Expected Return of a Portfolio

In one sentence: The expected return of a portfolio is the average of its parts’ expected returns, where each part counts in proportion to how much of the money it holds.

$E(R_p) = \sum_{i=1}^{n} w_i \, E(R_i)$

Read it out loud: “E of R p equals the sum, for i from 1 to n, of w i times E of R i. R p is the return of the whole portfolio, and n is the number of assets. w i is the share of the money in asset number i, and E of R i is that asset's expected return. Adding up the weighted returns gives the expected return of the portfolio.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Sigma and the sum: The big Greek letter sigma, $\sum$ , means add up. The index under it, $i = 1$ , says where to start, and the letter on top, $n$ , says where to stop. So $\sum_{i=1}^{n}$ means add one term for every asset, from the first to the last. It is a short way of writing a long addition.
Weight: A weight is the share of the whole that one part holds. Shares are written as fractions between 0 and 1, so 0.5 means half. The weights in a portfolio add to 1, which is the same as 100%, because all of the money is somewhere in the mix.
Expected return: The expected return is the average return you forecast, written as a rate per period. A forecast of 8% a year is written 0.08 in the formula. It is an expectation and not a guarantee, so the real return may differ.
Percent and decimal: Percent means out of 100. To use a percent in a formula, divide by 100, so 8% becomes 0.08 and 5.6% becomes 0.056. To go back, multiply the decimal by 100, so 0.056 becomes 5.6%.
Subscripts: A subscript is a small letter or number placed beside a symbol. In $w_i$ the subscript $i$ says which asset is meant, so $w_2$ is the weight of the second asset, and $n$ is the total count of assets.

What each symbol means

Symbol	Meaning
$E(R_p)$	The expected return of the whole portfolio, a rate per period, such as per year. It is the number the formula gives. A value of 0.056 means 5.6% a year.
$n$	The number of assets in the portfolio, a count with no unit. A portfolio of three funds has $n = 3$ .
$i$	The counter for the sum. It names each asset in turn, starting at 1 and ending at $n$ . It has no unit.
$w_i$	The weight of asset $i$ : its share of the total money, as a fraction with no unit. A weight of 0.5 means half of the money. All the weights add to 1, which is 100% of the money.
$E(R_i)$	The expected return of asset $i$ on its own, a rate per period written as a decimal. For example, 0.08 means 8% a year.

The formula piece by piece

$w_i$: The weight of asset $i$ : its share of the money, as a fraction. It is the amount of the mix that this asset fills.
$E(R_i)$: The expected return of asset $i$ on its own, as a rate per period. It is what this asset is forecast to earn.
$w_i \, E(R_i)$: The weighted return of asset $i$ : its share multiplied by its expected return. It is the piece of the portfolio return that this asset supplies.
$\sum_{i=1}^{n} w_i \, E(R_i)$: The sum over all assets: start at $i = 1$ , add the weighted returns one after another, and stop at the last asset, $i = n$ .
$E(R_p) = \sum_{i=1}^{n} w_i \, E(R_i)$: The expected return of the portfolio equals the sum, over all assets, of each asset’s weight multiplied by its expected return.

What the formula is doing, and why

A portfolio is a mix of investments held together, such as a stock fund, a bond fund and some cash. Each part has its own expected return, and the whole mix has one expected return. What goes in: the share of money in each asset and each asset’s expected return. What comes out: one number for the whole mix. The weights are shares, so they are fractions that add to 1, which means all of the money is invested.

Why multiply by the weight? An asset that holds half the money contributes half of its return to the total. Multiplying an expected return by its weight turns it into the piece that asset really adds to the whole. For example, $0.5 \times 0.08 = 0.04$ is the part of the portfolio return that comes from the half held in an asset returning 8%.

Why add? The pieces are parts of one total, so adding them gives the total. The sum sign $\sum$ says to do this for every asset, from $i = 1$ up to $i = n$ . The result is a weighted average: each return counts in proportion to its share of the money.

Why no extra terms? The expected return of a mix is the mix of the expected returns. How the assets move together does not change this average. Correlation matters for risk, which is the subject of the portfolio variance page, not for the expected return.

An everyday picture

Think of a cup of fruit punch made from three juices. If half the cup is orange juice with a sweetness score of 8, a third is pineapple juice with a score of 4, and a sixth is lime with a score of 2, the sweetness of the whole cup is the score of each juice counted in proportion to how much of the cup it fills. A portfolio works the same way: each asset’s expected return counts in proportion to how much of the money it fills, and the total is the expected return of the mix.

Worked example: Expected return of a mix of three funds

A portfolio holds 1000 dollars split across three places: 500 dollars in a stock fund, 300 dollars in a bond fund and 200 dollars in a cash account. The expected returns, in this made-up example, are 8% a year for the stock fund, 4% a year for the bond fund and 2% a year for the cash account. We want the expected return of the whole portfolio, and the expected gain in dollars over one year.

Turn the dollar amounts into weights. A weight is a share of the total, so we divide each amount by the total of 1000 dollars. The stock fund gets 500 divided by 1000, which is $w_1 = 0.5$ . The bond fund gets 300 divided by 1000, which is $w_2 = 0.3$ . The cash account gets 200 divided by 1000, which is $w_3 = 0.2$ .
Check that the weights add to 1. Every dollar sits in one of the three places, so the shares must add up to a whole: $0.5 + 0.3 + 0.2 = 1$ . If they added to something else, we would have made a mistake in the split or in the total.
Write each expected return as a decimal. The formula uses decimals, so 8% becomes 0.08, 4% becomes 0.04 and 2% becomes 0.02. These are the three values of $E(R_i)$ that go into the sum.
Multiply each weight by its expected return. This gives the piece each part adds to the whole: $0.5 \times 0.08 = 0.04$ from the stock fund, $0.3 \times 0.04 = 0.012$ from the bond fund, and $0.2 \times 0.02 = 0.004$ from the cash account. Each piece is the share of the portfolio return that one part supplies.
Add the three pieces. The sum sign says to add the pieces for $i = 1$ , $i = 2$ and $i = 3$ : $E(R_p) = 0.04 + 0.012 + 0.004 = 0.056$ . So the expected return of the whole portfolio is 0.056, which is 5.6% a year.
Turn the rate into dollars. A rate is a share of the money put in, so we multiply the rate by the 1000 dollars invested: $1000 \times 0.056 = 56\ \text{dollars}$ . The expected gain over one year is 56 dollars, so the portfolio would be expected to grow to 1056 dollars.

Answer: The expected return of the portfolio is 5.6% a year, which means an expected gain of 56 dollars on 1000 dollars over one year. The stock fund supplies most of it, because it holds the most money and has the highest expected return. As with every expected return, the real result may be higher or lower.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Forgetting that the weights must add to 1. If the shares add to 90%, part of the money is missing from the calculation and the answer is too small.
Using dollar amounts as weights without dividing by the total. A weight is a share, so 500 dollars out of 1000 dollars becomes 0.5 before it is used.
Taking a simple average of the returns. The average of 8%, 4% and 2% is 4.67%, but the 50-30-20 mix gives 5.6%, because the weights must be used.
Thinking that diversification changes the expected return. Spreading money across assets lowers risk, but the expected return is still the weighted average of the parts.

Where you will meet it

Planning a mix of stocks, bonds and cash, to estimate what the whole mix might earn in a year.
Pension and endowment funds, which estimate the expected return of their total holdings from the expected return of each asset class.
Mean-variance portfolio optimization, the framework introduced by Harry Markowitz in 1952, which starts from each asset’s expected return and the weights.
Blending the rates on several loans or deposits, to find the average rate on one total pot of money.

Questions people ask

Do the weights have to be positive?

A simple portfolio usually uses positive weights. A negative weight means borrowing an asset or betting against it, and the formula still works with negative weights, as long as the weights add to 1.

Does this give the return I will actually get?

No. It gives the expected return, the average outcome in the forecast. The real return over one year can be higher or lower.

Why are the weights fractions and not percentages?

A fraction multiplies a return directly. If you use percentages, divide each by 100 first, or the answer comes out 100 times too large.

Portfolio Variance (Two Assets): Portfolio variance says how much a two-asset mix’s risk comes from each part and from how the parts move together, and it shows that mixing assets can lower risk.
Capital Asset Pricing Model (CAPM): CAPM says the return you should expect from an investment is a safe rate plus extra return for the market risk it carries, scaled by its beta.
Weighted Average Cost of Capital (WACC): WACC is the average yearly price a company pays for all the money it uses, mixing the owners’ price and the lenders’ price by how much of each it has.