Futures

Question 1 (12)

(a) Derivation (5) No-arbitrage: replicate a long future by borrowing to buy the stock today and holding to expiry. (1)

• Buy stock now: cost $S$; finance by borrowing at $r$. (1)
• Holding cost over $T$: borrowed amount grows to $Se^{rT}$. (1)
• Stock pays dividends at yield $q$, reducing carry: net grows to $Se^{(r-q)T}$. (1)
• At expiry the replication must equal the futures price, else arbitrage: $F = S\,e^{(r-q)T}$ (1)

(b) Computation (4) $T = 90/365 = 0.24657$. (1) Exponent $= (0.08 - 0.02)(0.24657) = 0.06 \times 0.24657 = 0.0147945$. (1) $F = 2000\,e^{0.0147945} = 2000 \times 1.014904 = 2029.81$. (1) Basis $= F - S = 29.81$ points. (1)

(c) Arbitrage (3) Market $F=2045 >$ fair $2029.81$ and $F>S$ → contango. (1) Futures overpriced → cash-and-carry: short the future, borrow to buy spot, carry to expiry. (1) At expiry deliver stock against short future, capturing $\approx 2045-2029.81 = 15.2$ points risk-free (minus costs). (1)

Question 2 (12)

(a) SPAN vs exposure (3)

• SPAN = portfolio-risk-based margin computed by scanning worst-case price/volatility scenarios; covers one day's potential loss. (1.5)
• Exposure (additional) margin = flat buffer over SPAN against extreme gaps/tail moves. (1)
• Both upfront so clearing corp is covered before MTM collection. (0.5)

(b) Contract value & margin (4) Contract value per lot $= 50 \times 22000 = 1{,}100{,}000$. (1) Two lots $= 2 \times 1{,}100{,}000 = 2{,}200{,}000$. (1) Margin $= 12\% \times 2{,}200{,}000 = 264{,}000$. (2)

(c) MTM table (5) Position = long 100 units (2 lots × 50). P&L per day $= 100 \times (\text{today} - \text{prev})$.

(Table correctly built: 4; correct signs & cumulative: 1) Net P&L at Day 3 close $= +₹5{,}000$. Also equals $100\times(22050-22000)=5000$. ✓

Question 3 (10)

(a) Rollover meaning (3) Rollover = closing the expiring near-month position and simultaneously opening the same position in the next-month contract. (1.5) Done to maintain market exposure beyond expiry without taking physical/cash settlement or letting the position lapse. (1.5)

(b) Rollover cost (4) Rollover cost $= \text{far} - \text{near} = 2035 - 2010 = 25$ points. (2) As % of near $= 25/2010 = 1.244\%$. (1) Positive cost (far > near) = carry cost to roll → typical contango; not inherently bearish, but a rising rollover cost reflects higher cost of holding forward exposure (bullish carry / positive cost of carry). (1)

(c) Function (3)

def rollover_cost(near, far):
    points = far - near
    pct = (points / near) * 100
    return points, pct
# rollover_cost(2010, 2035) -> (25, 1.2437...)

(logic 2, correct pct formula 1)

Question 4 (10)

(a) Hedge ratio (5) Number of contracts $N = \dfrac{\beta \times \text{Portfolio Value}}{\text{Futures price} \times \text{lot size}}$. (2) $N = \frac{1.2 \times 5{,}000{,}000}{22{,}000 \times 50} = \frac{6{,}000{,}000}{1{,}100{,}000} = 5.45$ (2) Round to 5 contracts, SHORT (sell) to hedge a long portfolio. (1)

(b) Explanation (5) If NIFTY falls 5%: (1)

• Portfolio (beta 1.2) falls $\approx 1.2 \times 5\% = 6\%$ → loss $= 0.06 \times 5{,}000{,}000 = ₹300{,}000$. (2)
• Short futures gain: index drop of 5% on hedged notional $\beta\times PV = 6{,}000{,}000$ → gain $\approx 0.05 \times 6{,}000{,}000 = ₹300{,}000$. (2) Net ≈ 0: the short futures offset the portfolio loss (basis/rounding ignored).

Question 5 (8)

(a) Margin & leverage (3) Contract value $= 250 \times 800 = 200{,}000$. (1) Margin $= 15\% \times 200{,}000 = 30{,}000$. (1) Leverage $= 200{,}000 / 30{,}000 = 6.67\times$. (1)

(b) Return (5) Profit $= 250 \times (840 - 800) = 250 \times 40 = ₹10{,}000$. (1.5) Return on margin $= 10{,}000 / 30{,}000 = 33.33\%$. (1.5) Return on underlying $= 40/800 = 5\%$. (1) Leverage magnifies: a 5% move in price → 33.33% move on margin (≈ 6.67×). Same multiplier applies to losses — a 5% adverse move wipes ~33% of margin. (1)

Question 6 (8)

(a) Three differences (6, 2 each)

1. Settlement: Index futures are cash-settled (no delivery); stock futures may be physically settled (delivery of shares) on expiry in many markets.
2. Corporate actions/dividends: Single-stock futures are adjusted for dividends, splits, bonuses; index futures reflect the whole basket and dividends are diffused into the index level.
3. Diversification/risk: Index futures track a diversified basket → lower idiosyncratic risk and volatility; single-stock futures carry company-specific (idiosyncratic) risk and higher volatility.

(b) Margin reason (2) Single stocks have higher individual volatility and event/gap risk (results, news) with no diversification, so worst-case SPAN scenarios are larger → higher margin.

[
  {"claim":"Q1b fair futures ~2029.81 and basis ~29.81","code":"S=2000; r=0.08; q=0.02; T=90/365; F=S*exp((r-q)*T); result = abs(F-2029.81)<0.5 and abs((F-S)-29.81)<0.5"},
  {"claim":"Q2b margin blocked = 264000","code":"cv=2*50*22000; margin=0.12*cv; result = margin==264000"},
  {"claim":"Q2c net MTM P&L at day3 = 5000","code":"prices=[22000,21850,22100,22050]; units=100; mtm=[units*(prices[i]-prices[i-1]) for i in range(1,len(prices))]; result = sum(mtm)==5000"},
  {"claim":"Q3b rollover cost 25 pts and pct ~1.2437","code":"points=2035-2010; pct=points/2010*100; result = points==25 and abs(pct-1.2437)<0.01"},
  {"claim":"Q4a hedge contracts ~5.45 rounds to 5","code":"N=(1.2*5000000)/(22000*50); result = abs(N-5.4545)<0.01 and round(N)==5"},
  {"claim":"Q5 leverage 6.667x and return on margin 33.33%","code":"cv=250*800; margin=0.15*cv; lev=cv/margin; profit=250*(840-800); rom=profit/margin*100; result = abs(lev-6.6667)<0.01 and abs(rom-33.333)<0.01"}
]