Skip to content

📈 Returns & Growth Rates

This page covers the mathematical foundations of investment returns — how to measure, compare, and annualize growth rates. These concepts are used throughout LibreFolio's FX measurement tools and portfolio analytics.

📊 Simple (Discrete) Return

The simple return over a period is the percentage change:

\[ R_{simple} = \frac{P_{end} - P_{start}}{P_{start}} = \frac{P_{end}}{P_{start}} - 1 \]

Example

If EUR/USD moves from 1.10 to 1.14:

\[R = \frac{1.14 - 1.10}{1.10} = 0.0364 = 3.64\%\]

📊 Properties

  • Intuitive: directly represents "how much you gained/lost"
  • Not additive: you cannot simply sum simple returns across periods to get total return
  • Compounding: multi-period returns must be multiplied, not added
\[ R_{total} = (1 + R_1)(1 + R_2) \cdots (1 + R_n) - 1 \]

📐 Logarithmic (Continuous) Return

The log return is the natural logarithm of the price ratio:

\[ r_{log} = \ln\left(\frac{P_{end}}{P_{start}}\right) = \ln(P_{end}) - \ln(P_{start}) \]

📊 Properties

  • Additive across time: total log return = sum of sub-period log returns
\[ r_{total} = r_1 + r_2 + \cdots + r_n \]
  • Symmetric: a +5% move followed by a −5% move returns exactly to the starting point
  • Approximately equal to simple return for small values: \(r_{log} \approx R_{simple}\) when \(R_{simple}\) is small

🔄 Conversion

\[ r_{log} = \ln(1 + R_{simple}) \qquad R_{simple} = e^{r_{log}} - 1 \]

📅 Annualized Return

To compare returns across different time periods, we annualize them — projecting the observed growth rate to a full year.

📈 Compound Annual Growth Rate (CAGR)

The most common annualization method. Given a total return over \(d\) calendar days:

\[ R_{annual} = \left(\frac{P_{end}}{P_{start}}\right)^{365/d} - 1 \]

This is what LibreFolio's Measures tool displays.

Example

EUR/USD moves from 1.10 to 1.14 over 90 days:

\[R_{annual} = \left(\frac{1.14}{1.10}\right)^{365/90} - 1 = (1.0364)^{4.056} - 1 \approx 15.5\%\]

📐 Annualized Log Return

For log returns, annualization is simply scaling:

\[ r_{annual} = r_{log} \times \frac{365}{d} \]

This linearity is one of the key advantages of log returns in quantitative finance.

🔄 Relationship Between Simple and Log Returns

Property Simple Return \(R\) Log Return \(r\)
Compounding Multiplicative: \((1+R_1)(1+R_2)\) Additive: \(r_1 + r_2\)
Symmetry Asymmetric: +10% then −10% ≠ 0 Symmetric: +10% then −10% = 0
Annualization \((1+R)^{365/d} - 1\) \(r \times 365/d\)
Portfolio returns Weighted sum works ✅ Weighted sum doesn't work ❌
Time series Not additive ❌ Additive ✅
Interpretation "I gained 5%" "Log growth rate was 0.0488"

When to use which?

  • Simple returns for reporting to users and computing portfolio-level returns
  • Log returns for statistical analysis, volatility estimation, and time-series models

📏 Day Count Conventions

The number of days \(d\) can be computed differently depending on the convention:

  • Actual/365: Calendar days (what LibreFolio uses)
  • Actual/360: Calendar days over a 360-day year (common in money markets)
  • 30/360: Assumes 30-day months and 360-day year

For more details, see Day Count Conventions.

⚠️ Pitfalls

  1. Very short periods: Annualizing a 3-day return can produce misleading figures (e.g., a 0.1% 3-day move → 12.5% annualized)
  2. Negative prices: Log returns are undefined for negative values — not an issue for FX rates
  3. Compounding frequency: CAGR assumes continuous compounding; real-world instruments may compound daily, monthly, or quarterly