Level 4 — ApplicationFutures

Futures

⏱ 60 minutes50 marksprintable — key stays hidden on paper

Chapter: Futures (Stock-Market) Level: 4 — Application (novel problems, no hints) Time Limit: 60 minutes Total Marks: 50

Instructions: Attempt all questions. Show all working. Use $...$ notation where relevant. Round monetary answers to 2 decimal places unless stated otherwise.

Question 1 — Cost of Carry, Basis & Contango [10 marks]

Stock XYZ trades at a spot price of $₹2{,}400$ . The risk-free rate is $8\%$ per annum (continuous compounding is NOT used — use simple annualised carry). The stock pays no dividend. A futures contract expires in 73 days.

(a) Compute the fair (theoretical) futures price using the cost-of-carry model. [3]

(b) The futures actually trades at $₹2{,}450$ . Compute the basis and state whether the market is in contango or backwardation. [3]

(c) Explain, with a numeric arbitrage sketch, how a trader could profit if the market futures price ( $₹2{,}450$ ) deviates from your fair value from part (a). State the theoretical arbitrage profit per share. [4]

Question 2 — Margin & Mark-to-Market [12 marks]

A trader BUYS 2 lots of NIFTY futures. Contract details: lot size = $50$ , entry futures price = $22{,}000$ . SPAN margin = $9\%$ and exposure margin = $3\%$ of contract value.

(a) Compute the total contract value and total initial margin blocked. [4]

(b) Over the next 3 days the futures settlement prices are: Day 1 = $22{,}150$ , Day 2 = $21{,}900$ , Day 3 = $22{,}050$ . Prepare a day-wise MTM table showing daily profit/loss and cumulative P/L. [6]

(c) On which day (if any) would the trader receive an MTM credit and on which day face an MTM debit? State the net P/L over the 3 days. [2]

Question 3 — Hedging with Futures [10 marks]

A fund manager holds a portfolio worth $₹1{,}20{,}00{,}000$ with a beta of $1.4$ relative to NIFTY. NIFTY futures trade at $20{,}000$ with a lot size of $50$ .

(a) Compute the number of futures contracts required for a complete hedge, and state whether the manager should go long or short. [4]

(b) The market falls and NIFTY drops to $18{,}500$ . Assuming the portfolio moves exactly per its beta, compute the loss on the portfolio and the gain on the futures hedge. Comment on the net position. [4]

Question 4 — Rollover & Rollover Cost [8 marks]

A trader is long 1 lot (lot size $= 100$ ) of a stock future. Near expiry:

Near-month future = $₹1{,}280$
Next-month future = $₹1{,}298$

(a) Compute the rollover cost per share and the total rollover cost for the position. [3]

(b) Express the rollover cost as a percentage of the near-month price. [2]

(c) The trader expects the stock to rise to $₹1{,}350$ by next-month expiry. After paying the rollover cost, what is the net profit per share on the rolled position (measured from the next-month entry)? [3]

Question 5 — Speculation: Index vs Stock Futures [10 marks]

A speculator has $₹1{,}00{,}000$ of risk capital.

Option A: Stock future — price $₹500$ , lot size $1{,}000$ , total margin $20\%$ .
Option B: Index future — price $16{,}000$ , lot size $25$ , total margin $12\%$ .

(a) For each option, compute the margin required for ONE lot and how many lots the speculator can take with the capital (whole lots only). [4]

(b) If both underlyings rise by exactly $4\%$ , compute the total rupee profit under each option using the maximum whole lots from part (a). [4]

(c) Which option gives higher return on capital, and give one risk-based reason why a speculator might still prefer the index future. [2]

Answer keyMark scheme & solutions

Question 1 [10 marks]

(a) Cost of carry (simple): $F = S(1 + r \cdot t)$ where $t = 73/365 = 0.2$ yr. $F = 2400(1 + 0.08 \times 0.2) = 2400 \times 1.016 = ₹2{,}438.40$

Correct $t = 0.2$ [1]
Correct formula [1]
Answer $₹2{,}438.40$ [1]

(b) Basis is usually defined Spot − Futures: $\text{Basis} = S - F_{mkt} = 2400 - 2450 = -₹50$ Futures > Spot ⇒ Contango.

Basis value (−50, sign either convention accepted if labelled) [2]
Contango identification [1]

(c) Market future $₹2{,}450$ > fair $₹2{,}438.40$ ⇒ future is overpriced. Arbitrage (cash-and-carry): Sell futures at $2450$ , Buy stock at $2400$ borrowing at 8%.

At expiry deliver stock, receive $2450$ .
Carrying cost of stock = $2400 \times 0.016 = ₹38.40$ .
Total cost = $2400 + 38.40 = ₹2438.40$ .
Profit per share $= 2450 - 2438.40 = ₹11.60$ .
Correct strategy (sell fut / buy spot) [2]
Carry cost computation [1]
Profit $₹11.60$ [1]

Question 2 [12 marks]

(a) Contract value $= 2 \times 50 \times 22000 = ₹22{,}00{,}000$ . Total margin $= (9\% + 3\%) = 12\%$ of value $= 0.12 \times 2200000 = ₹2{,}64{,}000$ .

Contract value $₹22,00,000$ [2]
Margin $₹2,64,000$ [2]

(b) Position size $= 2 \times 50 = 100$ units. Long, so P/L $= 100 \times (\text{today} - \text{previous})$ .

Day	Settle	Δ vs prev	Daily P/L (₹)	Cumulative (₹)
Entry	22000	—	—	0
1	22150	+150	+15,000	+15,000
2	21900	−250	−25,000	−10,000
3	22050	+150	+15,000	+5,000

Day 1 = +15,000 [2]
Day 2 = −25,000 [2]
Day 3 = +15,000 & cumulative correct [2]

(c) MTM credit on Day 1 and Day 3 (price rose); MTM debit on Day 2 (price fell). Net P/L over 3 days $= ₹+5{,}000$ .

Correct credit/debit days [1]
Net P/L $₹5,000$ [1]

Question 3 [10 marks]

(a) Hedge contracts: $N = \frac{\text{Portfolio} \times \beta}{\text{Futures price} \times \text{lot}} = \frac{12000000 \times 1.4}{20000 \times 50} = \frac{16800000}{1000000} = 16.8 \approx 17$ Manager holds stock (long) ⇒ SHORT futures to hedge.

Formula with beta [1]
Value $16.8$ [1]
Round to 17 (accept 16.8) [1]
Short [1]

(b) NIFTY falls $20000 \to 18500$ , a drop of $7.5\%$ . Portfolio loss $= 12000000 \times 1.4 \times 0.075 = ₹12{,}60{,}000$ . Futures gain (short, using 17 lots) $= 17 \times 50 \times (20000-18500) = 850 \times 1500 = ₹12{,}75{,}000$ . Net $= +12,75,000 - 12,60,000 = ₹+15{,}000$ (slight over-hedge due to rounding up to 17).

Portfolio loss $₹12,60,000$ [2]
Futures gain $₹12,75,000$ (or $₹12,60,000$ if 16.8 used) [1]
Comment: nearly offset / small residual [1]

(c) Any one: beta is estimated/unstable; rounding to whole lots leaves residual exposure; basis risk between futures and spot; portfolio composition differs from index. [2]

Question 4 [8 marks]

(a) Rollover cost per share $= 1298 - 1280 = ₹18$ . Total $= 18 \times 100 = ₹1{,}800$ .

Per share $₹18$ [1]
Total $₹1,800$ [2]

(b) $\dfrac{18}{1280} \times 100 = 1.40625\% \approx 1.41\%$ .

Correct $\approx 1.41\%$ [2]

(c) Net entry (next-month) $= 1298$ . Target $= 1350$ . Net profit per share $= 1350 - 1298 = ₹52$ .

Recognising next-month entry 1298 [1]
Profit $₹52$ [2]

Question 5 [10 marks]

(a) Option A margin/lot $= 500 \times 1000 \times 0.20 = ₹1{,}00{,}000$ ⇒ 1 lot. Option B margin/lot $= 16000 \times 25 \times 0.12 = ₹48{,}000$ ⇒ $100000/48000 = 2.08$ ⇒ 2 lots.

A margin & lots [2]
B margin & lots [2]

(b) 4% rise: Option A: 1 lot, notional $= 500 \times 1000 = 500000$ ; profit $= 500000 \times 0.04 = ₹20{,}000$ . Option B: 2 lots, notional $= 16000 \times 25 \times 2 = 800000$ ; profit $= 800000 \times 0.04 = ₹32{,}000$ .

A profit $₹20,000$ [2]
B profit $₹32,000$ [2]

(c) Return on capital: A $= 20000/100000 = 20\%$ ; B $= 32000/96000 \approx 33.3\%$ (or on full ₹1,00,000 = 32%). Option B higher. Reason for preferring index: diversification ⇒ lower single-stock/idiosyncratic risk and lower volatility. [2]

[
  {"claim":"Q1a fair futures price = 2438.40","code":"S=2400; r=Rational(8,100); t=Rational(73,365); F=S*(1+r*t); result=(F==Rational(24384,10))"},
  {"claim":"Q1c arbitrage profit per share = 11.60","code":"S=2400; r=Rational(8,100); t=Rational(73,365); F=S*(1+r*t); profit=2450-F; result=(profit==Rational(116,10))"},
  {"claim":"Q2a total margin = 264000","code":"cv=2*50*22000; margin=cv*Rational(12,100); result=(margin==264000)"},
  {"claim":"Q2 net MTM P/L over 3 days = 5000","code":"units=100; net=units*((22150-22000)+(21900-22150)+(22050-21900)); result=(net==5000)"},
  {"claim":"Q3a hedge contracts = 16.8","code":"N=Rational(12000000*14,10)/(20000*50); result=(N==Rational(84,5))"},
  {"claim":"Q3b futures gain with 17 lots = 1275000","code":"g=17*50*(20000-18500); result=(g==1275000)"},
  {"claim":"Q4 rollover total = 1800 and net profit 52","code":"roll=(1298-1280)*100; netprofit=1350-1298; result=(roll==1800 and netprofit==52)"},
  {"claim":"Q5b Option B profit = 32000","code":"lots=2; notional=16000*25*lots; profit=notional*Rational(4,100); result=(profit==32000)"}
]