Return Calculations

Purpose

This skill enables Claude to compute, explain, and compare investment return metrics across different methodologies and time horizons. It covers the full spectrum from simple holding-period returns through time-weighted and money-weighted returns, ensuring the appropriate metric is selected for each analytical context.

Layer

0 — Mathematical Foundations

Direction

retrospective

When to Use

• User asks about calculating investment returns

• Comparing returns across different time periods

• Understanding arithmetic vs geometric vs log returns

• Computing CAGR, TWR, MWR/IRR

• Linking sub-period returns

Core Concepts

Simple (Holding Period) Return

The most basic measure of investment performance over a single period, capturing price change plus any income received.

$$R = \frac{V_{end} - V_{begin} + D}{V_{begin}}$$

where:

• V_end = ending value

• V_begin = beginning value

• D = distributions (dividends, interest) received during the period

Arithmetic Mean Return

The simple average of a series of periodic returns. Represents the expected return for any single period.

$$R_a = \frac{1}{n} \sum_{i=1}^{n} R_i$$

The arithmetic mean is always greater than or equal to the geometric mean. It is an unbiased estimator of the expected single-period return but overstates the compounded growth rate.

Geometric Mean Return

The constant rate that, if earned each period, would produce the same terminal wealth as the actual sequence of returns.

$$R_g = \left(\prod_{i=1}^{n}(1 + R_i)\right)^{1/n} - 1$$

The geometric mean captures the effects of compounding and volatility drag. It is always the correct choice for describing realized multi-period growth.

Log (Continuously Compounded) Return

The natural logarithm of the wealth ratio. Log returns are time-additive, making them convenient for multi-period aggregation and statistical modeling.

$$r = \ln\left(\frac{V_{end}}{V_{begin}}\right)$$

Properties:

• Time-additive: r_total = r_1 + r_2 + ... + r_n

• Conversion: R_simple = e^r - 1 and r = ln(1 + R_simple)

• More symmetric and closer to normally distributed than simple returns for small magnitudes

CAGR (Compound Annual Growth Rate)

The annualized geometric return that equates the beginning value to the ending value over a given number of years.

$$CAGR = \left(\frac{V_{end}}{V_{begin}}\right)^{1/n} - 1$$

where n is measured in years. CAGR smooths out volatility and provides a single annualized growth figure.

Time-Weighted Return (TWR)

Chain-links sub-period returns calculated between each external cash flow, thereby removing the effect of cash flow timing on the measured return. TWR measures the manager's investment skill independent of investor deposit/withdrawal decisions.

$$1 + R_{TWR} = \prod_{i=1}^{n}(1 + R_i)$$

where each sub-period return R_i is computed between consecutive cash flow dates:

$$R_i = \frac{V_{end,i}}{V_{begin,i} + CF_i}$$

In practice, exact TWR requires portfolio valuation on every cash flow date. The Modified Dietz method approximates TWR when daily valuations are unavailable.

Money-Weighted Return (MWR / IRR)

The internal rate of return that sets the net present value of all cash flows (contributions, withdrawals, and terminal value) to zero.

$$0 = \sum_{t=0}^{T} \frac{CF_t}{(1 + r)^t}$$

MWR reflects the actual investor experience because it is sensitive to the timing and magnitude of cash flows. It rewards (penalizes) investors who add capital before good (bad) periods.

Annualization

Converts a return measured over any holding period to an equivalent annual rate, assuming compounding.

$$R_{annual} = (1 + R_{period})^{periods_per_year} - 1$$

For example, a 2% quarterly return annualizes to (1.02)^4 - 1 = 8.24% .

Sub-Period Linking

Combines returns from contiguous sub-periods into a single cumulative return.

$$(1 + R_{total}) = \prod_{i=1}^{n}(1 + R_i)$$

This is the foundational identity behind TWR and CAGR calculations.

Key Formulas

Formula Expression Use Case

Holding Period Return R = (V_end - V_begin + D) / V_begin

Single-period total return

Arithmetic Mean R_a = (1/n) * sum(R_i)

Expected single-period return

Geometric Mean R_g = [prod(1+R_i)]^(1/n) - 1

Realized compound growth rate

Log Return r = ln(V_end / V_begin)

Time-additive return for modeling

CAGR (V_end / V_begin)^(1/n) - 1

Annualized growth over n years

TWR prod(1 + R_i) - 1

Manager performance (cash-flow neutral)

MWR / IRR sum(CF_t / (1+r)^t) = 0 , solve for r Investor-specific experience

Annualization (1 + R_period)^(periods/year) - 1

Standardize to annual basis

Sub-Period Linking (1 + R_total) = prod(1 + R_i)

Combine contiguous returns

Worked Examples

Example 1: Computing CAGR from a 5-Year Investment

Given: An investment of $10,000 grows to $16,105.10 over exactly 5 years with no intermediate cash flows.

Calculate: The compound annual growth rate (CAGR).

Solution:

CAGR = (V_end / V_begin)^(1/n) - 1 CAGR = (16,105.10 / 10,000)^(1/5) - 1 CAGR = (1.610510)^(0.2) - 1 CAGR = 1.10 - 1 CAGR = 0.10 = 10%

The investment grew at a compound annual rate of 10% per year.

Verification: $10,000 * (1.10)^5 = $10,000 * 1.61051 = $16,105.10

Example 2: TWR vs MWR Divergence with Poorly Timed Cash Flow

Given: A fund has the following history:

• Start of Year 1: Portfolio value = $100,000

• End of Year 1: Portfolio value = $120,000 (return = +20%)

• Start of Year 2: Investor deposits $100,000, bringing portfolio to $220,000

• End of Year 2: Portfolio value = $198,000 (return = -10%)

Calculate: Both TWR and MWR, and explain the divergence.

Solution:

Time-Weighted Return (TWR):

Sub-period 1 return: R_1 = (120,000 - 100,000) / 100,000 = +20% Sub-period 2 return: R_2 = (198,000 - 220,000) / 220,000 = -10%

TWR (cumulative) = (1 + 0.20) * (1 + (-0.10)) - 1 = 1.20 * 0.90 - 1 = 1.08 - 1 = +8.0%

TWR (annualized) = (1.08)^(1/2) - 1 = 3.92%

Money-Weighted Return (MWR / IRR): Cash flows from the investor's perspective:

• t=0: -$100,000 (initial investment)

• t=1: -$100,000 (additional deposit)

• t=2: +$198,000 (terminal value)

Solve: -100,000 + (-100,000)/(1+r) + 198,000/(1+r)^2 = 0

Testing r = -0.0051 (approximately -0.51%):

-100,000 + (-100,000)/0.9949 + 198,000/0.9899 = -100,000 - 100,512.6 + 200,020.2 approx -492.4 (close to zero; actual IRR approx -0.48%)

The MWR is approximately -0.48% annualized.

Interpretation: The TWR of +3.92% annualized reflects the manager's skill: the fund gained 20% then lost 10%, netting +8% over two years. The MWR of approximately -0.48% reflects the investor's experience: more money was at risk during the losing year (Year 2) because of the large deposit, so the investor's dollar-weighted outcome was slightly negative. This divergence highlights why TWR is preferred for evaluating manager performance, while MWR better describes the specific investor's realized result.

Common Pitfalls

• Confusing arithmetic and geometric means: the arithmetic mean is always greater than or equal to the geometric mean (AM-GM inequality). Using arithmetic mean to project compounded growth overstates terminal wealth.

• Using arithmetic mean for multi-period compounding: always use geometric mean or CAGR when describing compound growth over multiple periods.

• Annualizing returns from very short periods: annualizing a 2% weekly return yields (1.02)^52 - 1 = 180% , which amplifies noise and is misleading. Annualization is most meaningful for periods of at least one year.

• Ignoring cash flow timing when TWR is appropriate: MWR conflates manager skill with investor timing decisions. Use TWR for manager evaluation.

• Double-counting dividends: if the ending value V_end already includes reinvested dividends, do not add D separately in the holding period return formula.

Cross-References

• time-value-of-money (core plugin, Layer 0): NPV, IRR, and discounting concepts overlap with MWR calculations

• statistics-fundamentals (core plugin, Layer 0): Arithmetic and geometric means, return distribution analysis

Reference Implementation

See scripts/return_calculations.py for computational helpers.