# Tracking error and index tracking

Let's say I want to replicate an index like S&P500. People generally minimize a mathematical notion called tracking error which is nothing else than the standard deviation between the index's returns and our portfolio's returns. More precisely, let $$r_{pf}$$ be the returns of our portfolio and $$r_{idx}$$ be the returns of the index. We have $$TE = std(r_{pf}-r_{idx})$$.

I don't understand why minimizing this tracking error is sufficient for constructing a portfolio that tracks the index. Let's say we can minimize the tracking error very close to 0, that means the random variable $$X$$ defined such as $$X = r_{pf} - r_{idx}$$ will have a null variance, and hence $$X$$ is a almost surely constant random variable.

However, what tells us that the constant will be equal to 0 ? It can be equal to anything, since as long as $$X$$ is almost surely constant its variance is null. That doesn't answer our original problem of tracking the index at all., since we wanted $$r_{pf}$$ to approximately be equal to $$r_{idx}$$.

• Theoretically you may be right that you should minimize the sum of squares of return differences, instead of the variance of return differences. When you compute a variance the mean is subtracted out before taking the squares, so it is not the same thing as the sum of squares. But for financial returns the mean is close to zero and much smaller than the standard deviation, so variance and sum of squares are almost the same. Commented Sep 11, 2020 at 14:32
• A more fundamental question is "what do we mean by replication" ? If portfolio B underperforms portfolio A by the same number of basis points every day, would that be considered a perfect replication? By the Tracking Error criterion yes, by the "equal return every day" criterion, no. Commented Sep 11, 2020 at 14:46

Setup: Mean Difference

It seems like your complaint with that definition of tracking error is that it does not consider the mean difference between $$r_{pf}$$ and $$r_{idx}$$. There are a couple of ways to consider your complaint.

Let's consider two portfolios which track the index: portfolios $$A$$ and $$B$$ with returns $$r_{P_A}$$ and $$r_{P_B}$$. Your concern seems to be if $$\text{var}(r_{P_A}-r_{idx})=\text{var}(r_{P_B}-r_{idx})=0$$ but $$r_{P_A}=r_{idx}: both $$A$$ and $$B$$ have 0 tracking error, but $$r_{P_B}-r_{P_A}=c>0$$.

Which Portfolio is Better?

First, it is true that given a choice between $$A$$ and $$B$$, we would prefer portfolio $$B$$. $$B$$ does not track the index exactly; it returns more. In this way, you are right that the definition of tracking error seems to be missing an important detail.

Arbitrage Concerns

Second, however, you should consider what would probably happen in this scenario. I have two instruments which differ in returns by only a constant. This is textbook arbitrage: I short $$A$$ and use the money to buy $$B$$ earning $$c$$. How much do I do that? A lot -- as in "as much as possible." Who cares about index returns when I can earn a guaranteed $$c$$ without having to put down any money? Actions of arbitrageurs would close those prices until $$r_{idx}=r_{P_A}=r_{P_B}$$.

Practical Issues: Liquidity, Credit, and Risk Limits

Third: we should consider practicalities since this does actually happen. There are a number of index funds which all replicate the index but which have different expense ratios. What has happened in those scenarios? Well, many of the expensive (high expense) index funds are mutual funds and thus you cannot short them.

Index ETFs which have different expense ratios exist, however, and ETFs can be shorted. For example SPY has an expense ratio of about 0.09% while VOO (which also tracks the S&P 500 index) has an expense ratio of about 0.03%.

Why do people not arbitrage VOO versus SPY? They can and probably do to some extent; however the practicalities of liquidity, margin, and credit intrude:

• SPY typically has a \$0.01 bid-ask spread (about 0.003%) and VOO typically has a \$0.03 bid-ask spread (about 0.01%). Trading one versus the other would consume about 0.026% for entry and exit which reduces the profit to about 0.06% (being conservative).
• Trading requires some capital in an account at a brokerage; you cannot open an account with no money, short assets for funding, and trade using that capital alone.
• Shorting one of these does not give you full funding for the other position due to Reg T; therefore, you need to have some capital yourself to fund half of the long position.
• Finally, you cannot do this in huge size without hitting position limits at many brokers.

Summary

In summary, you are right that we would prefer a portfolio yielding more among choices with no tracking error versus an index. Some instances of worse portfolios (higher expense) exist -- and persist -- because they are mutual funds which cannot be shorted. However, most of the instances of more expensive ETFs are not much more expensive such that they are within the arbitrage bounds of bid-ask spreads plus loans to fund the long position.

Allied to many comments already made, you can almost be sure that the constant "missing" from the standard tracking error methodology will not be zero. Because the tracker costs something positive (usually much lower than active fund management costs in the same asset class) to hold, it is really "Benchmark index - Total Expense Ratio" in the first place.

Minimising the total sum of squared residuals (RSS) between tracker and index after costs would force the tracker to position in ways that would leave any user of the tracker with a different risk exposure with respect to bond, gold, credit etc. than they would have using the index itself in their models, comparing to other asset classes. So there is a sense in which min-TE and accepting some drag is, indeed, a more legitimate representation of the index than one which is just least-squared-errors. If min(RSS) is achieved, eg by holding different mixes of say Tech vs Energy stocks, then this could well cause the fund to behave differently from the index in diversified multi-asset portfolios. Because the Tech-vs-Energy mix has correlations to say Oil and Treasuries, that are different from those of the S&P500.

It is also assumed that just trying to hold the same mix as the market at any point in time, aka "full replication", which massive whales like SPY can credibly do at low cost, is a basic fair test of concept. So tracking error simply represents an intuitive efficiency metric (not a test of statistical purity). It is possible to construct an ETF that delivers pure tracking less a (small) fixed cost with NO tracking error. If the fund simply obtains obtains the relevant index swap from an investment bank... except then your near-zero tracking error and low RSS contains an immeasurable and irreportible counterparty credit risk (that might go to zero, creating a big tracking error) ;-) Easier and better then just to hold the same mix of stocks (and take the rough with smooth re tracking errors), than try to game the measurement game too hard?