Quantitative Finance FAQ

The long volatility premium is the claim that buying put options, when beta-adjusted to neutralize their embedded short-equity exposure, delivers a positive return over time rather than the negative return associated with raw put buying. Patrick Causley at One River Asset Management documented this over approximately 40 years of data, showing that a beta-1 portfolio with long volatility outperformed the S&P 500 while producing lower volatility and shallower drawdowns.

Raw put options embed a large short-beta position that bleeds value whenever the market rises. Beta-adjusting neutralizes this directional exposure by adding long equity to offset the put's delta, isolating the convexity (gamma) and volatility sensitivity (vega) components. The result is a position that still delivers explosive payoffs during crashes but no longer fights the equity risk premium during normal markets.

The variance tax is the hidden drag on compound returns caused by volatility. The compound growth rate is approximately the arithmetic mean minus half the variance (G ≈ μ − ½σ²). Because this penalty is quadratic, reducing drawdown severity has a nonlinear effect on terminal wealth. A portfolio that falls 50% needs 100% to recover. By truncating left-tail outcomes, even a costly tail hedge can increase compound wealth over time.

AQR's research shows the two approaches are complementary. Put strategies deliver spectacular returns in sudden crashes like COVID-19 but are expensive to maintain with negative long-run expected returns. Trend-following earns positive long-run returns and excels in protracted bear markets like the dot-com bust. Academic work combining both via portable alpha found statistically significant alpha of 0.25% per month after controlling for equity factors.

Most practitioners suggest 1 to 5% of portfolio value. The Wall Street Journal reported that a 3.3% allocation to Universa Investments with the rest in the S&P 500 achieved a 12.3% compound annual return over 10 years, beating the index itself. The optimal size is ultimately psychological rather than mathematical: it must be small enough to tolerate years of negative carry without abandoning the strategy.

A CAIA Association study found that several popular tail-risk strategies, including short-dated VIX futures and 1-month variance swaps, failed to beat a simple cash benchmark, with performance drags of 355 and 203 basis points respectively. The specific implementation matters a lot, and many approaches are structurally flawed by contango decay in the VIX term structure.

Variance drain is the gap between an investment's arithmetic mean return and its compound (geometric) growth rate. It equals approximately ½σ², where σ is the volatility of returns. Higher volatility means a larger gap between the average return you see reported and the actual wealth you accumulate.

Leverage L scales arithmetic return linearly (Lμ) but scales variance drain quadratically (½L²σ²). Doubling leverage quadruples the drag. This is why leveraged ETFs can underperform their target multiple over time, especially in volatile markets.

The Kelly criterion gives the leverage ratio that maximizes compound growth: L* = (μ − r) / σ². It falls directly out of the variance drain formula — it is the point where the marginal return from additional leverage exactly equals the marginal cost of additional variance drain.

Full Kelly assumes perfect knowledge of expected return (μ) and volatility (σ). In practice, both are estimated with error. Half-Kelly — sizing at L*/2 — sacrifices about 25% of theoretical growth but dramatically reduces the risk of overleveraging due to estimation error.

It is a mathematical relationship, not a physical force — the geometric mean is always less than or equal to the arithmetic mean (AM-GM inequality). But the P&L consequences are entirely real: two portfolios with the same average return but different volatilities will produce different terminal wealth.

AQR's 2026 capital market assumptions show U.S. buyouts returning 4.2% versus 3.9% for public equities. That's a 30 basis point premium for accepting years of lockup, unpredictable capital calls, and manager selection risk. For most investors outside top-quartile funds, the premium that justified alternatives allocation has been competed away.

According to AQR's 2026 capital market assumptions, expected real returns for U.S. buyouts are 4.2% over the next 5-10 years. Private credit is even lower at 2.6%, down 0.5 percentage points year-over-year as spreads narrowed. These returns barely exceed public market alternatives when adjusted for illiquidity and fees.

Manager selection is critical. In venture capital, top decile managers generate 31.7% IRR while bottom decile managers return negative 7%. The spread between winners and losers is enormous. But this dispersion is precisely why average returns have compressed: access to top-tier funds is limited, and everyone else fights over what remains.

Too much capital chased the same opportunities. When every pension fund, endowment, and sovereign wealth fund allocates 20-30% to alternatives, the returns that made them attractive get arbitraged away. The money didn't find alpha; it became beta with a lockup.

Access to companies you can't reach in public markets. 87% of U.S. companies with more than $100 million in revenue are now private. Value creation has shifted earlier: for 2020-2023 IPOs, 55% of median value was created in private markets versus just 12% for 2014-2019 IPOs. But access isn't the same as returns.

Yes, in the cryptographic sense. Statistical analysis of 20,000 rounds confirms the random number generator produces fair outcomes matching the stated 97% RTP. However, mathematical fairness doesn't ensure consumer safety, as the rapid pace of 179 rounds per hour means expected losses exceed 500% of amounts wagered per hour.

No. Monte Carlo simulations of 10,000 betting sessions across four strategies (1.5x, 2x, 3x, and 5x cash-outs) confirm every single strategy produces negative expected returns. The game is mathematically "strategy-proof" because the expected value equals RTP minus 1 regardless of cash-out timing.

For a 97% RTP crash game, the probability of reaching multiplier m before crashing equals 0.97/m. A 2x target succeeds about 48.5% of the time. A 10x target works only 9.7% of rounds. A 100x target succeeds just 1.1% of the time.

At 179 rounds per hour with 16-second median intervals and a 3% house edge per round, players face expected losses exceeding 500% of amounts wagered per hour of play. This is far faster than traditional casino games.