Understanding Absolute, Discrete, And Continuous Proportional Returns

In finance, few concepts are as fundamental yet as multifaceted as the notion of returns. Investment decisions, risk assessments, and portfolio strategies revolve around a single question: what return can we expect, and what risks do we face in pursuing that return? Yet this seemingly simple question masks a surprising complexity, because returns themselves come in multiple forms, each with its own mathematical properties, practical applications, and theoretical implications.

Whether you're a portfolio manager assessing daily performance, a risk manager calculating potential losses under stress scenarios, a financial analyst valuing derivatives, or an individual investor trying to understand your investment outcomes, you need to understand how returns are calculated, what they represent, and which type of return is appropriate for your specific purpose.

The challenge lies in recognizing that returns, or "shocks", are not a monolithic concept or a one-size-fits-all. The same change in asset price can be expressed in different ways, and these different expressions have profoundly different mathematical properties. A stock that rises from $100 to $120 has generated a return, but should we express this as a $20 gain, a 20% increase, or an 18.23% logarithmic return? Each of these expressions is correct, yet each serves different purposes and behaves differently in subsequent calculations

This section will explore three ways of measuring returns: absolute returns, discrete proportional returns, and continuous proportional returns. More importantly, we will examine why these different measures exist, when each should be used, and how they relate to the broader concepts. By understanding these distinctions, you will gain the analytical framework necessary to choose the appropriate return measure for any financial application.

What Are Returns?

At its most basic level, a return represents the financial outcome of an investment over a specific period. It captures the profit or loss generated by changes in asset prices, dividend payments, or other income-producing events. Returns answers a fundamental question: "How much money has been made or lost on this investment over a certain period of time?"

Returns can be measured in absolute dollar terms or as percentages. They can be calculated for any time period; daily, monthly, annual, or any other interval. They can include only price changes or incorporate dividends and other distributions. They can be measured using simple arithmetic or logarithmic calculations. Each of these choices affects not only the numerical value of the return but also its mathematical properties and practical applications. And by studying the patterns, distributions, and relationships among returns, we can build models that help us understand risk, price securities, and make informed investment decisions.

In risk management, returns take on additional significance and are often referred to as "shocks", sudden changes in asset prices that can affect portfolio values and risk exposures. This terminology reflects the risk manager's perspective that price changes represent "shock", resulting in gains or losses that must be managed and controlled.

The concept of shocks is particularly important in stress testing and scenario analysis. Risk managers need to understand how portfolios would respond to various market events- a sudden equity market crash, a rapid rise in interest rates, or a sharp movement in currency exchange rates. By modeling these events as shocks to asset prices, risk managers can quantify potential losses and ensure that portfolios can withstand adverse conditions.

The choice of how to measure these shocks, whether as absolute dollar changes, discrete percentage changes, or continuous logarithmic changes, has profound implications for risk models. Different shock measures have different statistical properties, different aggregation rules, and different relationships to the underlying probability distributions that describe market behavior. Understanding these differences is essential for building robust risk management frameworks.

1. Absolute Returns (Dollar Returns)

Absolute returns, often called dollar returns or nominal returns, represent the simplest and most intuitive way to measure investment performance. An absolute return is simply the difference between the ending value of an investment (or current price of an asset) and its beginning value (or previous price of an asset), expressed in currency units.

The mathematical formula for absolute return is straightforward:

Where:

𝑟Absolute represents an absolute return

S𝑡 = Price of an asset at time 𝑡 (current price)

S𝑡-1 = Price of an asset at time 𝑡-1 (previous price)

Example: You purchase shares of a company for $100 per share. Six months later, the share price has risen to $120. Your absolute return is simply $120 - $100 = $20 per share. If you owned 100 shares, your total absolute return would be $2,000. This measure does not consider the scale of the investment (relative percentage), but tells you exactly how many dollars you gained from the investment (the absolute increase or decrease).

The primary advantage of absolute returns lies in their simplicity and direct interpretability. When someone asks, "How much money did you make?" the absolute return provides a direct answer. There's no need for conversion, percentage calculations, or logarithmic transformations; the answer is simply the dollar amount gained or lost.

In portfolio management, absolute returns allow for direct aggregation across multiple assets. If you own ten different stocks, you can simply add up the absolute returns from each to determine your total dollar gain or loss. This additive property makes absolute returns convenient for certain types of portfolio accounting.

The most fundamental limitation is that absolute returns are not scale-independent. A $20 gain means something very different if you invested $100 versus $10,000. This scale dependency makes it impossible to compare returns across investments of different sizes.

Example: Investment A gains $1,000 from a $5,000 initial investment, while Investment B gains $1,000 from a $100,000 initial investment. Using absolute returns alone, these investments appear to perform identically; both generated $1,000. However, any rational investor would recognize that Investment A performed far better in relative terms, generating a 20% return versus only a 1% return for Investment B.

This scale dependency makes absolute returns inappropriate for performance comparison and benchmarking. Portfolio managers evaluated on absolute returns face perverse incentives; they can improve their measured performance simply by managing more money, regardless of their skill in generating returns relative to the capital employed.

2. Discrete Proportional Returns (Simple Returns)

Discrete proportional returns, commonly called simple returns or percentage returns, represent the most widely used measure of investment performance. These returns express the change in asset price as a percentage of the initial price, providing a scale-independent measure that allows for meaningful comparison across different investments and time periods.

The formula for discrete proportional returns is:

Where:

𝑟discrete represents the discrete proportional return

S𝑡 = Price of an asset at time 𝑡 (current price)

S𝑡-1 = Price of an asset at time 𝑡-1 (previous price)

Using our previous example, where a stock price rises from $100 to $120, the discrete proportional return would be:

rdiscrete = ($120 - $100) / $100 = $20 / $100 = 0.20 = 20%

This return can be expressed as either a decimal (0.20) or a percentage (20%), with both representations conveying the same information.

Discrete proportional returns possess several important mathematical properties that make them useful for financial analysis. First and most importantly, they are scale-independent. A 20% return means the same thing whether you invested $100 or $1,000,000; your investment grew by one-fifth of its initial value. This property enables meaningful comparison across investments of different sizes.

Discrete returns are also bounded below at -100% (representing total loss of investment) but unbounded above. You cannot lose more than 100% of your investment (absent leverage), but there's theoretically no limit to how much you can gain. This asymmetry reflects the mathematical reality that prices cannot go below zero but can rise indefinitely.

An important property of discrete returns is their relationship to portfolio aggregation. If you hold a portfolio of multiple assets, the portfolio's discrete return equals the weighted average of the individual asset returns, where the weights represent the proportion of the portfolio invested in each asset. This property makes discrete returns particularly convenient for portfolio management and performance attribution.

Investment and risk managers use discrete returns for performance measurement and risk attribution analysis. Because portfolio returns can be calculated as weighted averages of individual asset returns, discrete returns facilitate the decomposition of portfolio performance into contributions from different assets or sectors.

However, discrete returns have a crucial limitation when dealing with multiple time periods: they are not additive over time. You cannot simply add the returns from different periods to get the total return over multiple periods. Instead, you must compound them multiplicatively.

To understand why discrete returns don't simply add up over time, consider a simple example.

Example: A stock has the following returns over three consecutive periods: +10%, -5%, +8%.

What is the total return over all three periods?

Your intuition might suggest adding them: 10% - 5% + 8% = 13%. However, this is incorrect because it fails to account for the compounding effect, the fact that each period's return is earned on a different base amount.

The correct calculation requires multiplicative compounding:

• After period 1: $100 x (1 + 0.10) = $110

• After period 2: $110 x (1 - 0.05) = $104.50

• After period 3: $104.50 x (1 + 0.08) = $112.86

The actual total return is ($112.86 - $100) / $100 = 12.86%, not 13%.

This compounding can be expressed more formally as:

(1 + total return) = (1 + r1) x (1 + r2) x (1 + r3) = (1 + 0.10) x (1 - 0.05) x (1 + 0.08)

This multiplicative structure creates complexity in statistical analysis and modeling. When returns are not additive, calculating expected returns over multiple periods requires careful attention to the effects of volatility on compound growth. This limitation of discrete returns motivates the use of continuous returns in many theoretical and empirical applications.

3. Continuous Proportional Returns (Logarithmic Returns)

Continuous proportional returns, also called logarithmic returns or log returns, represent a more sophisticated approach to measuring investment performance. Rather than expressing returns as simple percentage changes, continuous returns use the natural logarithm of the price ratio. This seemingly complex transformation creates a return measure with mathematical properties that make it indispensable for theoretical modeling and certain practical applications.

The formula for continuous proportional returns is:

Where:

𝑟continuous or 𝑟log represents the continuous proportional (log) return

LN denotes the natural logarithm

S𝑡 = Price of an asset at time 𝑡 (current price)

S𝑡-1 = Price of an asset at time 𝑡-1 (previous price)

Using our continuing example where a stock rises from $100 to $120:

rcontinuous = ln(120/100) = ln(1.20) ≈ 0.1823 = 18.23%

Notice that this differs from the discrete return of 20%. Continuous returns are always slightly smaller in magnitude than corresponding discrete returns for positive price changes, and slightly larger in absolute magnitude for negative price changes.

To understand continuous returns, we must first understand continuous compounding. While discrete returns assume that compounding occurs at discrete intervals (daily, monthly, annually), continuous returns assume that compounding occurs continuously, at every instant in time.

Imagine an investment that pays interest. With annual compounding, you receive interest once per year. With monthly compounding, you receive interest twelve times per year, earning "interest on interest" more frequently. With daily compounding, this effect intensifies further. Continuous compounding represents the mathematical limit as the compounding frequency approaches infinity, compounding occurring at every instant.

The relationship between discrete and continuous returns reflects this difference in compounding assumptions. If you earn a 20% discrete return, this is equivalent to earning approximately 18.23% continuously compounded. The difference arises because continuous compounding generates slightly more growth from the same nominal rate due to the infinite frequency of compounding.

Mathematically, continuous compounding follows the exponential function. An investment of $100 growing at a continuous rate of r for time t will be worth:

St = S0 x e^(r x t)

Where:

e is Euler's number (approximately 2.71828).

This exponential growth model underlies much of modern financial theory.

The most important property of continuous returns is that they are additive over time. Unlike discrete returns, which must be compounded multiplicatively, continuous returns can simply be added together to calculate multi-period returns.

Using our earlier example where a stock has returns of +10%, -5%, and +8% in discrete terms, let's convert these to continuous returns:

• Period 1: ln(1.10) ≈ 0.0953 or 9.53%

• Period 2: ln(0.95) ≈ -0.0513 or -5.13%

• Period 3: ln(1.08) ≈ 0.0770 or 7.70%

The total continuous return is simply: 0.0953 - 0.0513 + 0.0770 = 0.1210 or 12.10%. Converting this back to a discrete return: e^0.1210 - 1 ≈ 0.1286 or 12.86%

This matches the result we obtained earlier through multiplicative compounding of discrete returns!

This additive property dramatically simplifies many calculations and theoretical derivations. When building statistical models or conducting time-series analysis, the ability to add returns across periods rather than multiplying them provides enormous mathematical convenience.

Continuous proportional returns form the foundation of many important financial models and theories. The Black-Scholes option pricing model, perhaps the most famous financial model of all time, assumes that stock prices follow a geometric Brownian motion with continuous returns that are normally distributed.

Practical Implementation

Let's work through a practical example that illustrates how the different return measures are calculated and applied in practice. Consider an investment analyst evaluating a stock's performance over a five-day period:

Day 0 (Start): $100.00 Day 1: $102.50 Day 2: $101.80 Day 3: $105.00 Day 4: $103.50 Day 5: $107.25

Absolute Returns:

• Day 1: $102.50 - $100.00 = $2.50

• Day 2: $101.80 - $102.50 = -$0.70

• Day 3: $105.00 - $101.80 = $3.20

• Day 4: $103.50 - $105.00 = -$1.50

• Day 5: $107.25 - $103.50 = $3.75

Total absolute return: $107.25 - $100.00 = $7.25

Discrete Returns:

• Day 1: ($102.50 - $100.00) / $100.00 = 2.50%

• Day 2: ($101.80 - $102.50) / $102.50 = -0.68%

• Day 3: ($105.00 - $101.80) / $101.80 = 3.14%

• Day 4: ($103.50 - $105.00) / $105.00 = -1.43%

• Day 5: ($107.25 - $103.50) / $103.50 = 3.62%

Total discrete return (compounded): (1.025) x (0.9932) x (1.0314) x (0.9857) x (1.0362) - 1 = 7.25%

Note: Simply adding the daily returns (2.50% - 0.68% + 3.14% - 1.43% + 3.62% = 7.15%) gives a slightly incorrect result due to the compounding effect.

Continuous Returns:

• Day 1: ln(102.50/100.00) = 2.47%

• Day 2: ln(101.80/102.50) = -0.68%

• Day 3: ln(105.00/101.80) = 3.09%

• Day 4: ln(103.50/105.00) = -1.44%

• Day 5: ln(107.25/103.50) = 3.56%

Total continuous return (additive): 2.47% - 0.68% + 3.09% - 1.44% + 3.56% = 7.00%

Note: You can convert continuous return back to discrete: e^0.07 - 1 = 7.25%

Relationships, Conversions, Comparisons, and Aggregations

Understanding how the three return measures relate to each other is essential for choosing the appropriate measure and converting between different formats. The relationships among absolute, discrete, and continuous returns follow from their mathematical definitions.

Starting from absolute returns, we can derive discrete returns by dividing by the initial price:

rdiscrete = rabsolute / (S𝑡-1)

Converting between discrete and continuous returns uses the logarithmic and exponential functions:

rcontinuous = LN (1 + rdiscrete)

rdiscrete = e^(rcontinuous) - 1

These conversion formulas allow us to move between return measures as needed for different applications. However, it's important to understand that the conversion is not merely a cosmetic change, different return measures have different mathematical properties and serve different analytical purposes.

For small returns (roughly between -10% and +10%), discrete and continuous returns are approximately equal. As returns become larger in magnitude, the difference between discrete and continuous returns increases. This relationship can be approximated using a Taylor series expansion:

rcontinuous ≈ rdiscrete - (1/2)rdiscrete^2 + (1/3)rdiscrete^3 - ... so on!

For small values, the higher-order terms become negligible, and rcontinuous ≈ rdiscrete

The numerical differences among return measures are important to understand. For positive returns, the magnitude ordering is always:

rdiscrete > rcontinuous < rabsolute / initial price

For example, with a price increase from $100 to $120:

• Discrete return: 20.00%

• Continuous return: 18.23%

• Absolute return: $20 (which equals 20% when scaled by initial price)

For negative returns, the relationship in absolute magnitude reverses:

rdiscrete < rcontinuous > rabsolute / initial price

For example, with a price decrease from $100 to $80:

• Discrete return: -20.00%

• Continuous return: -22.31%

This asymmetry reflects the mathematical properties of logarithmic functions. The practical implication is that continuous returns show slightly smaller gains and slightly larger losses than discrete returns, creating a conservative bias that can be useful in risk management.

The behavior of returns across multiple time periods differs fundamentally among the three measures, with important implications for practical applications.

• Absolute returns aggregate linearly across assets within a period but not across time periods. If you own multiple stocks, you can add their absolute dollar returns to get your total dollar gain or loss. However, you cannot meaningfully add absolute returns from different time periods without accounting for changing capital bases.

• Discrete returns aggregate as weighted averages across assets within a period. The portfolio return equals the weighted sum of individual asset returns, where weights represent portfolio allocations. Across time periods, discrete returns must be compounded multiplicatively:

(1 + total return) = (1 + r1) x (1 + r2) x ... x (1 + rn)

• Continuous returns aggregate additively both across time and (approximately) across assets. The total continuous return over multiple periods simply equals the sum of the individual period returns:

total return = r1 + r2 + ... + rn

This additive property provides substantial mathematical convenience for multi-period analysis.