Risk & Money Management

Question 1 (12)

(a) Risk budget $= 1.5\% \times 8{,}00{,}000 = ₹12{,}000$. Stop distance per share $= 640 - 608 = ₹32$. Shares $= 12{,}000 / 32 = 375$ shares. (4) (1 for risk budget, 1 for stop distance, 2 for division & answer)

(b) Round lots of 50 not exceeding 375 → $350$ shares (7 lots), since 400 shares would risk $400\times32 = ₹12{,}800 > 12{,}000$. Actual risk $= 350 \times 32 = ₹11{,}200$. (4) (2 for choosing 350, 1 for rejecting 400, 1 for actual risk)

(c) New stop distance $= 640 - 624 = ₹16$. Shares $= 12{,}000 / 16 = 750$ (position size doubles). (2) Trade-off: tighter stop allows a larger position (more shares/leverage on the same risk), but the price has less room to fluctuate, so probability of being stopped out on normal noise rises. (2)

Question 2 (12)

(a) After a $28\%$ loss, capital $= 0.72$ of peak. Required gain $= \dfrac{1}{0.72} - 1 = 0.3889 = 38.89\%$. (4) (2 for 0.72 factor, 2 for gain computation)

(b) After $20\%$ drawdown, capital $= 8{,}00{,}000$. Need to grow back to $10{,}00{,}000$. Growth factor needed $= 10/8 = 1.25$. Each win $= \times 1.02$. Need $1.02^n \ge 1.25$. $n \ge \dfrac{\ln 1.25}{\ln 1.02} = \dfrac{0.22314}{0.01980} = 11.27$ → $n = 12$ trades. (5) (1 for target factor 1.25, 2 for log setup, 2 for n=12)

(c) A daily loss limit caps losses within a single session (e.g., a bad day of overtrading/tilt), stopping the trader before one session compounds into disaster — it is a short-horizon circuit breaker. A max drawdown limit governs the cumulative peak-to-trough decline over many sessions and triggers strategy-level changes (reduced sizing/stop trading). They protect against different failure modes: intraday emotional blowups vs. long-run capital erosion. (3)

Question 3 (14)

(a) Kelly: $f^* = \dfrac{bp - q}{b} = p - \dfrac{q}{b}$, where $b$ = win/loss payoff ratio, $q = 1-p$. (2) Here $p = 0.55$, $q = 0.45$, $b = 1.6$. $f^* = \dfrac{1.6(0.55) - 0.45}{1.6} = \dfrac{0.88 - 0.45}{1.6} = \dfrac{0.43}{1.6} = 0.26875 \approx 26.9\%$. (4)

(b) Half-Kelly $= 0.5 \times 0.26875 = 0.1344 \approx 13.4\%$ of capital. (2) Reason: full Kelly is very volatile and highly sensitive to errors in estimated $p$ and $b$; fractional Kelly greatly reduces drawdown volatility for only a modest reduction in long-run growth. (2)

(c) With $p = 0.5$, $q = 0.5$: $f^* = \dfrac{1.6(0.5) - 0.5}{1.6} = \dfrac{0.8 - 0.5}{1.6} = \dfrac{0.3}{1.6} = 0.1875 \approx 18.75\%$. (2) Interpretation: still positive because the payoff ratio $b=1.6>1$ gives an edge even at 50% win rate. (A negative/zero Kelly would mean no edge — the trader should not bet at all.) (2)

Question 4 (12)

(a) With correlation $+0.9$, A and B move almost together, so their risks add rather than diversify — in a drawdown both lose simultaneously. Effective combined risk of A+B is close to a single doubled ($\approx 4\%$) position, not two independent $2\%$ bets. C, being uncorrelated, genuinely diversifies. (4)

(b) Sector rule: A + B share a sector = $2\% + 2\% = 4\%$ sector exposure, which exceeds the $3\%$ single-sector cap → non-compliant on the sector limit (even though total heat $6\%$ meets the portfolio cap). (4)

(c) Acceptable hedging actions (any one, with mechanism): reduce A or B size; buy a put on the sector/one holding to cap downside; short a correlated sector ETF or index future against the pair to neutralize the shared systematic move while keeping his higher-conviction single-name upside. Mechanism: the hedge gains when the correlated positions fall, offsetting the doubled directional exposure. (4)

Question 5 (10)

(a) Larger fractional bets increase the variance of the equity curve. Even with a positive expected value, a run of consecutive losses at large size can drop capital below the recovery threshold; because losses compound multiplicatively, oversizing past the Kelly optimum reduces long-run growth and can drive geometric growth negative — so ruin probability rises sharply despite positive edge. (4)

(b) Increasing capital from 10 to 20 units (same 1-unit bet) lowers risk of ruin: more "buffer" units mean a longer adverse streak is required to reach zero, and with a positive per-trade edge ($p=0.6$) the ruin probability falls roughly exponentially with the number of units of capital. (3)

(c) Anti-martingale rule: after wins, increase size (risk a larger fraction as equity grows / add units); after losing streaks, cut size (halve risk until recovery). This ties bet size to current equity, so losses are taken at smaller absolute size — shrinking the tail toward ruin — while wins are pressed at larger size. It contrasts with martingale (increasing size after losses), which dramatically raises risk of ruin. (3)

[
  {"claim":"Q1a shares = 375","code":"risk=0.015*800000; dist=640-608; result=(risk/dist==375)"},
  {"claim":"Q1b actual risk at 350 shares = 11200 and 400 exceeds budget","code":"budget=12000; result=(350*32==11200 and 400*32>budget)"},
  {"claim":"Q2a recovery gain approx 38.89%","code":"g=1/Rational(72,100)-1; result=(abs(float(g)-0.388888)<1e-4)"},
  {"claim":"Q2b needs 12 winning trades","code":"import math; n=math.ceil(math.log(1.25)/math.log(1.02)); result=(n==12)"},
  {"claim":"Q3a full Kelly = 0.26875","code":"f=(1.6*0.55-0.45)/1.6; result=(abs(f-0.26875)<1e-9)"},
  {"claim":"Q3c Kelly at p=0.5 = 0.1875","code":"f=(1.6*0.5-0.5)/1.6; result=(abs(f-0.1875)<1e-9)"}
]