required rate of return), the investment adds value, i.e. the net present value of cash flows, discounted at the cost of capital, is greater than zero. Otherwise, the investment does not add value. Note that there is not always an internal rate of return for a particular set of cash flows (i.e. the existence of a real solution to the equation depends on the pattern of cash flows). There may also be more than one real solution to the equation, requiring some interpretation to determine the most appropriate one. Money-weighted return over multiple sub-periods Note that the money-weighted return over multiple sub-periods is generally not equal to the result of combining together the money-weighted returns within the sub-periods using the method described above, unlike time-weighted returns. Comparing ordinary return with logarithmic return The value of an investment is doubled if the return = +100%, that is, if = ln($200 / $100) = ln(2) = 69.3%. The value falls to zero when = -100%. The ordinary return can be calculated for any non-zero initial investment value, and any final value, positive or negative, but the logarithmic return can only be calculated when . Ordinary returns and logarithmic 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 percent changes are high. For example, an arithmetic return of +50% is equivalent to a logarithmic return of 40.55%, while an arithmetic return of −50% is equivalent to a logarithmic return of −69.31%. Advantages of logarithmic return: Logarithmic returns are symmetric, while ordinary returns are not: positive and negative percent ordinary returns of equal magnitude do not cancel each other out and result in a net change, but logarithmic returns of equal magnitude but opposite signs will cancel each other out. This means that an investment of $100 that yields an arithmetic return of 50% followed by an arithmetic return of −50% will result in $75, while an investment of $100 that yields a logarithmic return of 50% followed by a logarithmic return of −50% will come back to $100. Logarithmic return is also called the continuously compounded return. This means that the frequency of compounding does not matter, making returns of different assets easier to compare. Logarithmic returns are time-additive, meaning that if and are logarithmic returns in successive periods, then the overall logarithmic return is the sum