In this article, we’re going to learn how to calculate portfolio returns from scratch.
Portfolio returns can be calculated using this equation:
Where represents the weight or proportion invested in an asset , and reflects the return on an asset .
You could of course also estimate portfolio returns using expected return instead of return; in that case, you’d have the portfolio expected return which can be estimated as…
Pretty much identical to the one above, of course. The only difference is that one case (the first one) explores realised returns while the other looks at expected return.
If the equation above is freaking you out, or if you don’t quite understand why it works, and how it works, then keep reading.
Recap / Fundamentals: Individual Returns
First, a quick recap – recall that we calculate individual stock returns using this formula right here:
This is equivalent to writing it like this…
The denominator is still the older price (aka purchase price / beginning value), and the numerator is the newer price (aka selling price / ending value).
Now going forward, we’re going to assume that you’re quite happy with this formula and that you’re comfortable with calculating individual stock returns. If you’re not, please read this article first.
All right. So let’s go ahead now and learn how to calculate portfolio returns.
Portfolio Returns Walkthrough
Consider an example.
What is your profit and what is your return if you sold Facebook for $172.47, and if you sold Apple for $188.83 cents?
Hopefully, the first part of this question is already quite straightforward.
To calculate the profit, all we need to do is subtract the purchase price from the selling price for each stock individually. And then just add the two profits together.
So in this example, your total profit would be $21.30. How do we get that?
You take the selling price of Facebook, subtract the purchase price of Facebook as…
$172.47 – $160 = $12.47
Then do exactly the same thing for Apple as…
$188.83 – $180 = $8.83
Finally, add the two together to get your total profit…
$12.47 + $8.83 = $21.30
In this context, the profit is identical to because we’re essentially just comparing the with the of the . That is identical to how we measure .
To calculate the portfolio return, you’d need to start by calculating the individual returns.
So your return on Facebook would be 7.79%. And your return on Apple would be 4.91%.
How do we get that?
We literally apply the formula to calculate the return of an individual stock as…
In the case of Facebook, we have…
And for Apple, we have…
Calculating Investment Portfolio Return
Now that we have the individual returns, we can think about how we calculate the portfolio return.
Perhaps you’re thinking, that’s easy – just add the two individual returns together and you’ve got your portfolio return!
Calculating portfolio returns is not just a simple case of adding individual asset returns.
When you invest in two or more asset classes, your total return is a function of the:
- return that you earn from each individual asset, and
- amount you invested in each individual asset
In other words, portfolio returns depend on the return of each asset, and the proportion invested in each asset.
This combined dependence is sometimes referred to as , because the returns are weighted based on the total .
In the example that we’re looking at, you invested $160 in Facebook and $180 in Apple.
Your total investment was thus $340 (being $160 + $180).
Given that you haven’t invested all of your money in Facebook, naturally, you can’t expect to take all of the return.
And similarly, because you haven’t invested all of your money in Apple, we can’t say that the return from Apple is your total return from Apple.
This Article features a concept that is covered extensively in our course on Investment Analysis & Portfolio Management (with Excel®).
If you’re interested in learning how to quantify risk for stocks and portfolios, working with real world data, then you should definitely check out the course.
Thus, although Facebook’s return is 7.79%, you only invested about 47% of your money in Facebook, not a hundred percent. And therefore you essentially only earned 47% off the 7.79%.
Because recall, we invested $160 in Facebook and $340 in total.
$160 relative to the total investment of $340 is approximately equal to 47%
You’ve invested 47% of your money in Facebook, and thus you essentially only earn 47% of the 7.79% total return.
In other words, your share of return of Facebook is 3.67% (that being 47% 7.79% 3.67%).
Put differently, we multiply 47%, which is our investment in Facebook, by the individual return of Facebook (7.79%), to get our share of the return of Facebook, or the weighted return of Facebook.
Similarly, we’re interested in the share of return from Apple.
And in our case, that would equate to 2.6% because we invested $180 in Apple, $340 in total and $180 divided by $340 is approximately equal to 53%.
So we essentially only get 53% off the 4.91% return of Apple. And therefore, our share of return of Apple (or weighted return) is just 2.6% (that being 53% 4.91% 2.6%).
Visually, it looks like this…
Given these two individual share of returns, we can say that our portfolio return is 6.27%.
How do we get that?
We take the share of Facebook return and add that to the share of Apple’s return to get our total portfolio return.
This is equivalent to writing it as…
So that’s how we go about calculating portfolio returns.
Exploring the Portfolio Return Formula
Let’s now take a moment to make sure we really understand this formula.
The equation above essentially has four components, right?
We’ve got the…
- return on Facebook ()
- return on Apple ()
- proportion invested in Facebook (), and
- proportion invested in Apple (
Calculating Weights / Proportions
Notice that we’re defining the proportions invested as omega ().
This is the generally accepted notation for “weights”, or the proportions invested in a given security.
So , pronounced “omega Facebook” refers to the proportion invested in Facebook.
And refers to the proportion invested in Apple.
Thus, if we take the proportion invested in Facebook, or the “weight of Facebook”, and multiply that by the return on Facebook, then we get the share of Facebook return.
Similarly, if we take the weight of Apple and multiply it by the return on Apple, we get the share of Apple’s return.
If we then add these two, then you’ll end up with the return on the portfolio!
So we now have a formula for the return on our portfolio which we can write as this…
Importantly of course, this formula is specific to our portfolio. We’ll talk about a more generalised formula further down.
For now, let’s think about omega a little more.
Omega, remember, reflects the weight or proportion invested in each individual asset.
We can estimate the weight of Facebook as…
Here reflects the total dollar amount invested in Facebook, and reflects the total amount invested across all securities.
And naturally, if you want to calculate the weight of Apple, you would just take the dollar invested in Apple and divide it by the total dollars invested.
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Portfolio Return as a Percentage of Investment
Now, if you’ll recall, when we first looked at calculating individual stock returns, we said that return is simply the amount of money that you’ve made expressed in percentage terms.
So we can cross verify, or cross-check the results from this formula by simply expressing our profit figure as a percentage of the total investment!
Recall that we said the profit on our investment was $21.30.
If we express this $21.30, as a percentage of $340 (the total investment), then you’ll find that it’s equal to approximately 6.26%, which is pretty much the same as what we got when we calculated it using the formula above!
Now of course, there is a slight difference, but that’s only because we’ve rounded off some numbers.
We rounded off the weight for instance; it wasn’t strictly 47% or even 46.07%. It had a lot more decimals.
And so, because we rounded off some numbers, we’ve ended up with a slightly different result.
Does that make sense?
If any part of this example, doesn’t quite make sense, please re-read this article up until this point before reading any further.
Because it’s important that you understand the process behind calculating portfolio returns.
We’re going to assume that you’re more or less alright.
Generalised Portfolio Returns
Let’s go ahead now and think about a more generalised equation for the return of a portfolio.
Remember, up until now, we’ve only really considered the specific case of our investment in Facebook and Apple. But naturally, those aren’t the only 2 stocks / securities in the whole world!
If we think about generalising it, we could write it out like this…
Where the subscripts 1 and 2 reflect “asset 1” and “asset 2” respectively. “Asset 1” could be whatever stock / security you fancy, as could “asset 2”.
Alternatively, we could express it like this…
Where the subscripts and reflect some generic / random assets and .
Regardless of whether you express them as “assets 1 and 2” or “assets j and k”, fundamentally, calculating portfolio returns is about adding the share of returns of individual assets that make up the portfolio.
We multiply the weight of asset 1 by the return of asset 1 to get the share of return from asset 1.
And then you add that to the share of return from asset 2.
The case of Weights and 1
Now, crucially, a portfolio strictly comprises of all of your investments.
The collective weights across all your investments must, by definition, equate to one.
Because you invest a hundred percent in a portfolio, a portfolio comprises of a hundred percent of your individual assets.
The sum of your weights, by definition, must equate to 1 (where 1 is 100%).
We can thus say…
If we’ve got multiple assets, if you take the individual weights and add all of the weights together, you must end up with a value of one.
This fact, or this quality or characteristic of weights is something that’s crucially important for portfolio management.
We exploit this fact and use this fact to manage portfolios well; and indeed, to make our life easier!
To give you an example, given that the sum of weights must always equate to one, we can say that for a two asset portfolio…
The weight of “asset one”, plus the weight of “asset two” must be equal to 1.
And given this, we can say that the weights of “asset 1” must be equal to…
Why is this important?
Well, because if we know the amount invested in asset 1, then we can save ourselves some time because we don’t need to calculate the weight / proportion invested in asset 2.
Alternative Expressions for Portfolio Return
Now, if we think about calculating portfolio returns, there are a few ways in which we can do it.
Profit over Investment
We can say that the return on the portfolio is equal to the profit of the portfolio divided by the total amount invested.
Thus, for a 2 asset portfolio, we can say…
Where reflects the profit earned.
So this version of portfolio returns is essentially calculating the return on the portfolio using our original definition of returns. It’s the amount of money you’ve earned relative to the initial investment expressed in percentage terms.
Sum of Share of Individual Returns
Alternatively, of course we can say that the portfolio return is the sum of the share of individual returns as…
And of course, if you were to multiply the return by the investment, then you’ll end up with the profit. So these two formulas are of course equivalent.
Share of Individual Returns and Unified Weight
Lastly, given the fact that weights must always sum up to one, we can say the return on a 2 asset portfolio is…
Because, remember, the sum of the weights must equate to one!
Multi-asset general case
In the general case, we can say that the return on a portfolio is equal to this beauty right here…
So we’re saying it’s the sum of the individual returns on assets multiplied by the individual weights.
Put differently, it’s simply the sum of the individual shares of returns.
So we add all of the share of returns, starting from the first observation where , and going all the way up to and including the th observation.
You can think of as the total number of assets held in the portfolio.
If we now were to open up this equation, then we can say that the return of portfolio is…
Now, we fully appreciate that the last few parts of this article have been quite math heavy, with lots of equations.
But it is indeed important for you to understand the different ways of writing out the return on the portfolio equation.
Because it can actually be written in many forms.
And the reason it’s important to know the many forms is because sometimes you just won’t have one type of data / information, but you will have another type of data / information.
By knowing the different ways of writing it out the equation for portfolio return, you can work with different sorts of data / information to achieve the same objective.
That being said, this will all make a lot more sense when we look at it with an example. So let’s go ahead and do that now.
Portfolio Return Example
Consider, Wizard Inc., which maintains an investment portfolio comprising of £30,000 invested in a FTSE 100 index fund, £15,000 invested in Stark Inc. (“ST”), and £5,000 invested in IronHero Plc (“IH”).
This business or investment company essentially has three different types of investment.
We’re also told that the investments have returned the following over the past year…
The FTSE returned 12%; Stark Inc., returned 22.5%; and IronHero Plc has returned -11%.
Given this information, what is Wizard Inc.’s portfolio return?
Setting up to solve
We’re dealing with three assets, but to calculate the return, the portfolio we use exactly the same generalised formula…
Opening it up, in this case, since we have…
Substituting the subscripts with tickers / symbols for the 3 securities we have…
Plugging in the return figures into the equation, we have…
Calculating individual weights
Finally, we just need to figure out what the weights are. Recall that the weight in any given asset can be estimated as…
Thus, to get the weight of the FTSE investment, we’d have…
Given that Wizard Inc. invested £30,000 in the FTSE and £50,000 in total, we’d have…
Thus, is equal to 0.6, or 60%.
We can estimate and using exactly the same approach as above. Doing so would give us and
Solving the question
Plugging in these values into our portfolio return equation above gives us…
Solve for that, and you’ll find that the portfolio return is equal to 12.85%
Does that make sense?
If any part of this example is not clear, please read it again and make sure that you really do understand how to calculate portfolio returns.
If you found the math a little bit too much, or if you still get freaked out by equations, you should explore our course on Financial Math Primer for Absolute Beginners. It’s designed for you specifically, and will help you get over your fear of math.
Alright, in summary…
You learned that portfolio returns depend on individual asset returns, as well as the weights or proportions of money invested in assets individually.
You also learned that the sum of weights invested in any portfolio must always equate to 1 (which is 100%). And that’s because your portfolio forms a hundred percent of your funds invested.
Importantly, of course, you learned that the return on a portfolio is calculated using this formula right here…
Of course, that’s not the only way to express the return on a portfolio. You’ve also explored the alternative expressions, but for the most part, the equation above works just fine.
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