Time-Weighted Return

I’ve been struggling a bit with keeping track of the performance of my portfolio and specific assets and asset classes that I own. I am not an accountant. My initial approach to measuring performance was to take the difference between current portfolio or holding value (V), subtract the amount I invested (I) and divide the difference with the amount I invested (I).

So if I invested USD 100 (I) and current value was USD 200 (V), then the return (=performance) is (200-100)/100 = 1 = 100%

That’s a valid approach, but the problem arises if there are multiple inflows and outflows of money. Deposits and withdrawals. Such events can make it quite tricky to keep track of the precise investment.

There are several methods to address this, like internal rate of return (IRR) and time-weighted return (TWR).

TWR negates the effects of cash flows (which distort growth rates). It does that by calculating returns for each holding period (i.e. a new holding period starts whenever a deposit or withdrawal happens) and link them together through multiplication to show how they compound over time. The rate for each sub-period is always isolated from the actual amounts that were deposited or withdrawn.

Let’s illustrate with a simple example.

Let assume the following deposits and withdrawals.

Date	Cash Flow	Portfolio Value
1/2019	10000	10000
2/2019	0	10420
3/2019	2000	12635
4/2019	0	12200
5/2019	-2500	10550
6/2019	0	10700

I can’t accurately calculate the return by taking the difference between initial portfolio value and current value, i.e. 10700 – 10000 / 10000 = 7%. The deposits and withdrawals distort the growth rate.

Now, what TWR does is calculate a return for each holding period. I will do that by calculating it for each month (regardless of whether there was a deposit/withdrawal or not). I use this monthly approach for each asset (class) that I hold. Only when using the same holding periods I can effectively compare performance among different assets.

Anyway, let’s look at the returns for each holding period. The formula for calculating it is as follows:

where:

• RN = Sub-period Return
• EMV = Ending Market Value
• BMV = Beginning Market Value
• CF = Cash Flow

Date	Cash Flow	Portfolio Value	RN
1/2019	10000	10000	N/A
2/2019	0	10420	4.2%
3/2019	2000	12635	1.7%
4/2019	0	12200	-3.4%
5/2019	-2500	10550	8.7%
6/2019	0	10700	1.4%

For example, the 1.7% (third row) is the result of the following calculation:

(12635 / (10420 + 2000)) – 1 = 1.7%

And the 8.7% (fifth row) is the result of the following calculation:

(10550 / (12200 – 2500)) – 1 = 8.7%

As you can see, the cash flows are being negated. The cool thing about it is that the returns for each holding period are solely the result of market conditions/developments and my own investment decisions (but excluding cash flows).

Now, from here we can calculate the TWR using the following formula:

TWR = [(1 + RN) × (1 + RN) × … − 1] × 100

TWR = [((1 + 4.2%) × (1 + 1.7%) × (1 – 3.4%) × (1 + 8.7%) × (1 + 1.4%)) – 1] × 100

TWR = 12%

This 12% is the result of compounding the return rates (for the sub periods).

To calculate a monthly performance (equivalent to annualizing a rate of return) I can use the following formula:

([(1 + RN) × (1 + RN) × … ] ^ (1/number of months)) – 1

If I calculate it for June 2019, the result is:

[1.12 ^ (1/6)] – 1 = 1.9%

The higher this monthly number, the better my portfolio performs.

What TWR doesn’t tell me

It is important to understand that TWR looks at the periodic returns after cash flows have been negated.

As such this is a good method for comparing how different investment managers are performing regardless of their clients’ deposits/withdrawals. Or for me: How my different asset classes are performing regardless of deposits/withdrawals.

The problem with TWR is that it can be a bit counter intuitive. Imagine that I made another deposit in July of $1,000,000.00 and that the last periodic return would be -2%. That would mean a loss of $20K.

Overall I would have had a loss. But TWR would still show me a positive number.

Should I use TWR at all?

I can use TWR to compare performance of different asset classes, but TWR does not reflect how much money I made or lost.

If I want to have a more accurate measure of how much I made I think I need to resort to the internal rate of return (IRR) method instead. I will dedicate another post to that. Until then you may find TWR numbers across my blog.

How about you?

This is a very mathematical approach to keeping track of performance. Do you use methods like this? If yes, how do you do it?

Any flaws in the above? I am not an accountant so any feedback would be highly appreciated.