Rate of return

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 (whether 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.

Contents

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 expected value must be clearly stated along with the rationale for this initial value. Similarly, 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.

The rate of return can be calculated over a single period, or expressed as an average over multiple periods of time.

Single-period

Arithmetic return

The arithmetic return is:

r_{arith}=\frac{V_f - V_i}{V_i}

r_{arith} is sometimes referred to as the yield. See also: effective interest rate, effective annual rate (EAR) or annual percentage yield (APY).

Logarithmic or continuously compounded return

The logarithmic return or continuously compounded return, also known as force of interest, is defined as:

r_{log} = \ln\left(\frac{V_f}{V_i}\right)

or R = P*e^{-rt} where: R = Return P = Principal amount r = rate t = time period It is the reciprocal of the e-folding time.

Multiperiod average returns

Arithmetic average rate of return

The arithmetic average rate of return over n periods is defined as:

\bar{r}_{arithmetic} = \frac{1}{n}\sum_{i=1}^n r_{arith,i} = \frac{1}{n} (r_{arith,1}%2B\cdots%2Br_{arith,n})

Geometric average rate of return

Main article: True Time-Weighted Rate of Return

The geometric average rate of return, also known as the True Time-Weighted Rate of Return, over n periods is defined as:

\bar{r}_{geometric} = \left({\prod_{i=1}^n (1%2Br_{arith,i})}\right)^{1/n}-1

The geometric average rate of return calculated over n years is also known as the annualized return.

Time-weighted rates of return (TWRR) are important because they eliminate the impact of cash flows. This is helpful when assessing the job that a money manager did for his/her clients, where typically the clients control these cash flows.[1]

Internal rate of return

Main article: Internal rate of return

The internal rate of return (IRR), also known as the dollar-weighted rate of return or the money-weighted rate of return (MWRR), is defined as the value(s) of \bar{r} that satisfies the following equation:

\mbox{NPV} = \sum_{t=0}^{n} \frac{C_t}{(1%2B\bar{r})^{t}} =0

where:

When the cost of capital r is smaller than the IRR rate \bar{r}, the investment is profitable, i.e., NPV > 0. Otherwise, the investment is not profitable.

MWRR are helpful in that they take cash flows into consideration. This is especially helpful when evaluating cases where the money manager controls cash flows (for private equity investments, for example, as well as sub-portfolio rates of return) as well as to provide the investor with their return. Contrast with TWRR.

Comparisons between various rates of return

Arithmetic and logarithmic return

The value of an investment is doubled over a year if the annual ROR r_{arith} = +100%, that is, if r_{log} = ln($200 / $100) = ln(2) = 69.3%. The value falls to zero when r_{arith} = -100%, that is, if r_{log} = -∞.

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 a logarithmic return of -50% it will remain $100.

Comparison of arithmetic and logarithmic returns for initial investment of $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%

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.[2] 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% 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.

Example #1 Level Rates of Return
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) $5.00 $10.25 $15.76 $21.55
Compound Yield 5% 5.4%
Example #2 Volatile Rates of Return, including losses
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%
Example #3 Highly Volatile Rates of Return, including losses
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%

Annual returns and annualized returns

Care must be taken not to confuse annual and annualized returns. An annual rate of return is a single-period return, while an annualized rate of return is a multi-period, arithmetic average return.

An annual rate of return is the return on an investment over a one-year 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 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%.

Uses

Cash Flow Example on $1,000 Investment
Year 1 Year 2 Year 3 Year 4
Dollar Return $100 $55 $60 $50
ROI 10% 5.5% 6% 5%
0.05 x 0.15 = 0.0075
0.05 - 0.0075 = 0.0425 = 4.25%
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 non-taxable investment returns.

Cash or potential cash returns

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:

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.

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 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.

Compound Interest Example
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.

Returns when capital is at risk

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.

Reinvestment when capital is at risk: rate of return and yield

Example: Stock with low volatility and a regular quarterly dividend, reinvested
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.

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 U.S. 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:

\text{ROI}_\text{Log} = \ln\left(\frac{V_f}{V_i}\right) = \ln\left(\frac{103.02}{100}\right) = 2.98%.

Mutual fund and investment company returns

Mutual funds, exchange-traded funds (ETFs), and other equitized investments (such as unit investment trusts or UITs, insurance separate accounts and related variable products such as variable universal life insurance policies and variable annuity contracts, and bank-sponsored commingled funds, collective benefit funds or common trust funds) are essentially portfolios of various investment securities such as stocks, bonds and money market instruments which are equitized by selling shares or units to investors. Investors and other parties are interested to know how the investment has performed over various periods of time.

Performance is usually quantified by a fund's total return. In the 1990s, many different fund companies were advertising various total returns—some cumulative, some averaged, some with or without deduction of sales loads or commissions, etc. To level the playing field and help investors compare performance returns of one fund to another, the U.S. Securities and Exchange Commission (SEC) began requiring funds to compute and report total returns based upon a standardized formula—so called "SEC Standardized total return" which is the average annual total return assuming reinvestment of dividends and distributions and deduction of sales loads or charges. Funds may compute and advertise returns on other bases (so-called "non-standardized" returns), so long as they also publish no less prominently the "standardized" return data.

Subsequent to this, apparently investors who'd sold their fund shares after a large increase in the share price in the late 1990s and early 2000s were ignorant of how significant the impact of income/capital gain taxes was on their fund "gross" returns. That is, they had little idea how significant the difference could be between "gross" returns (returns before federal taxes) and "net" returns (after-tax returns). In reaction to this apparent investor ignorance, and perhaps for other reasons, the SEC made further rule-making to require mutual funds to publish in their annual prospectus, among other things, total returns before and after the impact of U.S federal individual income taxes. And further, the after-tax returns would include 1) returns on a hypothetical taxable account after deducting taxes on dividends and capital gain distributions received during the illustrated periods and 2) the impacts of the items in #1) as well as assuming the entire investment shares were sold at the end of the period (realizing capital gain/loss on liquidation of the shares). These after-tax returns would apply of course only to taxable accounts and not to tax-deferred or retirement accounts such as IRAs.

Lastly, in more recent years, "personalized" investment returns have been demanded by investors. In other words, investors are saying more or less the fund returns may not be what their actual account returns are based upon the actual investment account transaction history. This is because investments may have been made on various dates and additional purchases and withdrawals may have occurred which vary in amount and date and thus are unique to the particular account. More and more fund and brokerage firms have begun providing personalized account returns on investor's account statements in response to this need.

With that out of the way, here's how basic earnings and gains/losses work on a mutual fund. The fund records income for dividends and interest earned which typically increases the value of the mutual fund shares, while expenses set aside have an offsetting impact to share value. When the fund's investments increase in market value, so too does the value of the fund shares (or units) owned by the investors. When investments increase (decrease) in market value, so too the fund shares value increases (or decreases). When the fund sells investments at a profit, it turns or reclassifies that paper profit or unrealized gain into an actual or realized gain. The sale has no effect on the value of fund shares but it has reclassified a component of its value from one bucket to another on the fund books—which will have future impact to investors. At least annually, a fund usually pays dividends from its net income (income less expenses) and net capital gains realized out to shareholders as an IRS requirement. This way, the fund pays no taxes but rather all the investors in taxable accounts do. Mutual fund share prices are typically valued each day the stock or bond markets are open and typically the value of a share is the net asset value of the fund shares investors own.

Total returns

This section addresses only total returns without the impact of U.S. federal individual income and capital gains taxes.

Mutual funds report total returns assuming reinvestment of dividend and capital gain distributions. That is, the dollar amounts distributed are used to purchase additional shares of the funds as of the reinvestment/ex-dividend date. Reinvestment rates or factors are based on total distributions (dividends plus capital gains) during each period.

Average annual total return (geometric)

US mutual funds are to compute average annual total return as prescribed by the U.S. Securities and Exchange Commission (SEC) in instructions to form N-1A (the fund prospectus) as the average annual compounded rates of return for 1-year, 5-year and 10-year periods (or inception of the fund if shorter) as the "average annual total return" for each fund. The following formula is used:[5]

\mathrm {P \left( 1 %2B T \right) ^ 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 10-year periods at the end of the 1-, 5-, or 10-year periods (or fractional portion).

Solving for T gives

\mathrm {T = \left( \frac {ERV} {P} \right) ^ {1 / n} - 1}

Example

Example: Balanced mutual fund during boom times with regular annual dividends, reinvested at time of distribution, initial investment $1,000 at end of year 0, share price $14.21
Year 1 Year 2 Year 3 Year 4 Year 5
Dividend per share $0.26 $0.29 $0.30 $0.50 $0.53
Capital gain distribution per share $0.06 $0.39 $0.47 $1.86 $1.12
Total Distribution Per Share $0.32 $0.68 $0.77 $2.36 $1.65
Share Price At End Of Year $17.50 $19.49 $20.06 $20.62 $19.90
Reinvestment factor 1.01829 1.03553 1.03975 1.11900 1.09278
Shares owned before distribution 70.373 71.676 74.125 76.859 84.752
Total distribution $22.52 $48.73 $57.10 $181.73 $141.60
Share price at distribution $17.28 $19.90 $20.88 $22.98 $21.31
Shares purchased 1.303 2.449 2.734 7.893 6.562
Shares owned after distribution 71.676 74.125 76.859 84.752 91.314

Using a Holding Period Return calculation, after five years, an investor who reinvested owned 91.314 shares valued at $19.90 per share. ((($19.90 × 91.314) / $1,000) - 1) / 5 = 16.34% return. An investor who did not reinvest received total cash payments of $5.78 per share. ((($19.90 + $5.78) / $14.21) - 1) / 5 = 16.14% 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.

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.

See also

References

  1. ^ Strong, Robert (2009). Portfolio construction, management, and protection. Mason, Ohio: South-Western Cengage Learning. p. 527. ISBN 0324665105. 
  2. ^ Damato, Karen. Doing the Math: Tech Investors' Road to Recovery is Long. Wall Street Journal, pp.C1-C19, May 18, 2001
  3. ^ A. A. Groppelli and Ehsan Nikbakht (2000). Barron's Finance, 4th Edition. New York. pp. 442–456. ISBN 0-7641-1275-9. 
  4. ^ Barron's Finance. pp. 151–163. 
  5. ^ U.S. Securities and Exchange Commission (1998). "Final Rule: Registration Form Used by Open-End Management Investment Companies: Sample Form and instructions". http://www.sec.gov/rules/final/33-7512f.htm#E12E2. 

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