How do I compute PVBP for a specific bond and use it in a hedge?

Question

AcadiFi · Accepted Answer

PVBP (Price Value of a Basis Point) = change in full bond price for a 1bp yield change. You can compute it either analytically (using modified duration) or numerically (reprice at y + 1bp and y − 1bp, take the difference / 2).

**Numerical method** (most robust):
1. Compute current price at current yield (YTM implied by 98.50)
2. Reprice at YTM + 0.0001
3. Reprice at YTM − 0.0001
4. PVBP per $100 face = [P(y−) − P(y+)] / 2
5. Scale to full position: PVBP_total = PVBP_per_$100 × (Face / 100)

**Analytical shortcut**:
PVBP = MD × Market Price × 0.0001

For your bond: Face = $50M, Price = 98.50 per 100, so Market Value = $50M × 0.985 = $49.25M.

Estimate MD for 12yr 4.5% at reasonable YTM (~4.6% given sub-par price): roughly 9.0 (back-of-envelope Macaulay × 1/(1+y)).

PVBP_total ≈ 9.0 × $49.25M × 0.0001 = $44,325

So a 1bp yield rise costs ~$44,325; a 10bp rise costs ~$443K.

**Hedge sizing**:
Number of short futures = PVBP_bond / PVBP_futures = $44,325 / $95 = 467 contracts

Selling 467 10-year Treasury futures creates a hedge where the futures' $95 × 467 = $44,365 BPV approximately offsets the cash bond's $44,325 BPV.

**Refinements**:

1. **Yield beta**: 10-year futures yields move 1:1 with long-bond yields in parallel shifts but may differ under steepening. Estimate the yield beta from regression of 10yr vs 12yr moves — typically 0.95-1.05 — and multiply hedge ratio accordingly.

2. **Cheapest-to-deliver (CTD)**: Futures PVBP depends on the CTD bond. If CTD changes (as yields move across 6% threshold), futures PVBP jumps — rehedge.

3. **Convexity mismatch**: A linear hedge is exact only for tiny moves. For large moves, convexity of cash bond (positive) vs futures (approximately zero for futures outright) leaves residual P&L — negligible for 10-50bp moves, material for 200+ bp shifts.

4. **Time decay**: Futures' BPV shifts as delivery approaches due to changes in CTD and carry. Rehedge periodically — weekly at minimum.

5. **Cash-and-carry basis risk**: The hedge is imperfect because futures price reflects financing and repo — basis moves will not track yield 1:1.

**Practical rule**: PVBP hedging is the workhorse of Treasury desks. For more precise risk management, decompose into key rate PVBPs and hedge curve-wise with different-maturity futures — covered next in curve risk management.

How do I compute PVBP for a specific bond and use it in a hedge?

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