Answer: 3.05%

### Question Recap We are given annual returns for a fund and its benchmark over 5 years: | Year | Benchmark Return | Fund Return | Active Return (Fund – Benchmark) | |------|------------------|-------------|-----------------------------------| | 2005 | 9.00% | 1.00% | 1.00 – 9.00 = **–8.00%** | | 2006 | 7.00% | 3.00% | 3.00 – 7.00 = **–4.00%** | | 2007 | 7.00% | 5.00% | 5.00 – 7.00 = **–2.00%** | | 2008 | 5.00% | 4.00% | 4.00 – 5.00 = **–1.00%** | | 2009 | 2.00% | 1.50% | 1.50 – 2.00 = **–0.50%** | We need to calculate the **tracking error volatility** (also called tracking error standard deviation). **Definition (FRM context)** Tracking error volatility = standard deviation of the active returns Active return in each period = Portfolio return – Benchmark return ### Step-by-step Calculation 1. Active returns (already shown): **–8.00%, –4.00%, –2.00%, –1.00%, –0.50%** 2. These are the deviations we use to compute standard deviation. 3. Most common convention in performance reporting and risk management exams (including FRM / CFA): - Use **sample standard deviation** (divide by n–1, where n = 5 → divide by 4) → This is the unbiased estimator of volatility. 4. Calculation: - Mean active return = (–8 –4 –2 –1 –0.5) / 5 = –15.5 / 5 = **–3.1%** - Deviations from mean: –8 – (–3.1) = –4.9 –4 – (–3.1) = –0.9 –2 – (–3.1) = +1.1 –1 – (–3.1) = +2.1 –0.5 – (–3.1) = +2.6 - Squared deviations: 24.01 + 0.81 + 1.21 + 4.41 + 6.76 = **37.20** - Variance (sample) = 37.20 / 4 = **9.30** - Standard deviation = √9.30 ≈ **3.049%** → rounds to **3.05%** (If population standard deviation were used — divide by n=5 — result would be ≈ 2.73%, which does not match any option.) ### Explanation of Each Option - **A: 0.09%** **Incorrect**. This is far too small. A tracking error of only 9 basis points would mean the fund is essentially a perfect passive replica (like the lowest-cost ETFs). The active returns here vary significantly (from –0.5% to –8%), so volatility cannot be this low. - **B: 1.10%** **Incorrect**. This would represent very tight tracking — typical of enhanced index funds or very closet-indexers. Looking at the large underperformance in 2005 (–8%) and consistent underperformance, the dispersion is clearly much higher than 1.1%. - **C: 3.05%** **Correct**. This matches the sample standard deviation of the active returns (≈3.05%). It is a realistic value for many actively managed funds or funds with meaningful active bets (even if those bets were unsuccessful in this period, as shown by the consistent negative active returns). - **D: 4.09%** **Incorrect**. This is higher than the actual calculation. It might result from common mistakes such as: - Taking square root of variance without dividing by (n–1) - Incorrectly including/excluding the mean - Miscalculating one or more active returns - Using absolute deviations instead of squared ones ### Reference Answer **Correct answer: C: 3.05%** This question tests whether you know: - Tracking error volatility = **std dev of (Rp – Rb)** - The convention in risk/performance measurement usually uses the **sample standard deviation (ddof=1 / n–1)** - How to distinguish realistic values from distractors that are either unrealistically low or the result of a common calculation error Good luck with your FRM Part 1 preparation! Keep practicing these kinds of calculation-based questions — they appear frequently in the quant section.