This is the first post in a three-part series on investment performance evaluation. The series is broken up into three parts: 1) Performance Measurement, 2) Performance Attribution, and 3) Performance Appraisal.

Why all the fuss? Why not just look at a portfolio’s change in value and be done with it? Do we really need a three-part explanation? Maybe not, but performance evaluation allows us to examine the effectiveness of our investment process. This feedback loop gives us a systematic way of judging our decisions and improving on them, which is what investment theory is all about.

Today’s post deals with performance measurement, which is the procedure of calculating returns for an investment account. It’s the boring math part, but it is also the starting point for all other analysis.

### The Simple Case

When measuring investment performance, the key is to isolate the change in an account’s value due solely to investment-related sources, i.e. capital appreciation and income. If there are no external cash flows into or out of an account, i.e. a deposit into a savings account or a withdrawal for a purchase, then any change in the account value will already be isolated on investment results.

In this simple case, calculating the rate of return is simple. We just take the ending value minus the beginning value, divided by the beginning value:

r = PV

_{1}– PV_{0}/ PV_{0}

For example, if our portfolio is worth $1,000 at the beginning of the month and $1,090 at the end of the month, assuming no external cash flows and the reinvestment of all income, our rate of return would be:

r = 1090 – 1000 / 1000 = 0.09

What if we added or subtracted money from the account? For example, what if we withdrew or deposited $50 to the account during that month? These external cash flows complicate things, since we have to deal with the additional earnings or losses on the new asset base. Our simple formula no longer works. Luckily, if the cash flows occur at the start or end of the period, there is still a simple way to calculate the rate of return, just add or subtract from the beginning or ending balances:

r = (PV

_{1}+- CF) – PV_{0}/ PV_{0}

r = PV_{1}– (PV_{0}+- CF) / PV_{0}

So whenever possible, it make sense to limit contributions and withdraws to the beginning or end of an evaluation period. When cash flows are allowed to occur anytime within an evaluation period, the calculation becomes more tricky, as we will see.

### Time-Weighted Return

Remember that we are only interested in the investment-related growth in an account for any given evaluation period. If we deposit $1,000 dollars to an account that had a beginning balance of $1,000, and then look at the portfolios value in isolation, it would look like we had a 100% return for the period.

To get around this problem, we can look at the portfolios growth rate as being applied to a single dollar invested at the start of the period. This is the concept behind a time-weighted rate of return. This measure requires that our account be valued every time an external cash flow occurs. Once these sub-period returns are calculated, they are combined into one rate of return that the single dollar would have earned.

For example, say we started the evaluation period at $1,000 and ended at $1,090 again. However, 6 days into the month, we contributed $50 to the account and the account was worth $1,060 at the time. Let’s break the performance down for the two sub-periods, the first 6 days of the month and the rest of the month.

period1 = (1060 – 50) – 1000 / 1000 = 0.01

period2 = 1090 – 1060 / 1060 = 0.028

Now we can combine the sub-period returns using a process called chain-linking. This can be thought of as what one invested dollar would have earned in the first sub-period, and then what that same dollar plus appreciation would have earned in the next sub-period, and so on.

r = (1 + r

_{1}) * (1 + r_{2}) *… * (1 + r_{n}) -1

r = 1.01 * 1.028 – 1 = 0.038

### Money-Weighted Return

Unlike time-weighted return, the money-weighted return looks at the growth rate of all funds invested over the period. This is also known as the internal rate of return and takes into account the different amounts earned on the different asset bases throughout the period. Money-weighted return is calculated as the growth rate that will link the ending value to the beginning value plus all intermediate cash flows. Basically, we lay out all the information and then solve for R. Let’s look at the same set up as the above example and say that a month (m) has 30 days.

PV

_{1}= PV_{0}(1 + R)^{m }+ CF_{1}(1 + R)^{m-d }+ … + CF_{n}(1 + R)^{m-d }1090 = 1000(1 + R)^{30 }+ 50(1 + R)^{30-6}Through trial and error, we get that R = 0.00126

This is the value of the daily return for the month. To get the return for the full month we can just calculate:

r = (1 + 0.00126)

^{30 }– 1 = 0.038

### Comparing Time-Weighted and Money-Weighted

Recall that time-weighted return represents the growth of a single dollar and money-weighted return represents the average growth of all money in an account. This makes money-weighted returns highly sensitive to the timing and value of external cash flows. For example, if $1,000,000 was deposited into the account instead of $50 we would get:

TWR = 0.038

MWR = 0.035

This difference between the two measures becomes more extreme over longer time periods. Basically, if funds are contributed right before strong performance, it will positively affect the money-weighted return more than the time-weighted return. For this reason, time-weighted return is often the better choice when you have no control over the timing or amount of external cash flows into an account. If you do have control, for example, if you add a bunch of funds to the account to capitalize on a great investment opportunity, then money-weighted return makes more sense.

### Linked Internal Rate of Return

The main draw back of time-weighted returns is that it requires constant account valuations every time an external cash flow occurs. Money-weighted returns are simpler in that the portfolio only needs to be valued at the beginning and end of the period. Linked internal rate of return (LIRR) was developed to combine time-weighted’s immunity to cash flows with money-weighted’s ease of calculation.

LIRR attempts to approximate time-weighted returns by chain-linking money-weighted returns over reasonable time intervals. For example, if we calculate the money-weighted return every week in our month evaluation period, we could then just combine them for our month time-weighted return. This turns out to be a very accurate approximation of time-weighted returns, without valuing the portfolio at every cash flow.

### Annualized Return

To compare these results we need to standardize them, which is why returns are almost always reported in annualized return. For example, if we made 3.8% in that one month period, annualized it would be:

annualized return = (1 + 0.038)

^{12 }– 1 = 0.56

However, we are extrapolating our one month period into the future which probably doesn’t make much sense. For this reason, annualized returns are usually used as a way to smooth out yearly returns in order to show the compounded average annual rate of return earned over multiple periods. This allows us to compare two portfolios, one that did well the first year and bad the seconds year, with one that did well the second year and bad the first year.

### Conclusion

Performance measurement is only one component of overall performance evaluation. It answers the simple question of how much money we made in a period. Next, we need to know why the account produced the measured performance, and finally whether the performance was robust of simply due to luck.