Tracking Error

Tracking Error is a measure of how closely a portfolio follows the index to which it is benchmarked. It quantifies the volatility of the return differences between the portfolio and the benchmark over a specific period.

Understanding Tracking Error

Tracking error is crucial for investors, particularly those managing index funds or exchange-traded funds (ETFs), as it determines how effectively the fund mirrors its benchmark. A low tracking error indicates that the portfolio closely follows its benchmark, while a high tracking error shows substantial deviation.

Key Points to Consider

• Formula: Tracking error can be calculated using the formula:
  TE = √(Σ(Rp – Rb)² / (n-1))
  where Rp is the return of the portfolio, Rb is the return of the benchmark, and n is the number of observations.
• Standard Deviation: Tracking error is essentially the standard deviation of the return differences.
• Comparison: It is helpful for comparing different portfolios or funds and determining how much active risk a manager is taking.

Example of Tracking Error

Imagine a mutual fund designed to replicate the performance of the S&P 500 index. Over a five-month period, the returns of the mutual fund and the S&P 500 are as follows:

• Month 1: Fund Return = 2%, Benchmark Return = 2.5%
• Month 2: Fund Return = 1%, Benchmark Return = 1.2%
• Month 3: Fund Return = 3%, Benchmark Return = 3.5%
• Month 4: Fund Return = 2.5%, Benchmark Return = 3%
• Month 5: Fund Return = 1.5%, Benchmark Return = 2%

Calculating Tracking Error

First, calculate the differences between the fund and the benchmark returns:

• Month 1: 2% – 2.5% = -0.5%
• Month 2: 1% – 1.2% = -0.2%
• Month 3: 3% – 3.5% = -0.5%
• Month 4: 2.5% – 3% = -0.5%
• Month 5: 1.5% – 2% = -0.5%

Now, square each difference:

• Month 1: (-0.5%)² = 0.0025
• Month 2: (-0.2%)² = 0.0004
• Month 3: (-0.5%)² = 0.0025
• Month 4: (-0.5%)² = 0.0025
• Month 5: (-0.5%)² = 0.0025

Sum of the squared differences:

0.0025 + 0.0004 + 0.0025 + 0.0025 + 0.0025 = 0.0104

Now, divide by (n-1) which is (5-1) or 4:

0.0104/4 = 0.0026

Finally, take the square root to find the tracking error:

TE = √0.0026 ≈ 0.0510 or 5.1%

Real-World Application

Investors and fund managers use tracking error to assess the effectiveness of an index fund or ETF. For example, a tracking error of 5.1% suggests that the mutual fund’s performance is deviating from the S&P 500 index by that percentage, indicating how well it is achieving its goal of matching the benchmark.