Return Calculations

Return Calculations

Core Concepts

Simple (Holding Period) Return

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

where D = distributions (dividends, interest) received during the period. If V_end already reflects reinvested distributions, do not add D again.

Mean and Log Return Conventions

• Arithmetic mean R_a = (1/n) * sum(R_i) — unbiased estimate of the expected *single-period* return (use for forward-looking inputs, e.g., mean-variance optimization). Always >= geometric mean; overstates realized compound growth.
• Geometric mean R_g = [prod(1 + R_i)]^(1/n) - 1 — the correct measure of realized multi-period compound growth. The gap below the arithmetic mean approximates sigma^2 / 2 (volatility drag).
• Log return r = ln(V_end / V_begin) — time-additive (r_total = r_1 + ... + r_n), so preferred for statistical modeling and multi-period aggregation. Convert with R_simple = e^r - 1 and r = ln(1 + R_simple). Log returns are additive across time but NOT across assets.

CAGR (Compound Annual Growth Rate)

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

where n is measured in years. The annualized geometric growth rate between two valuations with no intermediate cash flows.

Time-Weighted Return (TWR)

Chain-links sub-period returns calculated between each external cash flow, removing the effect of cash flow timing. TWR measures the manager's investment skill independent of investor deposit/withdrawal decisions, and is the GIPS standard for manager performance.

$$1 + R_{TWR} = \prod_{i=1}^{n}(1 + R_i), \qquad R_i = \frac{V_{end,i}}{V_{begin,i} + CF_i} - 1$$

Exact TWR requires a portfolio valuation on every cash flow date.

Modified Dietz Return

When valuations on each cash flow date are unavailable, Modified Dietz approximates the period return by day-weighting each external cash flow within the period:

$$R_{MD} = \frac{V_{end} - V_{begin} - CF_{net}}{V_{begin} + \sum_i CF_i \times w_i}, \qquad w_i = \frac{CD - D_i}{CD}$$

where CF_net = sum of external cash flows, CD = calendar days in the period, and D_i = day of flow i (so w_i is the fraction of the period the flow was invested). It is a money-weighted approximation; chain-linking Modified Dietz sub-period returns approximates TWR. Accuracy degrades when flows are large relative to portfolio value or markets are volatile within the period — revalue on large-flow dates instead.

Money-Weighted Return (MWR / IRR)

The internal rate of return that sets the NPV of all investor 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. Solved numerically (Newton-Raphson or bisection).

Annualization

$$R_{annual} = (1 + R_{period})^{periods\_per\_year} - 1$$

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

Sub-Period Linking

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

The foundational identity behind TWR and CAGR.

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

This is quadratic in x = 1/(1+r); the positive root gives r = -0.66815% (verifiable with the bundled script or any IRR solver).

NPV check at r = -0.0066815:

-100,000 + (-100,000)/0.9933185 + 198,000/0.9933185^2
= -100,000 - 100,672.65 + 200,672.65
= 0.00  (exact)

The MWR is approximately -0.67% 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 -0.67% 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.
• Trusting Modified Dietz with large intra-period flows: when a single flow exceeds roughly 10% of portfolio value, revalue the portfolio on the flow date rather than day-weighting.

Running the Script

scripts/return_calculations.py provides a Returns class with static methods for every formula above (holding period return, TWR, MWR/IRR via Newton's method, Modified Dietz is straightforward to compose from these, CAGR, annualization, linking, arithmetic/geometric means, log-return conversions).

• Run: uv run scripts/return_calculations.py (PEP 723 inline metadata resolves numpy automatically), or python3 scripts/return_calculations.py with numpy installed.
• Bare invocation (or --verify) prints a demo of all functions and asserts the worked-example values above (Example 1 CAGR = 10%, Example 2 TWR = +8.0% cumulative / 3.92% annualized, MWR = -0.6682%), exiting nonzero on any mismatch.
• --help lists the available functions and import usage.
• For programmatic use, import rather than run: from return_calculations import Returns.

Cross-References

• time-value-of-money (core plugin): NPV, IRR, and discounting concepts overlap with MWR calculations; owns project/loan IRR
• statistics-fundamentals (core plugin): Arithmetic and geometric means, return distribution analysis