businesses use the same idea when they face uncertain outcomes. A new product might be a hit, do moderately well or flop. By putting a rough chance on each and averaging, managers get a fair single figure for comparison, while remembering that the actual result will be one of the possibilities, not the average.

For a set of possible outcomes, the expected value is:

• $E(X)$ is the expected value of the uncertain amount $X$.
• $x_i$ is the value of outcome $i$.
• $p_i$ is the probability of outcome $i$; the probabilities must add up to 1.
• $n$ is the number of possible outcomes.

A company is considering a product launch with three possible results: a 30 percent chance of a $500,000 profit, a 50 percent chance of a $200,000 profit and a 20 percent chance of a $300,000 loss. The expected value is $0.3 \times 500{,}000 + 0.5 \times 200{,}000 + 0.2 \times (-300{,}000)$, which is 150,000 plus 100,000 minus 60,000, or $190,000.

The spread around that figure is large. The variance is $0.3 \times 310{,}000^{2} + 0.5 \times 10{,}000^{2} + 0.2 \times 490{,}000^{2}$, about 76.9 billion, so the standard deviation is about $277,000. A manager comparing this launch with a safer project offering an expected $170,000 with little variation would weigh the higher average against the much greater risk, depending on the company's risk aversion.

• Capital budgeting. In capital budgeting, expected cash flows under several scenarios are often discounted to find an expected net present value.
• Decision trees. Sequential decisions, such as whether to test-market a product before a full launch, are analyzed by calculating the expected value at each branch.
• Revenue recognition. Under US GAAP, when a contract includes variable consideration such as rebates or performance bonuses, a company estimates it using either the expected value method or the most likely amount, whichever better predicts the result. The expected value approach suits contracts with many possible outcomes, as part of revenue recognition.
• Credit losses. The current expected credit loss model for loans and receivables measures the lifetime losses a lender expects. A simplified calculation multiplies the probability of default by the loss if default occurs and by the amount exposed: on $1,000,000 with a 2 percent default probability and a 40 percent loss rate, the expected loss is $8,000.
• Insurance and warranties. Premiums and warranty liabilities start from the expected cost of claims.

An expected value is only as good as the probabilities and outcomes fed into it, and in business these are often subjective estimates. It also hides the shape of the risk: a small chance of a catastrophic loss may be unacceptable even if the expected value is positive. That is why it is paired with measures of spread, such as standard deviation or value at risk, and with judgment about what the business can afford to lose. Expected value suits decisions that are repeated many times, where results average out, better than one-off, bet-the-company choices. In the accounting standards, not every uncertain amount is measured this way; a loss contingency, for example, is accrued at the best estimate within a range, or the low end of the range if no amount is more likely, rather than at an expected value, as explained under loss contingencies. Because the measure ignores the order and timing of outcomes, it is often combined with discounting, as in discounted cash flow analysis, or with probabilities derived from regression analysis of past data.