# two tricks of performance calculation arithmetic

measuring performance

The acid test of active management–both of our own efforts and of the professionals we may hire to invest for us–is whether they add value versus an appropriate index.  Picking the benchmark against which to measure results is a pretty straightforward task, though judgment issues do sometimes arise.  (For example, if all a manager’s outperformance of the S&P 500 over the past three years comes from holding a large position in Baidu (BIDU), the Chinese internet company listed on NASDAQ, is the S&P really the right index to be using?  But that’s a story for another post.)

What I want to point out here is a quirk in the way performance calculations are done:

–in a rising market, outperformance tends to look better than it really is;

–in a falling market, outperformance tends to look worse than it really is.

The opposite is true of underperformance.

in a rising market, underperformance tends to look worse than it really is;

–in a falling market, underperformance tends to look better than it really is.

Here’s what I mean:

Let’s take an example where the numbers are impossibly large, just to illustrate the point.

outperformance

We’ll suppose that on Day 1 of the measurement period the index is unchanged but our portfolio gains 50%.  At the end of Day 1 we’re 50 percentage points ahead of the index.

a.  rising market.  Suppose that for the rest of the year, our portfolio matches the market performance exactly and that the index doubles from Day 2 through the rest of the year.  How far ahead of the index is our portfolio for the year?

Your first instinct is probably to say “50 percentage points,” since we’ve made no further gains after Day 1  …but that’s wrong.  The actual outperformance is 100 percentage points.

If the index starts the year at 100, its ending value is 200.

If our portfolio starts the year at 100, we’re at 150 at the end of Day 1 and we double from there–meaning we’re at 300 on the final day, or 100 percentage points ahead of the market.  Whatever positive thing we did on Day 1 has been magnified by the rising market.

b.  falling market.  Let’s take the same portfolio, up 50% in a flat market on Day 1.  This time, let’s suppose our portfolio matches the index for the rest of the year, but that the index falls by 50% between Day 2 and the end.  How far ahead are we for the year?

Having seen a., you’re already going to guess that 50 percentage points is wrong.  …and 50 is wrong.  But what’s the right number?

Well, if the index starts at 100 and loses 50%, at the end of the year it’s at 50.  At the end of Day 1, we’re at 150, but we lose half that amount through yearend.  So we end up at 75, or 25 percentage points ahead of the index.

underperformance

Let’s start again with crazy numbers.  Assume Day 1 is the day from hell and we lose half our money in a flat market.  We’re 50 percentage points behind the index.

c.  rising market.  The market doubles from Day 2, going from 100 to 200 by yearend.  We match the market.  Our 50 goes to 100.  We’re 100 percentage points behind the market.

d.  falling market.  The market declines from Day 2 on, and drops from 100 to 50 by yearend.  Our 50 is cut in half to 25.  We’re 25 percentage points behind the market.

implications

There are all sorts of implications for professional investors, who tend to earn most of their compensation based on annual performance vs. an index.  You never want to get behind in a rising market, for instance.  Or, a falling market tends to compress out- and underperformance numbers closer to the index, so that’s the best time to play catch-up.

For the rest of us, the lessons are:

–don’t get too excited about the “phantom” outperformance that a rising market (2009, 2010) brings, and

–more important, a decline of 15% like the one we’ve been in will reduce your under- or outperformance by 15%.  Don’t think your stocks are suddenly doing better/worse than they are.  To see your real performance during the downturn, don’t check the year to date figures, check them from the start of the downturn until now.

NOTE:  If you’ve constructed a portfolio for a rising market, or if you were ahead year to date before the current decline began, you should expect some slippage in relative performance as the market sags.  Similarly, if your holdings are geared for a down market, you should now be seeing a pickup in relative performance.

How much relative gain or loss?  That’s another post-full.  A lot depends on the level of risk you’ve assumed and your skill in picking stocks.  But if you’ve battened down the hatches, you should be seeing at least some benefit.  If you’ve continued to keep a lot of sail let out (which is my usual position), you shouldn’t be surprised/dismayed by a modest relative loss.