Active Return and Tracking Error

/**
 * Fintech Math Bootcamp · Lesson 109 of 120
 * Active Return and Tracking Error
 * Module 11: Financial Risk and Performance Statistics
 *
 * Scenario: Measuring benchmark-relative consistency on matched observations
 * Rule:     activeₜ=rₜ−bₜ; TE=sampleSD(active)
 *
 * Try it:   Can tracking error be zero while the portfolio loses money?
 *
 * Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/financial-risk-and-performance-statistics/active-return-and-tracking-error/
 * Free course:    https://courses.thefintechbuilder.com
 * Synthetic teaching example, not financial advice or a production library.
 */

export function lesson109() {
  const r=[.01,.02,-.01,.03],b=[.005,.015,-.005,.02];
  const active=r.map((v,i)=>v-b[i]);
  const mean=active.reduce((s,v)=>s+v,0)/active.length;
  const variance=active.reduce((s,v)=>s+(v-mean)**2,0)/(active.length-1);
  const te=Math.sqrt(variance);
  const result={active,mean,te,informationRatio:mean/te};
  return result;
}

export const checkedResult = {"active":[0.005,0.005000000000000001,-0.005,0.009999999999999998],"mean":0.00375,"te":0.006291528696058958,"informationRatio":0.5960395606792697};

// Run this file directly: npx tsx lessons/11-financial-risk-and-performance-statistics/109-active-return-and-tracking-error.ts
if (process.argv[1] && import.meta.url.endsWith(process.argv[1].replace(/\\/g, "/").split("/").pop()!)) {
  console.log(JSON.stringify(lesson109(), null, 2));
}

# Fintech Math Bootcamp · Lesson 109 of 120
# Active Return and Tracking Error
# Module 11: Financial Risk and Performance Statistics
#
# Scenario: Measuring benchmark-relative consistency on matched observations
# Rule:     activeₜ=rₜ−bₜ; TE=sampleSD(active)
#
# Try it:   Can tracking error be zero while the portfolio loses money?
#
# Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/financial-risk-and-performance-statistics/active-return-and-tracking-error/
# Free course:    https://courses.thefintechbuilder.com
# Synthetic teaching example, not financial advice or a production library.
#
# Run: python main.py

import json
import math


def lesson_109():
    r = [0.01, 0.02, -0.01, 0.03]
    b = [0.005, 0.015, -0.005, 0.02]
    active = [v - b[i] for i, v in enumerate(r)]
    total = 0.0
    for v in active:
        total += v
    mean = total / len(active)
    squares = 0.0
    for v in active:
        squares += (v - mean) ** 2
    te = math.sqrt(squares / (len(active) - 1))
    return {"active": active, "mean": mean, "te": te, "informationRatio": mean / te}


if __name__ == "__main__":
    print(json.dumps(lesson_109(), indent=2))

/*
 * Fintech Math Bootcamp - Lesson 109 of 120
 * Active Return and Tracking Error
 * Module 11: Financial Risk and Performance Statistics
 *
 * Scenario: Measuring benchmark-relative consistency on matched observations
 * Rule:     active_t=r_t-b_t; TE=sampleSD(active)
 *
 * Try it:   Can tracking error be zero while the portfolio loses money?
 *
 * Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/financial-risk-and-performance-statistics/active-return-and-tracking-error/
 * Free course:    https://courses.thefintechbuilder.com
 * Synthetic teaching example, not financial advice or a production library.
 *
 * Run: javac Main.java && java Main
 */

import java.util.*;

public class Main {
    static Map<String, Object> lesson109() {
        double[] r = {.01, .02, -.01, .03};
        double[] b = {.005, .015, -.005, .02};
        double[] active = new double[r.length];
        double mean = 0;
        for (int i = 0; i < r.length; i++) {
            active[i] = r[i] - b[i];
            mean += active[i];
        }
        mean /= active.length;
        double variance = 0;
        for (double v : active) variance += Math.pow(v - mean, 2);
        variance /= active.length - 1;
        double te = Math.sqrt(variance);
        return obj("active", list(active), "mean", mean, "te", te, "informationRatio", mean / te);
    }

    public static void main(String[] args) {
        System.out.println(toJson(lesson109(), ""));
    }

    // Minimal JSON writer: insertion-ordered Map, List, Number, Boolean, String and null.
    static String toJson(Object value, String indent) {
        if (value == null) return "null";
        if (value instanceof String) return quote((String) value);
        if (value instanceof Boolean) return value.toString();
        if (value instanceof Double) return formatNumber((Double) value);
        if (value instanceof Number) return value.toString();
        String inner = indent + "  ";
        StringBuilder sb = new StringBuilder();
        if (value instanceof Map) {
            Map<?, ?> map = (Map<?, ?>) value;
            if (map.isEmpty()) return "{}";
            sb.append("{\n");
            int i = 0;
            for (Map.Entry<?, ?> e : map.entrySet()) {
                sb.append(inner).append(quote(e.getKey().toString())).append(": ").append(toJson(e.getValue(), inner));
                sb.append(++i < map.size() ? ",\n" : "\n");
            }
            return sb.append(indent).append("}").toString();
        }
        List<?> list = (List<?>) value;
        if (list.isEmpty()) return "[]";
        sb.append("[\n");
        for (int i = 0; i < list.size(); i++) {
            sb.append(inner).append(toJson(list.get(i), inner)).append(i + 1 < list.size() ? ",\n" : "\n");
        }
        return sb.append(indent).append("]").toString();
    }

    // Integral doubles print without ".0", as JavaScript does; others use Java's round-trip form.
    static String formatNumber(double x) {
        if (x == Math.rint(x) && Math.abs(x) < 1e15) return Long.toString((long) x);
        return Double.toString(x);
    }

    static String quote(String s) {
        StringBuilder sb = new StringBuilder("\"");
        for (char c : s.toCharArray()) {
            if (c == '"' || c == '\\') sb.append('\\').append(c);
            else if (c < 0x20) sb.append(String.format("\\u%04x", (int) c));
            else sb.append(c);
        }
        return sb.append('"').toString();
    }

    // Builds an insertion-ordered object from alternating keys and values.
    static Map<String, Object> obj(Object... keysAndValues) {
        Map<String, Object> map = new LinkedHashMap<String, Object>();
        for (int i = 0; i < keysAndValues.length; i += 2) map.put((String) keysAndValues[i], keysAndValues[i + 1]);
        return map;
    }

    static List<Object> list(double... values) {
        List<Object> out = new ArrayList<Object>();
        for (double v : values) out.add(v);
        return out;
    }
}

// Fintech Math Bootcamp · Lesson 109 of 120
// Active Return and Tracking Error
// Module 11: Financial Risk and Performance Statistics
//
// Scenario: Measuring benchmark-relative consistency on matched observations
// Rule:     activeₜ=rₜ−bₜ; TE=sampleSD(active)
//
// Try it:   Can tracking error be zero while the portfolio loses money?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/financial-risk-and-performance-statistics/active-return-and-tracking-error/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.
//
// Run: go run main.go

package main

import (
	"encoding/json"
	"fmt"
	"math"
)

type Result struct {
	Active           []float64 `json:"active"`
	Mean             float64   `json:"mean"`
	TE               float64   `json:"te"`
	InformationRatio float64   `json:"informationRatio"`
}

func lesson109() Result {
	r := []float64{.01, .02, -.01, .03}
	b := []float64{.005, .015, -.005, .02}
	active := make([]float64, len(r))
	mean := 0.0
	for i, v := range r {
		active[i] = v - b[i]
		mean += active[i]
	}
	mean /= float64(len(active))
	variance := 0.0
	for _, v := range active {
		variance += math.Pow(v-mean, 2)
	}
	variance /= float64(len(active) - 1)
	te := math.Sqrt(variance)
	return Result{Active: active, Mean: mean, TE: te, InformationRatio: mean / te}
}

func main() {
	out, err := json.MarshalIndent(lesson109(), "", "  ")
	if err != nil {
		panic(err)
	}
	fmt.Println(string(out))
}

// Fintech Math Bootcamp · Lesson 109 of 120
// Active Return and Tracking Error
// Module 11: Financial Risk and Performance Statistics
//
// Scenario: Measuring benchmark-relative consistency on matched observations
// Rule:     activeₜ=rₜ−bₜ; TE=sampleSD(active)
//
// Try it:   Can tracking error be zero while the portfolio loses money?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/financial-risk-and-performance-statistics/active-return-and-tracking-error/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.
//
// Run: g++ -std=c++17 -O1 -o lesson main.cpp && ./lesson

#include <algorithm>
#include <charconv>
#include <cmath>
#include <cstdint>
#include <cstdio>
#include <deque>
#include <iostream>
#include <limits>
#include <map>
#include <optional>
#include <stdexcept>
#include <string>
#include <utility>
#include <variant>
#include <vector>

// Minimal JSON value: objects keep insertion order; numbers print in shortest round-trip form.
struct Json {
    enum class Kind { Null, Bool, Number, String, Array, Object };
    Kind kind = Kind::Null;
    bool flag = false;
    double number = 0.0;
    std::string text;
    std::vector<std::string> keys;  // object keys, parallel to items
    std::vector<Json> items;        // array items, or object values
};

Json jnull() { return Json{}; }
Json jbool(bool value) { Json j; j.kind = Json::Kind::Bool; j.flag = value; return j; }
Json jnum(double value) { Json j; j.kind = Json::Kind::Number; j.number = value; return j; }
Json jstr(const std::string& value) { Json j; j.kind = Json::Kind::String; j.text = value; return j; }
Json jarr(std::vector<Json> items) { Json j; j.kind = Json::Kind::Array; j.items = std::move(items); return j; }

Json jobj(std::vector<std::pair<std::string, Json>> fields) {
    Json j;
    j.kind = Json::Kind::Object;
    for (auto& field : fields) {
        j.keys.push_back(field.first);
        j.items.push_back(std::move(field.second));
    }
    return j;
}

Json jnums(const std::vector<double>& values) {
    std::vector<Json> items;
    for (double v : values) items.push_back(jnum(v));
    return jarr(std::move(items));
}

Json jnums(const std::vector<std::optional<double>>& values) {
    std::vector<Json> items;
    for (const auto& v : values) items.push_back(v ? jnum(*v) : jnull());
    return jarr(std::move(items));
}

std::string formatNumber(double value) {
    char buffer[64];
    auto result = std::to_chars(buffer, buffer + sizeof buffer, value);  // shortest round-trip form
    return std::string(buffer, result.ptr);
}

std::string quote(const std::string& text) {
    std::string out = "\"";
    for (char c : text) {
        if (c == '"' || c == '\\') {
            out += '\\';
            out += c;
        } else if (static_cast<unsigned char>(c) < 0x20) {
            char buffer[8];
            std::snprintf(buffer, sizeof buffer, "\\u%04x", static_cast<unsigned>(c));
            out += buffer;
        } else {
            out += c;
        }
    }
    return out + "\"";
}

void writeJson(std::ostream& out, const Json& value, const std::string& indent = "") {
    const std::string inner = indent + "  ";
    switch (value.kind) {
        case Json::Kind::Null: out << "null"; return;
        case Json::Kind::Bool: out << (value.flag ? "true" : "false"); return;
        case Json::Kind::Number: out << formatNumber(value.number); return;
        case Json::Kind::String: out << quote(value.text); return;
        case Json::Kind::Array:
        case Json::Kind::Object: {
            const bool isObject = value.kind == Json::Kind::Object;
            if (value.items.empty()) {
                out << (isObject ? "{}" : "[]");
                return;
            }
            out << (isObject ? "{\n" : "[\n");
            for (std::size_t i = 0; i < value.items.size(); ++i) {
                out << inner;
                if (isObject) out << quote(value.keys[i]) << ": ";
                writeJson(out, value.items[i], inner);
                out << (i + 1 < value.items.size() ? ",\n" : "\n");
            }
            out << indent << (isObject ? "}" : "]");
        }
    }
}

Json lesson109() {
    const std::vector<double> r{.01, .02, -.01, .03};
    const std::vector<double> b{.005, .015, -.005, .02};
    std::vector<double> active;
    double mean = 0;
    for (std::size_t i = 0; i < r.size(); ++i) {
        active.push_back(r[i] - b[i]);
        mean += active.back();
    }
    mean /= static_cast<double>(active.size());
    double variance = 0;
    for (double v : active) variance += std::pow(v - mean, 2);
    variance /= static_cast<double>(active.size() - 1);
    const double te = std::sqrt(variance);
    return jobj({{"active", jnums(active)}, {"mean", jnum(mean)}, {"te", jnum(te)}, {"informationRatio", jnum(mean / te)}});
}

int main() {
    writeJson(std::cout, lesson109());
    std::cout << "\n";
    return 0;
}

// Fintech Math Bootcamp · Lesson 109 of 120
// Active Return and Tracking Error
// Module 11: Financial Risk and Performance Statistics
//
// Scenario: Measuring benchmark-relative consistency on matched observations
// Rule:     activeₜ=rₜ−bₜ; TE=sampleSD(active)
//
// Try it:   Can tracking error be zero while the portfolio loses money?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/financial-risk-and-performance-statistics/active-return-and-tracking-error/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.
//
// Run: rustc -O main.rs && ./main

#![allow(dead_code)]

/// Minimal JSON value; objects keep insertion order.
enum Json {
    Null,
    Bool(bool),
    Num(f64),
    Str(String),
    Arr(Vec<Json>),
    Obj(Vec<(String, Json)>),
}

fn obj(fields: Vec<(&str, Json)>) -> Json {
    Json::Obj(fields.into_iter().map(|(k, v)| (k.to_string(), v)).collect())
}

fn nums(values: &[f64]) -> Json {
    Json::Arr(values.iter().map(|&v| Json::Num(v)).collect())
}

fn optional_nums(values: &[Option<f64>]) -> Json {
    Json::Arr(values.iter().map(|v| v.map_or(Json::Null, Json::Num)).collect())
}

fn quote(text: &str) -> String {
    let mut out = String::from("\"");
    for c in text.chars() {
        match c {
            '"' => out.push_str("\\\""),
            '\\' => out.push_str("\\\\"),
            c if (c as u32) < 0x20 => out.push_str(&format!("\\u{:04x}", c as u32)),
            c => out.push(c),
        }
    }
    out.push('"');
    out
}

// Display for f64 prints the shortest string that round-trips, as JavaScript does.
fn write_json(value: &Json, indent: &str, out: &mut String) {
    let inner = format!("{}  ", indent);
    match value {
        Json::Null => out.push_str("null"),
        Json::Bool(b) => out.push_str(if *b { "true" } else { "false" }),
        Json::Num(n) => out.push_str(&format!("{}", n)),
        Json::Str(s) => out.push_str(&quote(s)),
        Json::Arr(items) if items.is_empty() => out.push_str("[]"),
        Json::Obj(fields) if fields.is_empty() => out.push_str("{}"),
        Json::Arr(items) => {
            out.push_str("[\n");
            for (i, item) in items.iter().enumerate() {
                out.push_str(&inner);
                write_json(item, &inner, out);
                out.push_str(if i + 1 < items.len() { ",\n" } else { "\n" });
            }
            out.push_str(indent);
            out.push(']');
        }
        Json::Obj(fields) => {
            out.push_str("{\n");
            for (i, (key, item)) in fields.iter().enumerate() {
                out.push_str(&inner);
                out.push_str(&quote(key));
                out.push_str(": ");
                write_json(item, &inner, out);
                out.push_str(if i + 1 < fields.len() { ",\n" } else { "\n" });
            }
            out.push_str(indent);
            out.push('}');
        }
    }
}

fn lesson_109() -> Json {
    let r: [f64; 4] = [0.01, 0.02, -0.01, 0.03];
    let b: [f64; 4] = [0.005, 0.015, -0.005, 0.02];
    let active: Vec<f64> = r.iter().zip(b.iter()).map(|(v, w): (&f64, &f64)| v - w).collect();
    let n = active.len() as f64;
    let mean = active.iter().fold(0.0, |s, v| s + v) / n;
    let variance = active.iter().fold(0.0, |s, v| s + (v - mean).powi(2)) / (n - 1.0);
    let te = variance.sqrt();
    obj(vec![
        ("active", nums(&active)),
        ("mean", Json::Num(mean)),
        ("te", Json::Num(te)),
        ("informationRatio", Json::Num(mean / te)),
    ])
}

fn main() {
    let mut out = String::new();
    write_json(&lesson_109(), "", &mut out);
    println!("{}", out);
}

// Fintech Math Bootcamp · Lesson 109 of 120
// Active Return and Tracking Error
// Module 11: Financial Risk and Performance Statistics
//
// Scenario: Measuring benchmark-relative consistency on matched observations
// Rule:     activeₜ=rₜ−bₜ; TE=sampleSD(active)
//
// Try it:   Can tracking error be zero while the portfolio loses money?
//
// Lesson article: https://thefintechbuilder.com/financial-mathematics-statistics-and-data-foundations/financial-risk-and-performance-statistics/active-return-and-tracking-error/
// Free course:    https://courses.thefintechbuilder.com
// Synthetic teaching example, not financial advice or a production library.
//
// Run: dotnet run

using System;
using System.Collections.Generic;
using System.Globalization;
using System.Linq;
using System.Text.Encodings.Web;
using System.Text.Json;

var options = new JsonSerializerOptions { WriteIndented = true, Encoder = JavaScriptEncoder.UnsafeRelaxedJsonEscaping };
Console.WriteLine(JsonSerializer.Serialize(Lesson109(), options));

static object Lesson109()
{
    double[] r = { .01, .02, -.01, .03 };
    double[] b = { .005, .015, -.005, .02 };
    double[] active = r.Select((v, i) => v - b[i]).ToArray();
    double mean = active.Aggregate(0.0, (s, v) => s + v) / active.Length;
    double variance = active.Aggregate(0.0, (s, v) => s + Math.Pow(v - mean, 2)) / (active.Length - 1);
    double te = Math.Sqrt(variance);
    return new { active, mean, te, informationRatio = mean / te };
}