Tracking error

Tracking error is a measure of the difference between the performance of an investment portfolio or fund and the performance of its benchmark. It is commonly used to evaluate how closely a portfolio follows the return characteristics of its benchmark.

Tracking error is particularly relevant to passive investment strategies, index funds, and exchange-traded funds (ETFs), where a fund may seek to replicate the performance of a particular market index.

Calculation

A simple measure of tracking error is the standard deviation of the differences between portfolio returns and benchmark returns.

First, the active return for period $t$ is:

$A_{t} = R_{p, t} - R_{b, t}$

where:

$A_{t}$ = active return in period $t$ ;
$R_{p, t}$ = portfolio return in period $t$ ;
$R_{b, t}$ = benchmark return in period $t$ .

Tracking error can then be expressed as:

$Tracking Error = σ (A_{t})$

or:

$Tracking Error = \sqrt{\frac{1}{n - 1} \sum_{t = 1}^{n} (A_{t} - \bar{A})^{2}}$

where $\bar{A}$ is the mean active return and $n$ is the number of observations.

Example

Suppose a portfolio and its benchmark produce the following annual returns:

Year	Portfolio return	Benchmark return	Active return
1	10%	9%	1%
2	7%	8%	-1%
3	12%	11%	1%
4	8%	9%	-1%

The active returns are:

$1 %, - 1 %, 1 %, - 1 %$

Tracking error measures the variability of these active returns rather than simply measuring their average difference.

Interpretation

A low tracking error indicates that the portfolio's returns have generally remained close to those of its benchmark.

A high tracking error indicates greater variation between portfolio returns and benchmark returns.

For a passive fund designed to replicate an index, a lower tracking error generally indicates closer replication of the benchmark.

Tracking error does not indicate whether a portfolio has outperformed or underperformed its benchmark on average. It measures the variability of the difference in returns.

Tracking difference

Tracking difference is different from tracking error.

Tracking difference generally refers to the difference between a portfolio's return and its benchmark's return over a specified period:

$Tracking Difference = R_{p} - R_{b}$

Tracking error instead measures the variability of the active returns over multiple periods:

$Tracking Error = σ (R_{p} - R_{b})$

Thus, tracking difference measures the magnitude of the return gap for a particular period or measurement interval, while tracking error measures the consistency of that gap over time.

Sources of tracking error

Tracking error can arise from factors including:

management fees;
transaction costs;
taxes;
cash holdings;
portfolio rebalancing;
differences in index composition;
corporate actions;
dividend treatment;
trading timing;
sampling strategies;
liquidity constraints;
index methodology changes.

The relative importance of these factors depends on the investment strategy and benchmark.

Passive management

Tracking error is particularly important in passive management.

A passive fund may seek to reproduce the performance of an index such as the S&P 500, Russell 1000, FTSE 100, or FTSE All-Share Index.

A fund that perfectly replicated the benchmark before costs would theoretically have zero tracking error. In practice, fees, trading costs, cash balances, taxes, and other factors can cause the fund's returns to differ from those of the benchmark.

Active management

Tracking error is also used in active management to measure the extent to which a portfolio's returns differ from those of its benchmark.

An actively managed portfolio may intentionally maintain a higher tracking error by taking positions that differ substantially from the benchmark.

A portfolio with a high tracking error can therefore have either higher or lower returns than its benchmark.

Annualization

Tracking error is often reported on an annualized basis.

If the tracking error calculated from monthly observations is $T E_{m}$ , a simplified annualization is:

$T E_{annual} = T E_{m} \sqrt{12}$

Similarly, for daily observations:

$T E_{annual} = T E_{d} \sqrt{N}$

where $N$ represents the approximate number of trading days in a year.

The appropriate annualization factor depends on the frequency and convention used.

Tracking error and portfolio management

Tracking error can help portfolio managers assess how closely a portfolio follows its benchmark.

It can be used alongside measures such as:

Tracking error alone does not indicate whether a portfolio has superior risk-adjusted performance.

Importance

Tracking error is an important measure in investment management because it quantifies the variability of a portfolio's performance relative to its benchmark. It is particularly useful for evaluating index-tracking strategies, assessing the degree of active management in a portfolio, and understanding the sources of differences between portfolio and benchmark returns.