of the individual logarithmic returns, i.e. . The use of logarithmic returns prevents investment prices in models from becoming negative. Comparing geometric with arithmetic average rates of return The geometric average rate of return is in general less than the arithmetic average return. The two averages are equal if (and only if) all the sub-period returns are equal. This is a consequence of the AM–GM inequality. The difference between the annualized return and average annual return increases with the variance of the returns – the more volatile the performance, the greater the difference. For example, a return of +10%, followed by −10%, gives an arithmetic average 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 occurs 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 return of +100%, followed by −100%, has an average return of 0% but an overall return of −100% since the final value is 0. In cases of leveraged investments, even more extreme results are possible: A return of +200%, followed by −200%, has an average return of 0% but an overall return of −300%. This pattern is not followed in the case of logarithmic returns, due to their symmetry, as noted above. A logarithmic return of +10%, followed by −10%, gives an overall return of 10% − 10% = 0% and an average rate of return of zero also. Average returns and overall returns Investment returns are often published as "average returns". In order to translate average returns into overall returns, compound the average returns over the number of periods. The geometric average rate of return was 5%. Over 4 years, this translates into an overall return of: The geometric average return over the 4-year period was −1.64%. Over 4 years, this translates into an overall return of: The geometric average return over the 4-year period was −42.74%. Over 4 years, this translates back into an overall return of: Annual returns and annualized returns Care must be taken not to confuse annual with annualized returns. An annual rate of return is a return over a period of one year, such as January 1 through December 31, or June 3, 2006, through June 2, 2007,