Risk and return

Every performance claim is a fraction: return over risk. Quoting either half alone is how investors get fooled, including by themselves. This doc is the measurement toolkit: what the standard numbers mean, how they are computed, and precisely where each one lies to you.

Returns first: the arithmetic that already misleads

Average vs compound (the volatility drag)

year 1: +50%   year 2: -33%
"average return" = (50 - 33) / 2 = +8.5% per year
actual outcome:  1.50 x 0.67 = 1.005  ->  +0.25% TOTAL

CAGR = (ending / beginning)^(1/years) - 1   <- the honest number
CAGR ≈ average return - volatility²/2       <- the drag, quantified

The gap between average and compound return grows with volatility, which is the mathematical reason wild rides underperform their marketing. Any track record quoted as an average of yearly returns deserves the CAGR question, and total return must always include dividends reinvested; price-only charts of high-yield assets are a standing deception.

Volatility: the standard risk unit, and its blind spots

Volatility is the annualized standard deviation of returns: how widely outcomes scatter around their average. Equities run roughly 15-20% a year, bonds 5-8%, bitcoin 60-80%. It is the industry's risk currency because it is computable, comparable and feeds every model downstream. Its three blind spots:

It is symmetric. Upside surprises count as "risk" the same as crashes, which offends common sense and motivates the downside-only measures below.
It assumes yesterday resembles tomorrow. Volatility clusters: calm regimes understate what the next regime does.
It misses the tails. Real markets deliver far more extreme days than the bell curve implies; measured volatility was low in June 2008.

The ratio family: return per unit of risk

Sharpe ratio, the lingua franca

Sharpe = (portfolio return - risk-free rate) / volatility

< 0.5  hard to distinguish from luck
~ 1    genuinely good over long periods
> 2    institutional-grade; audit it before believing it

Ratio	Denominator	Use when
Sharpe	Total volatility	The default; comparing anything to anything
Sortino	Downside deviation only	Strategies with asymmetric returns; upside noise should not be penalized
Calmar	Maximum drawdown	The investor's lived experience matters more than daily wiggle
Information ratio	Tracking error vs benchmark	Judging an active manager against the index they chose

Ratios are gameable, and sellers know it

Strategies that sell insurance (option writing, carry) post superb Sharpe ratios for years because the loss that defines them has not happened inside the sample yet. A high Sharpe with negative skew is not skill measured; it is a premium collected before the claim arrives. Always ask what the WORST plausible month looks like, and whether the sample contains one.

Drawdown: risk as investors actually experience it

The recovery arithmetic

maximum drawdown = largest peak-to-trough loss in the period

the asymmetry that makes it matter:
  -10% needs +11% to recover     -33% needs +50%
  -50% needs +100%               -90% needs +900%

Volatility is a statistician's number; drawdown is the one investors feel, quit over, and get margin-called on. Depth, length, and recovery time together describe the actual experience of holding a strategy, and the recovery arithmetic above is the quantitative case for defense: losses cost more than the same-sized gains earn, compounding's cruel asymmetry.

Beyond the average: skew, tails, and correlation's betrayal

Skew. Which tail is long: negative skew (carry, option selling) means many small wins and rare disasters; positive skew (trend following, long options) means many small losses and rare windfalls. Two strategies with identical Sharpe and opposite skew are opposite products.
Fat tails. Extreme days occur orders of magnitude more often than the normal distribution predicts, and returns compound THROUGH those days: miss the arithmetic of tails and every risk number downstream is optimistic.
Correlation is regime-dependent. The diversification measured in calm markets is not the diversification delivered in a crash, when correlations lurch toward one. Risk models built on full-sample correlations systematically flatter portfolios; professionals stress-test with crisis-period correlations instead.
Beta. A portfolio's sensitivity to the market is its single largest risk in most cases; return above what beta explains (alpha) is the part worth paying for, and most claimed alpha dissolves into beta on inspection.

VaR and expected shortfall, briefly and honestly

Value at risk answers "what loss is exceeded only 5% of days?" and became banking's standard gauge; its known flaw is saying nothing about how bad the exceeding days are. Expected shortfall repairs that by averaging the tail beyond VaR, and has largely replaced it in regulation. Both inherit whatever the input distribution missed, which in every crisis is the crisis.

Glossary for this guide

CAGR: Compound annual growth rate: the single yearly rate that turns the starting value into the ending one. The honest summary of a track record, unlike the average of yearly returns, which volatility inflates.
Volatility drag: The gap between average and compound returns, roughly half the variance: +50% then -33% averages +8.5% and compounds to nothing. The mathematical reason wild rides underperform their marketing.
Volatility: Annualized standard deviation of returns: how widely outcomes scatter. The industry's risk currency: computable and comparable, blind to direction, regime shifts and tails.
Sharpe ratio: Excess return over the risk-free rate, per unit of volatility. Around 1 is genuinely good long-run; above 2 deserves an audit. Gameable by strategies whose defining loss has not happened inside the sample yet.
Sortino ratio: Sharpe with only downside deviation in the denominator, so upside surprises are not punished as risk. Fairer to asymmetric strategies.
Calmar ratio: Annual return over maximum drawdown: return per unit of the pain investors actually remember.
Information ratio: Active return over tracking error: how much benchmark-beating a manager delivers per unit of benchmark-deviating. The fair exam for active management.
Maximum drawdown: The deepest peak-to-trough loss in a period. The risk number investors quit over, and the one with cruel arithmetic: -50% needs +100% to repair.
Skew: Which tail of the return distribution is long. Negative skew (carry, option selling): many small wins, rare disasters. Positive (trend, long options): many small losses, rare windfalls. Identical Sharpes with opposite skews are opposite products.
Fat tails: Extreme outcomes arriving far more often than the bell curve predicts, which in markets they reliably do. Every risk model built on normality is optimistic precisely when it matters.
Correlation: How much two assets move together, from -1 to +1. The input diversification lives on, and the one that betrays you: correlations measured in calm lurch toward one in a crash.
Alpha and beta: Beta is return explained by exposure to the market (or a factor); alpha is what remains. Most claimed alpha dissolves into beta on inspection, which is why the decomposition is the first test of any record.
Value at risk: The loss exceeded only X% of the time, banking's standard gauge. Says nothing about how bad the exceeding days are, which is what expected shortfall repairs.
Expected shortfall: The average loss across the tail beyond VaR: what the bad days cost when they come. The regulator's replacement for VaR, inheriting whatever the input distribution missed.
Negative skew: A return profile of many small gains and rare large losses, the shape of selling insurance. Carry trades and option selling share it; averages flatter it, and sizing must respect the tail rather than the average.