Exponentials, Logarithms and Compounding

1Introduction

Every return in finance, from a deposit rate to a fund's lifetime performance, describes one quantity: the growth factor, ending value divided by starting value. Growth compounds, so it multiplies, and the mathematics of repeated multiplication is the exponential function and its inverse, the logarithm. This module builds both from first principles, derives the number $e$ from compound interest, and turns them into conversions that practitioners perform every day. The same tools recur in every later module, from the Taylor approximations of LM3 to the lognormal prices of LM29.

Section 2 develops exponents, logarithms and log-scale charts. Section 3 turns periodic interest into effective annual rates, takes the limit of continuous compounding and arrives at $e$. Section 4 defines the three return measures, converts among them and resolves the case.

Section 5 shows how returns add across time and across positions. Section 6 solves growth equations for time, required return and drawdown recovery, using 0050's drawdowns of 2024 and 2025. Section 7 assembles the calculations in Python.

2Exponents and Logarithms

Growth multiplies

Wealth grows by factors. Returns $R_1$ in the first period and $R_2$ in the second turn $W_0$ into $W_0(1+R_1)(1+R_2)$. With the same return $R$ in each of $T$ periods,

The count $T$ sits in the exponent, so wealth is an exponential function of time. Doubling the number of periods squares the growth factor instead of doubling it, which is why long-horizon numbers defy intuition.

Laws of exponents

For a base $\beta >0$ and real exponents $x$ and $z$,

For whole-number exponents these are counting rules: ${\beta }^2{\beta }^3$ contains $2+3$ factors of $\beta$. Defining ${\beta }^{1/k}$ as the positive $k$-th root, and fractional powers as powers of such roots, extends them to rational exponents. Continuity extends them to all real exponents (stated without proof; see Stewart, Clegg and Watson, and LM2 for continuity). Exhibit 2 gives each law a financial reading.

Logarithms

Because ${\log }_{\beta }$ undoes ${\beta }^x$, every law of exponents becomes a law of logarithms. For $x,z>0$, write $x={\beta }^{{\log }_{\beta }x}$ and $z={\beta }^{{\log }_{\beta }z}$. The product law gives $xz={\beta }^{{\log }_{\beta }x+{\log }_{\beta }z}$, which by the definition means

The same steps give ${\log }_{\beta }(x/z)={\log }_{\beta }x-{\log }_{\beta }z$ and ${\log }_{\beta }1=0$. From $x^{\alpha }=({\beta }^{{\log }_{\beta }x}{)}^{\alpha }={\beta }^{\alpha {\log }_{\beta }x}$ follows the power rule ${\log }_{\beta }(x^{\alpha })=\alpha {\log }_{\beta }x$ for every real $\alpha$. A logarithm turns multiplication into addition and powers into multiples. That is why log returns add over time (Section 5) and why growth equations are solved with logs (Section 6).

Change of base

Any logarithm is the natural logarithm rescaled. Since $x={\beta }^{{\log }_{\beta }x}$, taking natural logs of both sides and applying the power rule gives $\ln x={\log }_{\beta }x\cdot \ln \beta$, so

The base is a unit of measurement, like kilometers and miles. For 0050, $\ln G=\ln 10.0225=2.3048$. In base 2 that is $2.3048/\ln 2=3.325$ doublings; in base 10 it is $2.3048/\ln 10=1.001$ tenfold increases. The fund grew almost exactly tenfold, which is why its log growth is close to $\ln 10=2.3026$.

Log-scale charts

A logarithmic axis places each value $W$ at a height that rises in step with $\ln W$. Equal changes in $\ln W$ are equal distances on the page, with two consequences.

By the quotient rule, $\ln W_2-\ln W_1=\ln (W_2/W_1)$, so equal vertical distances are equal ratios. The step from 1 to 2 is as tall as the step from 4 to 8. Constant growth $W_t=W_0{e}^{gt}$, with $t$ in years, plots as the straight line $\ln W_t=\ln W_0+gt$, by the product rule and $\ln e^{gt}=gt$. Its slope is the log growth rate $g$.

Exhibit 3 plots 0050's total-return index on both axes, with gridlines at successive doublings and the constant-rate path that ends at the same value.

On the linear axis the first six years look flat and the end looks explosive. Even the constant-rate path, which by construction grows at the same rate every year, appears to accelerate. On the log axis that path is a straight line, so only genuine changes in the growth rate bend the index away from it.

The doublings put numbers on the picture. The index first closed above 2 on 11 November 2020, above 4 on 4 July 2024 and above 8 on 27 April 2026. The three doublings took 6.0, 3.6 and 1.8 years.

Declines are distorted as well. The January–April 2025 decline cost 1.18 units of the index and the April–August 2015 decline 0.23 units, a ratio of five. In percentage terms they were −27.5% and −21.0%, or 0.32 and 0.24 log points.

3Compounding Frequency and the Number e

Simple and compound interest

Under simple interest, interest accrues on the principal only, so wealth grows linearly: $W_{\tau }=W_0(1+i\tau )$ after $\tau$ years at an interest rate $i$ per year. Under compound interest credited yearly, interest joins the balance and earns interest itself, which gives the exponential growth of (1.1): $W_{\tau }=W_0(1+i{)}^{\tau }$. For a positive rate the two agree at $\tau =0$ and $\tau =1$; simple interest is higher within the first year, compound interest after it. Both appear in Taiwan retail finance, as the examples below show.

Periodic, stated and effective rates

Banks quote rates per year but credit interest more often. A stated annual rate $r^{(n)}$ (名目年利率) credited $n$ times a year pays the periodic rate $i:=r^{(n)}/n$ each period; yearly crediting, $n=1$, gives the annual rate $i$ above. After one year a unit has grown to $(1+i{)}^n$.

The EAR is the common currency for comparing quotes: two rates are equivalent when their EARs match. Exhibit 4 applies (1.4) to two Taiwan benchmarks: Bank of Taiwan's rate for one-year time savings deposits (定期儲蓄存款) and the legal cap on credit-card revolving interest.

Two features stand out. The EAR rises with $n$ but by ever smaller steps, and it levels off at a finite limit. And frequency matters far more at high rates: moving from annual to continuous compounding adds 1.5 bp at 1.725% but 118 bp at 15%. The gap grows roughly with the square of the rate, a second-order effect that LM3 quantifies.

From discrete to continuous: the number $e$

Push the frequency to its limit. In 1683 Jacob Bernoulli asked what happens to one unit at 100% interest a year when interest is credited ever more often (O'Connor and Robertson). With $n$ credits the year-end value is $(1+1/n{)}^n$, and Exhibit 5 shows it rising but leveling off.

A limit is the value a sequence approaches, as closely as we like, once $n$ is large enough. LM2 makes limits and continuity precise; this section uses them informally.

The general limit follows from (1.5) by a change of variable.

Proof. For $x=0$ both sides equal 1. For $x>0$, put $m=n/x$, so that $x/n=1/m$ and $n=mx$:

For $x=-z<0$, note that $1-z/n=1/(1+\frac{z}{n-z})$ for $n>z$, so that

Applied to money, (1.6) says that a stated rate $r$ compounded continuously turns $W_0$ into

after $\tau$ years, so its effective annual rate, and the inverse conversion, are

The continuously compounded rate equivalent to an EAR $R$ is the natural logarithm of one plus the EAR, $r=\ln (1+R)$, not the logarithm of $R$ itself. Every stated rate with the same EAR lies between $R$ and $r$: $r^{(n)}$ falls from $r^{(1)}=R$ toward $r$ as $n$ grows. More frequent crediting needs a lower quote to deliver the same EAR.

4Holding-Period, Annualized and Continuously Compounded Returns

Holding-period return

The HPR carries no time unit: +902.25% describes twelve years, while +10% might describe a week. A price return omits income; a total return includes it. Fund comparisons should use total returns, because funds differ in how much they distribute.

For 0050 over the case period the price return was +604.13% and the total return +902.25% (Example 1). Taking the dividends in cash instead, NT$42.19 per 2014 share, and holding the cash idle would have returned $(459.80-65.30+42.19)/65.30=+668.74\%$.

Annualized return

To compare holdings of different lengths, express each as a constant annual rate. Measure the horizon in years as $\tau =$ calendar days ÷ 365.25, which averages the leap-year cycle. The annualized return $R_G$ is the return which, compounded once a year for $\tau$ years, produces the same growth factor:

The root law of Exhibit 2 does the work. For $\tau >1$ the exponent $1/\tau$ pulls the growth factor toward 1, shrinking gains and losses alike. For $\tau <1$ it pushes the factor away from 1 and compounds a short-period return forward, which is extrapolation rather than measurement. LM3 calls $R_G$ the geometric mean return and compares it with the arithmetic mean.

Continuously compounded return

The log return over the holding period is $r_{0,T}=\ln (1+R_{0,T})=\ln G$. Divided by the horizon, it gives the continuously compounded return per year, also called the log growth rate:

The second equality is the power rule, $\ln G/\tau =\ln \left(G^{1/\tau }\right)=\ln (1+R_G)$. Comparing (1.10) with (1.8) shows what $g$ is: the continuously compounded rate equivalent to the effective annual rate $R_G$. The annualized return and the continuously compounded return are one growth rate quoted under two compounding conventions. Exhibit 6 collects the four expressions of a growth factor with their values for 0050.

Annualizing across horizons and day-count conventions

A return over part of a year annualizes the same way; only the measure of $\tau$ changes. Over $d$ calendar days, $\tau =d/365.25$; over $T$ trading sessions, $\tau =T/A$, where $A$ is the number of sessions per year. Then

The log version is linear in $1/\tau$ because logs add over time. Scaling a simple return linearly, $R/\tau$, gives a stated rate compounded $1/\tau$ times a year, not an EAR. Instruments with their own day count keep it, such as the actual/365 deposit terms of Example 2.

Many analysts and software packages set $A=252$, but Taiwan's market is less busy. Counting the days on which 0050 or 00631L traded, the Taiwan Stock Exchange held between 239 and 247 sessions a year in 2015–2025. The yearly counts were 244, 244, 246, 247, 242, 245, 244, 246, 239, 242 and 243, an average of 243.8. Within one sample, the right $A$ is the number of return observations per year in that sample, as Example 6 shows.

Exhibit 7 applies the three measures to trailing windows ending on 8 October 2026, the format of a fund fact sheet.

Two patterns stand out. The continuously compounded figure is always below the annualized one, since $g=\ln (1+R_G)<R_G$ whenever $R_G\ne 0$ (Practice Problem 5). And for windows shorter than a year the exponent $1/\tau$ exceeds 1, so annualizing magnifies whatever happened in the window: the six-month 45.99% becomes 112.80%.

5Adding Returns Across Time and Across Assets

Across time, log returns add

Over $T$ consecutive periods growth factors multiply, $G={\prod }_t(1+R_t)$. The product rule (1.2) turns the product into a sum:

The identity is exact for any path. Simple returns do not add over time, and their sum has no meaning. Exhibit 8 splits 0050's twelve years into six two-year blocks.

The table also shows how simple returns distort comparisons. The last block's 149.606% looks more than ten times the second block's 14.063%; in log points it is seven times, 0.9147 against 0.1316. The final two blocks supplied 71% of the twelve-year log growth, the steep end of the log chart in Exhibit 3. Averaging the six simple returns would also overstate the typical compound return; LM3 derives that gap, volatility drag, and LM14 its general form, Jensen's inequality.

Across assets, simple returns add

A portfolio is a sum of money, not a product. Let $J$ positions start with values $V_{j,0}$, total value $V_0>0$ and weights $w_j=V_{j,0}/V_0$, which sum to 1; a short position has $V_{j,0}<0$ and $w_j<0$. If position $j$ returns $R_j$ over the period, the portfolio ends at ${\sum }_j{V}_{j,0}(1+R_j)$, so

exactly, with weights measured at the start of the period, whatever their signs. Log returns do not aggregate this way. The portfolio's log return is $r_p=\ln \left({\sum }_j{w}_j{e}^{r_j}\right)$.

Consider a long-only portfolio, with every $w_j\ge 0$ and ${\sum }_j{w}_j=1$. For positive $x_j$, the weighted arithmetic–geometric mean inequality states that ${\sum }_j{w}_j{x}_j\ge {\prod }_j{x}_j^{w_j}$, with equality only when all $x_j$ with $w_j>0$ are equal. It is stated here without proof; LM14 derives it from Jensen's inequality. Applied with $x_j=e^{r_j}$ and followed by logs, it gives

with equality only when every position with $w_j>0$ has the same log return. Averaging log returns across a long-only portfolio therefore understates its return, and the error grows with the dispersion of the positions' returns. With short positions the inequality can fail: weights $(1.5,-0.5)$ and log returns $(0.10,0.20)$ give $r_p=\ln (1.5e^{0.10}-0.5e^{0.20})=0.0460$, below ${\sum }_j{w}_j{r}_j=0.0500$.

Across multiplicative factors: currency and inflation

Some returns combine by multiplication rather than by weighting. A Taiwan investor in a US-dollar asset earns the asset's dollar return and the change in the TWD price of a dollar:

exactly in logs. In simple returns, $R^{\text{TWD}}=R^{\text{USD}}+R^{\text{FX}}+R^{\text{USD}}{R}^{\text{FX}}$ carries a cross term. Inflation works the same way: a real return satisfies $1+R^{\text{real}}=(1+R)/(1+\pi )$, so $r^{\text{real}}=r-\ln (1+\pi )$ exactly. The familiar shortcut $R^{\text{real}}\approx R-\pi$ drops a second-order term, which LM3 measures.

6Solving Growth Equations with Logarithms

One equation, four unknowns

With an annual rate $R$, equation (1.1) over $\tau$ years reads $W_{\tau }=W_0(1+R{)}^{\tau }$. It links four quantities, and any three determine the fourth. The ending value needs only multiplication; the starting value, a present value, is the subject of LM6. The other two need logarithms or roots:

The time formula takes logs of both sides and applies the power rule, $\ln (W_{\tau }/W_0)=\tau \ln (1+R)$. In log terms every growth problem is a division: the log growth required, divided by the growth rate available or by the time available.

Doubling time

Setting $W_{\tau }/W_0=2$ gives the doubling time ${\tau }_2=\ln 2/\ln (1+R)=\ln 2/g$. With a continuously compounded rate the formula is exact and simple, ${\tau }_2=\ln 2/g\approx 0.6931/g$. The familiar rule of 72 approximates $\ln 2/\ln (1+R)$ for annual rates; LM3 derives it from a Taylor expansion and explains why it works best near 8%. Exhibit 9 compares the two for rates that appear in this module.

Time to a goal and the required return

Drawdowns and recovery

A drawdown is the percentage decline from the highest value reached so far, ${\mathrm{DD}}_t:=1-W_t/\max (W_0,W_1,\dots ,W_t)$; the running peak includes the starting value $W_0$. Recovering from a drawdown $\mathrm{DD}$ requires a gain $R_{\text{rec}}$ with $(1-\mathrm{DD})(1+R_{\text{rec}})=1$:

The required gain grows faster than the loss: a 10% drawdown needs +11.1%, a 50% drawdown +100% and a 90% drawdown +900%. In logs the asymmetry disappears. A drawdown is a log return of $\ln (1-\mathrm{DD})$, and recovery is the same distance back, $-\ln (1-\mathrm{DD})$. At a log growth rate $g$, recovery therefore takes ${\tau }_{\text{rec}}=-\ln (1-\mathrm{DD})/g$ years by (1.16); Exhibit 10 tabulates both views.

Exhibit 11 and Exhibit 12 apply the arithmetic to 0050's own history.

Every recovery in the sample was faster than steady growth at the long-run rate would imply, because rebounds from troughs were steep. That is a description of 2015–2026, not a law. The 2022 episode also shows the other side: the deepest decline took 281 days to reach its trough and 478 more to recover.

7Return Calculations in Python

Three habits keep return code honest. Store prices and cash flows exactly as delivered and apply corporate actions explicitly. Compute log returns with log1p, which stays accurate for tiny returns. Keep every convention, such as the base date, the day count and $A$, in a named variable instead of a hard-coded number.

The first block rebuilds 0050's total-return series and reproduces the case. Run from the repository root, it reads the raw FinMind files that data/fetch_finmind.py downloads into data/raw/ (see the companion lab below).

from datetime import date

import numpy as np
import polars as pl

# 1. FinMind closes (TaiwanStockPrice), dividends (TaiwanStockDividendResult)
px = pl.scan_csv("data/raw/0050__*.csv").select("date", "close").collect()
div = (pl.read_csv("data/raw/0050_dividends.csv")
         .select("date", pl.col("cash_dividend").alias("D")))
SPLIT, RATIO = "2025-06-18", 4.0                # 4-for-1, effective date

# 2. R_t = (P_t + D_t) / (P_{t-1} / ratio_t) - 1, close restated per new share
P, D = pl.col("close"), pl.col("D")
ratio = pl.when(pl.col("date") == pl.lit(SPLIT)).then(RATIO).otherwise(1.0)
ret = (px.join(div, on="date", how="left")
         .sort("date")          # never rely on row order after a join
         .with_columns(D.fill_null(0.0), ratio.alias("ratio"))
         .with_columns(R=(P + D) / (P.shift(1) / pl.col("ratio")) - 1)
         .drop_nulls("R")
         .with_columns(pl.col("date").str.to_date(), r=pl.col("R").log1p()))

# 3. growth factor G, annualized R_G and continuous g: (1.9), (1.10), (1.12)
BASE, YEAR = date(2014, 10, 31), 365.25   # base close; days per year
G = float(np.exp(ret["r"].sum()))
tau = (ret["date"][-1] - BASE).days / YEAR
print(f"G = {G:.4f}  HPR = {G - 1:.2%}  "
      f"R_G = {G ** (1 / tau) - 1:.2%}  g = {np.log(G) / tau:.2%}")
# G = 10.0225  HPR = 902.25%  R_G = 21.30%  g = 19.31%

The second block computes drawdowns from the running peak and finds the 2025 recovery date of Example 10.

# 4. drawdown DD_t = 1 - W_t / running peak (incl. W_0 = 1), gain needed (1.17)
W = pl.col("W")
dd = (ret.select("date", W=(1 + pl.col("R")).cum_prod())
         .with_columns(DD=1 - W / pl.max_horizontal(W.cum_max(), 1.0)))
in_2025 = pl.col("date").is_between(date(2025, 1, 1), date(2025, 6, 30))
trough = dd.filter(in_2025).sort("DD", descending=True).row(0, named=True)
back = dd.filter(pl.col("date") > trough["date"], pl.col("DD") == 0)["date"][0]
need = trough["DD"] / (1 - trough["DD"])
print(trough["date"], f"DD {trough['DD']:.2%}", f"gain needed {need:.2%}",
      "recovered", back, (back - trough["date"]).days, "days")
# 2025-04-09 DD 27.49% gain needed 37.91% recovered 2025-07-17 99 days

The raw-file series counts 0050's own 2,906 sessions, four more than the aligned series of the daily example. It keeps the four March 2026 sessions on which 00631L did not trade. Growth factors, drawdowns and recovery dates are identical in both.