In this article, we'll go through:
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 = ( P_{current} – P_{initial} ) / P_{initial}
which can also be written:
Rc = ( P_{current} / P_{initial })-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:
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, R_{c}, was achieved and R_{a} is the annualized return, then:
( 1 + R_{a} ) ^ n = 1 + R_{c}
We can manipulate that equation to find Ra, which gives us:
R_{a} = ( (1 + R_{c}) ^ (1/n) ) – 1
If you've done a little statistics, you may recognize from this formula that the annualized return (R_{a}) is simply the geometric average of the cumulative return (R_{n}). 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 | |
---|---|---|
First day of trading |
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% |
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.)
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