Return can arise either from income (or cash receipts) or from capital gains. There are different definitions of return, including holding period return, arithmetic return, geometric return, etc.

## Holding period return

Holding period return is the total return earned on an investment over its whole holding period expressed as a percentage of the initial value of the investment. It is calculated as the sum capital gain and income divided by the opening value of the investment.

HPR=\frac{P_1-P_0+I_1}{P_0}

Holding period return is not a standardized measure of return. A given holding period return might be for a single day or 5 years. Holding period returns for multiple periods can be connected by chain-linking it as follows:

HPR=(1+{HPR}_1)(1+{HPR}_2)...(1+{HPR}_n)-1

## Arithmetic return

The arithmetic average return is the return on investment calculated by simply adding the returns for all sub-periods and then dividing it by the total number of periods. It overstates the true return and is only appropriate for shorter periods.

Arithmetic\ Return=\frac{R_1+R_1+...+R_n}{n}

## Geometric return

Geometric return is the average rate of return on an investment which is held for multiple periods such that any income is compounded. In other words, the geometric average return incorporates the compounding nature of an investment.

The arithmetic average return assumes that the initial investment is the same, which is not accurate because investment balance changes as we move from one period to another. Hence, the geometric mean is a better measure of multi-period return.

Geometric\ Return\ =\ \sqrt{(1\ +\ R_1)\ \times\ (1\ +\ R_2)\ \times\ ...\ \times\ (1\ +R_n)}\ -\ 1

## Annualized return

To compare return on different investments, we often need to quote it in annual terms. Following is the general expression for annualized return where rP is the periodic return, c is the total number of periods in a year:

r_{annual}={(1+r_P)}^c-1

However, such annualization assumes that the same pattern of return can be sustained over the whole year.

## Gross return vs net return

Gross return is the return earned by an investment vehicle (such a mutual fund) before deduction of management fees, custodial fees, and taxes, etc. However, trading expenses (transaction costs) are deducted.

Net return is a measure of what the investment vehicle has earned for the investor. It equals gross return minus all fees and administrative expenses.

## Pre-tax and after-tax nominal return

Most of the measures of return discussed above and quoted by investment professionals and investors do not deduct adjustment for inflation or income taxes. It is because taxes depend on the status of the investor, type of investment, investment horizon, etc. Such a return is called pre-tax nominal return.

When taxes are deducted, the return is called after-tax nominal return.

## Real returns

A nominal return (rn) can be thought of as a sum of risk-free rate (rf), expected inflation (π) and risk premium (RP). The risk-free interest rate is compensation for postponing consumption while expectation inflation is compensation for loss of purchasing power and risk premium accounts for any potential loss of investment.

Real return (r) measure return which an investor ultimately earns (after adjusting for any loss of purchasing power). The relationship between these returns is as follows:

(1\ +\ r_n)\ =\ (1\ +\ r_f)(1\ +\ RP)(1\ +\ \pi)

(1\ +\ r_n)\ =\ (1\ +\ r_f)(1\ +\ \pi)

r\ =\ (1\ +\ r_n)/(1\ +\ \pi)\ -\ 1

## Leveraged return

The measure of return discussed above assumed that the whole investment is made by the investor from its own money. However, if an investment strategy uses debt, the return to equity magnifies, i.e. both gains and losses are multiplied. Leverage return measures the return to equity in such situations.

## Historical return and expected return

Historical return is what has been earned by an asset class in the past while the expected return E(R) quantifies the minimum nominal return that would convince a marginal investor to invest in an asset. It is given by the following expression:

E(R)\ =\ [1\ +\ E(r_f)] × [1 + E(π)] × [1 + E(RP)] - 1

Even though historical return and expected return may not be equal in the short-run, in the long-run we can expect the future return to be equal to the historical return.