Return and excess return calculations — Make Your P&L Speak Human

If returns were a language, this is the Rosetta Stone. Learn it wrong and your portfolio whispers confusing things. Learn it right and your performance reports sing.

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What is return (and why we care)

Return is the basic answer to the question every investor asks at 3AM: did I make money or did I accidentally fund someone else’s yacht? In formula form, the simplest one-period return (also called holding-period return) is:

Return = (Ending Value + Cash Flows Received - Beginning Value) / Beginning Value

Key varieties you'll meet in the wild:

• Total return: includes price changes, dividends, coupons — the whole enchilada.
• Price return: only price appreciation/depreciation.
• Log returns (continuously compounded): convenient for time aggregation and statistical work: r = ln(Pt / Pt-1).

Why this matters beyond petty bragging rights: accurate returns are the building blocks for risk-adjusted metrics, attribution, and compensation. If you screw up returns, everything that follows (including excess return, alpha, and skill claims) is a house of cards.

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How do we calculate excess return?

Excess return = portfolio return minus benchmark or risk-free return, depending on context. It’s the part you can brag about, or explain away with fees.

Two common flavors:

1. Active excess return = Portfolio Return - Benchmark Return (e.g., S&P 500)
2. Risk-adjusted excess return = Portfolio Return - Risk-free Rate (e.g., Treasury), often used in Sharpe-type calculations

Example (simple):

Portfolio total return = 9%
Benchmark total return = 6%
Excess return = 9% - 6% = 3%

If you keep seeing 3% and think it's free money, congratulations: you might not be adjusting for fees, taxes, or survivorship bias yet.

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Real-world adjustments — the things that bite you if you're sloppy

Remember how we covered wrappers, fees, and fund structures? Good — this is where that becomes very relevant.

• Gross return vs net return: Gross ignores fees, net subtracts them. Always show both when evaluating a manager; net return is what the investor actually experiences.
• Cash flows and timing: Contributions/withdrawals change the math. Use time-weighted returns to neutralize investor timing, money-weighted returns (IRR) to capture investor experience.
• Corporate actions: Splits, dividends, rights issues — proper cleaning is mandatory (see previous module on data sourcing and cleaning).

Quick checklist before you compute returns:

• Are price series adjusted for dividends and splits? If not, your return is lying.
• Are returns reported gross of management fees? Mark that clearly.
• Are currency translations needed? Convert consistently.

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Arithmetic vs Geometric vs Log — pick your poison wisely

• Arithmetic mean: simple average of period returns. Good for expected single-period return estimation, bad for multi-period compounding.
• Geometric mean (CAGR): the proper multi-period compounded return. Use this for statements like 'this strategy returned X% per year over N years.'
• Log returns: approximately equal to arithmetic returns for small returns, additive over time, handy for statistical models.

Table: quick comparison

Measure	Use case	Aggregation property
Arithmetic mean	Short-run expected return	Not multiplicative
Geometric mean (CAGR)	Multi-period realized return	Multiplicative (compounding)
Log return	Modeling, normality approximations	Additive over time

Code-ish formulas:

Arithmetic = (r1 + r2 + ... + rn) / n
Geometric = ( (1+r1)*(1+r2)*...*(1+rn) )^(1/n) - 1
Log return period = ln(1 + r)

Ask yourself: are you reporting a ‘per-year’ figure or feeding data into a regression? Your choice decides the math.

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Examples: a walk-through with numbers

Scenario: You manage a fund that charges a 1% management fee and has the following annual gross returns for three years: 8%, 12%, -4%. The benchmark returned 6%, 10%, 0%.

Step 1: compute net returns (subtract fee annually):

• Year 1 net = 8% - 1% = 7%
• Year 2 net = 12% - 1% = 11%
• Year 3 net = -4% - 1% = -5%

Step 2: compute geometric (annualized) net return:

CAGR = (1.07 * 1.11 * 0.95)^(1/3) - 1 ≈ (1.128...)^(1/3) - 1 ≈ 4.07%

Step 3: compute benchmark CAGR:

Bench CAGR = (1.06 * 1.10 * 1.00)^(1/3) - 1 ≈ 5.31%

Step 4: excess return (annualized) ≈ 4.07% - 5.31% = -1.24% (ouch)

Interpretation: Gross returns might have looked fine, but after fees and compounding, the investor underperformed the benchmark.

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Common mistakes (and how to stop crying over Excel)

1. Not adjusting prices for dividends and splits. Result: phantom returns.
2. Confusing arithmetic average with geometric. Result: you claim nonsense annualized figures.
3. Subtracting fees once instead of at the correct frequency. Result: subtle but meaningful bias.
4. Comparing returns in different currencies without conversion. Result: misleading excess returns.
5. Using time-weighted returns when you need money-weighted (or vice versa). Result: blaming the manager for investor timing.

Pro tip: always annotate whether returns are gross/net, price/total, and the compounding basis.

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Quick Python pseudocode for calculating excess returns

# assume prices is a pandas.Series of adjusted close prices
returns = prices.pct_change().dropna()\nlog_returns = np.log(prices / prices.shift(1)).dropna()
# net returns: subtract fee as an approximation
net_returns = returns - fee_rate/periods_per_year
# excess over benchmark
excess = net_returns - benchmark_returns
# annualize geometric
annualized = (np.prod(1 + net_returns))**(1/years) - 1

(Yes, the real world requires more nuance. Use this as a starting ritual, not gospel.)

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Closing — how to not embarrass yourself in performance meetings

• Always clean your input data before calculating returns. Garbage in, poetic nonsense out.
• Report both gross and net returns, and label them boldly. Investors care about net.
• Use geometric mean/CAGR for multi-period realized returns; use arithmetic when forecasting a single period.
• Explicitly state the benchmark and whether excess returns are vs benchmark or vs risk-free.

A small truth: good analytics doesn't make a bad strategy good. But sloppy analytics can make a good strategy look criminal. Do the math right, then argue about markets.

Key takeaways:

• Return answers whether you made money; excess return answers whether you outperformed someone or something.
• Fee and wrapper adjustments matter — remember the conversations about wrappers and fees from previous modules.
• Choose the right mean for the right job: arithmetic for expectation, geometric for compounding, log for modeling.

Now go compute, annotate, and never again call an unadjusted price series 'returns' at a risk committee meeting. Your future self, and your investors, will thank you (or at least stop sending angry emails).