What to Trade

Question 1

(a) Round-trip spread cost fraction $= \dfrac{2 \times \text{spread}}{\text{price}}$ (pay half-spread on entry, half on exit; conservatively use full spread each way ⇒ 2×spread). [method 1]

• A: $c_A = \dfrac{2\times 0.05}{500} = \dfrac{0.10}{500} = 0.0002 = 0.02\%$ [1]
• B: $c_B = \dfrac{2\times 1.20}{300} = \dfrac{2.40}{300} = 0.008 = 0.80\%$ [1]
• A is far more liquid: 0.02% vs 0.80%, a factor of 40× cheaper to trade. [1]

(b) Order size fraction $Q/ADV$: value ₹10,00,000.

• A: shares $= 1{,}000{,}000/500 = 2000$; $Q/ADV = 2000/4{,}200{,}000 = 4.76\times10^{-4}$. Impact $=0.15\times4.76\times10^{-4}=7.14\times10^{-5}=0.00714\%$. [1] $E_{net,A} = 0.9\% - 0.02\% - 0.00714\% \approx 0.873\%$. [2]
• B: shares $=1{,}000{,}000/300 = 3333$; $Q/ADV = 3333/90{,}000 = 0.0370$. Impact $=0.15\times0.0370 = 0.00556 = 0.556\%$. [1] $E_{net,B} = 0.9\% - 0.80\% - 0.556\% = -0.456\%$. [2]
• Decision: A tradeable (+0.873%); B has negative net edge — spread + impact destroy the edge. Trade A. [1] (Illiquid stock B eaten alive by costs — core lesson of 4.2.2.)

(c) Nifty spread cost fraction: $c_N = \dfrac{2\times 0.5}{22000} = \dfrac{1}{22000} = 4.55\times10^{-5} = 0.00455\%$. [1]

• $E_{net,N} \approx 0.9\% - 0.00455\% \approx 0.8955\%$. [1]
• $R_A = E_{net,A}/\sigma_A = 0.873/1.2 = 0.7275$. [1]
• $R_N = E_{net,N}/\sigma_N = 0.8955/0.9 = 0.9950$. [1]
• Since $R_N (0.995) > R_A (0.728)$, Nifty is preferred on a risk-adjusted basis despite similar gross edge — its lower $\sigma$ and near-zero cost dominate. [2] Proof: with $E_{net}$ near-equal, $R\propto 1/\sigma$, and $\sigma_N<\sigma_A$ ⇒ $R_N>R_A$.

(d) Any two: [2 each]

• Index cannot go to zero / is not subject to single-firm event risk (fraud, delisting) — diversified basket.
• Deeper, more consistent liquidity and tighter spreads across all market conditions.
• No single-stock news gaps; cleaner technical/statistical behaviour for systematic models.
• (Also acceptable: high-leverage F&O access, no borrow constraints for shorting.)

Question 2

(a) Means:

• $\bar X = (1.0-0.5+2.0+0.5+1.0)/5 = 4.0/5 = 0.8$. [1]
• $\bar Y = (0.8-0.2+1.6+0.6+0.7)/5 = 3.5/5 = 0.7$. [1]

Deviations $X-\bar X$: $[0.2, -1.3, 1.2, -0.3, 0.2]$; $Y-\bar Y$: $[0.1, -0.9, 0.9, -0.1, 0.0]$.

• Covariance sum $\sum(X-\bar X)(Y-\bar Y) = 0.02 + 1.17 + 1.08 + 0.03 + 0 = 2.30$. [2]
• $\sum(X-\bar X)^2 = 0.04+1.69+1.44+0.09+0.04 = 3.30$. [1]
• $\sum(Y-\bar Y)^2 = 0.01+0.81+0.81+0.01+0 = 1.64$. [1]
• $\rho = \dfrac{2.30}{\sqrt{3.30\times1.64}} = \dfrac{2.30}{\sqrt{5.412}} = \dfrac{2.30}{2.3264} = 0.9886$. [2]

(b) $\rho\approx 0.99$ ⇒ X and Y move almost identically; treating them as two independent signals doubles exposure to the same risk factor — false diversification. [2]

• Actual portfolio variance uses population $\sigma^2$: $\sigma_X^2 = 3.30/5 = 0.66$, $\sigma_Y^2 = 1.64/5 = 0.328$; $\sigma_X=0.8124,\sigma_Y=0.5727$. [1]
• $\sigma_P^2 = \tfrac14(0.66 + 0.328 + 2\times0.9886\times0.8124\times0.5727) = \tfrac14(0.66+0.328+0.9199)=\tfrac14(1.9079)=0.4770$. [1.5]
• Naive ($\rho=0$): $\tfrac14(0.66+0.328)=\tfrac14(0.988)=0.2470$. [1]
• The real variance (0.477) is nearly double the assumed-diversified (0.247) ⇒ almost no diversification benefit; keep only one on the watchlist. [0.5]

(c)

def relative_strength(stock_prices, index_prices):
    import numpy as np
    s = np.asarray(stock_prices, dtype=float)
    i = np.asarray(index_prices, dtype=float)
    ratio = s / i
    return 100.0 * ratio / ratio[0]   # normalised to 100 at t=0

[5 for correct ratio, normalisation, base-100]

• Sector-leader rule: a stock is a leader if its RS series is rising (RS_t > RS_{t-k}) i.e. it outperforms the sector index — slope of RS > 0, or RS above its own moving average. Leaders are chosen for long trades, laggards avoided/shorted. [3]

Question 3

(a)

• Notional $= 100 \times 6200 = ₹6{,}20{,}000$. [1]
• Margin $= 8\% \times 620000 = ₹49{,}600$. [1]
• Leverage $= 620000/49600 = 12.5\times$. [1]
• +2% price move ⇒ P&L $= 0.02\times620000 = ₹12{,}400$. Return on margin $= 12400/49600 = 25\%$. [2]
• Comment: 2% underlying move → 25% on capital = leverage amplifies both gain and loss ~12.5×; capital-efficient but a 2% adverse move wipes 25% of margin. [1]

(b)

• 1 pip = 0.0025 INR per USD. Rupee value per lot $= 1000 \times 0.0025 = ₹2.50$ per pip. [2]
• 10-pip stop risk per lot $= 10 \times 2.50 = ₹25$. [1]
• Lots for ₹5,000 budget $= 5000 / 25 = 200$ lots. [3]

(c)

def build_watchlist(universe):
    # Filter 1 (cheapest, most eliminative): liquidity
    liquid = [s for s in universe if s.avg_daily_value > 50e7]   # 50 cr
    # Filter 2: relative strength rank top 20%
    liquid.sort(key=lambda s: s.rs_rank, reverse=True)
    cutoff = int(0.20 * len(liquid))
    strong = liquid[:cutoff]
    # Filter 3: correlation de-duplication
    selected = []
    for s in strong:                      # already RS-ordered => best first
        if all(corr(s, k) < 0.7 for k in selected):
            selected.append(s)
    return selected

• Ordering rationale: apply the cheapest, most eliminative filter first (liquidity is a simple threshold that removes most of the universe), then RS ranking (single sort), then the expensive $O(n^2)$ pairwise-correlation check on the small survivor set — minimising total computation. [correct logic 4, ordering explanation 2]

[
  {"claim":"Round-trip spread cost fractions for A and B",
   "code":"cA=2*0.05/500; cB=2*1.20/300; result=(abs(cA-0.0002)<1e-9) and (abs(cB-0.008)<1e-9)"},
  {"claim":"Net edge B is negative, A positive",
   "code":"EnetA=0.009-0.0002-0.15*(2000/4200000); EnetB=0.009-0.008-0.15*(3333.333333/90000); result=(EnetA>0) and (EnetB<0)"},
  {"claim":"Risk-adjusted R: Nifty beats stock A",
   "code":"EnetA=0.873; EnetN=0.8955; RA=EnetA/1.2; RN=EnetN/0.9; result=(RN>RA) and (abs(RA-0.7275)<1e-3) and (abs(RN-0.995)<1e-3)"},
  {"claim":"Pearson correlation of X,Y approx 0.9886",
   "code":"X=[1.0,-0.5,2.0,0.5,1.0]; Y=[0.8,-0.2,1.6,0.6,0.7]; mx=sum(X)/5; my=sum(Y)/5; cov=sum((a-mx)*(b-my) for a,b in zip(X,Y)); sx=sum((a-mx)**2 for a in X); sy=sum((b-my)**2 for b in Y); rho=cov/(sx*sy)**0.5; result=abs(rho-0.9886)<1e-3"},
  {"claim":"Crude leverage 12.5x and 25% return on margin for 2% move",
   "code":"notional=100*6200; margin=0.08*notional; lev=notional/margin; ret=0.02*notional/margin; result=(abs(lev-12.5)<1e-9) and (abs(ret-0.25)<1e-9)"},
  {"claim":"USDINR pip value and lots for 5000 budget",
   "code":"pip=1000*0.0025; stop=10*pip; lots=5000/stop; result=(abs(pip-2.5)<1e-9) and (abs(lots-200)<1e-9)"}
]