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# Comparing Your Stock Market Results: A 7-Step Guide To Performance Measurement

*An explanation of “internal rate of return” and other tricky ways to evaluate portfolios.*

This guide to measuring investment results aims to do two things: (a) show you the arithmetic, which is sometimes messy; and (b) make you a little bit skeptical of any performance claim.

As to point (b): The problem is not that money managers lie, but rather that they fudge. There is sometimes more than one right answer to a question about performance. They pick the one that looks good.

**1. Percentage gain.**** **

In two years, your $100,000 turned into $206,000. Gain, 106%. So, you made 53% per year?

There is some, very faint, validity to that kind of average, but no stock picker would dare put it in a Securities & Exchange Commission filing. It would be considered deceptive.

**2. Percentages, year by year.**

In the first year you went, let us suppose, from $100,000 to $180,000, and in the second from $180,000 to $206,000. The two annual moves are 80% and 14.4%. Those two numbers average to 47.2%. Is that your average annual return? It is more defensible than 53%, but still not entirely honest.

With a more extreme example, you can see what is flaky with simple arithmetic averages. Suppose the two performances were +100% one year followed by -100% the next. The simple average would be 0%, implying a sideways move, but the cumulative result would be something else entirely. It would be a wipeout.

**3. Compound annual return.**

With this method, you disregard the $180,000 intermediate value en route from $100,000 to $206,000 and consider only what happened, over the course of two years, to a starting dollar. It was multiplied by a factor of 2.06. Take the square root of that quantity and subtract 1. Using the ^ character on your calculator, you get 2.06^(1/2) – 1 = 43.5%. This is a pretty honest number.

What if it took a decade to turn $100,000 into $206,000? Insert 1/10 where we had ½. Try this on your phone. You arrive at a compound annual return of 7.5%.

If money managers could get away with it, they’d quote the simple average set out in No. 2 rather than the compound annual return. The simple average is always a more flattering number. (The only exception is if the portfolio enjoys exactly the same gain each year.) Don’t let them get away with it.

**4. Compared to what?**

A common practice is to compare a U.S. stock portfolio to the S&P 500 index. But there’s a better ruler: an index of 3,500 stocks. The way to track that index is to keep an eye on the Vanguard Total Stock Market Index Fund, available as an exchange-traded fund with the ticker VTI.

Go to Morningstar.com, search for VTI and select the Performance tab. You’ll see returns (annualized for the longer periods) for the calendar year to date and the trailing 1, 3, 5, 10 and 15 years. You’re looking at the performance of a fund, not an index, but it’s good enough for most purposes. The difference between the two is tiny because the fund’s expenses are tiny.

The important thing about a Morningstar return is that it includes reinvested dividends. Whatever your yardstick is, make sure that it’s a fair match. Both the yardstick and the portfolio should include dividends.

And whenever a money manager compares past results to an index, think about how the index was selected. Did the manager make the selection at the end of the measurement period? Decide, late in the game, that he was trying to beat a small-stock index rather than a broad-market index? Be skeptical.

**5. Money added.**

Suppose you started with $100,000, added $50,000 over two years and ended with $206,000. So your profit is only $56,000. What’s the percentage return? There are several ways to calculate it.

(a) Subtract the added money from the ending value. Turning $100,000 into $156,000 means a gain of 56%, for a compound annual return of 24.5%.

(b) Add the added money to the starting value. Turning $150,000 into $206,000 represents a gain of 37.3%, or a compound annual 17.2%.

(c) Go halfsies. Add $25,000 to the starting value and subtract $25,000 from the ending value. You turned $125,000 into $181,000, for a gain of 44.8%, or a compound annual 20.3%.

Very different results.

If the $50,000 was injected close to the ending date, (a) is the fair way to calculate. If it was injected near the start, use (b). If it came either right in the middle, or in equal monthly contributions over the 24 months, method (c) is best.

What about pulling money out? Just reverse the signs. If you took out $50,000 while seeing the balance climb from $100,000 to $206,000, then your profit was $156,000. The performance can be judged as taking $100,000 to $256,000 or taking $50,000 to $206,000 or taking $75,000 to $231,000. The resulting annualized performance numbers are again very different: 60%, 103% and 75%.

**6. Internal rate of return.**

Choosing one of the three methods above seems so arbitrary. Do you like formulas that sound more deterministic? Then you’ll love IRR. This is defined as the discount rate that makes the present value of the cash flows on an investment sum up to zero.

For an example, let’s have that $50,000 arrive in the dead center of the two-year horizon. Our cash flows are: Year 0, $100,000 in; Year 1, $50,000 in; Year 2, $206,000 out.

What discount rate zeroes out this string of cash flows? Let’s try 20.7% and see what happens. The present value of today’s $100,000 is $100,000. The present value of next year’s contribution is equal to $50,000 divided by the sum of 1 and the discount rate. Discounting at 20.7% would make that future sum equivalent to $50,000/1.207 in today’s money, or $41,400.

The present value of the ending withdrawal would be calculated in a similar way, but with the 1.207 factor used twice in the divisor. We would have $206,000/(1.207^2) = $141,400. Adding the three present values we have: -$100,000 - $41,400 + $141,400 = 0. Check! The 20.7% IRR is not far from the 20.3% we had with the halfsies approach.

Now, there are all manner of experts who take IRRs as gospel. You shouldn’t. I’ll illustrate why.

For the next example, we’re going to have you open a brokerage account with $100,000. You’re on a hot streak and a year later you’ve got a ton of money. You pull out $206,000 for a vacation home and let the balance grow. Exactly two years in you add $105,280 from a Christmas bonus and then invest the whole account in one exciting stock. Which goes bust, leaving you with nothing.

Cash flows: -$100,000 at Year 0, + $206,000 at Year 1, -$105,280 at Year 2. What’s your annualized performance? You’ve netted $720, so presumably it’s a positive number.

Well, 12% solves the equation, because -$100,000 + $206,000/1.12 - $105,280/(1.12^2) = 0. Surprise: -6% also works, because -$100,000 + $206,000/0.94 - $105,280/(0.94^2) = 0.

So IRR is telling you that you are a good stock picker, with a 12% annual return, and simultaneously telling you that your are an idiot with a negative 6% annual return. Gospel truth? Not quite.

Here’s another perverse IRR example. Year 0, put in $100,000; Year 1, pull out $200,000; Year 2, put in $105,000, followed by another wipeout on a bad stock. You’re down $5,000 net. Presumably the internal rate of return is negative? Nope. Both solutions to the equation are imaginary.

How would that go at a bank? “What’s my checking balance?” “Can’t tell you, sir. It’s an imaginary sum of money.”

IRR has its uses, and it often yields a reasonable answer. Your broker will probably display an IRR for your portfolio if you click on the Performance tab. But it’s best to see it as no more than a useful tool, one of many ways to look at how well you’re doing.

You might be curious about what caused that last hypothetical portfolio to have imaginary IRRs. It has to do with the account going to $0. So long as the account starts and ends with a positive balance, there will always be at least one real IRR (for which fact we can thank the Bohemian Bernard Bolzano), but there is still the possibility of more than one correct answer.

If there can be multiple correct IRRs, there is not some absolute mathematical truth to any of them. Money managers, especially those doing private equity, often whip out past-performance IRRs when the numbers suit them. Listen, but maintain your skepticism.

**7. Time-weighted and dollar-weighted.**

We’ll illustrate the difference between these two ways of measuring performance with another example, involving a hypothetical hedge fund manager, John. For simplicity we’ll assume he charges no fees.

At the start of 2008 John has $1 billion under management. He makes one big bet against mortgages and ends the year with $5 billion, for a 400% return. Dazzled, his clients throw $10 billion of new money his way at the start of 2009. He makes one big bet on gold and ends 2009 down 50%. What’s John’s average return?

The time-weighted calculation gives equal weight to the two calendar years. To compute the compound annual return you add 1 to each performance, getting (1 + 400%) and (1 – 50%). Multiply those two factors and take the square root: (5*0.5)^(1/2) = 1.58. Subtract 1 to get the 58% compound annual return.

The clients have a different perspective. John has turned $11 billion of their money into $7.5 billion. To describe their experience you need what is often called the dollar-weighted return, in which the performance during 2009 gets a higher weight in the average than 2008 because more money was in play. The dollar-weighted return is just the IRR (see No. 6 above), and it comes to -30%.

Which performance is the right one? They both have validity in context. The time-weighted return is a fair measure of John’s portfolio skill; it’s not his fault that new money showed up at an inopportune moment. But wise investors keep an eye on the dollar-weighted return as well, since it reminds them that chasing past performance is not a ticket to wealth.

In sum, there is more than one way to calculate a performance and more than one index to use for comparison. Look at a portfolio from different angles before drawing any conclusions.

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