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📈 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 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.

💰 Portfolio Return Methods

When a portfolio has cash flows (deposits, withdrawals), a single return formula is not enough. Two methods exist to isolate performance from cash flow effects:

⏱️ Time-Weighted Return (TWR)

Eliminates the effect of cash flows by computing sub-period returns between each flow event and chaining them:

\[ R_{TWR} = \prod_{i=1}^{n} (1 + r_i) - 1 \]
  • Measures pure portfolio performance regardless of deposit/withdrawal timing
  • Used by fund managers for benchmarking (GIPS-compliant)
  • Not affected by investor behavior (adding money at peaks, withdrawing at lows)

💵 Money-Weighted Return (MWR / IRR)

Accounts for the timing and size of cash flows — the internal rate of return that sets the NPV of all flows to zero:

\[ 0 = \sum_{i=0}^{n} \frac{CF_i}{(1 + r)^{t_i}} \]

where \(CF_i\) is each cash flow (deposits positive, withdrawals negative, final portfolio value positive).

  • Measures investor-specific experience (your actual return given when you added/removed cash)
  • Penalizes bad timing (depositing at highs, withdrawing at lows)
  • Used for personal portfolio performance

Which one does LibreFolio use?

LibreFolio will compute both TWR and MWR in the portfolio analytics dashboard. TWR for benchmark comparison, MWR for personal performance assessment.

⚠️ 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