One would assume that determining the return in a portfolio would be simple and easy: Just divide the
change in value by the beginning value. In the most simple of the cases, this can
be true. But external cash flows (cash flows in and out of a portfolio, rather
than those generated by the investments themselves) can make things more
difficult.
Whenever there is an external cash flow such as a deposit to or
withdrawal from the portfolio, the return should be calculated. Then, each period
between cash flows (or ending at specified dates such as year-end) can be
linked together in a process called chain-linking. This process is used to
determine the time weighted rate of return (TWR).
Consider the following exhibit, which shows the change in
portfolio value before and after cash flows.
The portfolio starts the year at $100,000 and ends at $118,000 –
so its return is 18%, right? Not so fast! All during the year (for simplicity
it is assumed to be on the last day of the month after the ending value is
calculated) there are deposits and withdrawals. The 118,000 reflects not only
the investment return, but these external cash flows as well.
The proper way to calculate return in this case is to take the
change in value from the beginning to the end of each month (before the cash
flow). So, in the first month the return is (110,000 – 100,000)/100,000 = 10%.
Next, the cash flow is added or subtracted from the ending value
to arrive at the following month’s beginning value, and that month’s return is
calculated the same way.
Returns can be linked geometrically. To do this, 1 is added to
each return and they are multiplied together. At the end, 1 is subtracted from
the final product. So the linked return for the three months ending in March
are (1.10 X 1.043 X 0.965)Â – 1 = 10.7% (which is slightly off from the
10.8% in the exhibit due to rounding).