Rate of return

Overview

In finance, return is a profit on an investment. It comprises any change in value of the investment, and/or cash flows (or securities, or other investments) which the investor receives from that investment over a specified time period, such as interest payments, coupons, cash dividends and stock dividends. It may be measured either in absolute terms (e.g., dollars) or as a percentage of the amount invested. The latter is also called the holding period return. A loss instead of a profit is described as a negative return, assuming the amount invested is greater than zero. To compare returns over time periods of different lengths on an equal basis, it is useful to convert each return into a return over a period of time of a standard length. The result of the conversion is called the rate of return. Typically, the period of time is a year, in which case the rate of return is also called the annualized return, and the conversion process, described below, is called annualization. The return on investment (ROI) is return per dollar invested. It is a measure of investment performance, as opposed to size (cf. return on equity, return on assets, return on capital employed).

Calculation

The return, or the holding period return, can be calculated over a single period. The single period may last any length of time. The overall period may, however, instead be divided into contiguous subperiods. This means that there is more than one time period, each sub-period beginning at the point in time where the previous one ended. In such a case, where there are multiple contiguous subperiods, the return or the holding period return over the overall period can be calculated by combining the returns within each of the subperiods.

Return

The direct method to calculate the return or the holding period return

R

{\displaystyle R}

over a single period of any length of time is:

R =

V

f

−

V

i

V

i

{\displaystyle R={\frac {V_{f}-V_{i}}{V_{i}}}}

where:

V

f

{\displaystyle V_{f}}

= final value, including dividends and interest

V

i

Negative initial value

Return measures the increase in size of an asset or liability or short position. A negative initial value usually occurs for a liability or short position. If the initial value is negative, and the final value is more negative, then the return will be positive. In such a case, the positive return represents a loss rather than a profit. If the initial value is zero, then no return can be calculated.

Currency of measurement

The return, or rate of return, depends on the currency of measurement. For example, suppose a US$10,000 (US dollar) cash deposit earns 2% interest over a year, so its value at the end of the year is US$10,200 including interest. The return over the year is 2%, measured in USD. Let us suppose also that the exchange rate to Japanese yen at the start of the year is 120 yen per USD, and 132 yen per USD at the end of the year. The value in yen of one USD has increased by 10% over the period. The deposit is worth 1.2 million yen at the start of the year, and 10,200 x 132 = 1,346,400 yen at the end of the year. The return on the deposit over the year in yen terms is therefore:

1 , 346 , 400 − 1 , 200 , 000

1 , 200 , 000

= 12.2 %

{\displaystyle {\frac {1,346,400-1,200,000}{1,200,000}}=12.2\%}

This is the rate of return experienced either by an investor who starts with yen, converts to dollars, invests in the USD deposit, and converts the eventual proceeds back to yen; or for any investor, who wishes to measure the return in Japanese yen terms, for comparison purposes.

Annualization

Without any reinvestment, a return

R

{\displaystyle R}

over a period of time

t

{\displaystyle t}

corresponds to a rate of return

r

{\displaystyle r}

:

r =

R t

{\displaystyle r={\frac {R}{t}}}

For example, let us suppose that US$20,000 is returned on an initial investment of US$100,000. This is a return of US$20,000 divided by US$100,000, which equals 20 percent. The US$20,000 is paid in 5 irregularly-timed installments of US$4,000, with no reinvestment, over a 5-year period, and with no information provided about the timing of the installments. The rate of return is 4,000 / 100,000 = 4% per year. Assuming returns are reinvested however, due to the effect of compounding, the relationship between a rate of return

r

{\displaystyle r}

, and a return

R

{\displaystyle R}

over a length of time

Logarithmic or continuously compounded return

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

R

l o g

= ln ⁡

(

V

f

V

i

)

{\displaystyle R_{\mathrm {log} }=\ln \left({\frac {V_{f}}{V_{i}}}\right)}

and the logarithmic rate of return is:

r

l o g

=

ln ⁡

(

V

f

V

Annualization of logarithmic return

Under an assumption of reinvestment, the relationship between a logarithmic return

R

l o g

{\displaystyle R_{\mathrm {log} }}

and a logarithmic rate of return

r

l o g

{\displaystyle r_{\mathrm {log} }}

over a period of time of length

t

{\displaystyle t}

is:

R

l o g

=

r

l o g

t

{\displaystyle R_{\mathrm {log} }=r_{\mathrm {log} }t}

so

Returns over multiple periods

When the return is calculated over a series of sub-periods of time, the return in each sub-period is based on the investment value at the beginning of the sub-period. Suppose the value of the investment at the beginning is

A

{\displaystyle A}

, and at the end of the first period is

B

{\displaystyle B}

. If there are no inflows or outflows during the period, the holding period return

R

1

{\displaystyle R_{1}}

in the first period is:

R

1

=

B − A

A

{\displaystyle R_{1}={\frac {B-A}{A}}}

1 +

R

1

Arithmetic average rate of return

The arithmetic average rate of return over

n

{\displaystyle n}

time periods of equal length is defined as:

r ¯

=

1 n

∑

i = 1

n

r

i

=

1 n

(

r

1

+ ⋯ +

r

n

Geometric average rate of return

If compounding is performed, (i.e. if gains are reinvested and losses accumulated), and if all periods are of equal length, then using the time-weighted method, the appropriate average rate of return is the geometric mean of returns, which, over n periods, is:

r ¯

g e o m e t r i c

=

(

∏

i = 1

n

( 1 +

r

i

)

)

1 n

− 1 =

∏

i = 1

n

( 1 +

r

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.