QA specializing in statistical modeling, risk assessment, and decision theory in financial markets.
The **Expected Value Calculator** determines the weighted average outcome of a decision or investment by considering all possible scenarios (Value 1, Value 2) and their corresponding probabilities. It can solve for the missing outcome value, probability, or the overall expected value.
Expected Value Calculator (Two Outcomes)
Instructions: Enter values for any three of the four core parameters (Value 1, Probability 1, Value 2, Expected Value) to solve for the missing one. Probability 2 will be automatically derived as $1 – P_1$.
Outcome Parameters
Expected Value Formula
For two mutually exclusive events ($E_1$ and $E_2$), where $P_1 + P_2 = 1$ (100%), the Expected Value ($EV$) is calculated as:
Expected Value ($EV$):
$$EV = (V_1 \times P_1) + (V_2 \times P_2)$$Where $V$ is the dollar value of the outcome and $P$ is the probability (as a decimal) of that outcome occurring. This calculator can solve for any of the four core variables using basic algebra.
Formula Source: Corporate Finance Institute (CFI)Variables Explained (P, F, V, Q – Parameters)
- $V_1$ (Outcome Value 1, $P$): The net financial result if the first event occurs.
- $P_1$ (Probability 1, $F$): The likelihood (as a percentage) that the first event occurs.
- $V_2$ (Outcome Value 2, $V$): The net financial result if the second event occurs.
- $EV$ (Expected Value, $Q$): The long-term average value of the outcome if the event is repeated many times.
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What is Expected Value?
Expected Value (EV) is a fundamental concept in probability and decision theory. It represents the value one would expect to receive on average if a probabilistic event were repeated many times. In finance and insurance, EV is crucial for assessing the profitability and risk associated with investments, gambles, and insurance policies. A positive EV suggests a long-term profit, while a negative EV suggests a long-term loss.
EV is not the most likely outcome, but rather a weighted average. For instance, winning a lottery with a low probability but a high payout might still result in a negative EV when accounting for the ticket cost, indicating that playing consistently is a losing proposition over time.
How to Calculate Expected Value (Example)
Consider a coin flip where you win \$10 if it lands heads (50%) and lose \$8 if it lands tails (50%).
- Step 1: Define Outcomes and Probabilities
$V_1 = \$10$ (Heads), $P_1 = 0.5$ (50%). $V_2 = -\$8$ (Tails), $P_2 = 0.5$ (100% – 50%).
- Step 2: Calculate the Weighted Value of Each Outcome
Weighted Value 1: $\$10 \times 0.5 = \$5.00$
Weighted Value 2: $-\$8 \times 0.5 = -\$4.00$
- Step 3: Sum the Weighted Values for Total Expected Value ($EV$)
$EV = \$5.00 + (-\$4.00) = \mathbf{\$1.00}$.
The Expected Value of \$1.00 means that on average, you would expect to gain \$1.00 for every flip, making this a favorable proposition in the long run.
Frequently Asked Questions (FAQ)
EV helps investors quantify risk. By comparing the potential profit and loss scenarios, they can determine if an investment opportunity offers a positive expected return over the long term, making it statistically sound.
No. The most likely outcome is the mode. EV is the average outcome weighted by probability. For instance, if you have a 99% chance of winning \$1 and a 1% chance of losing \$100, the most likely outcome is winning \$1, but the EV is slightly negative.
NPV uses the concept of EV. When calculating NPV, future cash flows are often the “outcomes” ($V$) and the time value of money (discounting) acts as the “probability weighting” to find the present-day EV of the project.
In a formal EV calculation, all possible outcomes must be accounted for, meaning their probabilities must sum exactly to 1 (100%). If they don’t, you are missing a possible scenario.