Case 084Hedging with futuresCore

A pension trust holds Rs 1,000 crore of government bonds with modified duration 7.2 and wants duration 4.0 for three months. A bond future has a DV01 of Rs 1,850. How many contracts, which side, and what risk remains?

The situation

Savitri Pension Trust holds Rs 1,000 crore of government bonds spread across the 5 to 15 year part of the curve, with a portfolio modified duration of 7.2. Its investment committee expects a rate rise over the next quarter and wants the portfolio's duration brought down to 4.0 for three months without selling bonds, which would crystallise gains and take weeks to rebuild.

The exchange-traded government bond future, which settles on a ten-year benchmark, has a DV01 of Rs 1,850 per contract: each contract gains or loses Rs 1,850 for a one basis point move in its yield.

Your task

How many futures contracts does the trust trade, on which side, and what risks does the hedge leave open?

Quick check

Before computing: to cut duration, does the trust buy or sell futures?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

Sell about 1,730 contracts. The portfolio's DV01 is Rs 1,000 crore times 7.2 over 10,000, Rs 72 lakh a basis point; at duration 4.0 it would be Rs 40 lakh, so Rs 32 lakh must go. Rs 3,200,000 over Rs 1,850 per contract is 1,729.7, rounded to 1,730 contracts sold. What remains is curve risk, because the future tracks one ten-year point while the bonds sit across 5 to 15 years, plus basis risk, roll risk at expiry and daily margin calls.

Step 1Why work in DV01 rather than in duration?

A hedge is matched in rupees of loss, not in a ratio. Two people can both have a duration of 7, and the one with ten times the money loses ten times as much. DV01Rupee change in value for a one basis point move in yield. Price times modified duration divided by 10,000. converts duration into the rupees at risk per basis point, which is the unit the future is quoted in. For the trust, Rs 1,000 crore times 7.2 over 10,000 is Rs 72 lakh a basis point. The target of duration 4.0 is Rs 40 lakh. The hedge has to carry the Rs 32 lakh in between.

The relationship

N = \frac{(D_{\text{now}} - D_{\text{target}}) \times V / 10{,}000}{\text{DV01}_{\text{future}}} = \frac{(7.2 - 4.0) \times 1{,}000\text{ crore} / 10{,}000}{1{,}850} = \frac{32{,}00{,}000}{1{,}850} \approx 1,730

D now, D target	modified duration today and the duration wanted
V	market value of the portfolio, Rs 1,000 crore
DV01 future	rupees per basis point per contract, Rs 1,850

What it says in wordsThe number of contracts is the DV01 you want to remove divided by the DV01 of one contract, and the sign is short because the loss to be offset comes from rising yields.

Selling 1,730 futures at Rs 1,850 a basis point each removes Rs 32.0 lakh of the portfolio's Rs 72 lakh DV01, leaving Rs 40.0 lakh, which is a duration of 4.0, so a 50 basis point rise costs Rs 20 crore instead of Rs 36 crore.

Step 2What does the hedge do when yields move, and what does it miss?

Test it with a parallel shock. If every yield rises 50 basis points, the bonds lose about Rs 36 crore and the short futures gain about Rs 16 crore, so the trust loses Rs 20 crore, as a duration-4 portfolio should. Now break the parallel assumption: the future moves with the ten-year yield only, while the trust's bonds run from 5 to 15 years, so a steepening where long yields rise more than the ten-year leaves the long bonds under-hedged and a flattening over-hedges them. This is curve risk, and it is the main thing the hedge leaves behind. Second is basis risk: the future tracks its cheapest-to-deliver bond, whose yield can drift from the benchmark. Third is roll risk, because the contract expires inside the three-month window and the new contract may be priced differently. Fourth is cash: daily variation margin on 1,730 contracts means the trust must hold liquid cash for calls it will receive if yields fall.

Scenario, yields over the quarter	Bonds, Rs crore	Short futures, Rs crore	Net, Rs crore
All yields up 50 bp	-36	+16	-20
All yields down 50 bp	+36	-16	+20
Ten-year up 50 bp, long end up 80 bp	about -50	+16	about -18
Ten-year up 50 bp, short end flat	about -26	+16	about +6

Under a parallel move the hedge behaves like a duration-4 portfolio, but when the curve steepens or flattens the ten-year future and the 5 to 15 year bonds move by different amounts and the net result drifts either side of the target.

Step 3What would you tell the committee before dealing?

Three things, each with a number. The trade is 1,730 contracts sold, about Rs 35 crore of notional if each contract covers Rs 2 lakh of face, which is large for the Indian bond futures market and may need to be built over several days. The residual curve risk can be cut by splitting the hedge across contracts of different maturities if they exist and trade. And the hedge is a three-month rental of lower duration, not a sale: if yields fall instead, the trust gives up the gain it would have had, which is the price of the insurance and should be said before it happens, not after.

Where candidates lose it

The common loss is hedging in duration units, dividing 7.2 by something, instead of converting to rupees per basis point. The future's DV01 is in rupees; the portfolio's must be too, and the ratio of the two is the contract count.

The second is calling the hedged portfolio safe at duration 4. The future removes parallel risk only; the curve, the cheapest-to-deliver basis and the roll are all still open.

What the interviewer asks next

Yields fall 40 basis points in the first month. What has the hedge cost, and should the trust unwind it?
The ten-year future becomes illiquid. How else could the trust cut duration quickly?
How does the answer change if the trust uses an interest rate swap instead of futures?
Why might the trust prefer to over-hedge the long bonds and under-hedge the short ones?

Company names and figures are illustrative.