Taylor Series and Approximation

1Introduction

Almost every formula a trader uses daily is an approximation in disguise. Examples are “log return ≈ simple return”, “geometric mean ≈ arithmetic mean − σ²/2” and “bond price return ≈ −duration × yield change”. Each is the first or second term of a Taylor expansion. This module shows where they come from, how wrong they are and when they break, and then uses them on real money.

Exhibit 1 collects the numbers.

The tool that resolves the paradox is a second-order Taylor expansion of $\ln (1+x)$. Sections 2–4 build the tool and its error bound. Sections 5–7 apply it to returns, means and leverage. Section 8 shows the same idea in bond pricing, option hedging and, as a preview of LM28, Itô's lemma.

2The Taylor Polynomial: A Local Model

Tangent line, then curvature

Near a point $a$, a smooth function looks like a straight line: its tangent. The tangent line is the first-order model

For $f(x)=\ln (1+x)$ at $a=0$, $f(0)=0$ and $f^{\prime }(0)=1$, so $\ln (1+x)\approx x$. That is the familiar “log return ≈ simple return”. It is exact at $x=0$ and drifts away as $x$ grows, because $\ln (1+x)$ bends downward. A better model adds the bend, measured by the second derivative:

With $f^{\prime \prime }(x)=-1/(1+x{)}^2$ and $f^{\prime \prime }(0)=-1$ this gives $\ln (1+x)\approx x-x^2/2$. The $-x^2/2$ term is the curvature correction, and it is the source of every “$-{\sigma }^2/2$” in this book.

Why the coefficients are $f^{(k)}(a)/k!$

$T_n$ is built so that it agrees with $f$ at $a$ in value, slope, curvature and every higher derivative up to order $n$. Write a general polynomial $h(x)={\sum }_{k=0}^n{c}_k(x-a{)}^k$ and differentiate it $m$ times. Every term with $k<m$ disappears, every term with $k>m$ still contains a factor $(x-a)$ and vanishes at $x=a$, and the term $k=m$ becomes $m!c_m$. So

Requiring $h^{(m)}(a)=f^{(m)}(a)$ for $m=0,\dots ,n$ forces $c_m=f^{(m)}(a)/m!$. The factorial is not decoration: it undoes the factorial that repeated differentiation of $(x-a{)}^m$ produces.

Each term has a job. The constant fixes the level, the linear term the slope, the quadratic term the curvature, and higher terms correct asymmetry and tails. In finance the terms through second order usually suffice, which is why “level, slope, curvature” recurs in duration–convexity, delta–gamma and the volatility drag.

3Four Expansions That Run Finance

Three Maclaurin series, plus the geometric series they all lean on, generate most approximations in this book.

The exponential. Every derivative of $e^x$ is $e^x$, so $f^{(k)}(0)=1$ for all $k$ and

The logarithm. Differentiate $f(x)=\ln (1+x)$ repeatedly: $f^{\prime }(x)=(1+x{)}^{-1}$, $f^{\prime \prime }(x)=-(1+x{)}^{-2}$, $f^{\prime \prime \prime }(x)=2(1+x{)}^{-3}$, and in general $f^{(k)}(x)=(-1{)}^{k+1}(k-1)!(1+x{)}^{-k}$. At $0$ this gives $f^{(k)}(0)/k!=(-1{)}^{k+1}/k$, so

A second derivation explains the range. For $|t|<1$ the geometric series gives $\frac{1}{1+t}=1-t+t^2-t^3+\cdots$. Integrating from $0$ to $x$ term by term yields (3.3) for $|x|<1$. At $x=1$ the alternating series still converges, to $\ln 2$ (by Abel's theorem), which is why the range is $-1<x\le 1$.

The binomial series. For any real exponent $\alpha$, $f(x)=(1+x{)}^{\alpha }$ has $f^{(k)}(0)=\alpha (\alpha -1)\cdots (\alpha -k+1)$, so

where $\left(\genfrac{}{}{0ex}{}{\alpha }{k}\right)=\alpha (\alpha -1)\cdots (\alpha -k+1)/k!$. With $\alpha =-1$ it is the alternating geometric series $1/(1+x)=1-x+x^2-\cdots$. With $\alpha =\frac{1}{2}$ it gives $\sqrt{1+x}\approx 1+\frac{x}{2}-\frac{x^2}{8}$. When $\alpha$ is a positive integer the series stops and (3.4) is the ordinary binomial theorem.

Exhibit 2 collects the four series with their ranges and typical uses.

The radius of convergence has a financial meaning

The series (3.3) diverges for $x>1$ and the function itself does not exist at $x=-1$. Both edges matter. At $x=-1$, a return of −100%, the log of wealth is $-\infty$: that is ruin. It is why log-utility investors such as Kelly bettors (LM41) never accept a position that can lose everything.

For $x>1$, $\ln (1+x)$ is perfectly finite, but no number of series terms will reach it. A truncated series is a local tool; for large moves, compute the exact function.

4How Good Is “≈”? The Remainder Term

An approximation without an error estimate is a guess. Taylor's theorem supplies the estimate.

Read (3.5) as: the error is the next term of the series, evaluated at an unknown point $\xi$. We rarely know $\xi$, but we usually know how large $f^{(n+1)}$ can be on the interval, and that bounds the error. In big-O notation we write $E_n(x)=O(|x-a{|}^{n+1})$: halve the distance to $a$ and an order-$n$ error shrinks by a factor of about $2^{n+1}$.

The error of “log return ≈ $R-R^2/2$”

For $f(x)=\ln (1+x)$, $f^{\prime \prime \prime }(\xi )=2/(1+\xi {)}^3$, so with $n=2$ and $a=0$

For $x>0$, $\xi \in (0,x)$ gives $1+\xi >1$, and the error lies in $(0,x^3/3)$. For $-1<x<0$, $\xi \in (x,0)$ gives $1+\xi >1+x$, and the error is negative with size at most $|x{|}^3/(3(1+x{)}^3)$. Losses are approximated less accurately than gains of the same size, because $\ln (1+x)$ bends harder on the left. Exhibit 3 compares the actual errors with these bounds.

5Simple Returns and Log Returns

A holding-period (simple) return and a log return describe the same price move:

From (3.3) and (3.2), each is a series in the other:

Because $\ln (1+x)$ is concave, it lies below its tangent line at the origin, which is $x$ itself. Hence $\ln (1+x)\le x$ for every $x>-1$, with equality only at $0$ (Practice Problem 5 proves it with (3.5)). So the log return is always the smaller number, by about $R^2/2$ (Exhibit 4).

Two aggregation rules, two return types

Each definition is additive along a different dimension, which is why both are needed.

• Across time, log returns add. Over $T$ periods the growth factor is $G={\prod }_t(1+R_t)$, so $\ln G={\sum }_{t=1}^T{r}_t$, because the log of a product is a sum. Simple returns compound instead: $1+R_{0\to T}=G$.
• Across assets, simple returns add. A portfolio with weights $w_j$ earns $R_p={\sum }_j{w}_j{R}_j$ exactly. Log returns do not: $r_p=\ln \left({\sum }_j{w}_j{e}^{r_j}\right)\ne {\sum }_j{w}_j{r}_j$.

The working rule follows. Use log returns for research along the time axis — trends, volatility, cross-asset comparisons. Use simple returns whenever money is aggregated across positions — portfolio accounting, settlement, drawdown in currency.

The gap in real data

The sample is the 2,902 days on which both 0050 and 00631L traded, from November 2014 to October 2026. Each fund was halted for a few sessions around its split, so two of these returns span several sessions.

Over these days, 0050's mean daily simple return was 0.0871% and its mean daily log return 0.0794%. The difference, 0.0077% (0.77 bp) a day, is what (3.6) predicts. Half the mean squared return is 0.771 bp, and half the variance is 0.767 bp. Summed over the sample's $A=243.1$ returns a year, this “small” daily gap is roughly 1.9 percentage points of annual return.

import numpy as np

# ret: the lab's daily total returns (columns date, R_0050, R_00631L)
# on the 2,902 days when both funds traded; Part 2 of the lab builds it
# from the FinMind files in its data/ folder
R = ret["R_0050"].to_numpy()
r = np.log1p(R)                                    # exact log returns
gap = (R.mean() - r.mean()) * 1e4                  # in bp
half_m2 = (R**2).mean() / 2 * 1e4                  # in bp
print(f"mean R      = {R.mean():.6%}")             # 0.087120%
print(f"mean r      = {r.mean():.6%}")             # 0.079422%
print(f"gap         = {gap:.3f} bp")               # 0.770 bp
print(f"mean(R^2)/2 = {half_m2:.3f} bp")           # 0.771 bp, eq. (3.6)

6Arithmetic Mean, Geometric Mean and Volatility Drag

Given returns $R_1,\dots ,R_T$, three averages answer three different questions.

$\bar{R}$ is the average one-period outcome. $R_G$ is the constant return that produces the same ending wealth. $g$ is the same thing on the log scale, where it adds up over time. Wealth compounds, so $R_G$ and $g$, not $\bar{R}$, describe what a buy-and-hold investor actually earned.

Deriving the drag

Apply (3.3) to each period and average:

The mean of squares needs no probability theory. With the population variance $s^2=\frac{1}{T}{\sum }_t(R_t-\bar{R}{)}^2$, expanding the square gives the identity

Substituting and stopping at second order:

Converting back with (3.2), $R_G=e^g-1\approx g+\frac{g^2}{2}$, and $g^2\approx {\bar{R}}^2$ to this order, so the ${\bar{R}}^2$ terms cancel:

This is the familiar rule of thumb “geometric mean ≈ arithmetic mean − σ²/2”. The gap $\bar{R}-R_G\approx s^2/2$ is called volatility drag. Equations (3.8) and (3.9) differ: (3.9) approximates the geometric simple return, while (3.8) approximates the log growth rate and keeps an extra $-{\bar{R}}^2/2$. Both drop third- and higher-order terms, so both are approximations.

How accuracy depends on the measurement period

The approximation error depends on how large the per-period returns are, so it depends on the sampling frequency. Exhibit 5 measures it on the same 0050 history at five frequencies.

At daily frequency the approximation is exact to the displayed precision. At annual frequency, where single observations reach +48.7% and −21.3%, the error of (3.9) is 15 bp and that of (3.8) is 62 bp. The rule: second-order formulas are reliable when per-period standard deviations are a few percent, and need checking when they approach 20%.

7Leveraged ETFs: Power, Drag and Residual

The daily-reset identity

A daily-reset fund with leverage $L$ promises $L$ times the index's daily simple return. Ignore costs for now and write the fund's daily return as $L{R}_t$. Its log return minus $L$ times the index's log return is, by (3.3),

The series converges when $|LR|<1$ and $|R|<1$, which daily moves satisfy by a wide margin. The first-order terms cancel exactly — that is what “$L\times$ daily” buys you. What remains starts at second order.

Now sum over the $T$ days of a holding period and keep terms to second order. With $G$ the index's growth factor, $G_{\text{fund}}$ the fund's and $\mathrm{RV}={\sum }_t{R}_t^2$ the index's realized variance, this gives

This dissolves the paradox. “Twice the index's return” means $1+2(G-1)$, but for $L>1$ the power satisfies $G^L\ge 1+L(G-1)$, with equality only at $G=1$. This is Bernoulli's inequality, itself a statement that $x^L$ lies above its tangent line at $x=1$. So the fund has two forces: compounding of the trend, $G^L$ versus $1+L(G-1)$, which favours the fund, and variance drag, which hurts it.

Strong trends with low volatility favour the fund; choppy markets with high volatility punish it. Practice Problem 9 makes this precise. Before costs, a 2× fund beats twice the index's simple return roughly when the trend $|1-1/G|$ exceeds the realized volatility $\sqrt{\mathrm{RV}}$.

Twelve years of 00631L

Exhibit 6 compares three quantities built from the same 2,902 days. They are 0050's total-return growth, an exact daily 2× of 0050, ${\prod }_t(1+2R_t)$, and the Taylor formula $G^2{e}^{-\mathrm{RV}}$ along the path. 00631L's market price is plotted alongside.

Three conclusions follow from Exhibit 7. First, over the full sample the second-order term $-\mathrm{RV}$ comes within 0.0001 log points of the exact rebalancing effect. That is partly luck: the third-order term $2\sum R_t^3=+0.0027$ and the fourth-order term $-3.5\sum R_t^4=-0.0027$ cancel. The same cancellation explains why the third-order row alone lands farther away.

Single calendar years differ by up to 0.0025 log points (2026 to date), and the running gap reached 0.0046 on 9 April 2025. So (3.11) is accurate to a few thousandths of a log point, which against a drag of 0.447 still makes it an excellent model.

Second, the drag averaged $0.4473/11.94=0.0375$ log points a year, essentially 0050's annualized variance, $s^{2}A=(1.2385\%{)}^2\times 243.1=0.0373$; the small difference is ${\bar{R}}^{2}A=0.0002$. Third, 00631L still grew 44-fold against 10-fold for 0050, because $G^2\approx 100$ is a very large factor: the trend term overwhelmed the drag.

The year-by-year view in Exhibit 8 shows the two forces trading places.

• 2017, a steady uptrend with the lowest variance of any full year, is the textbook case for the power term. An exact daily 2× of 0050 would have returned 38.3% against twice the index's 36.3%.
• 2022 is the mirror image in a falling market. An exact daily 2× of 0050 would have lost 41.0%, less than twice 0050's 21.3% loss (42.7%). Compounding a downtrend also helps, because each day's 2× loss is taken on a smaller base. 00631L itself lost only 36.4%, helped by a positive residual.
• 2015, a choppy, slightly negative year, is where drag wins. An exact daily 2× would have lost 14.5% and 00631L lost 16.5%, against twice the index's −12.6%.

What Taylor cannot see: the residual

Over the whole sample the residual averaged $-0.3659/11.94=-0.031$ log points a year, about −3.0% of value. Exhibit 8 shows it swinging from +0.117 (2021) to −0.267 (2024), and January to 8 October 2026 alone contributed −0.183. Over the eleven full years 2015–2025 the average was −0.018. It is not a constant fee.

The residual contains management and trading costs, the financing embedded in the leverage, any gap between market price and NAV, and replication basis. 00631L builds most of its exposure with TAIEX futures rather than Taiwan 50 instruments. On 19 May 2026 it held TAIEX futures equal to 160.87% of net assets plus TSMC shares equal to 39.63%. TSMC's weight was then 43.7% in TAIEX and 60.9% in the Taiwan 50 index.

When TSMC leads, as in 2024–2026, a futures-heavy fund lags twice the Taiwan 50. In years when the broader market does relatively better, the residual can turn positive. A model explains only the terms it contains. Equation (3.11) prices the variance drag to within a few thousandths of a log point; the residual needs its own model.

The variance term in (3.11), $-\frac{L(L-1)}{2}{\sigma }^2$ per unit time, will reappear in LM42 as the curvature of the Kelly growth curve. A position held at a constant multiple of equity and rebalanced continuously gives up $\frac{1}{2}{\sigma }^2$ times the square of that multiple in log growth. A fund's leverage and a Kelly investor's exposure are the same variable seen from two sides.

8The Same Second-Order Idea Elsewhere

Bond prices: duration and convexity

A bond's price $P(y)$ is a smooth, convex function of its yield. A second-order expansion about the current yield gives the standard approximation

Modified duration is the slope term and convexity the curvature term: the same level–slope–curvature structure as Section 2.

Exhibit 9 shows the same comparison across all yields from 0% to 6%: the tangent line misses the curvature that the second-order term restores.

Options: delta and gamma

An option's value $V(S,t)$ expanded to second order in the underlying price and first order in time gives the trader's daily P&L approximation

where $\Delta =\partial V/\partial S$, $\Gamma ={\partial }^{2}V/\partial S^2$ and $\Theta =\partial V/\partial t$. A delta-hedged book removes the first term and keeps the gamma term: it earns $\frac{1}{2}\Gamma (\Delta S{)}^2$ from realized moves and pays $\Theta$ for the privilege. LM31 develops this for TXO and warrants.

A preview of Itô's lemma

Apply a second-order expansion to $\ln S$ when the price follows a random walk with per-period return $\mu \Delta t+\sigma \sqrt{\Delta t}Z$, with $Z$ a standard random shock:

In ordinary calculus the squared term is negligible, because $(\Delta x{)}^2$ is much smaller than $\Delta x$. Here it is not: $(\Delta S/S{)}^2\approx {\sigma }^2{Z}^2\Delta t$ is of the same order as the drift $\mu \Delta t$, and averages to ${\sigma }^2\Delta t$. As $\Delta t\to 0$ the shocks $\sqrt{\Delta t}Z$ become the increments $dB$ of a standard Brownian motion $B_t$ (LM27), and keeping the squared term yields

This is Itô's lemma for the logarithm, and the $-{\sigma }^2/2$ in it is the volatility drag of Section 6. In discrete time the drag was an approximation; in continuous time it becomes exact, because every higher-order term vanishes as $\Delta t\to 0$. LM28 makes this rigorous, and LM29 uses it to solve geometric Brownian motion.

9Taylor Approximations in Python

Two habits make approximations safe in code. First, compute the exact quantity alongside the approximation and report the gap. Second, prefer numerically stable library functions: np.log1p(x) and np.expm1(x) are accurate for tiny x, where np.log(1 + x) loses digits to rounding.

import math

def taylor_log1p(x, n):
    """Order-n Maclaurin polynomial of ln(1+x), eq. (3.3)."""
    return sum((-1) ** (k + 1) * x**k / k for k in range(1, n + 1))

def lagrange_bound_log1p(x, n):
    """Upper bound on |ln(1+x) - T_n(x)| from eq. (3.5)."""
    base = 1.0 if x >= 0 else 1.0 + x
    return abs(x) ** (n + 1) / ((n + 1) * base ** (n + 1))

for x in (-0.10, -0.0913, 0.05, 0.50):
    err = math.log1p(x) - taylor_log1p(x, 2)
    bound = lagrange_bound_log1p(x, 2)
    print(f"x={x:+.4f}  error={err:+.2e}  bound={bound:.2e}")

The decomposition of Section 7 takes a few vectorized lines for any index–fund pair with aligned daily simple returns:

import numpy as np

def leverage_decomposition(R, RF, L):
    """Split a daily-reset fund's log growth into power, drag and residual
    (eqs. 3.10-3.11). R, RF: aligned daily simple returns of the index and
    the fund, as NumPy arrays."""
    power = L * np.log1p(R).sum()                           # ln G^L
    exact_drag = (np.log1p(L * R) - L * np.log1p(R)).sum()  # exact effect
    taylor2 = -L * (L - 1) / 2 * (R**2).sum()               # to 2nd order
    taylor3 = taylor2 + (L**3 - L) / 3 * (R**3).sum()       # to 3rd order
    taylor4 = taylor3 - (L**4 - L) / 4 * (R**4).sum()       # to 4th order
    actual = np.log1p(RF).sum()
    return {"power": power, "exact_drag": exact_drag,
            "taylor2": taylor2, "taylor3": taylor3, "taylor4": taylor4,
            "residual": actual - power - exact_drag, "actual": actual}

# ret: the lab's aligned daily returns, built in its Part 2
# from the files in its data/ folder
R, RF = ret["R_0050"].to_numpy(), ret["R_00631L"].to_numpy()
d = leverage_decomposition(R, RF, L=2)
print({name: round(float(v), 5) for name, v in d.items()})
# power 4.60966, exact_drag -0.44729, taylor2 -0.44735, taylor3 -0.44461,
# taylor4 -0.44728, residual -0.36588, actual 3.79649