Expected Value: Definition, Formula, and Real-World Examples for Financial Decision-Making

Definition of Expected Value

The expected value is essentially the long-run average value of a random variable or an investment outcome. It is calculated by weighting each possible outcome by its probability and then summing these products. This means that EV represents a probability-weighted average of all possible outcomes, giving you an idea of what you can expect over many repetitions or in the long term.

To put it simply, if you were to repeat an investment many times, the average return you would get is roughly equal to the expected value. For instance, if you flip a coin where heads gives you $10 and tails gives you $0, with each outcome having a 50% chance, the expected value would be $5.

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Formula for Expected Value

The formula for calculating the expected value is straightforward:

[ E(X) = c1p1 + c2p2 + c3p3 + \ldots + cnpn ]

Here:

• ( c_i ) represents the i-th outcome.

• ( p_i ) represents the probability of the i-th outcome.

To calculate EV:

1. Multiply each outcome by its corresponding probability.

2. Sum these products.

For example, consider a discrete random variable where you have three possible outcomes: winning $100 with a probability of 20%, winning $50 with a probability of 30%, and losing $20 with a probability of 50%. The EV would be:

[ E(X) = (100 \times 0.20) + (50 \times 0.30) + (-20 \times 0.50) ]

[ E(X) = 20 + 15 – 10 ]

[ E(X) = 25 ]

This means that over many repetitions, you can expect an average gain of $25.

Real-World Examples in Finance

Project Evaluation

Imagine you are evaluating three projects: Project A, Project B, and Project C. Each project has different potential returns and associated probabilities.

• Project A: Returns $100,000 with a 30% chance; returns $50,000 with a 40% chance; loses $20,000 with a 30% chance.

• Project B: Returns $150,000 with a 20% chance; returns $75,000 with a 50% chance; loses $10,000 with a 30% chance.

• Project C: Returns $200,000 with a 10% chance; returns $100,000 with a 60% chance; loses $5,000 with a 30% chance.

Calculating the EV for each project:

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  Project A:

  [ E(A) = (100,000 \times 0.30) + (50,000 \times 0.40) + (-20,000 \times 0.30) ]

  [ E(A) = 30,000 + 20,000 – 6,000 ]

  [ E(A) = 44,000 ]

• Project B:

  [ E(B) = (150,000 \times 0.20) + (75,000 \times 0.50) + (-10,000 \times 0.30) ]

  [ E(B) = 30,000 + 37,500 – 3,000 ]

  [ E(B) = 64,500 ]

• Project C:

  [ E(C) = (200,000 \times 0.10) + (100,000 \times 0.60) + (-5,000 \times 0.30) ]

  [ E(C) = 20,000 + 60,000 – 1,500 ]

  [ E(C) = 78,500 ]

Based on these calculations, Project C has the highest expected value and would be chosen as the most promising project.

Investment Analysis

Suppose you are deciding between two investment funds: Fund A and Fund B.

• Fund A: Has an expected return of 8% with a probability of 70%; has an expected return of 5% with a probability of 30%.

• Fund B: Has an expected return of 10% with a probability of 40%; has an expected return of 6% with a probability of 60%.

Calculating the EV for each fund:

• Fund A:

  [ E(A) = (0.08 \times 0.70) + (0.05 \times 0.30) ]

  [ E(A) = 0.056 + 0.015 = 0.071 or 7.1% ]

• Fund B:

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  [ E(B) = (0.10 \times 0.40) + (0.06 \times 0.60) ]

  [ E(B) = 0.04 + 0.036 = 0.076 or 7.6% ]

Based on these calculations, Fund B has a higher expected return and would be preferable.

Scenario Analysis

Scenario analysis involves evaluating different possible outcomes of an investment by assigning probabilities to each scenario.

For example, consider investing in a new market where there are three possible scenarios:

• Scenario 1: Market growth leads to a 15% return with a probability of 40%.

• Scenario 2: Stable market conditions lead to a 10% return with a probability of 50%.

• Scenario 3: Market decline leads to a -5% return with a probability of 10%.

Calculating the EV:

[ E(X) = (0.15 \times 0.40) + (0.10 \times 0.50) + (-0.05 \times 0.10) ]

[ E(X) = 0.06 + 0.05 – 0.005 = 0.105 or 10.5% ]

This gives you an idea of what you can expect from this investment across various scenarios.

Applications in Financial Decision-Making

The expected value plays a crucial role in several aspects of financial decision-making.

Budgeting and Forecasting

EV provides a statistical basis for budgeting and forecasting by helping you anticipate average returns over time. This allows for more accurate financial planning and resource allocation.

Scenario Planning

In scenario planning, EV helps organizations anticipate and prepare for future changes by evaluating different possible outcomes based on their probabilities. This proactive approach enables better risk management and strategic planning.

Integration with Other Metrics

EV is often integrated with other financial metrics such as Net Present Value (NPV) to make comprehensive investment decisions. By considering both the expected returns and their present value, investors can make more informed choices about which investments are likely to yield the best results.

Additional Resources

For further reading or practical tools:

• Consider using scenario planning software like Monte Carlo simulations.

• Utilize financial templates available online that help calculate expected values.

• Refer to textbooks on financial analysis or online courses that delve deeper into the application of expected values in finance.

These resources will aid in applying expected values effectively in real-world scenarios.

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