# Annualized Return vs. Cumulative Return

### Cumulative return and annualized return are key tools in understanding investment performance. What are they and how does one calculate them?

Nov 3, 2015 at 7:04AM

1. What a cumulative return is and how to calculate it.

2. What the annualized return is, why it comes in handy, and how to calculate it.

What is a cumulative return, and how do you calculate it?
As the name suggests, the cumulative return indicates the aggregate effect of price change on the value of your investment. In effect, the cumulative return answers the question: What has this investment done for me?

To calculate a cumulative return, you need two pieces of data: the initial price, Pinitial, and the current price, Pcurrent (or the price at the end date of the period over which you wish to calculate the return).

The cumulative return is equal to your gain (or loss!) as a percentage of your original investment. Thus, the formula for cumulative return is:

Rc = ( Pcurrent – Pinitial ) / Pinitial

which can also be written:

Rc = ( Pcurrent / Pinitial )-1

First remark: Despite its name, the cumulative return doesn't always equate to an accumulation of wealth. A cumulative return can be negative: If you pay \$100 for a stock that's trading at \$50 a year later, your cumulative return is:

( \$50-\$100 ) / \$100 =-0.5 = (50%)

Second remark: You can calculate a cumulative return that's strictly due to price appreciation, or you can calculate a cumulative return that includes the effect of dividends. In the latter case, you use a dividend-adjusted price for your initial price.

Let's take a real-world example. What is the cumulative return on Microsoft's stock from the close of its first day of trading on March 13, 1986, through Sept. 30. 2015?

Looking the data up on Yahoo! Finance, you find:

• Closing price on 3/13/1986: \$28.00
• Closing price on 9/30/2015: \$44.26

Before we apply the formula for the cumulative return, we need to make one adjustment. The initial price, \$28.00, has not been adjusted for stock splits.

Since it has gone public, Microsoft has split its stock 2-for-1 seven times and 3-for-2 twice, such that one share bought at the IPO would leave you with ( 2 ^ 5 ) . ( 3/2 ) . ( 3/2 ) = 288 shares on Sept. 30, 2015 (excluding the effect of reinvesting the dividend).

Another way of putting it is that one share today is equivalent to 1/288th of a share when they started trading.

Our initial price is thus:

\$28.00 / 288 = \$0.09722 (after rounding to the fifth decimal)

We've now got our two prices; the cumulative return is:

( \$28.00 – \$0.09722 ) / \$0.09722 = 454.25 = 45,425%

Not a bad haul, but if we include dividends, which Microsoft began paying in February 2003, the return is even higher. The initial price, adjusted for splits and dividends, is \$0.06607 (this assumes that the cash dividend was reinvested in Microsoft shares).

The cumulative total return is then:

( \$44.26 – \$0.06607 ) / \$0.06607 = 668.90 = 66,890%

In mutual fund fact sheets and websites, the cumulative return can be quickly deduced from a graph that shows the growth of a hypothetical \$10,000 investment over time (usually starting at the fund's inception).

For example, the following graph was taken from a third-quarter 2015 portfolio manager commentary for the Thornburg Core Growth Fund:

Source: Thornburg Investment Management.

Because the starting value of \$10,000 is a multiple of 100, you can easily calculate the cumulative returns without the need for a calculator. Here:

Rc (A Shares without sales charge) = ( \$22,230 – \$10,000 ) / ( \$10,000 ) = \$12,230 / \$10,000 = 122.30%

Similarly,

Rc (A Shares with sales charge) = \$11,229 / \$10,000 = 112.29%

Rc (Russell 3000 Growth Index) = \$7,697 / \$10,000 = 76.97%

What is an annualized return, and why calculate it?
First, let's see how the need for an annualized return might arise.

We already calculated cumulative returns for Microsoft. Let's calculate the cumulative return from the first day of trading for another high-profile growth stock, Netflix. The company has never paid a dividend, so price return and total return are the same.

• Closing price on 5/23/2002: \$1.19643 (split-adjusted)
• Closing price on 9/30/2015: \$103.26

Cumulative return = ( \$103.26-\$1.19643 ) / \$1.19643 = 85.31 = 8,531%

Now, what if we want to try to compare the performance of Microsoft's stock to that of Netflix? Sure, Microsoft's cumulative return is a lot larger than Netflix's, but Microsoft had a 16-year head start. With the effect of compounding, that can make a huge difference.

This is where an annualized return can be helpful. In annualizing a return, you're answering the following question: What is the annual rate of return that would produce the same cumulative return if it's compounded over the same period? That annual rate of return is the annualized return.

Mathematically, if n is the number of years over which the cumulative return, Rc, was achieved and Ra is the annualized return, then:

( 1 + Ra ) ^ n = 1 + Rc

We can manipulate that equation to find Ra, which gives us:

Ra = ( (1 + Rc) ^ (1/n) ) – 1

If you've done a little statistics, you may recognize from this formula that the annualized return (Ra) is simply the geometric average of the cumulative return (Rn). A plain old arithmetic average won't do the trick, because it doesn't account for compounding.

Remark: You don't need the investment period to be a whole number of years to calculate the annualized return. The formula works just fine for periods that include a fractional part of a year. For example, for a 7 1/2-year period, you simply set n = 7.5 in the formula.

(Note that if the period is less than one year, it's good practice not to annualize a stock return (short-term debt securities are a different matter). If the period is short, with the effect of compounding, it can produce some very large (positive or negative) numbers that aren't meaningful.

Getting back to our example of Microsoft and Netflix: When we annualize their cumulative returns, we obtain the following results:

Microsoft

Netflix

March 13, 1986

May 23, 2002

End of measurement period

Sep. 30, 2015

Sep. 30, 2015

Measurement period length

29.55 years

13.36 years

Cumulative price return

45,425%

8,531%

Cumulative total return

66,890%

8,531%

Annualized price return

23.01%

39.61%

Annualized total return

24.63%

39.61%

Source: author's calculations, based on data from Microsoft, Netflix and Yahoo! Finance.

Expressing the cumulative rates of return in terms of annualized rates of return makes the performance comparison a bit more manageable, optically, but it isn't a panacea.

For example: Can we then conclude that, with an annualized return of 39.6% versus 24.6% for Microsoft, Netflix was much the superior investment?

No, we can't!

It's important to understand that it's not an apples-to-apples comparison: Netflix is at a much earlier stage of its growth path -- it won't be able to sustain a nearly 40% annualized rate of return for the next 16 years. If it did, Netflix would then be worth roughly \$9.8 trillion (assuming no change in share count) -- I think we can safely rule that out.

Conversely, Microsoft's annualized return over the first 13.36 years of its life as a public company -- the same period that Netflix has just reached -- was 58.77%. (It must be said that this was on July 20, 1999, the height of the technology bubble. Indeed, Microsoft's stock closed at essentially the same price on Oct. 5, 2015 – more than 16 years later.)

The \$15,978 Social Security bonus most retirees completely overlook
If you're like most Americans, you're a few years (or more) behind on your retirement savings. But a handful of little-known "Social Security secrets" could help ensure a boost in your retirement income. In fact, one MarketWatch reporter argues that if more Americans knew about this, the government would have to shell out an extra \$10 billion annually. For example: one easy, 17-minute trick could pay you as much as \$15,978 more... each year! Once you learn how to take advantage of all these loopholes, we think you could retire confidently with the peace of mind we're all after. Simply click here to discover how you can take advantage of these strategies.

Alex Dumortier, CFA has no position in any stocks mentioned. The Motley Fool owns shares of and recommends Netflix and Yahoo. The Motley Fool owns shares of Microsoft. Try any of our Foolish newsletter services free for 30 days. We Fools may not all hold the same opinions, but we all believe that considering a diverse range of insights makes us better investors. The Motley Fool has a disclosure policy.

# Money to your ears - A great FREE investing resource for you

### The best way to get your regular dose of market and money insights is our suite of free podcasts ... what we like to think of as “binge-worthy finance.”

Feb 1, 2016 at 5:03PM

Whether we're in the midst of earnings season or riding out the market's lulls, you want to know the best strategies for your money.

In short, you want a voice of reason you can count on.

A 2015 Business Insider article titled, "11 websites to bookmark if you want to get rich," rated The Motley Fool as the #1 place online to get smarter about investing.

And one of the easiest, most enjoyable, most valuable ways to get your regular dose of market and money insights is our suite of free podcasts ... what we like to think of as "binge-worthy finance."

Whether you make it part of your daily commute or you save up and listen to a handful of episodes for your 50-mile bike rides or long soaks in a bubble bath (or both!), the podcasts make sense of your money.

And unlike so many who want to make the subjects of personal finance and investing complicated and scary, our podcasts are clear, insightful, and (yes, it's true) fun.

## Our free suite of podcasts

Motley Fool Money features a team of our analysts discussing the week's top business and investing stories, interviews, and an inside look at the stocks on our radar. The show is also heard weekly on dozens of radio stations across the country.

The hosts of Motley Fool Answers challenge the conventional wisdom on life's biggest financial issues to reveal what you really need to know to make smart money moves.

David Gardner, co-founder of The Motley Fool, is among the most respected and trusted sources on investing. And he's the host of Rule Breaker Investing, in which he shares his insights into today's most innovative and disruptive companies ... and how to profit from them.

Market Foolery is our daily look at stocks in the news, as well as the top business and investing stories.

And Industry Focus offers a deeper dive into a specific industry and the stories making headlines. Healthcare, technology, energy, consumer goods, and other industries take turns in the spotlight.

They're all informative, entertaining, and eminently listenable. Rule Breaker Investing and Answers are timeless, so it's worth going back to and listening from the very start; the other three are focused more on today's events, so listen to the most recent first.