The rate of return is an indicator that measures the profitability of an investment.[1] It includes any change in the value of the investment and/or in the cash flow that the investor receives from their investment, such as interest or dividend payments. It can be measured either in absolute terms (such as in dollars) or as a percentage of the amount invested. The latter is also called holding period return.
A loss rather than a gain is described as a negative return&action=edit&redlink=1 "Negative return (finance) (not yet written)"), assuming the amount invested is greater than zero.
The rate of return is the profit on an investment over a period of time, expressed as a proportion of the original investment.[2] The time period is usually one year, in which case the rate of return is referred to as the annual return.
To compare returns over time periods of different lengths on an equal basis, it is useful to convert each return into an "annualized return." This conversion process, described later, is called "annualization."
The return on investment (also known as ROI) is the return for each monetary unit invested (for example, for each euro or each dollar). It is a measure of the performance of the investment, regardless of its size (unlike the concepts of financial profitability, economic profitability or return on capital employed).
Calculation
Contenido
El retorno o tasa de rendimiento se puede calcular en un único período de cualquier duración.
Sin embargo, el período global se puede dividir en subperíodos contiguos. Esto significa que hay más de un período de tiempo, cada subperíodo que comienza en el momento en el que finaliza el anterior. En tal caso, cuando hay varios subperíodos contiguos, se puede calcular el retorno y la tasa de retorno durante el período general, combinando los rendimientos dentro de cada uno de los subperíodos.
single period
The return during a single period of any length of time must:
where:.
For example, if someone buys 100 shares at an initial price of 10, the initial value is . If the shareholder then collects 0.50 per share in cash dividends, and the final share price is 9.80, then in the end the shareholder has cash, plus shares, totaling a final value of 1030. The change in value is , so the return is .
Project Profitability
Introduction
The rate of return is an indicator that measures the profitability of an investment.[1] It includes any change in the value of the investment and/or in the cash flow that the investor receives from their investment, such as interest or dividend payments. It can be measured either in absolute terms (such as in dollars) or as a percentage of the amount invested. The latter is also called holding period return.
A loss rather than a gain is described as a negative return&action=edit&redlink=1 "Negative return (finance) (not yet written)"), assuming the amount invested is greater than zero.
The rate of return is the profit on an investment over a period of time, expressed as a proportion of the original investment.[2] The time period is usually one year, in which case the rate of return is referred to as the annual return.
To compare returns over time periods of different lengths on an equal basis, it is useful to convert each return into an "annualized return." This conversion process, described later, is called "annualization."
The return on investment (also known as ROI) is the return for each monetary unit invested (for example, for each euro or each dollar). It is a measure of the performance of the investment, regardless of its size (unlike the concepts of financial profitability, economic profitability or return on capital employed).
Calculation
Contenido
El retorno o tasa de rendimiento se puede calcular en un único período de cualquier duración.
Sin embargo, el período global se puede dividir en subperíodos contiguos. Esto significa que hay más de un período de tiempo, cada subperíodo que comienza en el momento en el que finaliza el anterior. En tal caso, cuando hay varios subperíodos contiguos, se puede calcular el retorno y la tasa de retorno durante el período general, combinando los rendimientos dentro de cada uno de los subperíodos.
Return measures the increase in the size of an asset or liability, or a short-term position.
A negative initial value is usually given for a liability or short-term position. If the initial value is negative and the final value is more negative, the return will be positive. In such a case, the positive return represents a loss rather than a gain.
If the initial value is zero, then the return cannot be calculated.
The yield, or rate of return, may depend on the currency in which it is measured, due to the possible variation in the exchange rate between different currencies during the period considered. For example, suppose a cash deposit of US$10,000 earned 2% interest for one year, so its value at the end of the year in question is US$10,200, including interest. The annual return is 2%, measured in dollars.
Suppose also that the exchange rate of the Japanese yen at the beginning of the year is 120 yen per dollar and 132 yen per dollar at the end of the year. The yen value of a dollar has increased by 10% during the period. So, the deposit is worth 1.2 million yen at the beginning of the year and 10,200 x 132 = 1,346,400 yen at the end of the year. The return on the deposit in the year calculated in yen is therefore:.
This is the rate of return experienced by an investor who starts with yen, converts it to dollars, invests in the dollar deposit, and converts the final income back to yen. The result is the same if you want to measure performance in terms of Japanese yen, for comparison with other investments.
Without any reinvestment, a return over a duration period is equivalent to a rate of return of:.
For example, $20,000 returned on an initial investment of $100,000 is a 20% return. Suppose that the $20,000 is paid in 5 irregular installments of $4,000, without reinvestment, over a period of 5 years, and without information on the schedule of the installments, the rate of return is 4,000 / 100,000 = 4% per year.
However, assuming that the returns are reinvested, due to the effect of compound interest, the relationship between a rate of return of and a return of over a period of time is:.
which can be used to convert the return to a compound rate of return:.
For example, a 33.1% return over 3 months is equivalent to a rate of:.
per month with reinvestment.
Annualization is the process, described above, of converting a return to an annual rate of return, where the length of the period is measured in years and the rate of return is per year.
In accordance with the CFA Institute's Global Investment Performance Standards (GIPS),[3].
This is because an annualized rate of return over a period of less than a year is statistically unlikely to be indicative of the long-term annualized rate of return, in which there is risk involved.[4] Annualizing a return over a period of less than a year could be interpreted as suggesting that the remainder of the year is more likely to have the same rate of return, effectively projecting that rate of return throughout the year.
Please note that this does not apply to interest rates or returns where there is no significant risk. It is common practice to quote an annualized rate of return for borrowing or borrowing money for periods shorter than a year, such as overnight interbank rates.
The log return or compound return, also known as force of interest, is:.
and the log rate of return is:.
or equivalently it is the solution to the equation:.
where:.
For example, if a stock has a price of $3,570 at the close of one day and $3,575 at the close of the next day, the log return is: ln (3,575 / 3,570) = 0.0014 (0.14%).
Under a reinvestment assumption, the relationship between a log return and a log rate of return over a period of time is:.
so, trivially, is the annualized log rate of return for a return, where it is measured in years.
For example, if the log return of a security per trading day is 0.14%, assuming 250 trading days in a year, then the annualized log rate of return is.
Return over several periods
When performance is calculated over a series of sub-periods of time, the return in each sub-period is based on the value of the investment at the beginning of the sub-period.
Assuming that the returns are reinvested, if the returns over successive sub-periods of time are , then the cumulative return or overall return over the overall time period using the time-weighted method, is the result of compounding the returns together:.
However, if the returns are log returns, the log return over the overall time period is:.
This formula is applied by assuming reinvestment of returns and applying the time-weighted return method.
The average rate of return over time periods of equal length is defined as:.
If you have a sequence of logarithmic rates of return in equal successive periods, the appropriate method for finding their average is the arithmetic mean of the rates of return.
For ordinary returns, if there is no reinvestment, and losses are resolved by completing the invested capital, so that the value returns to its starting point at the beginning of each new subperiod, the arithmetic mean of the return is used.
However, with all gains and losses reinvested, the appropriate average rate of return is the geometric mean rate of return over n periods, which is:.
Note that the geometric mean return is equivalent to the cumulative return of all n periods, converted into a rate of return per period.
In the event that the periods are one year, and there is no reinvestment of returns, the annualized accumulated return is the arithmetic average return. When the individual subperiods are each year, and there is reinvestment of returns, the annualized cumulative return is the geometric mean rate of return.
For example, assuming that reinvestment occurs, the cumulative return of a series of annual returns of: 50%, -20%, 30% and -40%, is:.
and the geometric mean is:.
which is equal to the annualized cumulative return:.
Comparisons between various rates of return
External flows
In the presence of external flows, such as cash movements or securities moving into or out of the portfolio, performance must be calculated offsetting these movements. This is achieved using methods such as "time-weighted returns". Time-weighted returns offset the impact of cash flows. This is useful for evaluating the management of a money manager on behalf of its clients, where the clients typically control these cash flows.[5].
Fees
To measure the effect on net returns of fees, it is necessary that the value of the portfolio be reduced by the amount of fees applied. To calculate gross commission returns, they must be offset by treating them as an external flow, excluding accrued commissions from valuations.
Weighted rate of return
Like time-weighted return, money-weighted rate of return (MWRR) or dollar-weighted rate of return also takes cash flows into account. They are useful for 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 more applicable for evaluating the management of a money manager who has no control over external flows).
The internal rate of return (IRR) (which is a variant of the money-weighted rate of return), is the rate of return that makes the net present value of cash flows zero. It is a solution that satisfies the following equation:
where:.
and.
When the internal rate of return is greater than the cost of capital (also known as the required rate of return), the investment adds value, that is, the net present value of the cash flows, discounted at the cost of capital, is greater than zero. Otherwise, the investment adds no value.
Note that there is not always an internal rate of return for a particular set of cash flows (that is, the existence of a real solution to the equation depends on the pattern of the cash flows). There may also be more than one real solution to the equation, which requires some interpretation to determine the most appropriate one.
It should be noted that the money-weighted return over various sub-periods is generally not equal to the result of combining the money-weighted returns within the sub-periods using the method described above, unlike the time-weighted returns.
Comparison of ordinary return with logarithmic return
The value of an investment doubles if the return = + 100%, that is, if = ln (€200 / €100) = ln (2) = 69.3%. The value drops to zero when = -100%. The ordinary return can be calculated for any non-zero initial investment value and any ending value, positive or negative, but the log return can only be calculated when .
Ordinary and log 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 the percentage changes are high. For example, an arithmetic return of +50% is equivalent to a log return of 40.55%, while an arithmetic return of -50% is equivalent to a log return of -69.31%.
Symmetry of log returns
Log returns are useful for mathematical finance. One advantage is that log returns are symmetrical, whereas ordinary returns are not: positive and negative ordinary percentages of equal magnitude do not cancel each other and result in a net change, but log returns of equal magnitude but opposite signs cancel each other. This means that a $100 investment that produces a 50% arithmetic return followed by a -50% arithmetic return will return $75, while a $100 investment that produces a 50% log return followed by a -50% log return will return $100.
Comparison of geometric and arithmetic rates of return
The geometric mean rate of return is generally lower than the arithmetic mean rate of return. The two averages are equal if (and only if) all subperiod returns are equal. This is a consequence of the inequality of arithmetic and geometric means. The difference between the annualized return and the average annual return increases with the change in returns: the more volatile "Volatility (finances)") the return, the greater the difference.[note 1].
For example, a return of +10%, followed by −10%, gives an average arithmetic return of 0%, but the overall result over the 2 subperiods is 110% x 90% = 99% for an overall return of -1%. The order in which the loss and gain occur 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 performance of +100%, followed by −100%, has an average performance of 0%, but an overall performance of −100%, since the final value is 0.
In cases of leveraged investments, it is possible to obtain even more extreme results: a return of +200%, followed by −200%, has an average return of 0%, but an overall return of −300%.
This pattern does not follow in the case of logarithmic returns, due to their symmetry, as noted above. A log return of +10%, followed by −10%, gives an overall return of 10% - 10% = 0%, and also an average rate of return of zero.
Investment returns are often published as "average returns." To translate average returns into overall returns, average returns must be compounded over the number of periods.
The geometric average rate of return was 5%. Over 4 years, this translates to an overall performance of:.
The geometric average return over the 4-year period was -1.64%. Over 4 years, this translates to an overall performance of:.
The geometric average return over the 4-year period was -42.74%. Over 4 years, this again translates to an overall performance of:.
Annual returns and annualized returns
Care must be taken not to confuse annual returns with annualized returns. An annual rate of return is a return over a period of one year, such as January 1 to December 31 or June 3, 2006 to June 2, 2007, while an annualized rate of return is a rate of return per year, measured over a period either longer or shorter than a year, such as one month or two years, annualized for comparison with a one-year return.
The appropriate method of annualization depends on whether or not the returns are reinvested.
For example, a one-month return of 1% becomes an annualized rate of return of 12.7% = ((1 + 0.01) - 1). This means that if reinvested, earning a return of 1% each month, the return over 12 months would increase 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) - 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-out example shown below, the four-year dollar returns total $265. Assuming no reinvestment, the annualized rate of return for the four years is:
$265 ÷ ($1000 x 4 years) = 6.625% (per year).
time value of money
Investments generate returns to the investor as compensation for the time value of money.
Factors that investors can 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 a bank offers for deposit accounts, and also in the interest rate a bank charges for a loan, such as a mortgage. The "risk-free" rate on U.S. dollar investments is the U.S. Treasury Secured Bond rate), because it is the highest rate available without principal risk.
The rate of return that an investor requires from a particular investment is called the discount rate, and is also known as the cost of capital (or opportunity cost). The higher the risk, the higher the discount rate (rate of return) the investor will demand from the investment.
Compound interest (reinvestment)
The annualized return on an investment depends on whether 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 the invested capital for the next period (or reduces it, in the case of a negative return). Compound interest reflects the effect of the return in one period on the return in the next period, as a result of the change in the capital base at the beginning of this last period.
For example, if an investor places $1,000 in a 1-year certificate of deposit (CD) paying an annual interest rate of 4%, paid quarterly, they would earn 1% interest per quarter on the account balance. The account uses compound interest, meaning the account balance is cumulative, including interest reinvested and previously credited to the account. Unless interest is withdrawn at the end of each quarter, you will earn more interest in the next quarter.
At the beginning of the second quarter, the account balance is $1,010.00, which then earns a total interest of $10.10 during the second quarter. The extra cent was the interest on the additional investment of the $10 of 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 principal and then a return is earned. The yield (the annualized return) on the investment above is .
Return from currency exchange
Como se explicó anteriormente, el retorno, la tasa o el rendimiento, dependen de la moneda de medición. En el ejemplo anterior, un depósito en efectivo en dólares que repirtara un 2% durante un año (medido en dólares), devuelve un 12,2% medido en yenes japoneses durante el mismo período, si el dólar aumenta su valor en un 10% frente al yen durante el mismo periodo.
El rendimiento en yenes es el resultado de la combinación del rendimiento del 2% en dólares estadounidenses sobre el depósito en efectivo, con el rendimiento del 10% del dólar frente al yen japonés:.
En términos más generales, el rendimiento en una segunda moneda es el resultado de la combinación de los dos rendimientos:.
donde.
Esto es cierto si se utiliza el método de tiempo ponderado o si no hay flujos dentro o fuera durante el período. Si se utiliza uno de los métodos ponderados por el dinero, y hay flujos, es necesario volver a calcular el rendimiento en la segunda moneda utilizando uno de los métodos para compensar los flujos.
Returns in foreign currency in various periods
It makes no sense to combine returns for consecutive periods measured in different currencies. Before combining returns over consecutive periods, returns must be recalculated or adjusted using a single measurement currency.
A portfolio increases in value in Singapore dollars by 10% during the year 2015 (with no flows into or out of the portfolio during this period). In the first month of 2016, its value increased by another 7%, in US dollars. (Again, no entries or exits during the January 2016 period.)
What is the performance of the portfolio, from the beginning of 2015 to the end of January 2016?
The answer is that there is not enough data to calculate a return, in either currency, without knowing the exchange rate for both periods.
If the return in 2015 was 10% in Singapore dollars and the Singapore dollar increased by 5% against the US dollar compared to 2015, then as long as there are no flows in 2015, the return in 2015 in US dollars is d3:.
The return between the beginning of 2015 and the end of January 2016 in US dollars is:.
Return considering capital risk
Risk and volatility
Investments carry varying amounts of risk that the investor will lose some or all of the invested capital. For example, investments in a company's stock put capital at risk. Unlike capital invested in a savings account, the stock price, which is the market value of a stock at a given time, depends on what someone is willing to pay for it, and the price of a stock tends to change continually when the market for that stock is open. If the price is relatively stable, the stock is said to have "low volatility." If the price changes a lot often, the stock is said to have "high volatility."
United States income tax on investment returns
To the right is an example of the evolution of the value of an investment consisting of a share purchased at the beginning of the year for $100.
To calculate capital gain considering income tax (in this case, from the United States), reinvested dividends must be included in the cost basis. The investor received a total of $4.06 in dividends for the year, all of which were reinvested, so the basis increased by $4.06.
Therefore, for United States income tax purposes, the dividends were $4.06, the 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 funds and investments of large companies
Mutual funds, exchange-traded funds (ETFs) and other equity investments (such as unit investment trusts or UITs, separate insurance accounts and related variable products, such as variable universal life insurance policies and variable annuity contracts), and bank-sponsored pooled funds, collective benefit funds or common trust funds) are essentially portfolios of various investment securities, such as stocks, bonds and money market instruments that are realized through the sale of shares or units to investors. Investors and other interested parties are interested in knowing how investing has evolved over various periods of time.
Performance is usually quantified by a fund's total return. In the 1990s, many different fund companies advertised 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 fund-to-fund returns, the Securities and Exchange Commission (SEC) began requiring funds to calculate and report total returns based on a standardized formula, called "SEC standardized total return," which is the average annual total return, assuming the reinvestment of dividends and distributions and the deduction of sales loads or charges. Funds may calculate and announce returns on other bases (so-called "non-standardized" returns), provided that they also publish "standardized" return data no less prominently.
Subsequently, it appeared that investors who had sold their fund shares after a large rise in share prices in the late 1990s and early 2000s were unaware of how significant the impact of income/capital gains taxes was on their funds' "gross" returns. That is, they had little idea how significant the difference between "gross" returns (returns before federal taxes) and "net" returns (after-tax returns) could be. In reaction to this apparent investor ignorance, and perhaps for other reasons, the SEC developed further rules to require mutual funds to disclose in their annual prospectuses, among other things, total returns before and after the impact of U.S. individual federal income taxes. And in addition, the after-tax returns would include:.
The returns on a hypothetical taxable account after deducting fees on dividends and capital gains distributions received during the periods shown.
The impacts of the elements of the previous point, as well as assuming all of the investment shares that were sold at the end of the period (realization of capital gains/losses on the liquidation of the shares). These after-tax statements would apply, of course, only to taxable accounts and not to tax-deferred or retirement accounts.
Finally, in more recent years, investors have demanded "customized" brokerage account statements. In other words, they are pointing out that the fund's returns may not correspond to your actual account returns, depending on the trading history of the actual investment account. This is because investments may have been made on multiple dates and additional purchases and withdrawals may have been made that vary in amount and date and are therefore unique to the particular account. More and more funds and brokerage firms are providing personalized account returns on investors' account statements in response to this need.
With these differences resolved, the following describes how basic gains and gains/losses work in a mutual fund. The fund records dividend and interest income earned, which generally increases the value of the mutual fund's shares, while reserved expenses have an offsetting impact on sharing value. When the fund's investments increase (or decrease) in market value, the value of the fund's shares also increases (or decreases). When the fund sells investments at a gain, it converts or reclassifies that paper gain or unrealized gain into a real or realized gain. The sale has no effect on the value of the fund's shares, but has reclassified a component of its value from one group to another on the fund's books, which will have a future impact for investors. At least once a year, a fund generally pays dividends from its net income (income minus expenses) and net realized capital gains to shareholders as an internal revenue service requirement. In this way, the fund does not pay taxes, but all investors do in their taxable accounts. Share prices of mutual funds are generally valued each day that the stock or bond markets are open, and generally the value of a share is the net asset value) of the fund shares that investors they possess.
Mutual funds disclose their total returns assuming the reinvestment of dividends and capital gains distributions. That is, the dollar amounts distributed are used to purchase additional shares of the funds as of the reinvestment/since-dividend date. Reinvestment rates or factors are based on total distributions (dividends plus capital gains) during each period.
U.S. mutual funds must calculate the average annual total return as prescribed by the Securities and Exchange Commission (SEC) in the instructions for forming the N-1A (the fund's prospectus) as the average annual compounded rates of return for periods of 1 year, 5 years, and 10 years (or at the inception of the fund if shorter) as the "average annual total return" for each fund. The following formula is used:[8].
where:.
P = is a hypothetical down payment of $1000.
T = the average annual total return.
n = number of years.
ERV = final terminable value of a hypothetical payment of $1,000 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, we obtain:
Capital Gains Distributions in Mutual Funds
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 gains distribution is offset by an equal decrease in the value/price of the mutual fund shares. From the shareholder's perspective, a capital gain distribution is not a net gain in assets, but rather it is a realized capital gain (along with an equivalent decrease in the unrealized capital gain).
References
[6] ↑ Considérese la fórmula de la diferencia de cuadrados,
single period
The return during a single period of any length of time must:
where:.
For example, if someone buys 100 shares at an initial price of 10, the initial value is . If the shareholder then collects 0.50 per share in cash dividends, and the final share price is 9.80, then in the end the shareholder has cash, plus shares, totaling a final value of 1030. The change in value is , so the return is .
Return measures the increase in the size of an asset or liability, or a short-term position.
A negative initial value is usually given for a liability or short-term position. If the initial value is negative and the final value is more negative, the return will be positive. In such a case, the positive return represents a loss rather than a gain.
If the initial value is zero, then the return cannot be calculated.
The yield, or rate of return, may depend on the currency in which it is measured, due to the possible variation in the exchange rate between different currencies during the period considered. For example, suppose a cash deposit of US$10,000 earned 2% interest for one year, so its value at the end of the year in question is US$10,200, including interest. The annual return is 2%, measured in dollars.
Suppose also that the exchange rate of the Japanese yen at the beginning of the year is 120 yen per dollar and 132 yen per dollar at the end of the year. The yen value of a dollar has increased by 10% during the period. So, the deposit is worth 1.2 million yen at the beginning of the year and 10,200 x 132 = 1,346,400 yen at the end of the year. The return on the deposit in the year calculated in yen is therefore:.
This is the rate of return experienced by an investor who starts with yen, converts it to dollars, invests in the dollar deposit, and converts the final income back to yen. The result is the same if you want to measure performance in terms of Japanese yen, for comparison with other investments.
Without any reinvestment, a return over a duration period is equivalent to a rate of return of:.
For example, $20,000 returned on an initial investment of $100,000 is a 20% return. Suppose that the $20,000 is paid in 5 irregular installments of $4,000, without reinvestment, over a period of 5 years, and without information on the schedule of the installments, the rate of return is 4,000 / 100,000 = 4% per year.
However, assuming that the returns are reinvested, due to the effect of compound interest, the relationship between a rate of return of and a return of over a period of time is:.
which can be used to convert the return to a compound rate of return:.
For example, a 33.1% return over 3 months is equivalent to a rate of:.
per month with reinvestment.
Annualization is the process, described above, of converting a return to an annual rate of return, where the length of the period is measured in years and the rate of return is per year.
In accordance with the CFA Institute's Global Investment Performance Standards (GIPS),[3].
This is because an annualized rate of return over a period of less than a year is statistically unlikely to be indicative of the long-term annualized rate of return, in which there is risk involved.[4] Annualizing a return over a period of less than a year could be interpreted as suggesting that the remainder of the year is more likely to have the same rate of return, effectively projecting that rate of return throughout the year.
Please note that this does not apply to interest rates or returns where there is no significant risk. It is common practice to quote an annualized rate of return for borrowing or borrowing money for periods shorter than a year, such as overnight interbank rates.
The log return or compound return, also known as force of interest, is:.
and the log rate of return is:.
or equivalently it is the solution to the equation:.
where:.
For example, if a stock has a price of $3,570 at the close of one day and $3,575 at the close of the next day, the log return is: ln (3,575 / 3,570) = 0.0014 (0.14%).
Under a reinvestment assumption, the relationship between a log return and a log rate of return over a period of time is:.
so, trivially, is the annualized log rate of return for a return, where it is measured in years.
For example, if the log return of a security per trading day is 0.14%, assuming 250 trading days in a year, then the annualized log rate of return is.
Return over several periods
When performance is calculated over a series of sub-periods of time, the return in each sub-period is based on the value of the investment at the beginning of the sub-period.
Assuming that the returns are reinvested, if the returns over successive sub-periods of time are , then the cumulative return or overall return over the overall time period using the time-weighted method, is the result of compounding the returns together:.
However, if the returns are log returns, the log return over the overall time period is:.
This formula is applied by assuming reinvestment of returns and applying the time-weighted return method.
The average rate of return over time periods of equal length is defined as:.
If you have a sequence of logarithmic rates of return in equal successive periods, the appropriate method for finding their average is the arithmetic mean of the rates of return.
For ordinary returns, if there is no reinvestment, and losses are resolved by completing the invested capital, so that the value returns to its starting point at the beginning of each new subperiod, the arithmetic mean of the return is used.
However, with all gains and losses reinvested, the appropriate average rate of return is the geometric mean rate of return over n periods, which is:.
Note that the geometric mean return is equivalent to the cumulative return of all n periods, converted into a rate of return per period.
In the event that the periods are one year, and there is no reinvestment of returns, the annualized accumulated return is the arithmetic average return. When the individual subperiods are each year, and there is reinvestment of returns, the annualized cumulative return is the geometric mean rate of return.
For example, assuming that reinvestment occurs, the cumulative return of a series of annual returns of: 50%, -20%, 30% and -40%, is:.
and the geometric mean is:.
which is equal to the annualized cumulative return:.
Comparisons between various rates of return
External flows
In the presence of external flows, such as cash movements or securities moving into or out of the portfolio, performance must be calculated offsetting these movements. This is achieved using methods such as "time-weighted returns". Time-weighted returns offset the impact of cash flows. This is useful for evaluating the management of a money manager on behalf of its clients, where the clients typically control these cash flows.[5].
Fees
To measure the effect on net returns of fees, it is necessary that the value of the portfolio be reduced by the amount of fees applied. To calculate gross commission returns, they must be offset by treating them as an external flow, excluding accrued commissions from valuations.
Weighted rate of return
Like time-weighted return, money-weighted rate of return (MWRR) or dollar-weighted rate of return also takes cash flows into account. They are useful for 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 more applicable for evaluating the management of a money manager who has no control over external flows).
The internal rate of return (IRR) (which is a variant of the money-weighted rate of return), is the rate of return that makes the net present value of cash flows zero. It is a solution that satisfies the following equation:
where:.
and.
When the internal rate of return is greater than the cost of capital (also known as the required rate of return), the investment adds value, that is, the net present value of the cash flows, discounted at the cost of capital, is greater than zero. Otherwise, the investment adds no value.
Note that there is not always an internal rate of return for a particular set of cash flows (that is, the existence of a real solution to the equation depends on the pattern of the cash flows). There may also be more than one real solution to the equation, which requires some interpretation to determine the most appropriate one.
It should be noted that the money-weighted return over various sub-periods is generally not equal to the result of combining the money-weighted returns within the sub-periods using the method described above, unlike the time-weighted returns.
Comparison of ordinary return with logarithmic return
The value of an investment doubles if the return = + 100%, that is, if = ln (€200 / €100) = ln (2) = 69.3%. The value drops to zero when = -100%. The ordinary return can be calculated for any non-zero initial investment value and any ending value, positive or negative, but the log return can only be calculated when .
Ordinary and log 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 the percentage changes are high. For example, an arithmetic return of +50% is equivalent to a log return of 40.55%, while an arithmetic return of -50% is equivalent to a log return of -69.31%.
Symmetry of log returns
Log returns are useful for mathematical finance. One advantage is that log returns are symmetrical, whereas ordinary returns are not: positive and negative ordinary percentages of equal magnitude do not cancel each other and result in a net change, but log returns of equal magnitude but opposite signs cancel each other. This means that a $100 investment that produces a 50% arithmetic return followed by a -50% arithmetic return will return $75, while a $100 investment that produces a 50% log return followed by a -50% log return will return $100.
Comparison of geometric and arithmetic rates of return
The geometric mean rate of return is generally lower than the arithmetic mean rate of return. The two averages are equal if (and only if) all subperiod returns are equal. This is a consequence of the inequality of arithmetic and geometric means. The difference between the annualized return and the average annual return increases with the change in returns: the more volatile "Volatility (finances)") the return, the greater the difference.[note 1].
For example, a return of +10%, followed by −10%, gives an average arithmetic return of 0%, but the overall result over the 2 subperiods is 110% x 90% = 99% for an overall return of -1%. The order in which the loss and gain occur 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 performance of +100%, followed by −100%, has an average performance of 0%, but an overall performance of −100%, since the final value is 0.
In cases of leveraged investments, it is possible to obtain even more extreme results: a return of +200%, followed by −200%, has an average return of 0%, but an overall return of −300%.
This pattern does not follow in the case of logarithmic returns, due to their symmetry, as noted above. A log return of +10%, followed by −10%, gives an overall return of 10% - 10% = 0%, and also an average rate of return of zero.
Investment returns are often published as "average returns." To translate average returns into overall returns, average returns must be compounded over the number of periods.
The geometric average rate of return was 5%. Over 4 years, this translates to an overall performance of:.
The geometric average return over the 4-year period was -1.64%. Over 4 years, this translates to an overall performance of:.
The geometric average return over the 4-year period was -42.74%. Over 4 years, this again translates to an overall performance of:.
Annual returns and annualized returns
Care must be taken not to confuse annual returns with annualized returns. An annual rate of return is a return over a period of one year, such as January 1 to December 31 or June 3, 2006 to June 2, 2007, while an annualized rate of return is a rate of return per year, measured over a period either longer or shorter than a year, such as one month or two years, annualized for comparison with a one-year return.
The appropriate method of annualization depends on whether or not the returns are reinvested.
For example, a one-month return of 1% becomes an annualized rate of return of 12.7% = ((1 + 0.01) - 1). This means that if reinvested, earning a return of 1% each month, the return over 12 months would increase 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) - 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-out example shown below, the four-year dollar returns total $265. Assuming no reinvestment, the annualized rate of return for the four years is:
$265 ÷ ($1000 x 4 years) = 6.625% (per year).
time value of money
Investments generate returns to the investor as compensation for the time value of money.
Factors that investors can 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 a bank offers for deposit accounts, and also in the interest rate a bank charges for a loan, such as a mortgage. The "risk-free" rate on U.S. dollar investments is the U.S. Treasury Secured Bond rate), because it is the highest rate available without principal risk.
The rate of return that an investor requires from a particular investment is called the discount rate, and is also known as the cost of capital (or opportunity cost). The higher the risk, the higher the discount rate (rate of return) the investor will demand from the investment.
Compound interest (reinvestment)
The annualized return on an investment depends on whether 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 the invested capital for the next period (or reduces it, in the case of a negative return). Compound interest reflects the effect of the return in one period on the return in the next period, as a result of the change in the capital base at the beginning of this last period.
For example, if an investor places $1,000 in a 1-year certificate of deposit (CD) paying an annual interest rate of 4%, paid quarterly, they would earn 1% interest per quarter on the account balance. The account uses compound interest, meaning the account balance is cumulative, including interest reinvested and previously credited to the account. Unless interest is withdrawn at the end of each quarter, you will earn more interest in the next quarter.
At the beginning of the second quarter, the account balance is $1,010.00, which then earns a total interest of $10.10 during the second quarter. The extra cent was the interest on the additional investment of the $10 of 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 principal and then a return is earned. The yield (the annualized return) on the investment above is .
Return from currency exchange
Como se explicó anteriormente, el retorno, la tasa o el rendimiento, dependen de la moneda de medición. En el ejemplo anterior, un depósito en efectivo en dólares que repirtara un 2% durante un año (medido en dólares), devuelve un 12,2% medido en yenes japoneses durante el mismo período, si el dólar aumenta su valor en un 10% frente al yen durante el mismo periodo.
El rendimiento en yenes es el resultado de la combinación del rendimiento del 2% en dólares estadounidenses sobre el depósito en efectivo, con el rendimiento del 10% del dólar frente al yen japonés:.
En términos más generales, el rendimiento en una segunda moneda es el resultado de la combinación de los dos rendimientos:.
donde.
Esto es cierto si se utiliza el método de tiempo ponderado o si no hay flujos dentro o fuera durante el período. Si se utiliza uno de los métodos ponderados por el dinero, y hay flujos, es necesario volver a calcular el rendimiento en la segunda moneda utilizando uno de los métodos para compensar los flujos.
Returns in foreign currency in various periods
It makes no sense to combine returns for consecutive periods measured in different currencies. Before combining returns over consecutive periods, returns must be recalculated or adjusted using a single measurement currency.
A portfolio increases in value in Singapore dollars by 10% during the year 2015 (with no flows into or out of the portfolio during this period). In the first month of 2016, its value increased by another 7%, in US dollars. (Again, no entries or exits during the January 2016 period.)
What is the performance of the portfolio, from the beginning of 2015 to the end of January 2016?
The answer is that there is not enough data to calculate a return, in either currency, without knowing the exchange rate for both periods.
If the return in 2015 was 10% in Singapore dollars and the Singapore dollar increased by 5% against the US dollar compared to 2015, then as long as there are no flows in 2015, the return in 2015 in US dollars is d3:.
The return between the beginning of 2015 and the end of January 2016 in US dollars is:.
Return considering capital risk
Risk and volatility
Investments carry varying amounts of risk that the investor will lose some or all of the invested capital. For example, investments in a company's stock put capital at risk. Unlike capital invested in a savings account, the stock price, which is the market value of a stock at a given time, depends on what someone is willing to pay for it, and the price of a stock tends to change continually when the market for that stock is open. If the price is relatively stable, the stock is said to have "low volatility." If the price changes a lot often, the stock is said to have "high volatility."
United States income tax on investment returns
To the right is an example of the evolution of the value of an investment consisting of a share purchased at the beginning of the year for $100.
To calculate capital gain considering income tax (in this case, from the United States), reinvested dividends must be included in the cost basis. The investor received a total of $4.06 in dividends for the year, all of which were reinvested, so the basis increased by $4.06.
Therefore, for United States income tax purposes, the dividends were $4.06, the 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 funds and investments of large companies
Mutual funds, exchange-traded funds (ETFs) and other equity investments (such as unit investment trusts or UITs, separate insurance accounts and related variable products, such as variable universal life insurance policies and variable annuity contracts), and bank-sponsored pooled funds, collective benefit funds or common trust funds) are essentially portfolios of various investment securities, such as stocks, bonds and money market instruments that are realized through the sale of shares or units to investors. Investors and other interested parties are interested in knowing how investing has evolved over various periods of time.
Performance is usually quantified by a fund's total return. In the 1990s, many different fund companies advertised 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 fund-to-fund returns, the Securities and Exchange Commission (SEC) began requiring funds to calculate and report total returns based on a standardized formula, called "SEC standardized total return," which is the average annual total return, assuming the reinvestment of dividends and distributions and the deduction of sales loads or charges. Funds may calculate and announce returns on other bases (so-called "non-standardized" returns), provided that they also publish "standardized" return data no less prominently.
Subsequently, it appeared that investors who had sold their fund shares after a large rise in share prices in the late 1990s and early 2000s were unaware of how significant the impact of income/capital gains taxes was on their funds' "gross" returns. That is, they had little idea how significant the difference between "gross" returns (returns before federal taxes) and "net" returns (after-tax returns) could be. In reaction to this apparent investor ignorance, and perhaps for other reasons, the SEC developed further rules to require mutual funds to disclose in their annual prospectuses, among other things, total returns before and after the impact of U.S. individual federal income taxes. And in addition, the after-tax returns would include:.
The returns on a hypothetical taxable account after deducting fees on dividends and capital gains distributions received during the periods shown.
The impacts of the elements of the previous point, as well as assuming all of the investment shares that were sold at the end of the period (realization of capital gains/losses on the liquidation of the shares). These after-tax statements would apply, of course, only to taxable accounts and not to tax-deferred or retirement accounts.
Finally, in more recent years, investors have demanded "customized" brokerage account statements. In other words, they are pointing out that the fund's returns may not correspond to your actual account returns, depending on the trading history of the actual investment account. This is because investments may have been made on multiple dates and additional purchases and withdrawals may have been made that vary in amount and date and are therefore unique to the particular account. More and more funds and brokerage firms are providing personalized account returns on investors' account statements in response to this need.
With these differences resolved, the following describes how basic gains and gains/losses work in a mutual fund. The fund records dividend and interest income earned, which generally increases the value of the mutual fund's shares, while reserved expenses have an offsetting impact on sharing value. When the fund's investments increase (or decrease) in market value, the value of the fund's shares also increases (or decreases). When the fund sells investments at a gain, it converts or reclassifies that paper gain or unrealized gain into a real or realized gain. The sale has no effect on the value of the fund's shares, but has reclassified a component of its value from one group to another on the fund's books, which will have a future impact for investors. At least once a year, a fund generally pays dividends from its net income (income minus expenses) and net realized capital gains to shareholders as an internal revenue service requirement. In this way, the fund does not pay taxes, but all investors do in their taxable accounts. Share prices of mutual funds are generally valued each day that the stock or bond markets are open, and generally the value of a share is the net asset value) of the fund shares that investors they possess.
Mutual funds disclose their total returns assuming the reinvestment of dividends and capital gains distributions. That is, the dollar amounts distributed are used to purchase additional shares of the funds as of the reinvestment/since-dividend date. Reinvestment rates or factors are based on total distributions (dividends plus capital gains) during each period.
U.S. mutual funds must calculate the average annual total return as prescribed by the Securities and Exchange Commission (SEC) in the instructions for forming the N-1A (the fund's prospectus) as the average annual compounded rates of return for periods of 1 year, 5 years, and 10 years (or at the inception of the fund if shorter) as the "average annual total return" for each fund. The following formula is used:[8].
where:.
P = is a hypothetical down payment of $1000.
T = the average annual total return.
n = number of years.
ERV = final terminable value of a hypothetical payment of $1,000 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, we obtain:
Capital Gains Distributions in Mutual Funds
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 gains distribution is offset by an equal decrease in the value/price of the mutual fund shares. From the shareholder's perspective, a capital gain distribution is not a net gain in assets, but rather it is a realized capital gain (along with an equivalent decrease in the unrealized capital gain).
References
[6] ↑ Considérese la fórmula de la diferencia de cuadrados,