Reducing portfolio beta with index futures
Series XXI-B case study · set 1, case 5 of 6 · 5 questions · 2 marks each
Question 1 of 5Case practice
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A manager holds a ₹5 crore equity portfolio with estimated beta 1.20 to a broad index. For the next month the investment committee wants portfolio beta reduced to 0.40 without selling shares. A hypothetical index-futures contract has notional value ₹10 lakh and beta 1 to the index. Use the simple hedge rule: contracts = (current beta − target beta) × portfolio value / contract notional. Assume no rounding is needed, the portfolio follows its beta exactly, and futures and the index move together. For a stress illustration, the index falls 5% and the portfolio value used to size the hedge remains the opening value. Ignore costs, basis changes and margin interest.
All amounts and contract terms in this case are hypothetical.
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The case and its answers
Every question of this case, with its answer and why each other option is wrong.
A manager holds a ₹5 crore equity portfolio with estimated beta 1.20 to a broad index. For the next month the investment committee wants portfolio beta reduced to 0.40 without selling shares. A hypothetical index-futures contract has notional value ₹10 lakh and beta 1 to the index. Use the simple hedge rule: contracts = (current beta − target beta) × portfolio value / contract notional. Assume no rounding is needed, the portfolio follows its beta exactly, and futures and the index move together. For a stress illustration, the index falls 5% and the portfolio value used to size the hedge remains the opening value. Ignore costs, basis changes and margin interest.
All amounts and contract terms in this case are hypothetical.
Q1Which futures direction reduces this long equity portfolio’s market exposure?
Option A. Buy index futures
Why it is wrong: A long futures position increases exposure under these assumptions.
Option B. Buy more of the same shares
Why it is wrong: That does not reduce beta exposure.
Option C. Sell index futures — Correct answer
Option D. Hold only additional cash borrowed against the shares
Why it is wrong: That is not the stated futures hedge and introduces different financing effects.
Answer: C. A short index-futures position offsets part of the positive beta exposure of the shares. Remember: Match the direction of the hedge to the underlying exposure.
Hint, shown before you answer in practice mode: The hedge must gain when the index falls.
XXIB-S1-C05-Q1easyapplication
Q2How many futures contracts are required?
Option A. 40 contracts — Correct answer
Option B. 4 contracts
Why it is wrong: This mis-converts crore and lakh.
Option C. 20 contracts
Why it is wrong: This corresponds to only 0.40 beta units of reduction.
Option D. 60 contracts
Why it is wrong: This removes all beta rather than leaving target beta 0.40.
Answer: A. (1.20 − 0.40) × ₹500 lakh / ₹10 lakh = 40 contracts to sell. Remember: Calculate the change in exposure, not just the current beta.
Hint, shown before you answer in practice mode: Use ₹500 lakh as the portfolio value.
XXIB-S1-C05-Q2mediumnumerical
Q3What is the modelled loss on the shares alone when the index falls 5%?
Option A. ₹10 lakh
Why it is wrong: This is the intended residual hedged loss.
Option B. ₹25 lakh
Why it is wrong: This assumes beta one.
Option C. ₹30 lakh — Correct answer
Option D. ₹20 lakh
Why it is wrong: This is the futures hedge gain.
Answer: C. Share loss = ₹500 lakh × 1.20 × 5% = ₹30 lakh. Remember: Use beta to estimate market-linked portfolio sensitivity.
Hint, shown before you answer in practice mode: Portfolio percentage movement equals beta times the index movement in the model.
XXIB-S1-C05-Q3mediumnumerical
Q4What is the modelled gain on the short futures hedge?
Option A. ₹10 lakh
Why it is wrong: This is the remaining portfolio loss.
Option B. ₹2 lakh
Why it is wrong: This understates the contract count or notional by ten times.
Option C. ₹20 lakh — Correct answer
Option D. ₹30 lakh
Why it is wrong: That would fully offset the estimated share loss, not the partial hedge.
Answer: C. 40 × ₹10 lakh × 5% = ₹20 lakh of gain. Remember: Translate contract count into notional exposure before calculating profit or loss.
Hint, shown before you answer in practice mode: Apply the 5% fall to the total short futures notional.
XXIB-S1-C05-Q4mediumnumerical
Q5What is the combined portfolio change in this model?
Option A. A ₹10 lakh loss, or 2% of opening portfolio value — Correct answer
Option B. A ₹50 lakh loss
Why it is wrong: This adds the hedge gain as if it were a loss.
Option C. No loss because futures guarantee protection
Why it is wrong: This is a partial hedge, and real hedges also have risks.
Option D. A ₹10 lakh gain
Why it is wrong: This reverses the residual sign.
Answer: A. ₹30 lakh loss − ₹20 lakh gain = ₹10 lakh loss. Relative to ₹500 lakh, that is 2%, matching target beta 0.40 × the 5% index fall. Remember: A partial beta hedge deliberately leaves some market exposure.
Hint, shown before you answer in practice mode: Offset the shares’ loss with the futures gain.
XXIB-S1-C05-Q5hardnumerical
Allocating a partially filled order fairly
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