return. If the individual security return matches the share price return, the transaction timing effect is zero. See Example 4 below, which illustrates this feature of the time-weighted method. Example 4 Let us imagine an investor purchases 10 shares at 10 dollars per share. Then the investor adds another 5 shares in the same company bought at the market price of 12 dollars per share (ignoring transaction costs). The entire holding of 15 shares is then sold at 11 dollars per share. The second purchase appears to be badly timed, compared with the first. Is this poor timing apparent, from the time-weighted (holding-period) return of the shares, in isolation from the cash in the portfolio? To calculate the time-weighted return of these particular shareholdings, in isolation from the cash used to purchase the shares, treat the purchase of shares as an external inflow. Then the first sub-period growth factor, preceding the second purchase, when there are just the first 10 shares, is: and growth factor over the second sub-period, following the second purchase, when there are 15 shares altogether, is: so the overall period growth factor is: and the time-weighted holding-period return is: which is the same as the simple return calculated using the change in the share price: The poor timing of the second purchase has made no difference to the performance of the investment in shares, calculated using the time-weighted method, compared, for instance, with a pure buy-and-hold strategy (i.e., buying all the shares at the beginning and holding them until the end of the period). Comparison with other returns methods Other methods exist to compensate for external flows when calculating investment returns. Such methods are known as "money-weighted" or "dollar-weighted" methods. The time-weighted return is higher than the result of other methods of calculating the investment return when external flows are badly timed - refer to Example 4 above. Internal rate of return One of these methods is the internal rate of return. Like the true time-weighted return method, the internal rate of return is also based on a compounding principle. It is the discount rate that will set the net present value of all external flows and the terminal value equal to the value of the initial investment. However, solving the equation to find an estimate of the internal rate of return generally requires an iterative numerical method and sometimes returns multiple results. The internal rate