Risk & Money Management

Question 1

(a) (4 marks)

• Rule: risk a fixed small fraction (1–2%) of equity per trade so no single loss materially harms capital. (1)
• Dollar risk = 0.015 \times 50{,}000 = \750$. (1)
• Stop distance = 200 - 188 = \12$ per share. (1)
• Shares $= 750 / 12 = 62.5 \Rightarrow 62$ shares (round down to stay within budget). (1)

(b) (8 marks)

• $g(f)=p\ln(1+bf)+(1-p)\ln(1-f)$. (1)
• First derivative: $g'(f)=\dfrac{pb}{1+bf}-\dfrac{1-p}{1-f}$. (2)
• Set $g'(f)=0$: $pb(1-f)=(1-p)(1+bf)$. Expanding: $pb - pbf = (1-p) + (1-p)bf$. Collect: $pb-(1-p) = bf[p+(1-p)] = bf$. So $f^* = \frac{pb-(1-p)}{b}=\frac{p(b+1)-1}{b}.$ (2)
• Second derivative: $g''(f) = -\dfrac{pb^2}{(1+bf)^2} - \dfrac{1-p}{(1-f)^2} < 0$ for all admissible $f$ ⇒ concave ⇒ maximum. (2)
• For $b=2,p=0.55$: $f^*=\dfrac{0.55\cdot 3 -1}{2}=\dfrac{0.65}{2}=0.325$. (1)

(c) (6 marks)

• For $b=1$: $g(f)=p\ln(1+f)+(1-p)\ln(1-f)$, and $f^*=2p-1$. (1)
• Break-even $f_0$: solve $g(f_0)=0$ with $f_0\ne 0$. $p\ln(1+f_0) = -(1-p)\ln(1-f_0)$. For $p=0.55$: $0.55\ln(1+f_0)+0.45\ln(1-f_0)=0$. (2)
• Numerically $f_0 \approx 0.1974$, while $f^*=0.10$, so $f_0 \approx 1.97 f^* \approx 2f^*$. (2)
• Conclusion: for $f>f_0\approx 2f^*$, $g(f)<0$ — over-betting drives geometric decay of capital despite a positive edge. (1)

(d) (6 marks)

• Quantitative: near $f^*$, $g$ is flat (parabolic top), so half-Kelly captures ~75% of the growth rate with roughly half the volatility of returns — much better risk-adjusted growth. Also parameter uncertainty in $p,b$ means true $f^*$ may be lower; over-estimating it pushes you toward the $g<0$ region. (3)
• Behavioural: full-Kelly drawdowns are brutal (can lose 50%+ frequently), causing traders to abandon the system; fractional Kelly keeps drawdowns psychologically tolerable and reduces risk of ruin. (3)

Question 2

(a) (6 marks)

• $\sigma_P=\sqrt{\sigma_A^2+\sigma_B^2+2\rho\sigma_A\sigma_B}=1000\sqrt{2+2\rho}$. (2)
• $\rho=+1$: \sigma_P=1000\sqrt{4}=\2000$ (risks fully add — worst case, concentrated). (1)
• $\rho=0$: \sigma_P=1000\sqrt{2}\approx\1414$(diversification benefit; less than the$$2000$ naive sum). (1)
• $\rho=-1$: $\sigma_P=0$ (perfect hedge — positions cancel). (1)
• Interpretation: correlated positions understate true risk if summed naively; the effective portfolio risk depends on $\rho$. (1)

(b) (6 marks)

def max_positions(equity, risk_per_trade_pct, heat_cap_pct):
    risk_per_trade = equity * risk_per_trade_pct/100
    heat_cap = equity * heat_cap_pct/100
    return int(heat_cap // risk_per_trade)
 
max_positions(50000, 1.5, 6)  # -> 4

• Each position risk =\750$; cap$=$3000$;$3000/750=4$ positions. (marks: correct formula 3, correct floor/int 1, answer 4 → 2)

(c) (8 marks)

• Position dollar value = 500\times200=\100{,}000$. (1)
• Beta-adjusted exposure =100{,}000\times1.3=\130{,}000$. (2)
• Value per futures contract =50\times4000=\200{,}000$. (2)
• Contracts to short $= 130{,}000/200{,}000 = 0.65 \approx 1$ contract (or 0.65 if fractional/mini allowed). (2)
• Residual risk: hedge removes only market/systematic (beta) risk; idiosyncratic (stock-specific) risk remains, plus basis risk between stock and index and beta-estimation error. (1)

Question 3

(a) (4 marks)

• edge $=2(0.55)-1=0.10$. (1)
• $R=\left(\dfrac{1-0.10}{1+0.10}\right)^{20}=\left(\dfrac{0.9}{1.1}\right)^{20}=(0.8\overline{18})^{20}$. (1)
• $\approx 0.0181$, i.e. about 1.8% chance of ruin with 20 units of capital. (1)
• Interpretation: even a modest edge plus adequate capitalisation (many risk-units) makes ruin unlikely; thin capitalisation (small $U$) makes ruin dangerously high. (1)

(b) (6 marks)

import random
def risk_of_ruin(p, units, trials=100000):
    ruined = 0
    for _ in range(trials):
        cap = units
        while 0 < cap < 2*units:
            cap += 1 if random.random() < p else -1
        if cap <= 0:
            ruined += 1
    return ruined/trials

• Simulate fixed 1-unit bets, absorbing barriers at 0 (ruin) and $2U$ (target). (3)
• Validation: run with p=0.55, units=20; empirical result should approximate the gambler's-ruin closed form (bounded version). For the unbounded formula in (a), compare with large target and check convergence; expect ~0.018. Increase trials to shrink Monte-Carlo standard error ($\sim 1/\sqrt{N}$). (3)

(c) (6 marks)

• Proposed rule (base risk $r_0=1\%$): $k(D)=\begin{cases}1 & D<5\%\\ 0.5 & 5\%\le D<10\%\\ 0 & D\ge 10\%\ (\text{stop trading})\end{cases}$ Effective risk per trade $= k(D)\cdot r_0$. (2)
• Daily-stop proof: with base risk 1% and max 4 trades, worst-case daily loss $\le 4\times1\%=4\%$ — this exceeds a 3% daily stop, so we add a circuit breaker: halt trading once cumulative daily loss reaches 3%. Then max realised daily loss $\le 3\%$ (plus at most one in-flight trade; sizing the last permitted trade so remaining budget $\le3\%$ enforces the cap). Formally, track cumulative loss $L$; permit a new trade only if $L + \text{next risk} \le 3\%$, guaranteeing $L_{\max}\le 3\%$. (2)
• Why reduce size in drawdown: smaller bets in drawdown reduce the number of effective risk-units lost per adverse trade, increasing $U$ in the risk-of-ruin sense and shrinking geometric decay; it prevents the "doubling-down into ruin" failure mode and preserves capital to recover. (2)

[
{"claim":"Position size = 62 shares from $750 risk / $12 stop","code":"risk=0.015*50000; shares=int(risk/(200-188)); result = (shares==62)"},
{"claim":"Kelly fraction f*=0.325 for b=2,p=0.55","code":"p=Rational(55,100); b=2; f=(p*(b+1)-1)/b; result = (f==Rational(325,1000))"},
{"claim":"Even-money Kelly f*=0.10 for p=0.55","code":"p=Rational(55,100); f=2*p-1; result = (f==Rational(1,10))"},
{"claim":"Break-even f0 ~1.97*f* (approx 2*Kelly) for p=0.55 even money","code":"import mpmath as mp; f0=mp.findroot(lambda x: 0.55*mp.log(1+x)+0.45*mp.log(1-x), 0.2); result = (abs(f0/0.10-2)<0.05)"},
{"claim":"Portfolio sigma at rho=0 is 1000*sqrt(2)~1414","code":"import math; s=1000*math.sqrt(2+2*0); result = (abs(s-1414.2135)<0.01)"},
{"claim":"Max positions = 4 at 1.5% risk, 6% heat cap","code":"result = (int((50000*6/100)//(50000*1.5/100))==4)"},
{"claim":"Beta-neutral hedge ~0.65 contracts","code":"c=(500*200*1.3)/(50*4000); result = (abs(c-0.65)<1e-9)"},
{"claim":"Risk of ruin ~0.0181 for p=0.55,U=20","code":"R=(Rational(9,10)/Rational(11,10))**20; result = (abs(float(R)-0.0181)<0.001)"}
]