Ever wonder why your brokers profit and loss statements don’t always add up with your calculations? This normally due to the different methods of measuring overall portfolio returns.

In this article we are going to discuss two of the most commonly used methods which you can apply to your portfolio holdings, that of the money-weighted rate of return and that of the time weighted rate of return. Both methods are officially recognised by the CFA charter.

Let’s start with the **money-weighted rate of** **return**. This method takes into account all cash inflows and outflows of a portfolio and gives a defined measure of the internal rate of return (IRR). Any money added at the beginning of the account is considered a money inflow, and all money withdrawn from the account as *well as the remaining balance at the end of the period *are considered outflows.measuring overall portfolio returns

We can use a basic model to illustrate what this means. Suppose an investors buys 1000 stocks of Vertu Motors at the beginning of the period for 30p each. After 1 year the investor still likes the stock and decides to buy another 1000, only this time the price is now 35p a share. At the end of year two the investor sell all 2000 of the stocks for 45p a share, additionally at the end of each period Vertu paid a 3p dividend on each stock.measuring overall portfolio returns

So

At T=0 1000 shares were bought at 30p a share, meaning a cash inflow of +£300

At T=1 another 1000 shares were purchase for 35p a share, however a £30 dividend was also paid, meaning a cash inflow of (1000 x 0.35) – 30 = +£320

At T=2 they sold all their shares for 40p a share, and received a 3p dividend on the 2000 shares, meaning (2000 x 0.4) + (2000 x 0.03) = -£860

From this we have our cash flow time series

CF0 = +£300

CF1 = +£320

CF2 = -£860

The formula for working out the final return does look a little scary:

£300 + (£320/(1+r)) = (860/(1+r)²)

Fortunately we live in a world where there are plenty of online resources that will work this out for us. I personally like to use a financial calulator to input cash flows from a time series to work out the IRR. The answer to this formula is

**24.18%** overall return

Now let’s see what happens when we use the **time weighted rate of return** to measure the portfolio’s actual return. Time weighted method has the benefit of measuring compound growth.

To do this we need to measure each individual holding periods growth. In the above example we have 2 holdings periods, year 1 and year 2.

Holding period 1: Beginning value = £300

Dividends paid = £30

End Value = £350 (1000 shares x 35p)

HPR1 = ((£350 + £30) / £300) – 1 = 26.66%

Holding period 2: Beginning value = £700 (2000 x 35p)

Dividends paid = £60

End Value = £800 (2000 shares x 40p)

HPR2 = ((£800 + £60) / £700) – 1 = 22.86%

Now that we have both these figures we need to find the total compounded annual growth rate that would have produced the total return that equals the return over the 2 holding periods. The formula for this is:

(1+ time-weighted rate of return)² = (1.2666)(1.2286)

We can rearrange this to get:

time-weighted rate of return = √((1.2666)(1.2286)) – 1

= **24.47%** overall return

As we can see both methods are considered correct, yet they yield different results. Although the difference is negligible, this was only demonstrated across a small sample portfolio. The larger the cash flows, the wider the differences in measurements will be.

As a rule the time weighed rate of return is considered superior as money weighted returns can be influenced by timing. Measuring money weighted returns before a period of relatively high returns will return a higher figure and can therefore be ‘massaged’.