Part I · Returns and risk · Chapter 1E

Drawdown, leverage and margin

Why the path matters as much as the destination

1 · Drawdown: the fall from the last peak

A drawdown is the peak-to-trough loss in an account's value over a period, measured from the highest value reached so far rather than from where you started. An account that grows from $100,000 to $150,000 and then falls to $120,000 is in a 20% drawdown while still up 20% overall.

Write $W_t$ for the account's value at time $t$ and $P_t = \max_{s \le t} W_s$ for its running peak, so that the drawdown at time $t$ is the fraction of the peak lost since:

Applied to the example, the formula gives $1 - 120{,}000/150{,}000 = 0.20$, the 20% drawdown above. A whole record of drawdowns is summarized by two numbers:

• Maximum drawdown (MaxDD) is the largest $D_t$ in the record, the single worst peak-to-trough loss.
• Conditional drawdown (CDaR-α) is the average drawdown beyond the α percentile, so that at α = 95% it averages the worst 5% of drawdowns.

Volatility is symmetric, counting a surprise gain as heavily as a surprise loss, while drawdown is one-sided. A strategy can have low volatility and still a 40% maximum drawdown if it crashes occasionally. For whether a portfolio survives, drawdown is the binding constraint, not volatility.

Fat tails make drawdowns deeper than a normal model predicts (Chapter 1B). For a strategy with a 5% mean return and 10% volatility, naive Gaussian math says the maximum drawdown over 10 years should be around 25%. In reality, fat-tailed strategies often see drawdowns of 40% to 50%.

From 1990 to 2024, the S&P 500 index (SPX) held long-only returned about 10% a year with a maximum drawdown of about 55%. The PUT index (Chapter 1B), which sells SPX puts monthly, returned about 9.5% with a maximum drawdown of about 33%. Their Calmar ratios (Chapter 1D), return divided by maximum drawdown, are about 0.18 and 0.29. That meaningful improvement is paid for in path dependency, the subject of this chapter's last section.

2 · What it takes to recover

A loss leaves each dollar worth $1 - D$, so the gain $G$ that repairs it must satisfy $(1 - D)(1 + G) = 1$, which solves to

As $D$ approaches 1, the denominator shrinks toward zero and the gain needed grows without limit. If the account then grows at a steady rate $g$ a year, it regains its peak when $(1 - D)(1 + g)^t = 1$, which after taking logarithms (Chapter 1A) gives the years needed:

This asymmetry of drawdown is a real cost, not a curiosity. Recovery time is non-linear in loss size: doubling a loss from 20% to 40% more than doubles the years needed.

3 · Leverage: holding more than you own

Leverage $k$ is the ratio of total exposure, the market value of everything your positions hold or control, to your capital:

A $1M account holding $1.4M of SPX has 1.4× leverage, with the extra $0.4M borrowed directly or through a contract. If the exposure returns $R$ in a year and the borrowed $k - 1$ per dollar of capital costs a rate $r_b$, your return on capital is

With $k = 1.4$, $R = 10\%$ and a hypothetical borrowing rate of 4%, that is $14\% - 1.6\% = 12.4\%$. Leverage multiplies excess return and volatility alike, which is why it leaves the Sharpe ratio unchanged (Chapter 1D). It scales a trade, losses included, without improving it.

A fall of $1/k$ in the exposure wipes out the capital entirely: a fall of 71% at 1.4× leverage, but only 25% at 4×. Treat leverage like the dose of a cytotoxic drug: set it from the drawdown the account can survive, not from the return you hope for.

The figure runs four accounts through one made-up fall in the index. Drag the fall from 10% toward 55% and watch the 4× account reach zero at 25%. At 55% the 1.4× row gives the worked example's 77% loss, 335% repair and 2.7× leverage at the trough.

the index falls

55%

new path

Leverage	Maximum drawdown	Gain to recover	Leverage at the trough	Capital gone at a fall of
1×	55%	122%	1.0×	100%
1.4×	77%	335%	2.7×	71%
2×	100%	without limit	no capital left	50%
4×	100%	without limit	no capital left	25%

The index falls 55%. The 1.4× account loses 77%, needs a 335% gain to recover and ends the fall at 2.7× leverage without a trade. Wiped out on the way down: the 2× and 4× accounts.

Three ways to take leverage

There are three economically distinct ways to take leverage, each with its own cost:

1. Borrow to buy, with a margin loan or with box spread financing, at an explicit cost: the interest rate.
2. Use futures such as ES and MES, the S&P 500 futures of Chapter 2A. The implicit cost is the basis between spot and futures, roughly SOFR + 10 to 20 bps.
3. Use options for replication, such as LEAPS or deep in-the-money (ITM) calls. The implicit cost is embedded leverage drag, typically 50 to 150 bps annualized (Frazzini-Pedersen 2012, 2022).

A basis point (bp) is a hundredth of a percentage point, and SOFR (the Secured Overnight Financing Rate) is a benchmark rate for overnight borrowing.

The first two are usually cheap, while the third is expensive and has path-dependent properties: Greeks that shift as the market moves, margin that can expand, and assignment risk on short options.

4 · Margin: the deposit that can be called

Margin is the capital your broker requires you to hold to keep a position open, much like a deposit. Each day the broker values your positions at market prices, which is called mark-to-market (MTM). The gains and losses reach your equity whether or not you have closed anything.

If your equity falls below the requirement, the broker issues a margin call asking you to pay in money or reduce the position. If you do not do so in time, the broker closes positions for you at whatever prices the market offers, which is forced liquidation. Brokers set the requirement with one of two main models:

• Reg-T margin is a fixed percentage of position size, simple but capital-inefficient.
• Portfolio margin (PM) is based on stress-tested theoretical losses, typically from a ±15% SPX move plus shifts in implied volatility (IV).

Chapter 2E gives the rules for each account type and Chapter 12G the margin on each options structure. Under PM, margin requirements expand non-linearly in stress: in March 2020, many PM accounts saw their requirements double or triple in 1 to 3 days. Brokers also raised house requirements, their own add-ons above the minimum, during the crisis.

The two shocks arrive together because one market stress drives both. This combination is what creates path-dependent risk: even if your terminal payoff is acceptable, you can be forced to close the position before the path completes.

5 · Path dependency: surviving to the payoff

A strategy is path-dependent if its MTM trajectory matters as much as its terminal payoff. Most options strategies are path-dependent in some way, because an option's payoff is fixed only at expiry while its market price, which your margin follows, moves every day until then (Chapter 5A).

Path dependency is the single most underestimated risk in retail options structures. Consider a 1-year SPX put spread sold as "defined-risk".

In operational terms, "defined risk" describes the terminal loss but not the path loss. Size every options structure to survive the path, not just the terminal payoff. Path-dependent risk takes several forms in an options account:

• Margin expansion that forces liquidation before prices revert.
• Early assignment on short options, before the expiry you planned around.
• Time decay (theta) that cannot be recovered if the MTM deteriorates and you have to close.
• Forced rolling into a later expiry, which changes the strategy from what you intended.

A few strategies are not path-dependent: long stock with no margin, a cash-secured put held to expiry (no margin call is possible), and box spread financing (no MTM risk).

6 · Check yourself