acalculator

What is the expected return?

Pick how you want to estimate it. List the possible outcomes with their chances, use the capital asset pricing model (CAPM) with a beta, or enter the holdings of a portfolio. The calculator gives the expected return, with the spread of the outcomes or the risk premium.

Your numbers

Method
Scenarios
Row 1
Row 2
Row 3
The probabilities must add up to 100%.
Expected return
9.5%

The expected return is 9.5%.

Standard deviation of return
13.87%

Expected return: 9.5%. The expected return is 9.5%.

The results are estimates for information only. They are not financial, tax, or legal advice. Check the numbers with your lender or a qualified professional before you decide. Terms of use

How to calculate

Computes the expected return of an investment from possible returns and their probabilities, from CAPM (risk-free rate, beta and market return), or for a portfolio from each holding’s amount and expected return.

Example with the default inputs (Method Scenarios, Scenarios [Probability 30%, Return 25%; Probability 50%, Return 10%; Probability 20%, Return −15%]): The expected return is 9.5%.

Method: Scenarios: E(R) = Σ pᵢ × Rᵢ, σ = √(Σ pᵢ × (Rᵢ − E(R))²); CAPM: E(R) = Rf + β × (Rm − Rf); portfolio: E(R) = Σ amountᵢ × E(Rᵢ) ÷ Σ amountᵢ.

  • Scenario probabilities add up to 100% and each return is for the same period.
  • CAPM uses the rates you type for the same period, usually a year; it gives the return investors require for the investment’s market risk.
  • Arithmetic is exact on the typed decimals; only the standard deviation uses a floating-point square root.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. Method Scenarios, Scenarios 30 25; 50 10; 20 -15 gives Expected return 9.5%, Standard deviation of return 13.865425%.Source: OpenStax, Introductory Statistics 2e, 4.2 Mean or Expected Value and Standard Deviation (μ = Σ x × P(x); σ = √(Σ (x − μ)² × P(x))). https://openstax.org/books/introductory-statistics-2e/pages/4-2-mean-or-expected-value-and-standard-deviation
  2. Method CAPM, Risk-free rate 3.36%, Beta (β) 1.39, Expected market return 11.64% gives Expected return 14.8692%, Market risk premium 8.28%, Risk premium of the investment 11.5092%.Source: OpenStax, Principles of Finance, 15.3 The Capital Asset Pricing Model (CAPM) (Re = Rf + β × (Rm − Rf); Delta Air Lines: 3.36% + 1.39 × (11.64% − 3.36%) = 14.87%). https://openstax.org/books/principles-finance/pages/15-3-the-capital-asset-pricing-model-capm: 3.36% + 1.39 × 8.28% = 14.87%
  3. Method Portfolio, Holdings 25 3.3; 25 0.21; 25 3.8; 25 3.47 gives Expected return 2.695%, Total invested $100.00.Source: OpenStax, Principles of Finance, 15.5 Using Excel to Make Investment Decisions (portfolio return = the weighted sum of the holdings’ returns, 0.25 × each of four stocks). https://openstax.org/books/principles-finance/pages/15-5-using-excel-to-make-investment-decisions: (3.30% + 0.21% + 3.80% + 3.47%) ÷ 4 = 2.695% (OpenStax writes 0.0269)
  4. Method Portfolio, Holdings 60000 8; 30000 4; 10000 2 gives Expected return 6.2%, Total invested $100,000.00.Source: OpenStax, Principles of Finance, 15.5 Using Excel to Make Investment Decisions (portfolio return = the weighted sum of the holdings’ returns, 0.25 × each of four stocks). https://openstax.org/books/principles-finance/pages/15-5-using-excel-to-make-investment-decisions

How it works

Scenarios. For outcomes i = 1 to n with probabilities pᵢ (percents that add up to 100) and returns Rᵢ:

  • Expected return E(R) = Σ pᵢ × Rᵢ ÷ 100
  • Standard deviation σ = √(Σ pᵢ × (Rᵢ − E(R))² ÷ 100)

CAPM. With the risk-free rate Rf, the beta β and the expected market return Rm:

  • Market risk premium = Rm − Rf
  • Risk premium of the investment = β × (Rm − Rf)
  • Expected return E(R) = Rf + β × (Rm − Rf)

Portfolio. For holdings with amounts Aᵢ and expected returns E(Rᵢ):

  • Expected return E(R) = Σ Aᵢ × E(Rᵢ) ÷ Σ Aᵢ, the average weighted by each holding's share of the total

Rules:

  • Scenarios: 1 to 20 rows; each probability is from 0% to 100% and they must add up to exactly 100%, or there is no answer. Each return is from −100% to 10,000%.
  • CAPM: the risk-free rate and market return are from −100% to 1,000%, and beta from −100 to 100.
  • Portfolio: 1 to 50 rows; amounts are from $0 to $1 quadrillion and must not all be $0. Each expected return is from −100% to 10,000%.
  • The arithmetic is exact on the decimals you type; only the standard deviation uses a floating-point square root. That root is of the exact variance, never of one first rounded to a float, so a variance that is the square of a fraction has an exact root. Percents show to 2 decimals and money to the cent, with halves rounded up (away from 0).
  • A result that is not 0 but too close to 0 for a 64-bit floating-point number (about 2.5 × 10⁻³²⁴ or less) gives no answer and a message, in every method and every output, so no output is ever a silent 0.

Assumptions

  • All returns and rates are for the same period, usually a year.
  • The scenario standard deviation weights each outcome by its probability. It is not the sample standard deviation of past returns.

Worked examples by hand

Three scenarios. A 30% chance of +25%, 50% of +10% and 20% of −15%:

E(R) = 0.3 × 25 + 0.5 × 10 + 0.2 × (−15) = 7.5 + 5 − 3 = 9.50%

σ² = 0.3 × (25 − 9.5)² + 0.5 × (10 − 9.5)² + 0.2 × (−15 − 9.5)² = 72.075 + 0.125 + 120.05 = 192.25, so σ = √192.25 = 13.87%.

CAPM for Delta Air Lines (OpenStax). Rf = 3.36%, β = 1.39, Rm = 11.64%. Market risk premium = 11.64 − 3.36 = 8.28%. Risk premium = 1.39 × 8.28 = 11.51%. E(R) = 3.36 + 11.5092 = 14.87%.

OpenStax's four-stock portfolio. Equal amounts in stocks with average monthly returns of 3.30%, 0.21%, 3.80% and 3.47%: (3.30 + 0.21 + 3.80 + 3.47) ÷ 4 = 2.695% a month, shown as 2.70% (OpenStax writes 0.0269).

A simple portfolio. $60,000 at 8%, $30,000 at 4% and $10,000 at 2%: (480,000 + 120,000 + 20,000) ÷ 100,000 = 6.20%.

Other questions people ask

What is expected return?

The expected return is the average return you would get from an investment if you could make it many times over: each possible return weighted by how likely it is. It is a planning number, not a promise; any one year can land far from it.

How do I calculate expected return with probabilities?

Multiply each possible return by its probability and add the results. With a 30% chance of +25%, a 50% chance of +10% and a 20% chance of −15%: 0.3 × 25% + 0.5 × 10% + 0.2 × (−15%) = 9.5%. The probabilities must add up to 100%.

How do I calculate expected return with CAPM?

Add the risk-free rate to beta times the market risk premium (market return minus risk-free rate). With a 3.36% risk-free rate, a 1.39 beta and an 11.64% market return: 3.36% + 1.39 × 8.28% = 14.87%.

How do I calculate the expected return of a portfolio?

Weight each holding’s expected return by its share of the money. $60,000 at 8%, $30,000 at 4% and $10,000 at 2% give (60,000 × 8% + 30,000 × 4% + 10,000 × 2%) ÷ 100,000 = 6.2%.

What does the standard deviation tell me?

How far the possible returns spread around the expected return. Two investments with the same 9.5% expected return can differ a lot in risk: the one with the larger standard deviation has outcomes further from 9.5%, both up and down.

Which method should I use?

Use scenarios when you can list the outcomes and their chances. Use CAPM for the return investors require for an investment’s market risk, such as a company’s cost of equity. Use the portfolio method when you already have an expected return for each holding.