Week 41: Risk Management — Position Sizing, Stop Losses, Kelly, and Bankroll Preservation

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1. Why This Is Important

Every retail loss-blowup story you have ever read shares the same shape. The investor was usually right about direction and wildly wrong about size. The Reddit screenshot showing "$50k lost on weekly SPY puts" is rarely a story of a bad thesis; it is almost always a story of a fine thesis sized at four times the bankroll the investor could survive. Position sizing is the most under- rated discipline in all of investing. It is what separates people who compound for thirty years from people who blow up in three.

You need this material for four reasons.

Sizing is the only knob you fully control. You cannot make

the market trend up. You cannot force your edge to be real. But you absolutely choose, before you click, what fraction of your capital is in this trade. That single decision dominates long-run wealth more than any view, any pick, any timing.

The math of drawdowns is asymmetric. A -50% drawdown

requires a +100% recovery. A -75% drawdown requires +300%. A -90% drawdown requires +900%. Sizing dictates how deep your worst drawdown goes; recovery cost grows non-linearly. Staying solvent longer than the market stays irrational — is impossible without sizing.

Edge is uncertain. Kelly is uncertain too. The Kelly

criterion gives you the geometric-growth-maximising bet size if your edge estimate is correct. In equities, your edge estimate is never correct to the third decimal. Fractional Kelly — a quarter or even an eighth of full Kelly — is the honest answer to that uncertainty.

Leverage is the silent multiplier of every mistake. Two-

times leverage doubles your return and your variance. But for max drawdown, the relationship is closer to multiplicative: a -25% drawdown at 1x becomes roughly -50% at 2x. Three times leverage on a 16% vol equity book turns a survivable bear market into a margin-call event. The blowup is non-linear.

This lesson works through Kelly, fractional Kelly, the 1-2% per- trade rule, stop losses versus hedges, correlation-adjusted sizing for multi-position books, and the leverage-versus-drawdown math. It then ties the whole thing back to the lessons that volatility is fat-tailed and that solvency beats being right.

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2. What You Need to Know

2.1 The Kelly Criterion — Geometric-Growth-Optimal Bet Size

John Kelly (Bell Labs, 1956) asked a simple question: if I have a repeated bet with edge, what fraction of my bankroll should I risk each time to maximise the long-run geometric growth rate of my wealth? The answer for a binary win/lose payoff is:

$$ f^* = \frac{b p - q}{b} $$

where:

• $p$ = probability of winning,
• $q = 1 - p$ = probability of losing,
• $b$ = net odds received on a win (so a 1:1 bet has $b = 1$, a

2:1 bet has $b = 2$).

Plug in a coin-flip with 55% heads and 1:1 odds: $f^* = (1 \cdot 0.55 - 0.45) / 1 = 0.10$. Kelly says bet ten percent of your bankroll on each flip. Above ten percent, the geometric growth rate falls. Above twenty percent — twice Kelly — your expected long-run growth becomes zero. Above $f = 2 f^*$ you are mathematically guaranteed to ruin yourself given enough trials, even with a positive edge.

For equity strategies, the win rate is rarely above 55% and the average win-to-loss ratio is rarely above 1.5. A typical setup might be $p = 0.52$, $b = 1$, giving $f^* = 0.04$ — about four percent of bankroll per trade. That is the universe you are operating in. You are not in a 1956 horse-race. You are in a near-fair-coin world where Kelly outputs single-digit percentages.

The chart shows full-Kelly $f^*$ as a function of win rate $p$ across five payoff ratios. Notice three things. First, at the typical equity-edge point ($p \approx 0.52$, $b = 1$) full Kelly is around four percent. Second, the curve is steeper than people expect — moving win rate from 50% to 55% triples Kelly. Third, when $p < q / b$ (the no-edge boundary) Kelly goes negative, which is the maths politely telling you not to take the bet.

2.2 Fractional Kelly — The Honest Answer to Edge Uncertainty

Full Kelly assumes you know your edge. In practice you are estimating $p$ and $b$ from a finite track record, and the standard error of that estimate is large. Edward Thorp, the mathematician who beat Vegas blackjack and ran a hedge fund for two decades, settled on half-Kelly for everything he did. The general formula for $k$-fractional Kelly:

$$ f_k = k \cdot f^* $$

with $k$ commonly chosen as $0.5$ (half-Kelly), $0.25$ (quarter- Kelly), or $0.125$ (eighth-Kelly). Two facts make fractional Kelly the default rather than full Kelly.

First, the geometric-growth penalty for under-betting is mild. At half-Kelly you capture roughly 75% of the optimum growth rate. At quarter-Kelly you still capture about 44%. The penalty for over-betting, however, is severe and asymmetric: at twice Kelly you grow at zero, and beyond that you lose money in expectation. So shading toward smaller-than-optimal is cheap insurance.

Second, the variance reduction is enormous. Half-Kelly cuts your drawdowns roughly in half compared to full Kelly. Quarter- Kelly cuts them again. For a strategy with a 30% annualised volatility, full Kelly produces drawdowns that can hit 80%; quarter-Kelly keeps them under 25%. Staying solvent longer than the market stays irrational is the entire reason quarter-Kelly is the industry standard for stock strategies.

The honest rule for retail: when you have a directional view on an individual stock, size at quarter-Kelly or smaller. When you have an index-level view, the same rule applies. Full Kelly is for blackjack tables where you literally counted the deck. It is not for your tax-deferred account.

2.3 The 1-2% Per-Trade Loss Rule

Professional traders rarely think in Kelly terms day to day. They think in a much simpler frame: *what is the maximum dollar loss I am willing to take on this single position?* The canonical rule is one to two percent of total bankroll.

Mechanics: if your bankroll is \$100,000 and your max loss per trade is 1%, you are willing to lose \$1,000 on this trade. If your stop is \$5 below entry on a \$100 stock, your position size is $1{,}000 / 5 = 200$ shares — \$20,000 of stock. If your stop is \$2 below entry, your position size is $1{,}000 / 2 = 500$ shares — \$50,000. The dollar-at-risk drives the share count, not the share count driving the dollar-at-risk.

Why the 1-2% range works:

• At 2% per loss, twenty consecutive losses produce roughly a

$1 - 0.98^{20} \approx 33\%$ drawdown. Survivable.

• At 5% per loss, twenty consecutive losses produce a $1 -

0.95^{20} \approx 64\%$ drawdown. Career-ending.

• At 10% per loss, twenty consecutive losses leave you with

$0.9^{20} \approx 12\%$ of your starting capital. Game over.

The 1-2% rule does not assume you are a great trader. It assumes you are an average trader who will, sooner or later, encounter a losing streak. Sized at 2%, you survive that streak. Sized at 10%, you do not.

2.4 Stop Losses versus Hedges — Two Different Tools

A stop loss is an exit. You decide, before entering the trade, the price at which you will close the position and accept the loss. The trade is then over — no more upside, no more downside. Mechanically: a stop-loss order at the broker, or a mental stop you actually honour.

A hedge is a capped-loss overlay that *keeps the position open*. Long 100 SPY at \$520 with a \$500 protective put: your maximum loss is capped at \$20 per share plus the premium, and you keep all the upside above \$520 minus the premium. The position is alive; only the tail is amputated.

Three rules for choosing between them:

Use a stop when your conviction is event-dependent. If a

key earnings number, an FDA decision, or a Fed meeting could invalidate your thesis, stop loss the position when the news prints. There is no point holding a thesis that has been refuted.

**Use a hedge when the thesis is right but the path is

volatile.** If you think NVDA is fairly valued for a 12-month horizon but you cannot stomach a -30% interim drawdown, collar it (long stock + short OTM call + long OTM put). You keep the multi-year thesis alive while bounding the path- level pain.

**Use neither when the position is sized small enough that

you do not need either.** This is the most under-used option. A 1% position needs no stop and no hedge. A 25% position needs both. Position sizing is the cheapest insurance you have ever bought.

Stops have a known failure mode: gap risk. If overnight news opens the stock 20% below your stop, you fill at the open, not at the stop. Hedges with options do not have that failure mode because the put has already been bought.

2.5 Correlation-Adjusted Sizing for Multi-Position Books

The 1-2% rule per trade assumes trades are independent. They are not. If you are long ten tech stocks and the Nasdaq sells off 5%, all ten move together. Your effective single-trade exposure is much higher than the per-trade rule suggests.

The fix is to think in terms of **correlated dollar-risk buckets** rather than individual trades. A simple framework:

• All stocks in the same sector count as one bucket. Cap that

bucket at 6-10% total at-risk.

• All positions correlated to a single macro factor (rates,

USD, oil) count as one bucket. Same cap.

• Apply the 1-2% rule within each bucket, after the bucket

cap.

Concretely: if you want to be long six semiconductors, do not size each at 2%. Size the aggregate tech-sector at 8%, then divide that across the six names — about 1.3% each. When the sector rotates against you, your loss is bucket-capped, not position-summed.

The institutional version of this is risk parity: weight positions inverse to their volatility, then apply a covariance overlay to scale down correlated bets. Retail does not need the matrix algebra; the bucket rule captures 80% of the benefit.

2.6 Leverage and Drawdown — The Asymmetric Blowup

Leverage scales both return and variance, but its effect on drawdowns is non-linear and asymmetric. The intuition:

$$ \text{DD}_{\text{lev}} \approx (1 + L) \cdot \text{DD}_{\text{unlev}} \quad \text{(plus financing drag)} $$

A diversified equity book with a -25% peak-to-trough drawdown at 1x becomes roughly -50% at 2x leverage and -75% at 3x. Add the borrow cost (currently \~5% annual at retail margin rates) and the realised drawdown is even worse, because the drag eats into your equity during the drawdown, when you can least afford it.

The chart simulates expected max drawdown over a 30-year horizon for a strategy with 10% expected return and 16% annualised volatility, across leverage levels from 1.0 to 3.0. Notice the shape. From 1.0 to 1.5 the curve climbs roughly linearly. From 1.5 to 2.0 it bends upward. Beyond 2.0 it goes vertical: at 3x, the expected worst drawdown over thirty years is around -85%, which is a polite way of saying *margin call, position liquidated at the worst possible moment, account closed*.

The lesson: leverage above 1.5x is a different sport. You can do it, but you must size the underlying position much smaller to keep the levered drawdown survivable. That defeats the point of the leverage. Fat-tailed volatility is operating at full strength here: volatility is normally distributed in textbooks, fat-tailed in real life, and leverage makes the fat tail unsurvivable.

2.7 The Bankroll Preservation Mindset

Everything in this lesson sits on top of a single mental model: your bankroll is the asset. Trades are merely instruments through which the bankroll grows or shrinks. The job is not to "win the trade." The job is to keep playing — to remain solvent through any sequence of outcomes the market can produce, so the long-run edge has time to compound.

This is what it means when we say *the market can stay irrational longer than you can stay solvent*. The investor who blows up in 2022 is not the one who was wrong about the Fed. The investor who blows up in 2022 is the one who was right about the Fed but sized the bet at full Kelly, took 3x leverage, and did not put a hedge on. The thesis worked. The sizing did not.

The bankroll preservation rules, in priority order:

Never lose more than 1-2% of bankroll on any single trade.

Never lose more than 6-10% of bankroll on any single sector

or factor.

Never run leverage above 1.5x on equities, or 1.0x if the

underlying strategy is itself volatile (options, crypto, single names).

Always size at quarter-Kelly or smaller until you have ten

years of out-of-sample performance proving your edge estimate.

Always know what would have to happen for you to be ruined,

and price the option that would prevent it.

These five rules will not make you rich. Skill, time, and a working edge will. But these rules will *let you stay in the game long enough* for the skill, time, and edge to do their work. That is the entire point.

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3. Common Misconceptions

"Bigger size = bigger profit." Linear in expectation,

geometric in reality. Above $2 f^*$ you grow at zero. Above $3 f^*$ you lose money even with positive edge.

"I have an edge, so I should bet full Kelly." You do not

know your edge. Half-Kelly captures 75% of the growth and cuts drawdowns in half. Quarter-Kelly is the retail default.

"A stop loss means I lose money." A stop loss means you

control the size of the loss. The alternative is letting the market decide.

"Hedging is expensive." Compared to a margin call,

hedging is the cheapest line item on your statement. The right comparison is hedge cost vs probability-weighted blowup cost, not hedge cost vs zero.

"2x leverage doubles my return." It also doubles your

variance and roughly doubles your drawdowns. The expected geometric return is $L \mu - 0.5 L^2 \sigma^2$ — not linear in $L$.

"Leveraged ETFs are 2x exposure forever." They are 2x

daily exposure. Over time, vol decay subtracts roughly $0.5 (L^2 - L) \sigma^2$ per year. See Week 37.

"My positions are uncorrelated." Stocks in the same

sector correlate at 0.6-0.9 in normal times and at 0.95+ in crashes. Correlations rise to one in the moments you most need diversification.

"I can use mental stops." Mental stops fail when the

trade is losing money, which is the only time they matter. Use the broker's stop or do not use one.

"Risk management is for losers." Risk management is what

keeps the winners winning. The hedge funds that have been open for thirty years are the ones with risk officers who say no.

"I'll add risk after a winning streak." This is the

inverse of the right rule. Add risk after *bankroll has grown, never after streak has happened*. The two are different signals.

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4. Q&A Section

**Q1: I have $50,000 and want to risk 1% per trade. The stock is \$80 with a stop at \$76. How many shares?** Risk per trade = \$500. Loss per share at stop = \$4. Position = $500 / 4 = 125$ shares = \$10,000 notional (20% of bankroll). The position is 20% of the account but the risk is 1%. That is the distinction sizing makes.

Q2: Why quarter-Kelly and not half-Kelly? Half-Kelly assumes your edge estimate has modest error. For discretionary equity trades the edge estimate has large error (you are guessing 52%-54% win rates from small samples). Quarter- Kelly bakes that uncertainty in. If you have a true mechanical strategy with thousands of trades and stable parameters, half-Kelly is defensible.

Q3: Should I use a trailing stop or a fixed stop? Fixed stops are cleaner for short-term trades — you defined the thesis, you defined the invalidation, you exit at invalidation. Trailing stops are better for trend-following positions where you want to let winners run but lock in gains as the position moves your way. They have a known failure mode: noise stops you out before the trend resumes.

Q4: Is selling cash-secured puts a "hedge"? No. Selling a put is a short volatility income trade with defined upside (the premium) and large defined downside (strike minus premium). It is not a hedge — it adds risk. Buying a put is a hedge.

Q5: How do I size options trades? Options sizing should be based on maximum loss, not notional. A long call has max loss = premium. Size the premium at 1-2% of bankroll. A short put has max loss = strike minus premium. Size that at 1-2% of bankroll, which is much smaller than 1-2% of premium received.

Q6: My friend doubled his account on TSLA. Should I size up? No. He took an unsized bet and got lucky on a single draw. The correct comparison is over a thousand draws, where the unsized strategy goes to zero with probability one. Use his result as a survivorship-bias data point, not a sizing prescription.

**Q7: What's a reasonable max leverage for a buy-and-hold investor?** 1.0x in tax-deferred accounts; 1.0x to 1.25x in taxable margin accounts using box-spread financing; 1.5x only with a written drawdown plan and an explicit liquidity reserve. Above 1.5x is leveraged trading, not buy-and-hold investing.

Q8: How do I know if I'm over-sizing? The sleep test. If a -3% day on the position would change your mood for the evening, you are over-sized. The position should be small enough that the worst plausible day is annoying, not distressing.

**Q9: What about Kelly when I have multiple simultaneous positions?** For independent positions, sum of $f^*$ values across positions. For correlated positions, sum of $f^*$ divided by an effective- correlation factor. The bucket rule from §2.5 is the practical shortcut.

Q10: Does the 1-2% rule apply to long-term index investing? No — it applies to active trades with explicit invalidation points. A 100% index allocation in your retirement account is not "violating the rule"; it is a different category of capital with a different time horizon. The rule applies to your trading sleeve, not your buy-and-hold sleeve. The barbell keeps these two sleeves separate by design.

Q11: How does this connect to the four-tranche framework? Each tranche has its own sizing rule. Growth: index-tracking, no per-trade rule. Income: position size = yield target / asset yield. Stores of value: fixed allocation, rebalanced. Opt: this is where Kelly, fractional Kelly, and the 1-2% rule live. Risk management discipline is loudest in the Opt tranche because that is where unsized blowups happen.

Q12: Can I use Kelly for crypto? You can compute it. The problem is your $p$ and $b$ estimates are far less reliable than for equities, and crypto's drawdown distribution has a fatter left tail than the normal model underlying Kelly. Use eighth-Kelly or smaller, and treat all sizing as provisional until you have a multi-cycle track record.