Case 002Fixed income, credit and LDICore

A bond fund holds Rs 800 crore of government bonds at a modified duration of 6.5 and wants 4.0. One bond futures contract has a DV01 of Rs 1,400. How many contracts does it sell?

PIMCOLondon · 2022

The situation

Sindhughosh Dynamic Bond Fund holds Rs 800 crore of government bonds with a modified duration of 6.5. The manager expects yields to rise and wants the fund's duration down to 4.0 for the next quarter, without selling bonds, which would cost dealing spreads and realise gains for investors.

The fund can sell government bond futures. The case sheet gives one contract's DV01 as Rs 1,400: the contract's value moves by Rs 1,400 for a one basis point change in yield, because it tracks a cheapest-to-deliver bond with a duration of about 7.0 on a notional of Rs 20 lakh.

Your task

How many contracts does the fund sell, what happens if yields rise 50 basis points, and what can make the hedge miss?

Quick check

How many contracts should Sindhughosh sell?

Worked solution

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

30-second answerThe answer to give first

Sell about 1,429 contracts. The fund's DV01 is Rs 800 crore times 6.5 times 0.0001, Rs 52 lakh a basis point; at duration 4.0 it would be Rs 32 lakh. The short must remove Rs 20 lakh a basis point, and Rs 20,00,000 divided by Rs 1,400 is 1,428.6. If yields then rise 50 basis points, the fund loses about Rs 16 crore instead of Rs 26 crore.

Step 1Why do you size a duration hedge in DV01 and not in rupees of notional?

Because the thing you are trying to cancel is rupees lost per basis point, and two positions with the same notional can lose very different amounts. DV01The rupee change in a position value for a one basis point, 0.01%, change in yield. Also called PV01 or the dollar value of a basis point. converts every position into that one currency. A hedge is right when the rupees the futures gain per basis point equal the rupees the bonds lose per basis point that you want to remove. It is like balancing a see-saw: what matters is weight times distance from the pivot, not weight alone.

Step 2How do the numbers work?

Start with the fund. Rs 800 crore times a modified duration of 6.5 means a 1% rise in yields costs 6.5%, Rs 52 crore, so one basis point costs one hundredth of that: Rs 52 lakh a basis point today, and Rs 32 lakh at the target duration of 4.0. The short must remove the Rs 20 lakh difference. Each contract removes Rs 1,400, so 1,428.6 contracts, rounded to 1,429.

The relationship

N = \frac{\text{DV01}_{\text{now}} - \text{DV01}_{\text{target}}}{\text{DV01}_{\text{future}}} = \frac{52{,}00{,}000 - 32{,}00{,}000}{1{,}400} \approx 1{,}429

52,00,000	Rs 800 crore x 6.5 x 0.0001, the fund's rupees per basis point today
32,00,000	Rs 800 crore x 4.0 x 0.0001, rupees per basis point at the target
1,400	rupees per basis point on one futures contract

What it says in wordsThe number of contracts is the DV01 you want to remove divided by the DV01 of one contract.

Sindhughosh starts at Rs 52 lakh a basis point; selling 1,429 futures at Rs 1,400 each removes Rs 20.0 lakh and leaves Rs 32.0 lakh, the DV01 of a duration of 4.0.

Step 3What happens if yields rise 50 basis points?

Check the hedge by running it. The bonds lose Rs 52 lakh times 50, Rs 26.0 crore. The futures gain 1,429 times Rs 1,400 times 50, Rs 10.0 crore. The net loss is Rs 16.0 crore, 2.0% of the fund, exactly what a 4.0-duration fund would lose, against 3.25% unhedged. The hedge has not removed the risk; it has cut it to the size the manager chose.

Step 4What do candidates get wrong, and what can make the hedge miss?

The common error is sizing by notional. Cutting duration by 2.5 out of 6.5 sounds like shorting 38% of Rs 800 crore, Rs 308 crore of futures, or 1,538 contracts. That leaves duration at 3.81, not 4.0, because the futures carry a duration of about 7.0, not 6.5. Shorting the full Rs 800 crore is worse still: 4,000 contracts take the fund to -0.50, a fund that now profits when yields rise.

Sizing the short in DV01 gives exactly the 4.0 target; sizing by notional ratio gives 3.81 because the futures are longer than the fund; shorting against the whole notional gives -0.50, turning a bond fund into a bet on rising yields.

Even the right number drifts. Duration changes as yields move, so the short must be re-sized; the contract's DV01 follows its cheapest-to-deliver bond, which can switch; and the fund's bonds sit across many maturities while the future sits near one point on the curve. A DV01 hedge protects against a parallel shift in yields, not against the curve twisting, so a manager would also check the hedge bucket by bucket.

Where candidates lose it

Candidates divide notional by notional. They reason that 2.5 of 6.5 duration is 38% of the fund and short that much, missing that the futures have their own duration, so the answer lands at 1,538 contracts and a duration of 3.81.

The second loss is a units slip: forgetting that one basis point is 0.0001, which turns Rs 20 lakh into Rs 2 lakh and the answer into 143 contracts. Say the units at every step and the slip shows itself.

What the interviewer asks next

Yields fall 60 basis points in a month. Is the fund still at duration 4.0, and what do you do?
Why might a manager hedge with an interest rate swap instead of futures?
The fund's bonds are mostly five-year paper and the future tracks a ten-year bond. What risk remains?
How would you cut duration to 4.0 by selling bonds instead, and what would it cost?

Asked at PIMCO, Sales, London, 2022 (Wall Street Oasis): Typically the product interview was toughest with questions regarding applications of duration

Company names and figures are illustrative.