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What does "time weighted average capital" mean in Newsletter No 39?

If we want to measure our return for our portfolio over the year, we will take the increase in its value over the year as a percentage of the capital at the start of the year.

So, if we invest $100,000 and by year end it has grown to $110,000, our return is

10,000/100,00 x 100 = 10%

Now, there is an assumption here, which is that we did not add or subtract any capital from our portfolio during the year.

Suppose that we invested $100,000 at the start of the year, but exactly half way through the year we withdrew $50,000. In this situation, we have $100,000 working for us for six months and only $50,000 working for us for the other six months. If the portfolio still increased by $10,000 over the year, unless we change the starting capital in some way it will appear as though our return is the same at 10%, yet we have clearly done better, since we only needed $50,000 for the second six months to achieve that return.

This is a very simple case. It can get far more complicated if capital is added and subtracted several times through the year. However, let us look at the simple case first. What we need to calculate is the average capital for the year. Since the reduction took place exactly half way through the year, we could simply add $100,000 and $50,000 and divide by two to get the average of $75,000.

What we have done in this simple case is to time weight the average capital, except that the example is so simple that we can do it mentally. In fact what we are doing is to take the capital in each half of the year and divide it by the fraction of the year for which it is employed and then the two products together to get a time weighted average capital. All time weighting means is that we give the capital employed in each part of the year an importance related to the time for which it was employed.

So, the long hand way to have done our simple calculation would look like this:

First six months capital = $100,000 x 0.5 = $50,000

Second six months capital = $ 50,000 x 0.5 = $25,000

Time Weighted Average capital = $75,000

Now consider a more complex case. Suppose we start the year with $100,000, take out $50,000 after two months and then add $300,000 after eight months. At the end of the year, the portfolio is worth $325,000. What is the return?

The calculation would be like this:

100,000 x 2/12 = 16,667

50,000 x 6/12 = 25,000

300,000 x 4/12 = 100,000

TWAC = 141,667

Return = 25,000/141,667 x 100 = 17.65%

Of course, this is still a very simple case, because it assumed that we made changes in the capital at the exact ends of a month. In a real-life situation, this may not be the case. In those situations, the time weighting will be done in days capital was employed divided by 365 (or 366 in a leap year). It can be argued that even the monthly calculation above should have been done in days, since all months are not the same length.