Rate of return

In finance, return is a profit on an investment.[1] It comprises any change in value, and interest or dividends or other such cash flows which the investor receives from the investment.

Ambiguously, return is also used to refer to a profit on an investment, expressed as a proportion of the amount invested. This is also called the holding period return.

A loss instead of a profit is described as a negative return.

Rate of return is a profit on an investment over a period of time, expressed as a proportion of the original investment.[2] The time period is typically a year, in which case the rate of return is referred to as annual return.

Return, in the second sense, and rate of return, are commonly presented as a percentage.

ROI is an abbreviation of return on investment, i.e. return per dollar invested. It is a measure of investment performance, as opposed to size (c.f. return on equity, return on assets, return on capital employed).

Calculation

The return, or rate of return, can be calculated over a single period, or where there is more than one time period, the return and rate of return over the overall period can be calculated, based upon the return within each sub-period.

Single-period

Return

The return over a single period is:

r=\frac{V_f - V_i}{V_i}

where:

V_f = final value, including dividends and interest
V_i = initial value

For example, if you hold 100 shares, with a starting price of 10, then the starting value is 100 x 10 = 1,000. If you then collect 0.50 per share in cash dividends, and the ending share price is 9.80, then at the end you have 100 x 0.50 = 50 in cash, plus 100 x 9.80 = 980 in shares, totalling a final value of 1,030. The change in value is 1,030 - 1,000 = 30, so the return is 30 / 1,000 = 3%.

Annualisation

Without any reinvestment, a return R over a period of length t is equivalent to a rate of return:

\frac {R}{t}

For example, 20,000 USD returned on an initial investment of 100,000 USD is a return of 20%. If the 20,000 USD is paid in 5 annual installments of 4,000 USD per year, with no reinvestment, the rate of return is 4,000 / 100,000 = 20% / 5 = 4% per year.

Assuming returns are reinvested however, due to the effect of compounding, the relationship between a rate of return r, and a return R over a period of length t is:

1 + R = (1 + r)^t

which can be used to convert the return R to a rate of return r:

r = (1 + R)^{1/t} - 1

For example, a 33.1% return over 3 months is equivalent to a rate of:

1.331^{1/3} - 1 = 10\%

per month with reinvestment.

Annualisation is the process of converting a return R to an annual rate of return r, where the length of the period t is measured in years and the rate of return r is per year.

Logarithmic or continuously compounded return

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

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

and the logarithmic rate of return is:

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

or equivalently it is the solution r to the equation:

V_f = V_i e^{rt}

where:

r = logarithmic rate of return
t = length of time period

For example, if a stock is priced at 3.570 USD per share at the close on one day, and at 3.575 USD per share at the close the next day, then the logarithmic return is: ln(3.575/3.570) = 0.0014, or 0.14%.

Annualisation of logarithmic return

Under an assumption of reinvestment, the relationship between a logarithmic return R and a logarithmic rate of return r over a period of time of length t is:

R = rt

so trivially r = \frac{R}{t} is the annualised logarithmic rate of return for a return R, where t is measured in years.

For example, if the logarithmic return of a security per trading day is 0.14%, assuming 250 trading days in a year, then the annualised logarithmic rate of return is 0.14%/(1/250) = 0.14% x 250 = 35%

Returns over multiple periods

When returns are calculated over more than one time period, they are based on the investment value at the beginning of each period.

If the returns over n successive time sub-periods are r_1, r_2, r_3, \cdots, r_n, then the cumulative return or overall return over the overall time period is:

(1 + r_1)(1 + r_2) \cdots (1 + r_n) - 1

If the returns are logarithmic returns however, the logarithmic return over the overall time period is:

\sum_{i=1}^n {r_i} = r_1 + r_2 + r_3 + \cdots + r_n

Arithmetic average rate of return

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

\bar{r} = \frac{1}{n}\sum_{i=1}^n {r_i} = \frac{1}{n} (r_1+\cdots+r_n)

If you have a sequence of logarithmic rates of return over equal successive periods, the appropriate method of finding their average is the arithmetic average rate of return.

Geometric average rate of return

For ordinary returns, if there is no reinvestment, and losses are made good by topping up the capital invested, so that the value is brought back to its starting-point at the beginning of each new sub-period, use the arithmetic average return.

With reinvestment of all gains and losses however, the appropriate average rate of return is the geometric average rate of return over n periods, which is:

\bar{r}_{\mathrm{geometric}} = \left({\prod_{i=1}^n (1+r_i)}\right)^{1/n}-1

Note that the geometric average return is equivalent to the cumulative return over the whole n periods, converted into a rate of return per period.

In the case where the periods are each a year long, and there is no reinvestment of returns, the annualized cumulative return is the arithmetic average return. Where the individual sub-periods are each a year, and there is reinvestment of returns, the annualized cumulative return is the geometric average rate of return.

For example, assuming reinvestment, the cumulative return for annual returns: 50%, -20%, 30% and -40% is:

(1+0.50)(1-0.20)(1+0.30)(1-0.40)-1=-0.0640=-6.40\%

and the geometric average is:

((1+0.50)(1-0.20)(1+0.30)(1-0.40))^{1/4}-1=-0.0164=-1.64\%

which is equal to the annualized cumulative return:

(1-0.0640)^{1/4}-1=-0.0164

Comparisons between various rates of return

External flows

In the presence of external flows, such as cash or securities moving into or out of the portfolio, the return should be calculated by compensating for these movements. This is achieved using methods such as the time-weighted return. Time-weighted returns compensate for the impact of cash flows. This is useful to assess the performance of a money manager on behalf of his/her clients, where typically the clients control these cash flows.[3]

Fees

To measure returns net of fees, allow the value of the portfolio to be reduced by the amount of the fees. To calculate returns gross of fees, compensate for them by treating them as an external flow, and exclude accrued fees from valuations.

Money-weighted rate of return

Like the time-weighted return, the money-weighted rate of return (MWRR) or dollar-weighted rate of return also take cash flows into consideration. They are useful evaluating and comparing cases where the money manager controls cash flows, for example private equity. (Contrast with the true time-weighted rate of return, which is most applicable to measure the performance of a money manager who does not have control over external flows.)

Internal rate of return

The internal rate of return (IRR) (which is a variety of money-weighted rate of return) is the rate of return which makes the net present value of cash flows zero. It is a solution r satisfying the following equation:

\mbox{NPV} = \sum_{t=0}^{n} \frac{C_t}{(1+r)^{t}} =0

where:

NPV = net present value

and

{C_t} = net cash flow at time {t}, including the initial value {C_0} and final value {C_n}, net of any other flows at the beginning and at the end respectively. (The initial value is treated as an inflow, and the final value as an outflow.)

When the internal rate of return is greater than the cost of capital, (which is also referred to as the required rate of return), the investment adds value, i.e. the net present value of cash flows, discounted at the cost of capital, is greater than zero. Otherwise, the investment does not add value.

Note that there is not always an internal rate of return for a particular set of cash flows (i.e. the existence of a real solution to the equation \mbox{NPV} = 0 depends on the pattern of cash flows). There may also be more than one real solution to the equation, requiring some interpretation to determine the most appropriate one.

Comparing ordinary return with logarithmic return

The value of an investment is doubled if the return r = +100%, that is, if r_{\mathrm{log}} = ln($200 / $100) = ln(2) = 69.3%. The value falls to zero when r = -100%. The ordinary return can be calculated for any non-zero initial investment value, and any final value, positive or negative, but the logarithmic return can only be calculated when V_f/V_i > 0.

Ordinary returns and logarithmic returns are only equal when they are zero, but they are approximately equal when they are small. 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%.

Comparison of ordinary returns and logarithmic returns for an initial investment of $100
Initial investment, V_i $100 $100 $100 $100 $100 $100 $100
Final investment, V_f $0 $50 $99 $100 $101 $150 $200
Profit/loss, V_f - V_i −$100 −$50 −$1 $0 $1 $50 $100
Ordinary return, r −100% −50% −1% 0% 1% 50% 100%
Logarithmic return, r_{\mathrm{log}} −∞ −69.31% −1.005% 0% 0.995% 40.55% 69.31%

Symmetry of logarithmic returns

Logarithmic returns are useful for mathematical finance. One of the advantages is that the logarithmic returns are symmetric, while ordinary returns are not: positive and negative percent ordinary returns of equal magnitude do not cancel each other out and result in a net change, but logarithmic returns of equal magnitude but opposite signs will cancel each other out. 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% will come back to $100.

Comparing geometric with arithmetic average rates of return

The geometric average rate of return is in general less than the arithmetic average return. The two averages are equal if (and only if) all the sub-period returns are equal. This is a consequence of the AM–GM inequality. The difference between the annualized return and average annual return increases with the variance of the returns – the more volatile the performance, the greater the difference.[note 1]

For example, a return of +10%, followed by −10%, gives an arithmetic average return of 0%, but the overall result over the 2 sub-periods is 110% x 90% = 99% for an overall return of −1%. The order in which the loss and gain occurs does not affect the result.

For a return of +20%, followed by −20%, this again has an average return of 0%, but an overall return of −4%.

A return of +100%, followed by −100%, has an average return of 0%, but an overall return of −100%, as the final value is 0.

In cases of leveraged investments, even more extreme results are possible: a return of +200%, followed by −200%, has an average return of 0%, but an overall return of −300%.

This pattern is not followed in the case of logarithmic returns, due to their symmetry, as noted above. A logarithmic return of +10%, followed by −10%, gives an overall return of 10% - 10% = 0%, and an average rate of return of zero also.

Average returns and overall returns

Investment returns are often published as "average returns". In order to translate average returns into overall returns, compound the average returns over the number of periods.

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) $21.55

The geometric average rate of return was 5%. Over 4 years, this translates into an overall return of:

1.05^4-1=21.55\%
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)

The geometric average return over the 4 year period was -1.64%. Over 4 years, this translates into an overall return of:

(1-0.0164)^4-1=-6.4\%
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)

The geometric average return over the 4 year period was -42.74%. Over 4 years, this translates back into an overall return of:

(1-0.4274)^4-1=-89.25\%

Annual returns and annualized returns

Care must be taken not to confuse annual with annualized returns. An annual rate of return is a return over a period of one year, such as January 1 through December 31, or June 3, 2006 through June 2, 2007, whereas an annualized rate of return is a rate of return per year, measured over a period either longer or shorter than one year, such as a month, or two years, annualised for comparison with a one-year return.

The appropriate method of annualization depends on whether returns are reinvested or not.

For example, a return over one month of 1% converts to an annualized rate of return of 12.7% = ((1+0.01)12 - 1). This means if reinvested, the return over 12 months would compound to give a return of 12.7%.

As another example, a two-year return of 10% converts to an annualized rate of return of 4.88% = ((1+0.1)(12/24) - 1), assuming reinvestment at the end of the first year. In other words, the geometric average return per year is 4.88%.

In the cash flow example below, the dollar returns for the four years add up to $265. Assuming no reinvestment, the annualized rate of return for the four years is: $265 ÷ ($1,000 x 4 years) = 6.625%.

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%

Uses

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, and the proper way to compare returns taxed at different rates of tax is after tax, from the end-investor's perspective.

Time value of money

Main article: time value of money

Investments generate returns to the investor to compensate the investor for the time value of money.

Factors that investors may use 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 rate that a bank offers for deposit accounts, and also in the interest rate that a bank charges for a loan such as a home mortgage. The “risk-free” rate on US dollar investments 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 requires from a particular investment is called the discount rate, and is also referred to as the (opportunity) cost of capital. 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

Main article: compound interest

The annualized return of an investment depends on whether or not the return, including interest and dividends, from one period is reinvested in the next period. If the return is reinvested, it contributes to the starting value of capital invested for the next period (or reduces it, in the case of a negative return). Compounding reflects the effect of the return in the next period on the return from the previous period.

For example, if an investor puts $1,000 in a 1-year certificate of deposit (CD) that pays an annual interest rate of 4%, paid quarterly, the CD would earn 1% interest per quarter on the account balance. The account uses compound interest, meaning the account balance is cumulative, including interest previously reinvested and credited to the account. Unless the interest is withdrawn at the end of each quarter, it will earn more interest in the next quarter.

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 return 1% 1% 1% 1%

At the beginning of the second quarter, the account balance is $1,010.00, which then earns $10.10 interest altogether during the second quarter. The extra dime was interest on the additional $10 investment from the previous interest accumulated in the account. The annualized return (annual percentage yield, compound interest) is higher than for simple interest, because the interest is reinvested as capital and then itself earns interest. The yield or annualized return on the above investment is 4.06\% = (1.01)^4-1.

Returns when capital is at risk

Risk and volatility

Main article: volatility (finance)

Investments carry varying amounts of risk that the investor will lose some or all of the invested capital. For example, investments in company stock shares put capital at risk. Unlike capital invested in a savings account, the share price, which is the market value of a stock share at a certain point in time, depends on what someone is willing to pay for it, and the price of a stock share tends to change constantly when the market for that share is open. 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.”

US income tax on investment returns

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%

To the right is an example of a stock investment of one share purchased at the beginning of the year for $100.

\ln\left(\frac{103.02}{100}\right) = 2.98\%.

To calculate the capital gain for US income tax purposes, include the reinvested dividends in the cost basis. The investor received a total of $4.06 in dividends over the year, all of which were reinvested, so the cost basis increased by $4.06.

For U.S. income tax purposes therefore, dividends were $4.06, the cost basis of the investment was $104.06 and if the shares were sold at the end of the year, the sale value would be $103.02, and the capital loss would be $1.04.

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 had 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" brokerage account statements have been demanded by investors. In other words, the investors are saying more or less that 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 funds and brokerage firms are now 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 (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

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:[6]

\mathrm {P \left( 1 + 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}

Mutual fund capital gain distributions

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 (coupled with an equivalent decrease in unrealized capital gain).

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
Shares owned before distribution 70.373 71.676 74.125 76.859 84.752
Total distribution (distribution per share x shares owned) $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 (total distribution / price) 1.303 2.449 2.734 7.893 6.562
Shares owned after distribution 71.676 74.125 76.859 84.752 91.314

See also

Notes

  1. ↑ Consider the difference of squares formula, (x+y)(x-y)=x^2-y^2. For x=100\% (i.e. x = 1) the terms have an arithmetic average of 100% but product less than 100%.

References

  1. ↑ "return: definition of return in Oxford dictionary (British & World English)".
  2. ↑ "rate of return: definition of rate of return in Oxford dictionary (British & World English)".
  3. ↑ Strong, Robert (2009). Portfolio construction, management, and protection. Mason, Ohio: South-Western Cengage Learning. p. 527. ISBN 0-324-66510-5.
  4. ↑ A. A. Groppelli and Ehsan Nikbakht (2000). Barron's Finance, 4th Edition. New York. pp. 442–456. ISBN 0-7641-1275-9.
  5. ↑ Barron's Finance. pp. 151–163.
  6. ↑ U.S. Securities and Exchange Commission (1998). "Final Rule: Registration Form Used by Open-End Management Investment Companies: Sample Form and instructions".

Further reading

External links