Price Value of a Basis Point

Modified duration tells you a bond will move about 5% for a 1% change in yield. That is a percentage, and a percentage cannot be hedged. A desk needs a number of rupees.

PV01 is that number. It is the money a position gains or loses when yields move by one basis point — a hundredth of one per cent. The workbook's intuition is exactly this: if a price change of 0.05 (from 100 to 99.95) corresponds to a 1 bp change in yield, the PV01 is Rs 0.05, or five paise, for that bond.

Everything a risk manager does with interest rates is expressed in this unit. A book is "long Rs 25,000 of PV01". A hedge is sized so the PV01 of the futures offsets the PV01 of the bonds.

PV01 is modified duration converted from a percentage into money, then scaled from 100 basis points down to one:

• modified duration gives the percentage price change for 100 bp
• dividing by 100 gives the change for 1 bp
• multiplying by the price turns the percentage into rupees

That is the 10,000 in the denominator — 100 for the percent, 100 more for the basis point.

The price used is the dirty price, clean price plus accrued interest, because that is the full amount the investor has at stake and what actually moves with yield.

Two properties follow. PV01 falls as yields rise, because both the price and the modified duration fall. And PV01 is linear — it assumes the price–yield line is straight, which it is not, so the estimate degrades as the yield move grows. That error is convexity.

PV01 = 0.01 × (Modified duration ÷ 100) × Bond price

     = (Dirty price × Modified duration) ÷ 10,000

And the hedge that PV01 exists to size:

Lots of IRF = Position PV01 ÷ PV01 of one futures lot

which is the same as the duration-based hedge ratio:

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

A fund holds Rs 5 crore face value of a GOI security. Its Macaulay duration is 5.34 and its yield 4.5%, so its modified duration is 5.34 ÷ 1.045 = 5.11. The dirty price is Rs 100.

PV01 per Rs 100 of face = (100 × 5.11) ÷ 10,000 = Rs 0.0511

Five paise per hundred rupees of face value — the workbook's five paise, arrived at properly. Scale it:

Cross-check against modified duration the long way: 5.11% of Rs 5 crore is Rs 25.55 lakh for 100 bp, so 25 bp is Rs 6,38,750. The two routes agree, as they must.

Sizing the hedge. Bond futures with a modified duration of 4.7 are trading at Rs 98.50.

PV01 per 100 of futures = (98.50 × 4.7) ÷ 10,000 = Rs 0.046295
PV01 per lot (2,000 units)                        = Rs 92.59

Lots required = 25,550 ÷ 92.59 = 276 lots

The workbook's duration-based hedge ratio, applied to the same numbers, gives (5,00,00,000 × 5.11) ÷ (98.50 × 4.7 × 2,000) = 276 lots. Same answer — because the hedge ratio is a PV01 ratio with the arithmetic rearranged.

Different contracts, different bp values. One basis point on a 91-day T-bill futures lot is 2,000 × 0.25 × 0.01 = Rs 5. One basis point on an overnight MIBOR futures contract is Rs 5 crore × 0.01% × 30/365 = Rs 411.

Chapter 1, section 1.12.7 (Price Value of Basis Point) gives the formula and the five-paise intuition. It is then used twice more: in Chapter 5 for the duration-based hedge ratio that sizes a portfolio hedge, and in Chapter 7 where the margining framework values risk in basis points.

Questions ask you to compute PV01 from a dirty price and a modified duration, to turn PV01 into a rupee profit or loss for a stated yield move, or to state the value of one basis point on a given contract — the Rs 411 on MIBOR futures is spelled out in the contract specification and is asked directly.

• Divide by 10,000, not by 100. The extra hundred is what turns a percentage point into a basis point. Halving that denominator inflates every answer a hundredfold.
• Use the dirty price. Modified duration measures the sensitivity of the total amount paid, which includes accrued interest — the workbook is explicit about this.
• PV01 is per unit of face value. Rs 0.0511 per Rs 100 becomes Rs 25,550 only after multiplying by the 5,00,000 units held.
• PV01 shrinks as yields rise. Both price and modified duration fall, so the same basis point is worth less money at a higher yield — which is why a PV01 hedge has to be rebalanced.
• PV01 is a straight-line estimate. It understates the gain when yields fall and overstates the loss when they rise; the gap is convexity, and it grows with the square of the move.
• Do not carry one contract's bp value to another. Rs 5 for a T-bill futures lot and Rs 411 for a MIBOR futures contract come from entirely different contract-value formulas.