nautilus_analysis/statistics/
tracking_error.rs

1// -------------------------------------------------------------------------------------------------
2//  Copyright (C) 2015-2026 Nautech Systems Pty Ltd. All rights reserved.
3//  https://nautechsystems.io
4//
5//  Licensed under the GNU Lesser General Public License Version 3.0 (the "License");
6//  You may not use this file except in compliance with the License.
7//  You may obtain a copy of the License at https://www.gnu.org/licenses/lgpl-3.0.en.html
8//
9//  Unless required by applicable law or agreed to in writing, software
10//  distributed under the License is distributed on an "AS IS" BASIS,
11//  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
12//  See the License for the specific language governing permissions and
13//  limitations under the License.
14// -------------------------------------------------------------------------------------------------
15
16//! Tracking error statistic (benchmark-relative).
17
18use std::fmt::Display;
19
20use nautilus_model::position::Position;
21
22use crate::{Returns, statistic::PortfolioStatistic};
23
24/// Calculates the tracking error of portfolio returns relative to a benchmark.
25///
26/// Tracking error is the volatility of the active return (portfolio minus benchmark):
27///
28/// `TE = std(active) * sqrt(period)`
29///
30/// where `active_i = portfolio_i - benchmark_i`, `std` uses Bessel's correction
31/// (`ddof = 1`), and the result is annualized by the square root of the specified period
32/// (default: 252 trading days).
33///
34/// # References
35///
36/// - Roll, R. (1992). "A Mean/Variance Analysis of Tracking Error".
37///   *Journal of Portfolio Management*, 18(4), 13-22.
38/// - CFA Institute Investment Foundations, 3rd Edition
39#[repr(C)]
40#[derive(Debug, Clone)]
41#[cfg_attr(
42    feature = "python",
43    pyo3::pyclass(module = "nautilus_trader.core.nautilus_pyo3.analysis", from_py_object)
44)]
45#[cfg_attr(
46    feature = "python",
47    pyo3_stub_gen::derive::gen_stub_pyclass(module = "nautilus_trader.analysis")
48)]
49pub struct TrackingError {
50    /// The annualization period (default: 252 for daily data).
51    period: usize,
52}
53
54impl TrackingError {
55    /// Creates a new [`TrackingError`] instance.
56    #[must_use]
57    pub fn new(period: Option<usize>) -> Self {
58        Self {
59            period: period.unwrap_or(252),
60        }
61    }
62}
63
64impl Display for TrackingError {
65    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
66        write!(f, "Tracking Error ({} days)", self.period)
67    }
68}
69
70impl PortfolioStatistic for TrackingError {
71    type Item = f64;
72
73    fn name(&self) -> String {
74        self.to_string()
75    }
76
77    fn calculate_from_returns(&self, _returns: &Returns) -> Option<Self::Item> {
78        None
79    }
80
81    fn calculate_from_realized_pnls(&self, _realized_pnls: &[f64]) -> Option<Self::Item> {
82        None
83    }
84
85    fn calculate_from_positions(&self, _positions: &[Position]) -> Option<Self::Item> {
86        None
87    }
88
89    fn calculate_from_returns_with_benchmark(
90        &self,
91        returns: &Returns,
92        benchmark: &Returns,
93    ) -> Option<Self::Item> {
94        let (r, b) = self.align_returns(returns, benchmark);
95        if r.len() < 2 {
96            return Some(f64::NAN);
97        }
98
99        let (_, std_active) = active_return_stats(&r, &b);
100        Some(std_active * (self.period as f64).sqrt())
101    }
102}
103
104/// Computes the mean and sample (`ddof = 1`) standard deviation of the active
105/// return series `r - b`.
106///
107/// Callers must ensure `r.len() == b.len() >= 2`.
108pub(crate) fn active_return_stats(r: &[f64], b: &[f64]) -> (f64, f64) {
109    let n = r.len() as f64;
110    let mean = r
111        .iter()
112        .zip(b.iter())
113        .map(|(&ri, &bi)| ri - bi)
114        .sum::<f64>()
115        / n;
116    let variance = r
117        .iter()
118        .zip(b.iter())
119        .map(|(&ri, &bi)| (ri - bi - mean).powi(2))
120        .sum::<f64>()
121        / (n - 1.0);
122    (mean, variance.sqrt())
123}
124
125#[cfg(test)]
126mod tests {
127    use std::collections::BTreeMap;
128
129    use nautilus_core::{UnixNanos, approx_eq};
130    use rstest::rstest;
131
132    use super::*;
133
134    fn create_returns(values: &[f64]) -> BTreeMap<UnixNanos, f64> {
135        let mut new_return = BTreeMap::new();
136        let one_day_in_nanos = 86_400_000_000_000;
137        let start_time = 1_600_000_000_000_000_000;
138
139        for (i, &value) in values.iter().enumerate() {
140            let timestamp = start_time + i as u64 * one_day_in_nanos;
141            new_return.insert(UnixNanos::from(timestamp), value);
142        }
143
144        new_return
145    }
146
147    #[rstest]
148    fn test_name() {
149        let stat = TrackingError::new(None);
150        assert_eq!(stat.name(), "Tracking Error (252 days)");
151    }
152
153    #[rstest]
154    fn test_name_non_default_period() {
155        let stat = TrackingError::new(Some(63));
156        assert_eq!(stat.name(), "Tracking Error (63 days)");
157    }
158
159    #[rstest]
160    fn test_known_value_nonzero_benchmark() {
161        // Both strategy and benchmark non-zero so the (r - b) subtraction is exercised.
162        //   r       = [0.03, -0.01, 0.02, 0.04]
163        //   b       = [0.01, 0.005, 0.005, 0.01]
164        //   active  = [0.02, -0.015, 0.015, 0.03]
165        //   std(active, ddof=1) = 0.019364916731037084
166        //   TE = std(active, ddof=1) * sqrt(252) = 0.30740852297878796
167        // A formula dropping the benchmark (active = r) would yield 0.34292856398964494,
168        // so this value discriminates against that bug.
169        let returns = create_returns(&[0.03, -0.01, 0.02, 0.04]);
170        let benchmark = create_returns(&[0.01, 0.005, 0.005, 0.01]);
171        let stat = TrackingError::new(Some(252));
172        let result = stat
173            .calculate_from_returns_with_benchmark(&returns, &benchmark)
174            .unwrap();
175        assert!(approx_eq!(f64, result, 0.30740852297878796, epsilon = 1e-9));
176    }
177
178    #[rstest]
179    fn test_known_value() {
180        // active = [0.01, 0.02, 0.03], sample std (ddof=1) = 0.01,
181        // annualized = 0.01 * sqrt(4) = 0.02.
182        let benchmark = create_returns(&[0.00, 0.00, 0.00]);
183        let returns = create_returns(&[0.01, 0.02, 0.03]);
184        let stat = TrackingError::new(Some(4));
185        let result = stat
186            .calculate_from_returns_with_benchmark(&returns, &benchmark)
187            .unwrap();
188        assert!(approx_eq!(f64, result, 0.02, epsilon = 1e-12));
189    }
190
191    #[rstest]
192    fn test_zero_active_is_zero() {
193        // Identical series -> active all zero -> TE = 0 (not NaN).
194        let benchmark = create_returns(&[0.01, -0.02, 0.015, -0.005]);
195        let returns = create_returns(&[0.01, -0.02, 0.015, -0.005]);
196        let stat = TrackingError::new(None);
197        let result = stat
198            .calculate_from_returns_with_benchmark(&returns, &benchmark)
199            .unwrap();
200        assert!(approx_eq!(f64, result, 0.0, epsilon = 1e-12));
201    }
202
203    #[rstest]
204    fn test_empty_returns_is_nan() {
205        let stat = TrackingError::new(None);
206        let result = stat
207            .calculate_from_returns_with_benchmark(&create_returns(&[]), &create_returns(&[]))
208            .unwrap();
209        assert!(result.is_nan());
210    }
211
212    #[rstest]
213    fn test_single_overlap_is_nan() {
214        let benchmark = create_returns(&[0.01, -0.02, 0.015]);
215        let returns = create_returns(&[0.02]);
216        let stat = TrackingError::new(None);
217        let result = stat
218            .calculate_from_returns_with_benchmark(&returns, &benchmark)
219            .unwrap();
220        assert!(result.is_nan());
221    }
222}
nautilus_analysis/statistics/tracking_error.rs

nautilus_analysis/statistics/
tracking_error.rs
