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).

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.

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 .

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:.

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:.

1. The returns on a hypothetical taxable account after deducting fees on dividends and capital gains distributions received during the periods shown.

2. 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).

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).

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.

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 .

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:.

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:.

1. The returns on a hypothetical taxable account after deducting fees on dividends and capital gains distributions received during the periods shown.

2. 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).