Rate of return
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In finance, rate of return (ROR), also known as return on investment (ROI), rate of profit or sometimes just return, is the ratio of money gained or lost (realized or unrealized) on an investment relative to the amount of money invested. The amount of money gained or lost may be referred to as interest, profit/loss, gain/loss, or net income/loss. The money invested may be referred to as the asset, capital, principal, or the cost basis of the investment. ROI is usually expressed as a percentage rather than a fraction.
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[edit] Calculation
The initial value of an investment, V_{i}, does not always have a clearly defined monetary value, but for purposes of measuring ROI, the initial value must be clearly stated along with the rationale for this initial value. The final value of an investment, V_{f}, also does not always have a clearly defined monetary value, but for purposes of measuring ROI, the final value must be clearly stated along with the rationale for this final value.^{[citation needed]}
The rate of return can be calculated over a single period, or expressed as an average over multiple periods.
[edit] Singleperiod
[edit] Arithmetic return
The arithmetic return is:
r_{arith} is sometimes referred to as the yield. See also: effective interest rate, effective annual rate (EAR) or annual percentage yield (APY).
[edit] Logarithmic or continuously compounded return
The logarithmic return or continuously compounded return, also as force of interest, is defined as:
 .
It is the reciprocal of the efolding time.
[edit] Multiperiod average returns
[edit] Arithmetic average rate of return
The arithmetic average rate of return over n periods is defined as:
[edit] Geometric average rate of return
The geometric average rate of return, also known as the timeweighted rate of return, over n periods is defined as:
The geometric average rate of return calculated over n years is also known as the annualized return.
[edit] Internal rate of return
The internal rate of return, also known as the dollarweighted rate of return, is defined as the value(s) of that satisfies the following equation:
where:
 NPV = net present value of the investment
[edit] Comparisons between various rates of return
[edit] Arithmetic and logarithmic return
For both arithmetic returns and logarithmic returns, an investment is profitable when either or > 0, and unprofitable when either or < 0.
The value of an investment is doubled over a year if the annual ROR or . The value falls to zero 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.
Initial investment, V_{i}  $100  $100  $100  $100  $100 

Final investment, V_{f}  $0  $50  $100  $150  $200 
Profit/loss, V_{f} − V_{i}  $100  $50  $0  $50  $100 
Arithmetic return, r_{arith}  100%  50%  0%  50%  100% 
Logarithmic return, r_{log}  69.31%  0%  40.55%  69.31% 
[edit] Arithmetic average and geometric average rates of return
Both arithmetic and geometric average rates of returns are averages of periodic percentage returns. Neither will accurately translate to the actual dollar amounts gained or lost if percent gains are averaged with percent losses. ^{[1]} A 10% loss on a $100 investment is a $10 loss, and a 10% gain on a $100 investment is a $10 gain. When percentage returns on investments are calculated, they are calculated for a period of time – not based on original investment dollars, but based on the dollars in the investment at the beginning and end of the period. So if an investment of $100 loses 10% in the first period, the investment amount is then $90. If the investment then gains 10% in the next period, the investment amount is $99.
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 
Dollar Profit/(Loss)  $21.55  
Compound Yield  5.4% 
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 
Dollar Profit/(Loss)  ($6.40)  
Compound Yield  1.6% 
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 
Dollar Profit/(Loss)  ($89.25)  
Compound Yield  22.3% 
[edit] Annual returns and annualized returns
Care must be taken not to confuse annual and annualized returns. An annual rate of return is a singleperiod return, while an annualized rate of return is a multiperiod, geometric average return.
An annual rate of return is the return on an investment over a oneyear period, such as January 1 through December 31, or June 3 2006 through June 2 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 oneyear return. For instance, a onemonth ROI of 1% could be stated as an annualized rate of return of 12%. Or a twoyear ROI of 10% could be stated as an annualized rate of return of 5%. **For GIPS compliance: you do not annualize portfolios or composites for periods of less than one year. You start on the 13th month.
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%.
[edit] Uses
 ROI is a measure of cash^{[citation needed]} generated by an investment, or the cash lost due to the investment. It measures the cash flow or income stream from the investment to the investor. Cash flow to the investor can be in the form of profit, interest, dividends, or capital gain/loss. Capital gain/loss occurs when the market value or resale value of the investment increases or decreases. Cash flow here does not include the return of invested capital.
Year 1  Year 2  Year 3  Year 4  

Dollar Return  $100  $55  $60  $50 
ROI  10%  5.5%  6%  5% 
 ROI values typically used for personal financial decisions include Annual Rate of Return and Annualized Rate of Return. For nominal risk investments such as savings accounts or Certificates of Deposit, the personal investor considers the effects of reinvesting/compounding on increasing savings balances over time. For investments in which capital is at risk, such as stock shares, mutual fund shares and home purchases, the personal investor considers the effects of price volatility and capital gain/loss on returns.
 Profitability ratios typically used by financial analysts to compare a company’s profitability over time or compare profitability between companies include Gross Profit Margin, Operating Profit Margin, ROI ratio, Dividend yield, Net profit margin, Return on equity, and Return on assets.^{[2]}
 During capital budgeting, companies compare the rates of return of different projects to select which projects to pursue in order to generate maximum return or wealth for the company's stockholders. Companies do so by considering the average rate of return, payback period, net present value, profitability index, and internal rate of return for various projects. ^{[3]}
 A return may be adjusted for taxes to give the aftertax rate of return. This is done in geographical areas or historical times in which taxes consumed or consume a significant portion of profits or income. The aftertax rate of return is calculated by multiplying the rate of return by the tax rate, then subtracting that percentage from the rate of return.
 A return of 5% taxed at 15% gives an aftertax return of 4.25%

 0.05 x 0.15 = 0.0075
 0.05  0.0075 = 0.0425 = 4.25%
 A return of 10% taxed at 25% gives an aftertax return of 7.5%

 0.10 x 0.25 = 0.025
 0.10  0.025 = 0.075 = 7.5%
Investors usually seek a higher rate of return on taxable investment returns than on nontaxable investment returns.
 A return may be adjusted for inflation to better indicate its true value in purchasing power. Any investment with a nominal rate of return less than the annual inflation rate represents a loss of value, even though the nominal rate of return might well be greater than 0%. When ROI is adjusted for inflation, the resulting return is considered an increase or decrease in purchasing power. If an ROI value is adjusted for inflation, it is stated explicitly, such as “The return, adjusted for inflation, was 2%.”
 Many online poker tools include ROI in a player's tracked statistics, assisting users in evaluating an opponent's profitability.
[edit] Cash or potential cash returns
[edit] Time value of money
Investments generate cash flow to the investor to compensate the investor for the time value of money.
Except for rare periods of significant deflation where the opposite may be 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:
 estimates of future inflation rates
 estimates regarding the risk of the investment (e.g. how likely it is that investors will receive regular interest/dividend payments and the return of their full capital)
 whether or not the investors want the money available (“liquid”) for other uses.
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 “riskfree” 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.
[edit] Compounding or reinvesting
Compound interest or other reinvestment of cash returns (such as interest and dividends) does not affect the discount rate of an investment, but it does affect the Annual Percentage Yield, because compounding/reinvestment increases the capital invested.
For example, if an investor put $1,000 in a 1year 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 
Quarterly ROI  1%  1%  1%  1% 
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. Because investors are not risking losing their capital on a bad investment, they earn a quite low rate of return. But their capital steadily increases.
[edit] Returns when capital is at risk
[edit] Capital gains and losses
Many investments carry significant risk that the investor will lose some or all of the invested capital. For example, investments in company stock shares put capital at risk. The value of a stock share depends on what someone is willing to pay for it at a certain point in time. Unlike capital invested in a savings account, the capital value (price) of a stock share constantly changes. If the price is relatively stable, the stock is said to have “low volatility.” If the price often changes a great deal, the stock has “high volatility.” All stock shares have some volatility, and the change in price directly affects ROI for stock investments.
Stock returns are usually calculated for holding periods such as a month, a quarter or a year.
[edit] Reinvestment when capital is at risk: rate of return and yield
End of:  1st Quarter  2nd Quarter  3rd Quarter  4th Quarter 

Dividend  $1  $1.01  $1.02  $1.03 
Stock Price  $98  $101  $102  $99 
Shares Purchased  0.010204  0.01  0.01  0.010404 
Total Shares Held  1.010204  1.020204  1.030204  1.040608 
Investment Value  $99  $103.04  $105.08  $103.02 
Quarterly ROI  1%  4.08%  1.98%  1.96% 
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 $100.
 The quarterly dividend is reinvested at the quarterend stock price.
 The number of shares purchased each quarter = ($ Dividend)/($ Stock Price).
 The final investment value of $103.02 is a 3.02% Yield on the initial investment of $100. This is the compound yield, and this return can be considered to be the return on the investment of $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.
 Total Investment = Cost Basis = $100 + $4.06 = $104.06.
 Capital gain/loss = $103.02  $104.06 = $1.04 (a capital loss)
 ($4.06 dividends  $1.04 capital loss ) / $104.06 total investment = 2.9% ROI
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:
 V_{i} is the initial investment ($100)
 V_{f} is the final value ($103.02)
 .
[edit] Mutual fund returns
Mutual funds and exchangetraded funds (ETFs) hold portfolios of various companies' stock shares. When the companies pay a dividend, and when the fund trades shares, dividends and capital gains are distributed to the mutual fund shareholders. Mutual funds trade at the net asset value of the stock shares.
[edit] Total returns
Mutual funds report total returns based on reinvestment factors. Reinvestment factors are based on total distributions (dividends plus capital gains) during each period.
 .
 .
 .
 .
 .
Total Return = ((Final Price x Last Reinvestment Factor)  Beginning Price) / Beginning Price
[edit] Average annual return (geometric)
Average Annual Return (geometric) US mutual funds use SEC form N1A to report the average annual compounded rates of return for 1year, 5year and 10year periods as the "average annual total return" for each fund. The following formula is used:^{[4]}
P(1+T)n = ERV
Where:
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 10year periods at the end of the 1, 5, or 10year periods (or fractional portion).
=
[edit] Example
End of:  Year 1  Year 2  Year 3  Year 4  Year 5 

Dividend  $5  $5  $5  $5  $5 
Capital Gain Distribution  $2  
Total Distribution  $5  $5  $7  $5  $5 
Share Price  $98  $101  $102  $99  $101 
Shares Purchased  0.05102  0.04950  0.06863  0.05051  0.04950 
Shares Owned  1.05102  1.10053  1.16915  1.21966  1.26916 
Reinvestment Factor  1.05102  1.05203  1.07220  1.05415  1.05219 
 Total Return = (($101 x 1.05219)  $100) / $100 = 6.27% (net of expenses)
 Average Annual Return (geometric) = (((28.19)/100)+1) ^ (1/5)) – 1) x 100 = 5.09%
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.
[edit] Summary: overall rate of return
Rate of Return and Return on Investment indicate cash flow from an investment to the investor over a specified period of time, usually a year.
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.
[edit] See also
 Average for a discussion of annualization of returns.
 Compound interest
 Capital budgeting
 Compound annual growth rate
 Dollar cost averaging
 Expected return
 Internal rate of return
 Net Present Value
 Rate of profit
 Return on assets
 Return on capital
 Return of capital
 Value investing
[edit] References
 ^ Damato,Karen. Doing the Math: Tech Investors' Road to Recovery is Long. Wall Street Journal, pp.C1C19, May 18, 2001
 ^ A. A. Groppelli and Ehsan Nikbakht (2000). Barron's Finance, 4th Edition. New York. pp. 442–456. ISBN 0764112759.
 ^ Barron's Finance. pp. 151–163.
 ^ SEC (1998). [http://www.sec.gov/rules/final/337512f.htm#E12E2 "Final Rule: Registration Form Used by OpenEnd Management Investment Companies: Sample Form and instructions"]. http://www.sec.gov/rules/final/337512f.htm#E12E2.
[edit] Further reading
 A. A. Groppelli and Ehsan Nikbakht. Barron’s Finance, 4th Edition. New York: Barron’s Educational Series, Inc., 2000. ISBN 0764112759
 Zvi Bodie, Alex Kane and Alan J. Marcus. Essentials of Investments, 5th Edition. New York: McGrawHill/Irwin, 2004. ISBN 0072510773
 Richard A. Brealey, Stewart C. Myers and Franklin Allen. Principals of Corporate Finance, 8th Edition. McGrawHill/Irwin, 2006
 Walter B. Meigs and Robert F. Meigs. Financial Accounting, 4th Edition. New York: McGrawHill Book Company, 1970. ISBN 007041534X
 Bruce J. Feibel. Investment Performance Measurement. New York: Wiley, 2003. ISBN 0471268496
[edit] External links
