ROI does not indicate how long an investment is held. However, ROI is most often stated as an annual or annualized rate of return, and it is most often stated for a calendar or fiscal year. In this article, "ROI" indicates an annual or annualized rate of return, unless otherwise noted.
ROI is used to compare returns on investments where the money gained or lost — or the money invested — are not easily compared using monetary values. For instance, a $1,000 investment that earns $50 in interest generates more cash than a $100 investment that earns $20 in interest, but the $100 investment earns a higher return on investment.
When considering a continuous process of gaining or losing money with a constant rate of return, the annual rate of return is any value greater than -100%; a positive percentage corresponds to exponential growth of the capital, a value between -100% and 0% exponential decay.
The rate of return can be calculated over a single period, or expressed as an average over multiple periods.
If the final investment value includes both capital gains and intermediate income such as dividends, then is the holding period return.
It is the reciprocal of the e-folding time.
The geometric average rate of return calculated over n years is also known as the annualized return.
The final value of an investment is twice the initial value when or . The final value falls to zero, i.e., the initial value can no longer be recovered) when or .
Arithmetic and logarithmic returns are not equal, but are approximately equal for small returns. The difference between them is large only when percent changes are high. For example, an arithmetic return of +50% is equivalent to a logarithmic return of 40.55%, while an arithmetic return of -50% is equivalent to a logarithmic return of -69.31%.
Logarithmic returns are often used by academics in their research. The main advantage is that the continuously compounded return is symmetric, while the arithmetic return is not: positive and negative percent arithmetic returns are not equal. This means that an investment of $100 that yields an arithmetic return of 50% followed by an arithmetic return of -50% will result in $75, while an investment of $100 that yields a logarithmic return of 50% followed by an logarithmic return of -50% it will remain $100.
A 10% gain followed by a 10% loss is a 1% dollar loss. The order in which the loss and gain occurs does not affect the result. A 50% gain and a 50% loss is a 25% loss. An 80% gain plus an 80% loss is a 64% loss. To recover from a 50% loss, a 100% gain is required. The mathematics of this are beyond the scope of this article, but since investment returns are often published as "average returns", it is important to note that average returns do not always translate into dollar returns.
|Year 1||Year 2||Year 3||Year 4|
|Rate of Return||5%||5%||5%||5%|
|Geometric Average at End of Year||5%||5%||5%||5%|
|Capital at End of Year||$105.00||$110.25||$115.76||$121.55|
|Year 1||Year 2||Year 3||Year 4|
|Rate of Return||50%||-20%||30%||-40%|
|Geometric Average at End of Year||50%||9.5%||16%||-1.6%|
|Capital at End of Year||$150.00||$120.00||$156.00||$93.60|
|Year 1||Year 2||Year 3||Year 4|
|Rate of Return||-95%||0%||0%||115%|
|Geometric Average at End of Year||-95%||-77.6%||-63.2%||-42.7%|
|Capital at End of Year||$5.00||$5.00||$5.00||$10.75|
An annual rate of return is the return on an investment over a one-year period, such as January 1 through December 31st, or June 3rd 2006 through June 2nd 2007. Each ROI in the cash flow example above is an annual rate of return.
An annualized rate of return is the return on an investment over a period other than one year (such as a month, or two years) multiplied or divided to give a comparable one-year return. For instance, a one-month ROI of 1% could be stated as an annualized rate of return of 12%. Or a two-year ROI of 10% could be stated as an annualized rate of return of 5%.
In the cash flow example below, the dollar returns for the four years add up to $265. The annualized rate of return for the four years is: $265 ÷ ($1,000 x 4 years) = 6.625%.
|Year 1||Year 2||Year 3||Year 4|
Except for rare periods of deflation where the opposite is true, a dollar in cash is worth less today than it was yesterday, and worth more today than it will be worth tomorrow. The main factors that are used by investors to determine the rate of return at which they are willing to invest money include:
The time value of money is reflected in the interest rates that banks offer for deposits, and also in the interest rates that banks charge for loans such as home mortgages. The “risk-free” rate is the rate on U.S. Treasury Bills, because this is the highest rate available without risking capital.
The rate of return which an investor expects from an investment is called the Discount Rate. Each investment has a different discount rate, based on the cash flow expected in future from the investment. The higher the risk, the higher the discount rate (rate of return) the investor will demand from the investment.
For example, if an investor put $1,000 in a 1-year Certificate of Deposit (CD) that paid an annual interest rate of 4%, compounded quarterly, the CD would earn 1% interest per quarter on the account balance. The account balance includes interest previously credited to the account.
|1st Quarter||2nd Quarter||3rd Quarter||4th Quarter|
|Capital at the beginning of the period||$1,000||$1,010||$1,020.10||$1,030.30|
|Dollar return for the period||$10||$10.10||$10.20||$10.30|
|Account Balance at end of the period||$1,010.00||$1,020.10||$1,030.30||$1,040.60|
The concept of 'income stream' may express this more clearly. At the beginning of the year, the investor took $1,000 out of his pocket (or checking account) to invest in a CD at the bank. The money was still his, but it was no longer available for buying groceries. The investment provided a cash flow of $10.00, $10.10, $10.20 and $10.30. At the end of the year, the investor got $1,040.60 back from the bank. $1,000 was return of capital.
Once interest is earned by an investor it becomes capital. Compound interest involves reinvestment of capital; the interest earned during each quarter is reinvested. At the end of the first quarter the investor had capital of $1,010.00, which then earned $10.10 during the second quarter. The extra dime was interest on his additional $10 investment. The Annual Percentage Yield or Future value for compound interest is higher than for simple interest because the interest is reinvested as capital and earns interest. The yield on the above investment was 4.06%.
Bank accounts offer contractually guaranteed returns, so investors cannot lose their capital. Investors/Depositors lend money to the bank, and the bank is obligated to give investors back their capital plus all earned interest. Since investors are not risking losing their capital on a bad investment, they earn a quite low rate of return. But their capital steadily increases.
Stock returns are usually calculated for holding periods such as a month, a quarter or a year.
|End of:||1st Quarter||2nd Quarter||3rd Quarter||4th Quarter|
|Total Shares Held||1.010204||1.020204||1.030204||1.040608|
Yield is the compound rate of return that includes the effect of reinvesting interest or dividends.
To the right is an example of a stock investment of one share purchased at the beginning of the year for $100.
To calculate the rate of return, the investor includes the reinvested dividends in the total investment. The investor received a total of $4.06 in dividends over the year, all of which were reinvested, so the investment amount increased by $4.06.
The disadvantage of this ROI calculation is that it does not take into account the fact that not all the money was invested during the entire year (the dividend reinvestments occurred throughout the year). The advantages are: (1) it uses the cost basis of the investment, (2) it clearly shows which gains are due to dividends and which gains/losses are due to capital gains/losses, and (3) the actual dollar return of $3.02 is compared to the actual dollar investment of $104.06.
For American income tax purposes, if the shares were sold at the end of the year, dividends would be $4.06, cost basis of the investment would be $104.06, sale price would be $103.02, and the capital loss would be $1.04.
Since all returns were reinvested, the ROI might also be calculated as a continuously compounded return or logarithmic return. The effective continuously compounded rate of return is the natural log of the final investment value divided by the initial investment value:
Total Return = ((Final Price x Last Reinvestment Factor) - Beginning Price) / Beginning Price
Average Annual Return (geometric) US mutual funds use SEC form N-1A to report the average annual compounded rates of return for 1-year, 5-year and 10-year periods as the "average annual total return" for each fund. The following formula is used:
P(1+T)n = ERV
P = a hypothetical initial payment of $1,000.
T = average annual total return.
n = number of years.
ERV = ending redeemable value of a hypothetical $1,000 payment made at the beginning of the 1-, 5-, or 10-year periods at the end of the 1-, 5-, or 10-year periods (or fractional portion).
|End of:||Year 1||Year 2||Year 3||Year 4||Year 5|
|Capital Gain Distribution||$2|
Using a Holding Period Return calculation, after 5 years, an investor who reinvested owned 1.26916 share valued at $101 per share ($128.19 in value). ($128.19-$100)/$100/5 = 5.638% yield. An investor who did not reinvest received a total of $27 in dividends and $1 in capital gain. ($27+$1)/$100/5 = 5.600% return.
Mutual funds include capital gains as well as dividends in their return calculations. Since the market price of a mutual fund share is based on net asset value, a capital gain distribution is offset by an equal decrease in mutual fund share value/price. From the shareholder's perspective, a capital gain distribution is not a net gain in assets, but it is a realized capital gain.
ROI is a measure of investment profitability, not a measure of investment size. While compound interest and dividend reinvestment can increase the size of the investment (thus potentially yielding a higher dollar return to the investor), Return on Investment is a percentage return based on capital invested.
In general, the higher the investment risk, the greater the potential investment return, and the greater the potential investment loss.