Risk & Money Management

Question 1 (10 marks)

(a) Position size: $\text{Shares} = \frac{\text{Equity} \times \text{Risk fraction}}{|\text{Entry} - \text{Stop}|}$

• 1 mark numerator (dollar risk), 1 mark denominator (per-share risk), 1 mark correct assembly.

(b)

• Dollar risk = 50{,}000 \times 0.01 = \500$. (1)
• Per-share risk = 120 - 114 = \6$. (1)
• Shares $= 500 / 6 = 83.33 \Rightarrow 83$ shares (round down to not exceed risk). (2)

(c) Sizing off stop distance means a wider stop → fewer shares, a tighter stop → more shares, so the dollar loss if stopped out is always the fixed $500 (1% of equity). Fixed share count would let dollar risk swing wildly with volatility/stop width. (3 for the constant-dollar-risk reasoning)

Question 2 (12 marks)

(a) Derivation: $G(f)=p\ln(1+bf)+q\ln(1-f)$ Differentiate and set to zero: $G'(f)=\frac{pb}{1+bf}-\frac{q}{1-f}=0$ Cross-multiply: $pb(1-f)=q(1+bf)$ $pb - pbf = q + qbf$ $pb - q = pbf + qbf = bf(p+q)=bf$ $f^*=\frac{pb-q}{b}=\frac{p(b+1)-1}{b}$

• 2 marks objective/derivative, 2 marks algebra, 2 marks final closed form. Second-derivative < 0 (concave) confirms max.

(b) $p=0.55,\ q=0.45,\ b=1.5$: $f^*=\frac{0.55(1.5)-0.45}{1.5}=\frac{0.825-0.45}{1.5}=\frac{0.375}{1.5}=0.25$ So full Kelly = 25% of equity. (3)

(c) Half-Kelly $=0.125=12.5\%$. (1) Reason: full Kelly maximises long-run growth but has large drawdowns and is highly sensitive to estimation error in $p,b$; fractional Kelly sacrifices a little growth for much lower volatility / drawdown. (2)

Question 3 (10 marks)

(a) Maximum drawdown: $\text{MDD}=\max_{t}\left(\frac{\max_{s\le t}E_s - E_t}{\max_{s\le t}E_s}\right)$ i.e. the largest peak-to-trough decline as a fraction of the running peak. (3)

(b) $\dfrac{80{,}000-62{,}000}{80{,}000}=\dfrac{18{,}000}{80{,}000}=0.225=22.5\%$. (3)

(c) Daily limit =3\%\times50{,}000=\1{,}500$; each loss$=1%=$500$;$1500/500=3$ losing trades trigger the halt. (2) Out-loud: per-trade sizing controls one trade, but a bad day (tilt, correlated stops, news) can stack many losses; a daily circuit breaker caps cumulative damage and forces a cooldown before emotional revenge-trading compounds losses. (2)

Question 4 (10 marks)

(a) $R=\left(\dfrac{1-A}{1+A}\right)^{U}$, where $A$ = the trader's edge/advantage per unit ($A=p-q$ in a simple even-money model), and $U$ = number of betting units of capital (capital ÷ risk-per-bet). (4)

(b) $A=0.1,\ U=20$: $R=\left(\frac{0.9}{1.1}\right)^{20}=(0.81818\ldots)^{20}\approx 0.0184 \approx 1.8\%$ (3) (accept ~2%).

(c) Increasing $U$ (smaller bets per unit of capital) raises the exponent, driving $R$ toward $0$ exponentially — so with a positive edge, betting smaller dramatically cuts ruin probability. The 1–2% rule keeps $U$ large (50–100 units), making ruin negligible while preserving the edge. (3)

Question 5 (10 marks)

(a) With correlation $\approx 1$, the three positions move together, so they act like one large position of $3\%$. Effective risk $\approx 3\% (\text{correlated block}) + 1\% (\text{independent}) = 4\%$ nominal — but the simultaneous loss risk of the block is $3\%$ in one event, not diversified. Naive sum of $4\%$ understates the concentrated downside because correlated risks don't diversify away. (4)

(b) $\sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\rho\sigma_1\sigma_2$ As $\rho\to-1$: $\sigma_p^2 \to (w_1\sigma_1 - w_2\sigma_2)^2$, which can reach $0$ if $w_1\sigma_1=w_2\sigma_2$ — perfect hedge, risk cancels. (4)

(c) Buy index put options / short index futures (or an inverse ETF) against a long book: the hedge has negative correlation to the long exposure, reducing the cross term $2w_1w_2\rho\sigma_1\sigma_2$ and thus portfolio variance. (2)

Question 6 (8 marks)

def position_size(equity, risk_pct, entry, stop):
    risk_dollars = equity * risk_pct           # e.g. risk_pct = 0.01
    per_share = abs(entry - stop)
    if per_share == 0:                         # guard div-by-zero
        return 0
    shares = int(risk_dollars // per_share)    # round down
    return max(shares, 0)                       # no negative sizes
 
def adjust_risk(base_risk, win_streak, loss_streak):
    # size up with wins, down with losses; clamp to sane band
    factor = 1 + 0.1 * win_streak - 0.2 * loss_streak
    factor = max(0.25, min(factor, 2.0))       # floor/ceiling
    return base_risk * factor

Mark scheme: correct dollar-risk & per-share (2), div-by-zero guard (1), int floor (1), non-negative guard (1), adjust scales up on wins / down on losses (2), clamping band (1).

[
  {"claim":"Position size for Q1b is 83 shares", "code":"risk=50000*0.01; per=120-114; shares=int(risk//per); result=(shares==83)"},
  {"claim":"Full Kelly fraction is 0.25 for p=0.55,b=1.5", "code":"p=Rational(55,100); q=1-p; b=Rational(3,2); f=(p*b-q)/b; result=(f==Rational(1,4))"},
  {"claim":"Drawdown from 80000 to 62000 is 22.5%", "code":"dd=(80000-62000)/80000; result=(dd==Rational(225,1000))"},
  {"claim":"Daily 3% limit / 1% risk = 3 losing trades", "code":"n=(0.03*50000)/(0.01*50000); result=(int(n)==3)"},
  {"claim":"Risk of ruin approx 1.8% for A=0.1,U=20", "code":"R=(Rational(9,10)/Rational(11,10))**20; result=(abs(float(R)-0.0184)<0.002)"}
]