Duration-based hedge ratio

A hedger holding one bond can hedge it with futures on that bond: divide the face value by Rs 2 lakh and sell that many lots.

Real portfolios are not like that. A fund holds thirty securities of assorted maturities and coupons, and futures exist on only a handful of government bonds. There is no contract for most of what it owns.

The duration-based hedge ratio solves it by hedging the characteristic rather than the security. Two positions with the same modified duration and the same market value have the same interest rate sensitivity, whatever they are made of. So compute the portfolio's sensitivity, compute one futures lot's sensitivity, and sell as many lots as it takes for the second to cancel the first.

What comes out is a position with a duration of zero — a book that, to a first approximation, no longer cares which way rates move.

The formula is a ratio of two rupee sensitivities, with the arithmetic rearranged.

The numerator, portfolio modified duration × market value, is what the portfolio loses for a 100 basis point move. The denominator is the same figure for one lot of futures: futures modified duration × futures price ÷ PAR, where PAR is 100 and the futures price per Rs 100 of face is scaled by the 2,000 units in a lot.

Divide one by the other and you have the number of lots. It is the same calculation as dividing the portfolio's PV01 by one lot's PV01 — the workbook simply expresses it in duration terms rather than basis point terms.

The direction follows the exposure. A portfolio holder is long bonds and short interest rates; he loses when rates rise; so he sells futures. The workbook frames the result as "the maximum extent of short position that may be taken in IRFs to hedge interest rate risk of the portfolio, or part of the portfolio" — a ceiling, and a regulatory one, not merely a suggestion.

The workbook is explicit about the limits. Duration is accurate only for small yield changes; for large changes the price-yield relationship is convex rather than linear, and using the strategy through a big move results in underhedging. And because the portfolio and the futures contract do not move in lockstep, what is left over is basis risk.

                  Portfolio modified duration × Market value of portfolio
Number of lots = ─────────────────────────────────────────────────────────
                  Futures modified duration × (Futures price ÷ PAR) × Lot

   PAR = 100
   Lot = 2,000 units, so the denominator is  MD_fut × Futures price × 2,000 ÷ 100

Equivalently, in basis point terms:

                  PV01 of the portfolio
Number of lots = ───────────────────────
                  PV01 of one futures lot

The workbook's case. An investor holds a Rs 26 crore portfolio of GOI bonds with a portfolio duration of 6.1. The bond futures have a duration of 4.7 and the one-month contract is priced at Rs 98.50.

                26,00,00,000 × 6.1        1,58,60,00,000
Number of lots = ───────────────────── = ─────────────── = 1,713 lots
                98.50 × 4.7 × 2,000          9,25,900

Sell 1,713 lots and the portfolio's duration goes to zero.

Check that it works. Yields rise 50 basis points across the curve:

Portfolio loss  = 6.1 × 0.50% × Rs 26,00,00,000        = Rs 79,30,000
Futures gain    = 4.7 × 0.50% × 98.50 × 2,000 × 1,713  = Rs 79,30,700
Net                                                     = Rs      700

Rs 700 on a Rs 26 crore book — the rounding from 1,712.9 lots to 1,713.

Now make the move large. Yields fall 150 basis points, and the workbook's warning bites. The portfolio is now worth roughly 26 × (1 + 6.1 × 1.5%) = Rs 28.38 crore and the futures have risen to about 98.50 × (1 + 4.7 × 1.5%) = Rs 105.44. Recompute the ratio at the new levels:

                28,38,00,000 × 6.1
Required lots = ────────────────────── = 1,746 lots
                105.44 × 4.7 × 2,000

The hedge is 33 lots short — about 2% underhedged, and that is before accounting for the fact that duration itself rises as yields fall, which widens the gap further. A duration hedge is a snapshot, and it decays.

Chapter 5, section 5.2.2 (Portfolio Based Hedging), poses the problem — an investor holds multiple bonds and futures are not available on all of them — gives the formula, works the Rs 26 crore example to 1,713 lots, and closes with the two limitations: duration is accurate only for small yield changes, so large moves cause underhedging, and portfolio and futures prices may not move in tandem, which is basis risk. Chapter 1, section 1.12.7, supplies the PV01 form of the same ratio.

Questions hand you a portfolio value, a portfolio duration, a futures duration and a futures price and ask for the number of contracts — with the Rs 2 lakh lot and the division by PAR doing the work.

• The lot is 2,000 units, and the price is per Rs 100. Dividing the futures price by PAR and multiplying by 2,000 is one step; omitting it changes the answer by a factor of 100 or 2,000.
• The hedger sells. A bond portfolio loses when rates rise, so the hedge is short futures — the formula gives a magnitude, not a direction.
• The workbook mixes "duration" and "modified duration" in the same example. Section 5.2.2 states the formula in modified durations and then labels its inputs 6.1 and 4.7 as plain "duration". Use whichever figures the question supplies, in the formula as written.
• This hedges parallel shifts only. A steepening or flattening can hurt a duration-neutral book badly, since duration says nothing about curve shape.
• Large moves underhedge. Duration is a straight line through a convex curve, and the workbook names underhedging as the specific failure mode.
• The result is a ceiling. The workbook describes it as the maximum short position that may be taken to hedge the portfolio, not simply the arithmetically convenient number.