*I don’t see it.*

*Here’s a simple example where this method seems to be telling you that you are a good investor while you’re losing money. I’m following the method as described in the Knowledgebase, Part 3 (https://boards.fool.com/knowledgebase-newly-revised-part-3-3…).*

Hi IGU,

OK, I just worked through Saul’s calculation method and can say that the return number it kicks out is identical to 1 + TWR, but without the % symbol. Let me work this through, demonstrating this, and then I’ll turn to your question.

Quoting from that page: *Here’s how to calculate your overall returns ignoring cash flow in or out. Say you start the year with $14,000. You want to equate that with 100% and calculate gains and losses from there. So you ask yourself “What number (factor) would I multiply $14,000 by to get 100?”*

*By simple arithmetic we have 14000 x F = 100*

*And thus F = 100/14000 = .0071428*

*Sure enough 14,000 x .0071428 = 100*

What this is doing is converting the $14,000 into 100 (which is equivalent to setting it to 100% but without the % symbol) and finding the multiplier (what Saul labels the “factor”). If you start with $14,000 and equate that to 100, then what must you multiply $14,000 by to get 100? F = 0.0071428571 (it goes on, but he rounded it to 0.0071428).

*Now say three weeks later you have $14,740 and you want to see how you are doing, you multiply that number by .0071428 and you get 105.3 (so you are up 5.3%).*

This is fine. 105.3 / 100 - 1 = 5.3% growth. Same as $14,740 / $14,000 - 1 = 5.3% growth.

So, without adding cash, you just take your ending balance, divide by your beginning balance and subtract one to get the return. Or you multiply your ending balance by F and then divide by the beginning 100 and subtract one to get the return, which is the same number. Two ways to do the same thing.

Now here’s the cash flow correction part: *Now say you add $2300 of fresh money, but you don’t want that to screw up your estimate of how well you are doing.*

*You add the $2300 to the $14,740 and get $17,040 which is your new balance that you are investing with. That’s your new starting point. It doesn’t affect how you’ve done up to here. You haven’t suddenly done better because you added money. You can’t still multiply by .0071428 because you’d get 121.7 and it would look as if you were up 21.7%, when you are really only up 5.3%.*

*So you need to change your factor to make it smaller so it will still reflect just the 5.3% gain you’ve made so far.*

*F x 17,040 = 105.3*

*F = 105.3/17,040 = .0061795*

*And that’s your new factor. If you multiply it by 17,040, sure enough you get 105.3. Now you continue to see how you will do for the rest of the year.*

After adding the cash – because you can’t count it as investment return – the balance is bigger, so the factor has to be smaller in order to remain at 105.3. This method corrects for the cash addition here.

That making F smaller so that you still have 5.3% return at this point is exactly what the TWR calculation is doing. Except TWR corrects for it by adding the cash to the balance – after calculating the return – and creates “$$ After CF” to use as the starting point for the next holding period return.

Finally, here’s the proof that both methods give the same answer:

*If a little later you are at $18,000, you multiply 18,000 by .0061795 and you get 111.2, so you know that your investing is now up ***11.2%** for the year.

Note, that’s the total return over the entire period, what TWR itself produces. Does TWR give the same result? Yes.

```
Period $$ Before CF CF In/Out $$ After CF HPR TWR
0 $ 0 $14,000 $14,000 N/A 0.0%
1 $14,740 $ 2,300 $17,040 5.3% 5.3%
2 $18,000 $ 0 $18,000 5.6% **11.2%**
```

Remember, HPR = (Previous $$ After CF) / (Current $$ Before CF) - 1 and TWR is the chaining of the individual (1 + HPRs) and minus 1 at the end. (1.053)*(1.056)-1 = 11.2%

Now, let’s look at your example.

*Start with $100, in what we like to call a starter position. Through a combination of brilliant choices and dumb luck you double your money to $200 in a few months.*

*Your initial factor is 1. At this point your factor is .5.*

I’m sorry to say that you’ve made a mistake, here. Your factor at this point is still 1. You only change the factor at points where you have a cash flow event and, at this point, you don’t have that. Therefore, F is 1 and what I’ll call the F-return number is 200, not 100. Remember, you just doubled your money.

*You decide you like this result so you go bigger by adding $1000 to the account. You now have $1200 and as you must continue to show a 100% gain, your factor becomes 200 / 1200 = 1/6 = .16666…*

Here, you’re right. The previous F-return number, 200, is divided by the after-cash-flow balance of $1,200 to give a new F = 0.1666667.

Now, to the crux of your question. *But over the next few days your luck runs out and you lose 1/4 of your money, leaving you $900. But you aren’t discouraged because you can see that 900 * 1/6 = 150, so you still have a ***50% return on your money**. And the index was up only 5%! Brilliant you!

*So why do you have only [$900] left when you put in $1100?*

The weird thing is that you can end up with less money, but have a positive TWR (or F-return, if we can use that phrase for the 200 or the 150, subtracting 100%). Why?

Well look at it. TWR = (1 + 100% gain) * (1 - 25% loss) - 1 = 2 * 0.75 - 1 = 0.5 or 50% return.

You were brilliant during the first period and earned 100%. You had bad luck and lost 25% the second period – both of these results of your investment **decisions**, not the cash deposited – and you ended up with a TWR of 50%. The reason you ended up with less money at the end is because your decisions had bad results after adding a big chunk of money to the account. That’s all.

You didn’t have a “50% return on your money” (the part I bolded), you had 50% time-weighted returns on your investments. You got a 100% return on your investments in the first period and a (25%) return on your investments in the second period. They’re not returns on your money, they’re returns on your investments.

It’s just your misfortune to have gotten bad results in the second period when you had more money to invest.

Suppose you added only $200 instead $1,000. With nothing else changed, you would have begun the second investment period with $400 and lost 25% of that, ending with $300.

The after-cash-flow F would be 200 / 400 = 0.5 (instead of 0.1666667). And the F-return would be $300 * 0.5 = 150, or 50% gain overall.

Here, you’ve put in a total of $300 (the original $100 and the $200 addition) and ended up with that same amount at the end, yet you still have an overall TWR or F-return of 50% on your investments to date.

Work it through with adding only $50 cash instead of $1,000 or $200. F = 200 / 250 = 0.8. You lose 25% and end with $187.50. $187.50 * 0.8 = 150 or 50%. It’s always the same total TWR or F-return of 50% on your investments to date, no matter how much cash you add. But here, you’ve put in $150 and end with a cash gain of $37.50.

It’s a subtle distinction, but one with very big consequences. TWR and the F-return are returns on investments, not returns on money.

What you’re calling “return on money” could be called a cash-on-cash return, comparing the end result to the total amount deposited. This type of return depends on **both** the amounts deposited *and* the performance the investments. In your example, you have a $900 / $1100 - 1 = (18.1%) return; in my first example we have a $300 / $300 - 1 = 0% return; and in my second example we have a $187.50 / $150 - 1 = 25% return. TWR and F-return are independent of the cash flows.

If the same investment **choices** were made and the same investment results obtained, then you will always end with the same 50% overall total TWR or F-return, no matter what amounts were deposited or withdrawn. That’s why these numbers answer the question, “How good am I at investing?”

I hope that clears it up.

Cheers,

Jim