"What are Returns - Really and how not to lie to yourself?"

Part 1: Returns from annual timeframes.

Compounding Returns (Don’t Just Add Returns!)

Returns stack multiplicatively, not additively.
For example:

• Day 1: +10%

• Day 2: -3%

You might think:
10% - 3% = 7% → Not true.

Real return is:
(1 + 0.10) × (1 - 0.03) - 1 = 6.7%

That’s because once you gain 10%, your base changes. So any future returns apply to the new amount.

Another Example:

• Day 1: +20% → value becomes 1.2x

• Day 2: -20% → value becomes 1.2 × 0.8 = 0.96
  Total return = -4%, not 0%!

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Annualized Returns – Why We Use Them

If an investment grows 1% a month, it’s tempting to just say:
1% × 12 = 12% – but that’s too simplistic.

Each month builds on the last. So:

(1 + 0.01)^12 - 1 = 12.68%

That little difference can really add up over time.

Example:

• Monthly return = 2%

• Annualized = (1.02)^12 - 1 ≈ 26.8%
  Much higher than just 2% × 12 = 24%

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Converting to Annual Terms

Just plug into this formula:
(1 + periodic_return)^periods_per_year - 1

Quick guide:

• Daily → Annual: ^252 (trading days)

• Weekly → Annual: ^52

• Monthly → Annual: ^12

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Comparing Investments Fairly

Let’s say:

• Strategy A: 1% return every month

• Strategy B: 5% return weekly

They don’t look that different until we anualize:

• A: (1.01)^12 - 1 = 12.68%

• B: (1.05)^52 - 1 ≈ 1039%

Same initial impression, completely different long-term outcome.
Always annualize if you’re comparing.

Example:

• Investment 1: +6% over 6 months → Anualized = (1.06)^2 - 1 ≈ 12.36%

• Investment 2: +4% over 3 months → Anualized = (1.04)^4 - 1 ≈ 16.99%

Investment 2 has a better anualized return despite lower short-term gain.

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Log Returns – Why Nerds Use Them

Log returns (aka continuously compounded returns) make life easier when modeling stuff.

Instead of multiplying returns across periods, you add them.

Formula:

log_return = ln(P_t / P_t-1)

Example:
From 100 → 110

• Simple return = 10%

• Log return = ln(1.1) ≈ 9.53%

Not a big deal over 1 day, but over 1000s of days, the math makes a difference.

Multi-period Example:

• Day 1: 5% up → log = ln(1.05) ≈ 4.88%

• Day 2: 4% down → log = ln(0.96) ≈ -4.08%

• Total log = 0.8% → Actual = exp(0.008) - 1 ≈ 0.8%

Bonus:
Annualized log return = daily_log × 252

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Risk-Adjusted Returns (Because Raw Return ≠ Good Return)

Returns alone don’t tell the full story—you’ve gotta ask: How much risk did I take to get that?

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Sharpe Ratio

Formula:
(Portfolio return - Risk-free rate) / Std dev of returns

Higher = better. Tells you how much bang you’re getting per unit of risk.

Example:

• A: 15% return, 10% volatility → Sharpe = 1.5

• B: 20% return, 25% volatility → Sharpe = 0.8

A is better on a risk-adjusted basis.

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Sortino Ratio

Like Sharpe, but only punishes downside volatility.
Formula:
(Return - Risk-free rate) / Downside deviation

More aligned with how humans actually think about risk (we don’t mind upside swings, right?).

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Max Drawdown

The scariest stat:
How far your portfolio drops from a peak before recovering.

Formula:
(Bottom - Peak) / Peak

Example:
Peak = 100, Drops to 70 → Max Drawdown = (70 - 100)/100 = -30%

Useful for understanding worst-case scenarios. Some high-return strategies also crash hard.

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Deeper Dive: More Ways to Understand Returns (Important Stuff…)

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CAGR – Compound Annual Growth Rate

This tells you the smoothed annual return as if your investment grew at a steady pace.

Formula:
CAGR = (Ending Value / Beginning Value)^(1 / n) - 1

Example:

• 100 → 150 in 3 years

• CAGR = (150/100)^(1/3) - 1 ≈ 14.47%

Great for understanding long-term performance, especially when yearly returns are uneven.

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Total Return vs. Price Return

• Price Return is just the stock price movement.

• Total Return includes dividends (assuming reinvestment).

Example:
Stock rises 8%, dividend yield 2% → Total return = 10%

Always prefer Total Return when analyzing true performance.

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Real Return vs. Nominal Return

• Nominal = what you see on paper.

• Real = adjusted for inflation.

Formula:
Real ≈ (1 + Nominal) / (1 + Inflation) - 1

Example:

• Nominal return = 8%

• Inflation = 5%

• Real = (1.08 / 1.05) - 1 ≈ 2.86%

Real return tells you if your wealth is actually growing.

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Volatility Drag

Volatility hurts compounding. Even with the same average return, higher volatility leads to lower final value.

Approx formula:
Geometric Return ≈ Arithmetic Return - 0.5 × Variance

Example:

• +20% one year, -20% next → Final value = 1.2 × 0.8 = 0.96 (i.e., -4%)

• Average return = 0%, but actual wealth declined.

More swings = less money over time.

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Time-Weighted vs. Money-Weighted Returns

• Time-Weighted (TWR): Neutral to deposits/withdrawals. Good for comparing managers.

• Money-Weighted (MWR) = IRR: Reflects actual investor experience (includes cash flows).

Example: If you invest a lot right before a downturn, MWR will be worse than TWR.

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Internal Rate of Return (IRR)

Used in private equity, VC, real estate. Tells you the effective annual return, considering all cash inflows and outflows.

Example:
You invest $1000,get$200 back yearly for 6 years → IRR = return rate that makes NPV = 0

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Rolling Returns

Instead of looking at fixed periods, rolling returns measure performance over moving windows (e.g., 3-year return calculated every month).

Helpful to understand consistency and timing sensitivity.

Example:

• Fund A: Consistently 7–9% every 3-year window

• Fund B: Swings from -5% to +20%

Fund A shows stability, B shows timing risk.

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Alpha & Beta

• Alpha: How much you beat the market.

• Beta: How much your portfolio reacts to market moves.

Example:

• Beta = 1 → matches market

• Beta > 1 → more volatile

• Alpha = +3% → you outperformed after adjusting for market exposure

Helps you understand active skill (alpha) vs market exposure (beta).

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Downside Capture Ratio

Measures how much you lose when the market falls.

• < 100% → You’re losing less than market.

• 100% → You’re losing more than market.

Example:

• Market down 10%, Fund down 7% → Downside capture = 70%

Useful to see how protective your strategy is during bear markets.

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Return Path Matters

Two portfolios can start and end at the same point, but the journey might be very different.

One might be smooth and steady, another could be a rollercoaster.

Example:

• Portfolio A: +1% per month

• Portfolio B: +20%, -15%, +10%, -12%, etc.

Both may end up at similar values, but B causes way more anxiety.

Visualizing the return path (e.g., via equity curves) tells you more than just start-to-end return.

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TL;DR

• Returns compound — don’t just add them up. Multiplication matters.

• Always annualize returns for apples-to-apples comparison.

• Use log returns for better math modeling and easier aggregation.

• Risk-adjusted metrics (Sharpe, Sortino, Max Drawdown) > raw returns.

• CAGR smooths uneven performance; tells true long-term growth.

• Total > Price return — dividends matter!

• Volatility hurts compounding — higher swings = less money.

• Real return = nominal minus inflation. Only real growth counts.

• TWR vs. MWR: Know whether you’re judging the investment or the investor.

• IRR helps evaluate performance when there are cash flows.

• Rolling returns show consistency, not just endpoint luck.

• Downside capture and drawdowns show how painful losses can be.

• Alpha & Beta: Skill vs. exposure. Know what you’re really earning.

• Return path matters — not just the destination, but the journey.

Part 2: Since I am getting old, I wanna get rich quickly. How to Look at return daily, weekly or monthly timeframe (7 din mei paisa double…).

You don’t always need to annualize—sometimes it’s more useful to analyze returns in their natural timeframe, especially for short- or medium-term trends.

What’s the difference?

• Daily Return: Measures change from one day to the next
  Return = (P_today / P_yesterday) - 1

• Weekly Return: From previous week’s close to current week’s close
  Return = (P_week_end / P_week_start) - 1

• Monthly Return: Same idea, just measured monthly
  Return = (P_month_end / P_month_start) - 1

These returns can be:

• Simple (arithmetic): Just % change

• Log (continuous): ln(P_t / P_t-1) – better for adding across time

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Metrics by Timeframe

You can extract insights from each period without converting to annual terms. Some useful metrics:

Rolling Returns

• Rolling 7-day, 30-day, or 90-day returns show short-term performance momentum

• Example: Fund X had positive returns in 18 of the last 20 rolling 7-day periods → Strong short-term trend

Average Periodic Return

• Mean return over each week or month
  Helpful to understand “typical” performance

Hit Ratio

• % of days/weeks/months with positive return
  e.g., Stock has 65% positive days → tends to move upward often

Max/Min Period Return

• Shows volatility in that timeframe
  e.g., biggest weekly drawdown or best monthly rally

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Want to Compare Across Timeframes? (Read Robert Carver ch 1 again if this looks confusing in future)

You can still convert between them:

From → To	Formula
Daily → Weekly	(1 + daily_return)^5 - 1
Daily → Monthly	(1 + daily_return)^21 - 1
Weekly → Monthly	(1 + weekly_return)^4 - 1
Monthly → Weekly	(1 + monthly_return)^(1/4) - 1

These let you translate returns into different periods without annualizing.

• Skewness: Measures asymmetry.

  • Positive skew → more frequent small losses, rare big gains.

  • Negative skew → more frequent small gains, rare big losses.

  Example:

  • Strategy A: Mostly gains, occasional big loss → Negative skew.

  • Strategy B: Mostly flat/slightly down, rare big spike → Positive skew.

• Kurtosis: Measures “fat tails” or extreme events.

  • High kurtosis → More outliers (more big shocks).

  • Low kurtosis → More “normal” distribution.

  Example:

  • Market crash strategy: Rare events dominate → High kurtosis.

Together, skewness and kurtosis help evaluate tail risk and return asymmetry, which average returns and volatility might hide.

Important Note to Self:

Don’t get too smart and use all return variants as features — more noise than signal; daily modeling’s a grind solo, better for risk monitoring.

so again:

• Returns compound, not add.

• Always annualize to compare. (This is important)

• Use log returns in models.

• Look at risk-adjusted stats like Sharpe, Sortino.

• Analyze daily, weekly, monthly returns properly using compounding and proper annualization formulas.

• Rolling and path-dependent metrics give better insight than point-to-point snapshots.

• Understand return shape via skewness and kurtosis for better risk awareness.