14.18 The mathematics of leverage: drawdowns and margin calls

Motivation

Leverage multiplies returns and also multiplies drawdowns, but not linearly, and the margin call arrives before the drawdown you planned for. This section derives the three numbers a leveraged trader must know before entering: the drawdown multiplier, the price at which the margin call comes, and the probability that it comes within the holding period.

14.18.1Static and constant leverage#

Setup

Equity $E_0$ buys stock worth $V_0 = \lambda E_0$ , borrowing $B_0 = (\lambda - 1)E_0$ at rate $c$ . Write $x = P_t/P_0$ for the price ratio.

Proposition 14.25static leverage

If the position is not rebalanced,

\frac{E_t}{E_0} = 1 + \lambda(x - 1) - (\lambda - 1)ct, \tag{14.18}

using simple interest. A price drawdown $D$ produces an equity drawdown of $\lambda D$ (plus interest), and equity is wiped out at $x = 1 - 1/\lambda$ (ignoring interest).

Proof

$E_t = V_0x - B_0(1 + ct) = \lambda E_0x - (\lambda - 1)E_0(1 + ct)$ .∎

Proposition 14.26constant leverage

If the position is rebalanced continuously to keep $V_t = \lambda E_t$ , the stock follows geometric Brownian motion with volatility $\sigma$ , and borrowing costs $c$ continuously, then

\ln\frac{E_T}{E_0} = \lambda\ln\frac{P_T}{P_0} - \tfrac12\lambda(\lambda - 1)\sigma^2T - (\lambda - 1)cT. \tag{14.19}

Equity never reaches zero, but a price drawdown $D$ over a period $\tau$ produces an equity drawdown of $1 - (1 - D)^\lambda e^{-\frac12\lambda(\lambda-1)\sigma^2\tau - (\lambda-1)c\tau}$ .

Proof

$dE/E = \lambda\,dP/P - (\lambda - 1)c\,dt$ with $dP/P = \mu\,dt + \sigma\,dW$ . By Itô, $d\ln E = [\lambda\mu - (\lambda - 1)c - \tfrac12\lambda^2\sigma^2]dt + \lambda\sigma\,dW$ , and $d\ln P = (\mu - \tfrac12\sigma^2)dt + \sigma\,dW$ . Subtract $\lambda\,d\ln P$ : the $dW$ and $\mu$ terms cancel, leaving $[-(\lambda - 1)c - \tfrac12\lambda(\lambda - 1)\sigma^2]dt$ . Integrate.∎

Two consequences. First, with $\lambda = 2$ and a 30% price fall, static leverage loses 60% and constant leverage 51% (52% with $\sigma = 20\%$ over six months), because rebalancing sells on the way down. Second, the expected log growth rate of equity is $c + \lambda(\mu - c) - \tfrac12\lambda^2\sigma^2$ , maximised at $\lambda^* = (\mu - c)/\sigma^2$ : the continuous Kelly fraction of M03 · 3.4. Any leverage above $2\lambda^*$ has a lower growth rate than holding cash at $c$ , since $c + \lambda(\mu - c) - \tfrac12\lambda^2\sigma^2 < c \iff \lambda > 2\lambda^*$ . At an MTF rate of 15% and an expected stock return below 15%, $\mu - c < 0$ and no borrowing is justified. Even funded at 10%, a single stock with an expected return of 13% and volatility 30% has $\lambda^* = 0.03/0.09 = 0.33$ : Kelly says buy a third of a position, not two and a half.

14.18.2The margin-call trigger#

Setup

A funded position worth $V_t = V_0x$ with loan $B_t$ ; optional collateral (pledged shares) worth $K_t$ with haircut $h$ ; margin rate $m$ required on the funded position (for MTF, VaR + 3 ELM or VaR + 5 ELM). The client's margin is

M_t = (V_t - B_t) + (1 - h)K_t, \tag{14.20}

and a call is made when $M_t < mV_t$ .

Proposition 14.27trigger price

(a) Collateral independent of the stock ( $K_t = K_0$ ): the call comes when

x < x^* = \frac{B_t - (1 - h)K_0}{(1 - m)V_0}.

(b) No collateral, initial margin fraction $e_0 = (V_0 - B_0)/V_0 \ge m$ , loan growing at simple rate $c$ : the call comes after a price fall of

d^* = 1 - \frac{(1 - e_0)(1 + ct)}{1 - m}, \qquad d^* = \frac{e_0 - m}{1 - m}\ \text{ when } c = 0. \tag{14.21}

Proof

Substitute into $M_t < mV_t$ and solve for $x$ . (a) $V_0x - B_t + (1 - h)K_0 < mV_0x \iff x(1 - m)V_0 < B_t - (1 - h)K_0$ . (b) Set $K_0 = 0$ , $B_t = (1 - e_0)V_0(1 + ct)$ , $d^* = 1 - x^*$ . (c) $V_0x - B_t + (1 - h)K_0x < mV_0x \iff x[(1 - m)V_0 + (1 - h)K_0] < B_t$ .∎

If you post exactly the minimum, $e_0 = m$ and $d^* = 0$ : the first down-tick is a margin call. The buffer between your initial margin and the requirement is the only thing between you and a call, and (14.21) converts it into a price distance. In case (c) the collateral falls with the position, so the trigger is closer than in (a) for the same initial margin.

Proposition 14.28curing a call by selling

At a call, selling funded stock worth $y$ and repaying $y$ of the loan restores the requirement iff

y \ge V_t - \frac{M_t}{m} = \frac{mV_t - M_t}{m}. \tag{14.22}

Proof

Selling $y$ and repaying $y$ leaves $V_t - B_t$ , hence $M_t$ , unchanged, while the requirement falls to $m(V_t - y)$ . Solve $M_t \ge m(V_t - y)$ .∎

The cure by selling is $1/m$ times the cash shortfall: at $m = 27.5\%$ , every ₹1 of shortfall forces ₹3.64 of sales at the low. This is the arithmetic of a liquidation cascade.

Worked example 14.18.1

Buy ₹10 lakh of an F&O Group I stock with VaR 12.5% and ELM 5% ( $m = 27.5\%$ ), paying 40% (₹4 lakh) and funding ₹6 lakh by MTF at an assumed 15% a year.

Leverage $\lambda = 2.5$ . By (14.21) with no interest, the call comes after a fall of $d^* = (0.40 - 0.275)/0.725 = 17.24\%$ , at which point equity has fallen $2.5 \times 17.24\% = 43.1\%$ .
After 60 days of interest the loan is ₹6,14,795 and the trigger is a fall of 15.20%.
If at day 60 the stock is down 20% ( $V = \text{₹}8$ lakh), margin is ₹1,85,205 against a requirement of ₹2,20,000. The cure is ₹34,795 of fresh cash, or by (14.22) selling ₹1,26,526 of stock.
If instead the ₹4 lakh margin had been posted as pledged shares of the same stock (₹5 lakh at a 20% haircut) with the full ₹10 lakh funded, the trigger is $x^* = 10/(7.25 + 4) = 0.889$ , an 11.1% fall, against 17.24% if the pledged shares were uncorrelated with the position.

14.18.3How likely is the call?#

Proposition 14.29first-passage probability

If $\ln(P_t/P_0) = \nu t + \sigma W_t$ with $\nu = \mu - \tfrac12\sigma^2$ , the probability that the price ratio touches $1 - d^*$ (with $b = \ln(1 - d^*) < 0$ ) at some time before $T$ is

P(\tau_b \le T) = \Phi\Big(\frac{b - \nu T}{\sigma\sqrt T}\Big) + e^{2\nu b/\sigma^2}\,\Phi\Big(\frac{b + \nu T}{\sigma\sqrt T}\Big). \tag{14.23}

Proof sketch

For $\nu = 0$ the reflection principle gives $P(\min_{t\le T} W_t \le b/\sigma) = 2\Phi(b/(\sigma\sqrt T))$ . For $\nu \ne 0$ , change measure by Girsanov so that $\nu t + \sigma W_t$ is a driftless Brownian motion, apply the reflection principle to the joint law of the minimum and the endpoint, and change back; the Radon–Nikodym factor produces the term $e^{2\nu b/\sigma^2}$ . See Shreve, Stochastic Calculus for Finance II (2004), 7.2, or Harrison, Brownian Motion and Stochastic Flow Systems (1985), ch. 1.∎

For the worked example ( $d^* = 17.24\%$ , $b = -0.1892$ ) with $\mu = 12\%$ and $\sigma = 30\%$ a year, (14.23) gives 2.5% within one month, 17.6% within three months, 31.6% within six months and 44.6% within a year. A Monte Carlo with daily monitoring gives 15.2% at three months: discrete monitoring misses intraday touches, but a broker monitors intraday, so the continuous figure is the relevant one. The model also ignores gaps; an overnight gap through the trigger turns the call into a loss larger than the cushion. A position that has a one-in-six chance of a margin call within a quarter is not a long-term investment, whatever the intention at purchase.

Practitioner notes

Compute $d^*$ and the probability (14.23) before entering, using the margin rate you would face in a stressed market (VaR margins rise with volatility), not today's.
Keep the cash to meet a call outside the trading account and decide in advance whether you will meet it or cut. Deciding at the low is the disposition effect (14.9) under time pressure.
Never pledge shares correlated with the leveraged position. Case (c) of Proposition 14.27 is the retail version of every collateral spiral from LTCM to promoter-pledge collapses.
Jhunjhunwala's reported practice of keeping the investing book separate from the trading book (14.12.3) also keeps the investing book from being pledged against trading losses.

Facts used in 14.16–14.18#

Fact	Value	As of	Source
MTF eligible securities	Group I equity shares and equity ETF units	text in force before the June 2026 proposals	SEBI consultation paper, 18 Jun 2026 (reproducing master circular para 4.2–4.3)
MTF margin, F&O Group I stocks	VaR + 3 × ELM (cash-segment rates)	same	same; original circular CIR/MRD/DP/54/2017, 13 Jun 2017
MTF margin, other Group I stocks and ETFs	VaR + 5 × ELM	same	same
MTF consultation (net worth ₹3 cr → ₹5 cr proposed; margins unchanged)	proposed, comments due 9 Jul 2026	18 Jun 2026	Taxmann (secondary); adoption not verified
Pledge/re-pledge framework	SEBI circular 25 Feb 2020; mandatory from 1 Sep 2020	2020	broker notices (secondary)
Short-selling framework update	SEBI/HO/MRD/MRD-PoD-3/P/CIR/2024/1	5 Jan 2024	secondary summaries (TaxGuru, Business Standard)
SLB tenure	1 to 12 months; shorter-tenure contracts from 17 Aug 2026	2026	NSE SLBS features page (search summary); secondary news (verify with the exchange)
Delivery STT; futures STT (sell)	0.1% each side; 0.05%	from 1 Apr 2026	NSE circular NSE/FATAX/73524, 31 Mar 2026 (fetched 6 Nov 2026)