An expected return calculator takes a set of possible outcomes with the weight you assign to each and collapses them into one number: the probability-weighted mean. It is the oldest idea in decision theory applied to money, and it is useful precisely because it forces the assumptions into the open. You cannot compute an expected return without first writing down what you think could happen and how likely you think each outcome is.
Arb Digital publishes this in its free tools library alongside the annualized return calculator, which measures a return that has already happened, and the CAPM calculator, which derives a required return from a single risk factor instead of from your own scenarios. Where the expected value calculator handles the general probability case in any units, this page is built for returns specifically and adds the dispersion measures that make an expected return readable. Those are different jobs, and this one assumes percentages throughout.
What This Expected Return Calculator Does
It accepts up to five rows. Each row carries a weight and a return. The weights are normalised to their own total, so entering 3, 5 and 2 works exactly like entering 30, 50 and 20 — useful when you are sketching relative likelihoods rather than committing to a precise probability distribution.
The headline figure is the weighted mean of the returns. Beneath it the page reports the variance, which is the weighted mean of the squared deviations from that mean, and its square root, the standard deviation. It also reports the coefficient of variation — dispersion divided by expected return, a scale-free way of comparing spread between opportunities of different sizes — and the total weight sitting on outcomes below zero.
The same arithmetic serves two different readings. If the rows are mutually exclusive scenarios with probabilities, the output is a genuine statistical expectation and the variance is the correct variance of that distribution. If the rows are holdings with portfolio weights, the expected return is still exactly right, because expectation is linear and always passes through a weighted sum. The dispersion figures are not: they measure how far the individual assets' returns sit from the portfolio mean, which is a different quantity from portfolio risk.
How to Use It
- Decide first whether your rows are scenarios or holdings. Scenarios are mutually exclusive and one of them happens. Holdings all happen at once. The arithmetic is identical and the interpretation of the risk numbers is not.
- Write the scenarios before the probabilities. People who set probabilities first tend to anchor on round numbers and then invent outcomes to fit them. Describe the states of the world, then assign weight.
- Include a genuinely bad outcome. A five-row distribution whose worst case is a small loss is not a distribution, it is an optimism exercise. The most common failure in scenario analysis is a truncated left tail.
- Zero out rows you are not using. A weight of zero removes a row entirely from both the mean and the variance rather than dragging a spurious return into the calculation.
- Read the standard deviation next to the mean, not after it. An expected return of 7% with a standard deviation of 13 points and an expected return of 7% with a standard deviation of 2 points are entirely different propositions that this page deliberately shows side by side.
The Formula and How It Is Calculated
With weights pi normalised so they sum to one, and returns ri:
E(r) = Σ pi ri
Var(r) = Σ pi (ri − E(r))2, and σ = √Var(r)
The coefficient of variation is σ ÷ E(r), which is undefined when the expected return is zero and misleading when it is negative, so the page reports it only where it carries meaning.
Worked example, matching the values the page loads with. Five scenarios carry weights of 15%, 25%, 30%, 20% and 10% against returns of 28%, 14%, 7%, −3% and −18%. The weighted contributions are 4.20, 3.50, 2.10, −0.60 and −1.80, which sum to an expected return of 7.40%. The deviations from that mean are 20.6, 6.6, −0.4, −10.4 and −25.4 points. Squaring and weighting gives 63.654, 10.890, 0.048, 21.632 and 64.516, summing to a variance of 160.74 percentage points squared. The standard deviation is the square root of that, 12.68 percentage points. The coefficient of variation is 12.68 ÷ 7.40 = 1.71, and 30% of the total weight sits on outcomes below zero.
The Expected Value Is a Number Nobody Receives
This is the most important thing to understand about the headline figure, and it is routinely forgotten.
In the worked example the expected return is 7.40%. Look at the five scenarios: 28%, 14%, 7%, −3% and −18%. None of them is 7.40%. The mean of a distribution need not be a member of the distribution, and in scenario analysis it usually is not. What actually happens is one of the five listed outcomes, and an investor experiences that one, not the average of all of them.
This matters in three practical ways. First, a single decision is not a long-run average. Expected value arithmetic describes what happens over many independent repetitions; a one-off allocation gets one draw. Second, the distance between the mean and the worst case — here, 7.40% versus −18% — is the number that determines whether an outcome is survivable, and it never appears in the headline. Third, when losses are not recoverable, the arithmetic mean systematically overstates what a sequence of such bets delivers, because compounding is multiplicative: a 50% loss followed by a 50% gain leaves you down 25%, even though the arithmetic mean of those two returns is zero.
The general rule is that the geometric mean of a series of returns is always at or below the arithmetic mean, and the gap widens with volatility. The CAGR calculator computes the geometric version for a realised series, and comparing the two for the same data is the fastest way to see how much volatility costs a compounding portfolio.
Why Portfolio Weights Do Not Give You Portfolio Risk
Feed portfolio weights into this page and the expected return is exactly correct. Expectation is linear: the expected return of a portfolio is genuinely the weighted average of its holdings' expected returns, with no correction for correlation, no matter how the assets move together.
Variance is not linear, and this is where the arithmetic stops transferring. The variance of a two-asset portfolio is:
σ2p = w12σ12 + w22σ22 + 2w1w2ρσ1σ2
That final term is the whole of diversification. When the correlation is below one, portfolio volatility is lower than the weighted average of the individual volatilities, and at a correlation of −1 with the right weights it can fall to zero. Nothing in a list of weights and expected returns contains this information, which is why the dispersion figures on this page describe the spread of your inputs rather than the risk of a portfolio built from them.
The practical consequence is worth stating plainly: two portfolios with identical expected returns and identical holding-level volatilities can have very different real risk, decided entirely by correlations you have not entered. The correlation coefficient calculator measures the missing input from return series, and the US Securities and Exchange Commission's guide to asset allocation and diversification on Investor.gov explains why the relationship between holdings does more work than the holdings themselves.
Where the Probabilities Come From
The output of this page is only as good as the weights entered, and there is no objective source for them. That is not a flaw in the method; it is the method being honest about what it is.
Three approaches are common. Historical frequency uses how often something happened in the past as an estimate of how often it will happen — defensible for phenomena with stable distributions and a long record, and treacherous for anything structurally new. Aswath Damodaran's historical returns on stocks, bonds and bills at NYU Stern is a long-running public series for this purpose, though the page itself is careful about the limits of extrapolating it.
Subjective probability is a considered judgement expressed as a number. It is what most scenario analysis actually uses, and its main virtue is that it is checkable: writing 20% down commits you to something a vaguer statement does not.
Market-implied probability backs the distribution out of prices, most commonly from option prices across strikes. It has the advantage of aggregating many participants' views and the disadvantage of embedding a risk premium, so implied probabilities are not the same as real-world ones.
Whichever you use, the honest presentation is a range of expected returns from a range of weights, not a single figure. If moving one probability by five points changes the answer materially, the answer was never a point estimate.
Arb Digital builds acquisition forecasts with the same discipline — explicit scenarios, stated assumptions and a downside case that is allowed to be genuinely bad.
See Web Growth Services Talk to Arb DigitalCommon Mistakes to Avoid
- Probabilities that do not describe mutually exclusive states — if two scenarios can both happen, the weights are not probabilities and the variance figure means nothing.
- Truncating the downside — a worst case chosen for comfort rather than analysis moves the mean up and the standard deviation down at the same time, which flatters both headline numbers.
- Treating the mean as an outcome — nobody receives the expected return. One of the listed scenarios happens, and the gap between the mean and the worst case is what determines survivability.
- Reading dispersion from portfolio weights as portfolio risk — correlations decide portfolio variance and no list of weights contains them.
- Comparing coefficients of variation across a zero — the ratio is undefined at a zero expected return and changes sign below it, so it only compares opportunities that are all expected to be positive.
Related Free Tools From Arb Digital
Measure a return that has already happened with the annualized return calculator or the holding period return calculator, derive a required return from market risk with the CAPM calculator, and compare return against volatility with the Sharpe ratio calculator. For the underlying statistics, the standard deviation calculator and the variance calculator work on raw data sets rather than weighted scenarios, and the portfolio rebalancing calculator handles the weights themselves. Everything else sits in the free online tools hub.
Frequently Asked Questions
It is the weighted average of all the returns you consider possible, using the probability of each as its weight. Written formally it is the sum of each probability multiplied by its return. It describes the centre of a distribution you have specified, not a return anyone will actually receive in any single period.
No. The page normalises the weights to their own total, so entering three rows as 3, 5 and 2 gives the same result as 30, 50 and 20. That makes it practical for sketching relative likelihoods. If you intend them as true probabilities, they should still sum to 100 in your own working.
Yes for the expected return, which is exactly correct either way because expectation is linear. No for the risk figures. The standard deviation shown then measures how far individual holdings sit from the portfolio mean, which is not portfolio volatility. Real portfolio variance depends on correlations between holdings and needs a covariance matrix.
Because the mean of a distribution need not be a member of it. With scenario returns of 28, 14, 7, minus 3 and minus 18 per cent, an expected return of 7.4% is simply the balance point of those five outcomes. What happens in reality is one of the five, and the average is a summary of the set rather than a prediction of the draw.
It divides standard deviation by expected return to give a scale-free measure of dispersion per unit of expected return. That makes it useful for comparing opportunities of very different sizes. It becomes meaningless at a zero expected return and misleading below zero, so it only compares alternatives that are all expected to be positive.
The page makes no such claim and cannot. Expected return says nothing about the dispersion around it, the size of the worst case, whether losses are recoverable, or how the holding fits anything else you own. Two positions with the same expected return can be entirely different propositions, which is exactly why the dispersion figures are shown beside the mean.
There is no objective source. Common approaches are historical frequency, considered subjective judgement, and probabilities implied by option prices. Each has known weaknesses: history assumes the future resembles the past, judgement is subject to bias, and market-implied figures embed a risk premium. Running a range of weights is more informative than defending one set.
Because compounding is multiplicative rather than additive. A 50% loss followed by a 50% gain leaves a portfolio 25% down even though the arithmetic mean of those returns is zero. The geometric mean of a return series is therefore always at or below the arithmetic mean, and the gap grows as volatility rises.
This tool performs a probability-weighted arithmetic calculation on figures you supply. It is not investment advice, not a recommendation to buy, sell or hold any security, and its output is not a forecast of any actual return. Every probability and return on this page is your own estimate. Decisions about your money should involve a licensed financial adviser who is regulated to advise on them.
