Risk Management

Risk management is arithmetic, not temperament. The formulas chain: a measured edge sets the size, the size sets the loss path, the loss path sets the recovery burden, and the leverage behind the size sets the price at which a broker starts selling.

What does risk management actually decide?

Six calculations, in sequence.

The first asks whether an edge exists at all. That is trading expectancy, assembled from win rate and the risk-reward ratio. The second asks how much capital one occurrence of that edge justifies: position sizing, bounded above by the Kelly criterion. The third asks what loss path the chosen size implies, which is drawdown and its convex recovery arithmetic.

The fourth asks how the whole book is exposed rather than each trade alone: exposure, correlation, and beta. The fifth asks how bad the far end of the distribution gets: value at risk, conditional value at risk, and tail risk. The sixth asks what can close the account without any decision from the trader, which is margin and the margin call.

Skipping one of them does not remove it. A trader who never computes a maintenance trigger price still has one, and the broker knows where it is.

How is expectancy calculated from win rate and risk-reward?

Expectancy is the probability-weighted average outcome of one repeated trade: win rate multiplied by average win, minus loss rate multiplied by average loss magnitude, where loss rate is one minus win rate.

One R equals the initial planned dollar risk on a trade. Expressing results in R separates strategy quality from account size, so a 40 percent win rate that pays 2R on winners and loses 1R on losers has a gross expectancy of (0.40 × 2.0R) − (0.60 × 1.0R) = +0.20R per trade before costs.

The related figure is the break-even win rate, the average loss divided by the sum of the average win and the average loss. At a 2-to-1 payoff that threshold is 450 / (900 + 450) = 33.33 percent. Any win rate above it produces positive gross expectancy at that payoff, and any below it does not, no matter how comfortable the frequency feels.

Quote both numbers with their trade count and the convention for scratches, because six winners in ten trades is consistent with a wide band of underlying probabilities.

The worked example used through the rest of this page

A $60,000 account caps planned risk at 0.75 percent per trade, so one R is $60,000 × 0.0075 = $450. The strategy has 200 completed trades: 80 winners averaging $900 net, 120 losers averaging $450 net.

The strategy loses more often than it wins and still returns $90 per trade, because the average winner is twice the average loser. That $450 unit is what every later section sizes against.

How does position sizing convert a stop distance into a quantity?

Risk-based sizing divides permitted dollar risk by risk per unit. For long stock, risk per share is the entry price minus the planned invalidation price.

On the same account, a long entry at $48.00 with invalidation at $45.00 carries $48.00 − $45.00 = $3.00 of risk per share. Permitted risk is $450, so the size is $450 / $3.00 = 150 shares controlling 150 × $48.00 = $7,200 of stock. Position value is $7,200 while planned loss is $450, which is why the two are different quantities.

The stop distance, not conviction, drives the count. Holding the $450 budget fixed, a $1.50 stop permits 300 shares and a $6.00 stop permits 75, and the planned loss stays $450 in each case.

Options require the 100-share contract multiplier. Premium is quoted per share, so a $2.40 option costs $240 per contract, and a defined-risk vertical with a $300 maximum loss is sized from $300 per spread unless a tested earlier exit is part of the plan.

What does the Kelly criterion say, and why do traders size below it?

For a binary bet, the Kelly fraction is f* = W - (1 - W) / R, where W is the win probability and R is the reward-to-risk ratio. The numerator written out, W x R - (1 - W), is the expectancy of the trade in units of the amount risked, so the Kelly fraction is simply expectancy divided by payoff.

The example strategy has W = 0.40 and R = 2, so f* = 0.40 − (0.60 / 2) = 0.40 − 0.30 = 0.10. Full Kelly is 10 percent of capital per occurrence, half Kelly is 5 percent, and the 0.75 percent risk rule above is 0.15 of half Kelly, since 0.75 / 5 = 0.15.

Kelly maximizes the long-run compounded growth rate when W and R are exact. Its fraction sizes the capital at risk, because the binary formula assumes a loss forfeits the amount risked. Whatever fraction you take, run it through the stop, the way the $450 budget on a $3.00 stop becomes 150 shares, then test that size against a gap through the stop and a slow fill.

At a 60% loss rate, any given block of eight independent trades loses all eight with probability 0.60^8 = 1.68%, and a 200-trade record holds many such blocks, so plan for the streak. The table shows what one eight-loss run does to the account at each size.

The arithmetic compounds rather than adds: $60,000 × (1 − 0.10)^8 = $60,000 × 0.4305 = $25,828. Growth falls on both sides of the Kelly optimum, but asymmetrically. Underbetting reduces growth gradually, overbetting reduces it faster, and beyond roughly twice the Kelly fraction the long-run growth rate turns negative even when the edge is real. Because W and R are always estimates, an overstated edge translates directly into an overbet.

Why does drawdown recovery punish large losses disproportionately?

Drawdown is measured from a running peak, so the reference resets only when a new high is made. An account that rises to $80,000, falls to $60,000, and later recovers has a peak-to-trough drawdown of ($60,000 − $80,000) / $80,000 = −25 percent and needs $80,000 / $60,000 − 1 = 33.33 percent from the trough to get back.

The asymmetry comes from the changing base. The decline is measured against the peak, the recovery against the trough.

Depth is one dimension. Two strategies with identical maximum drawdown impose very different burdens if one recovers in weeks and the other in years. The statistic is also sample dependent: a shorter history, a start date just after a crisis, or daily closes that miss an intraday low all improve it without improving the strategy.

Which risk-adjusted measures describe what?

Variance is the mean squared deviation of returns from their average, and standard deviation is its square root, back in the original units. Variance adds cleanly across independent periods, which is why volatility scales with the square root of time rather than linearly.

The Sharpe ratio divides mean return above the risk-free rate by the standard deviation of excess returns. The Sortino ratio divides mean return above a chosen minimum acceptable return by downside deviation. Downside deviation is the square root of the average squared shortfall below that target, with returns above the target contributing zero. A higher target lowers the Sortino from both ends, with a smaller numerator and more returns counted as shortfalls, so quote the ratio with its target.

Two rows deserve arithmetic. A portfolio returning 12 percent against a 3 percent risk-free rate at 15 percent volatility scores 9% / 15% = 0.60, while one earning the same 12 percent at 24 percent volatility scores 9% / 24% = 0.375. Beta equals correlation multiplied by the ratio of standard deviations, so 30 percent volatility against a 20 percent market at 0.8 correlation gives 0.8 × (0.30 / 0.20) = 1.2.

For two assets, portfolio variance is wA² × sigmaA² + wB² × sigmaB² + 2 × wA × wB × correlation × sigmaA × sigmaB. A 60/40 mix with 18% and 6% volatilities has 13.20% volatility at correlation +1.00 and 10.58% at correlation −0.20. Compare the result across correlations to see how shared moves change the portfolio.

What do VaR and CVaR measure, and where do they stop?

Every value at risk statement has three parts: a horizon, a confidence level, and a loss amount. Parametric one-period VaR is V x (z x sigma - mu), with z near 1.645 at 95 percent confidence and 2.326 at 99 percent.

For a $500,000 book with 1.5 percent daily return volatility and an expected daily return near zero, the one-day 95 percent VaR is 1.645 × 1.5% × $500,000 = $12,337.50. Scaling to ten days under an independence assumption multiplies by the square root of ten, giving $39,015.

Expected shortfall averages the worst 5% of modeled losses at 95% confidence. For a continuous normal distribution with zero mean, its volatility multiplier is the standard normal density at the 95% quantile divided by 0.05, approximately 2.063. The example gives 2.063 × 1.5% × $500,000 = $15,472.50. For discrete losses, form a probability-weighted worst-5% tail rather than averaging every outcome equal to or greater than VaR.

Both numbers inherit their distribution assumption. Real return tails are fatter than a normal curve implies, so parametric estimates understate the frequency and the depth of extreme moves. Federal Reserve research on models in use at large banks found VaR estimates that looked conservative on average yet were breached in clusters during stress, the failure pattern a single confidence number hides. Hedging transfers part of that tail for a known premium: a 95-strike protective put paid at $3.00 against stock bought at $100 caps the loss at ($100 − $95) + $3.00 = $8.00 per share, or $800 on 100 shares, and leaves upside open above $103.

How does risk management read on a live tape?

Three surfaces show the arithmetic in real time.

The exit orders. A stop order is a trigger that releases a market order, so it controls timing and not price. A trader long 400 shares from $52.00 with a sell stop at $49.50 plans $2.50 × 400 = $1,000 of risk, but if news leaves the next executable bids at $49.20 and $49.05, an average fill near $49.10 turns the realized loss into ($52.00 − $49.10) × 400 = $1,160. A stop-limit order trades that slippage for nonexecution risk: a 300-share sell stop at $76 with a $75.50 limit fills only the 100 shares bid at $75.70 and leaves 200 exposed when the book drops to $75.40. The stop versus stop-limit comparison lays out that tradeoff. A trailing stop order ratchets one way only: a 6 percent trail on a $40 entry starts at $37.60, and once the tracked reference peaks at $47.00 the trigger locks at $47.00 × 0.94 = $44.18.

The exposure pane. Convert option positions into delta-equivalent dollars before combining them with shares. A portfolio long $60,000 of stock and short $20,000 of an index ETF has $40,000 of net stock exposure. Ten calls at 0.40 delta on a $150 underlying add 0.40 × 10 × 100 × $150 = $60,000, bringing combined delta-equivalent exposure to $100,000. Compare that exposure with account equity, which includes cash, liabilities, and the current market value of the options.

The chain and the buying power line. Buying power reduction is the collateral a structure ties up plus any debit paid. With a stock at $100, one naked 95-strike put sold for $2.00 carries a strategy-based requirement of 0.20 × $10,000 − $500 + $200 = $1,700 against a floor of 0.10 × $9,500 + $200 = $1,150, so the net reduction is $1,700 − $200 = $1,500. The same view as a 95/90 put credit spread sold for $1.50 reduces buying power by $500 − $150 = $350 and stays fixed, while the naked requirement floats with the underlying.

What margin mechanics can end a position without the trader deciding?

Margin is a collateralized loan. Equity is market value minus the loan balance, and because the balance does not move when prices do, the entire decline is absorbed by equity. Watching equity as a percentage of market value, rather than profit and loss, is what makes this section visible in advance.

Regulation T generally permits a firm to lend up to 50 percent of the purchase price of an eligible margin stock. FINRA Rule 4210 requires equity in an account holding only margin stock to stay at or above 25 percent of current long market value, with a higher schedule for short stock. House maintenance rates sit above that 25 percent floor, commonly at 30 to 40 percent, and can be raised at any time without advance notice.

A trader who buys 200 shares at $100 for $20,000 and borrows the $10,000 Regulation T maximum can compute the trigger before entering. For a long position, the maintenance trigger price is the loan per share divided by one minus the maintenance rate: ($10,000 / 200) / (1 − 0.25) = $50 / 0.75 = $66.67. At $60, market value is $12,000 and equity is $12,000 − $10,000 = $2,000, or 16.7 percent, against a required 0.25 × $12,000 = $3,000. The deficiency is $1,000.

What happens next is not the trader’s choice. Firms are not required to issue a call before selling, may sell enough to retire the entire loan rather than only the deficiency, and select which positions are closed.

Portfolio margin reprices the full portfolio across stress scenarios. Consider 1,000 shares at $50 and ten standard $45 puts initially valued at $1. In a down-15% scenario, the shares fall to $42.50 and lose $7,500. If the puts reprice to $3, they gain ($3 − $1) × 100 × 10 = $2,000, leaving a $5,500 scenario loss. Repeat the valuation across the required scenarios and apply the account’s offsets, minimums, and house requirements to calculate margin.

What are the standard mistakes and why is each wrong?

Reading win rate as edge. Measure edge with expectancy, which combines win rate with payoff. Nine $100 gains against one $1,200 loss is a 90 percent win rate and a $300 net loss: frequency describes how often, never how much.

Widening a stop after entry. Hold the original invalidation and measure R against it. Realized R divides the actual result by the original planned risk. Moving invalidation farther away converts a planned 1R loss into something larger, while the journal still reports 1R if the denominator moves with the stop.

Treating a stop price as a maximum loss. Size the position with an allowance for fills beyond the stop. The trigger is an activation level; the realized loss is the weighted average fill, which gaps, halts, and thin books can push well past it.

Reading VaR as a worst case. Pair VaR with conditional VaR to size the tail. A 95 percent VaR is breached about one day in twenty by construction, and conditional VaR measures how deep those breaches run on average.

Treating correlated trades as separate risks. Group positions by market, sector, and catalyst, then test their simultaneous price moves and execution scenarios. Five trades with planned risk of 0.5% each total 2.5% of planned account risk, and the positions’ maximum contractual loss is a separate calculation from that stop-based figure.

Applying full Kelly to estimated inputs. Size at a fraction of Kelly. The formula is optimal only when W and R are exact, and estimated inputs never are. Overbetting is punished harder than underbetting, and past roughly twice the fraction the long-run growth rate is negative despite a real edge.

Treating buying power as the risk limit. Set the risk limit from the loss the account can absorb. Buying power measures collateral remaining, and a naked short option carries a requirement that floats with the underlying, so an adverse move raises the reduction while equity is falling.

Comparing ratios computed differently. Match the return frequency and the target before comparing. A monthly Sharpe ratio and an annualized one differ by the square root of twelve, and a Sortino ratio measured against a zero target answers a different question from one measured against a benchmark.

What to learn next

Read the inputs before the outputs. Win rate and the risk-reward ratio feed everything downstream, and trading expectancy is where they combine. From there, position sizing and the Kelly criterion turn an edge into a quantity, and drawdown shows what that quantity costs along the way.

For portfolio work, standard deviation and variance supply the dispersion, beta and correlation explain how positions interact, and the Sharpe ratio and Sortino ratio score reward against different definitions of risk. Tail risk, value at risk, and conditional value at risk describe what those ratios smooth over, and hedging transfers it.

For execution and account mechanics, stop orders, stop-limit orders, and trailing stop orders implement invalidation, while exposure, buying power reduction, margin, the margin call, and portfolio margin set how much of that plan the account can carry.