3.9 Position sizing in practice · M03 Kelly and ruin

Motivation

The theory gives a fraction $f$ of capital to put at risk. A trader needs a number of shares or lots, and needs to know what can make the realised loss larger than planned.

Percent-risk sizing#

Setup

With equity $E_q$ , risk fraction $f$ , entry price $P_e$ , stop $P_s$ and lot size $\ell$ (1 for cash equity; 65 for Nifty futures and options as of Nov 2026), the percent-risk (fixed-fractional) position size is

Q = \ell\left\lfloor\frac{1}{\ell}\left\lfloor\frac{fE_q}{|P_e - P_s|}\right\rfloor\right\rfloor, \tag{3.23}

the largest whole number of lots whose loss at the stop does not exceed $fE_q$ .

Worked example 3.9.1cash equity

Account ₹10 lakh, risk 1% (₹10,000). A stock is bought at ₹1,640 with a stop at ₹1,598 (₹42 below entry). $Q = \lfloor 10{,}000/42\rfloor = 238$ shares, a position of ₹3.90 lakh, 39% of the account. The position's notional size is set by the stop distance, not by conviction.

Worked example 3.9.2index futures

Same account, long one Nifty futures lot at 24,000 with a 120-point stop. Risk per lot $= 120 \times 65 = \text{₹}7{,}800 \le \text{₹}10{,}000$ , so $Q = 65$ (one lot). Notional exposure is $24{,}000 \times 65 = \text{₹}15.6$ lakh, 1.56 times the account. At 0.5% risk (₹5,000) the answer is zero lots: the minimum contract size of index derivatives (₹15 lakh at launch under SEBI's October 2024 framework) puts index futures out of reach of small accounts at prudent risk levels.

Gaps and option positions#

Gap risk A stop is an order, not a guarantee. If the market opens beyond the stop, the fill is at the open or worse.

23 Mar 2020: Nifty opened at 7,945.70 against the previous close of 8,745.45, a gap of $-9.1\%$ (Nifty sample, Yahoo open prices). A long futures position with a 120-point stop lost about 800 points, $6.7R$ .
4 Jun 2024: Nifty opened only 0.4% lower at 23,179.50 but fell to 21,281.45 intraday ( $-8.5\%$ ). A stop-market order would have filled near its level; a stop-limit order could have been skipped in a fast market.
In the Nifty sample, 5.9% of daily opens were more than 1% away from the previous close and 1.3% more than 2% away.

Gaps make the R distribution's left tail longer than $-1$ . Model it explicitly: add a small probability of $-3R$ to $-7R$ outcomes in simulations (Problem 9), or size with the stop at the "gap-adjusted" distance.

Option positions For a bought option the maximum loss is the premium paid, so $1R$ can be set to the premium without a stop: one Nifty lot of a ₹100 option is ₹6,500 of risk. For a sold option the loss is unbounded and stop orders on options fill badly in fast markets; size short options by stress loss (a $-10\%$ index move with implied volatility up 50%, M09), not by a stop.

Volatility sizing and bootstrapped drawdowns#

Volatility-based sizing Setting the stop at a multiple of a volatility measure (for example $2\times$ the average true range, ATR, which is the average daily high–low range adjusted for gaps) and applying (3.23) makes the rupee volatility of every position roughly equal. Combined with a portfolio-level cap on total open risk (often called "heat"), this is a crude risk-parity rule.

Kelly, R-multiples and the bootstrap For a trade list, the risk fraction $f$ in (3.23) is the Kelly variable of 3.2. The table resamples the swing system's 100 trades (40 of $+2.5R$ , 60 of $-1R$ ) with replacement into 10,000 sequences of 250 trades (bootstrap_mdd in the reference solution of Exercise m03_sizing):

Risk per trade	Median max drawdown	95th percentile	$P(\text{MDD} > 30\%)$
0.5%	5.6%	9.2%	0.0000
1%	11.0%	17.7%	0.0002
2%	20.9%	32.9%	0.093
5%	46.0%	65.3%	0.985
16% (Kelly)	91.4%	98.9%	1.000

The Kelly fraction maximises growth and nearly guarantees a 90% drawdown within 250 trades. Practitioner rules of 0.5–1% risk per trade sit at roughly $c = 1/32$ to $1/16$ of this system's Kelly fraction. Proposition 3.9 alone justifies $k^* = t^2/(1+t^2) = 0.84$ for $t = 2.3$ , so the rest of the gap is not estimation error in $E$ ; it is aversion to drawdown, uncertainty about the shape of the R distribution (gaps), correlation across simultaneous trades, and the knowledge that edges decay.

Practitioner notes

Decide the stop from the trade's logic (where the setup is invalidated), then compute size. Never move the stop to fit a desired size.
Risk per trade and number of simultaneous positions jointly set portfolio risk. Five open trades at 1% each in correlated stocks is a 5% bet on one factor.
Reduce $f$ after drawdowns only if the drawdown is evidence that the edge has changed. Fixed-fractional sizing already reduces rupee risk automatically as equity falls.