based on the investment value at the beginning of the sub-period. Suppose the value of the investment at the beginning is , and at the end of the first period is . If there are no inflows or outflows during the period, the holding period return in the first period is: is the growth factor in the first period. If the gains and losses are reinvested, i.e. they are not withdrawn or paid out, then the value of the investment at the start of the second period is , i.e. the same as the value at the end of the first period. If the value of the investment at the end of the second period is , the holding period return in the second period is: Multiplying together the growth factors in each period and : is the holding period return over the two successive periods. This method is called the time-weighted method, or geometric linking, or compounding together the holding period returns in the two successive subperiods. Extending this method to periods, assuming returns are reinvested, if the returns over successive time subperiods 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: If the returns are logarithmic returns, however, the logarithmic return over the overall time period is: This formula applies with an assumption of reinvestment of returns and it means that successive logarithmic returns can be summed, i.e. that logarithmic returns are additive. In cases where there are inflows and outflows, the formula applies by definition for time-weighted returns, but not in general for money-weighted returns (combining the logarithms of the growth factors based on money-weighted returns over successive periods does not generally conform to this formula). Arithmetic average rate of return The arithmetic average rate of return over time periods of equal length is defined as: This formula can be used on a sequence of logarithmic rates of return over equal successive periods. This formula can also be used when there is no reinvestment of returns, any losses are made good by topping up the capital investment and all periods are of equal length. 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