Key Rate Duration: Why a DV01-Neutral Hedge Can Still Lose

A hedge can cancel the portfolio’s total DV01 and still lose six figures when the yield curve changes shape. That is not necessarily a failed calculation. It may be a successful hedge of the wrong risk: a parallel shift, when the market actually delivered a twist.

Key rate duration separates interest-rate exposure by maturity. The sum answers how much risk the portfolio carries; the distribution answers where that risk sits. A usable hedge needs both, measured on the same curve and in the same currency.

Two portfolios, one duration, different weak spots

Consider two hypothetical portfolios, each worth $10 million. A holds only a five-year zero-coupon bond. B puts 62.5% of its initial market value in a two-year zero and 37.5% in a ten-year zero. All three annual-compounded spot rates start at 4%. These are market-value weights, not face-value weights. Payments are fixed, with no embedded options or credit changes.

For a zero maturing in t years, modified duration is t ÷ 1.04. A therefore has duration 4.807692. B has 0.625 × 2 ÷ 1.04 + 0.375 × 10 ÷ 1.04 = 4.807692 as well. Multiply by $10 million and 0.0001: both have total DV01 of $4,807.69 per basis point. We quote a positive DV01 as the approximate loss when the relevant rate rises one basis point.

Yet A’s entire exposure is at five years. B’s node DV01 is $1,201.92 at two years and $3,605.77 at ten years. Matching the totals has not matched the cash-flow locations. An unchanged five-year yield can leave A unmoved while B falls.

How to calculate the node sensitivity

Bump one curve node up and down, keep the agreed treatment of the other nodes unchanged, and reprice. For node i, KRDᵢ ≈ [P(r − hᵢ) − P(r + hᵢ)] ÷ (2P₀h), with h = 0.0001 for one basis point. Dollar node DV01 is P₀ × KRDᵢ × 0.0001. Summing these node sensitivities approximates parallel-shift duration when the bump functions together reproduce the same parallel shift.

The curve definition is part of the number. Spot-rate bumps, par-yield bumps and swap-quote bumps are not interchangeable. Neither are two systems using different nodes or interpolation. A coupon bond can have exposure between several nodes; allocating its entire DV01 to its final maturity misses its earlier payments. Our zeros mature exactly at the three chosen nodes, so those complications are deliberately absent.

Reprice the twist instead of guessing from the total

The following shocks occur instantaneously: time does not pass, and funding, dealing costs, taxes and currency changes are excluded. For each zero, new value equals its original market value multiplied by [1.04 ÷ (1.04 + Δr)]ᵗ. Add the two components of B. The table reports changes relative to each portfolio’s own $10 million starting value, not returns over a year.

The parallel 25-basis-point rise produces nearly equal losses. The small difference is nonlinear price sensitivity, not a contradiction of equal first-order duration. With the short rate down 50 basis points and the long rate up 50, however, A is unchanged and B loses about $115,079. Its ten-year exposure dominates the benefit at two years.

The third row changes the middle of the curve relative to both ends. A loses about $236,956 while B gains about $121,504. A headline such as “yields rose” cannot describe that outcome. The relevant question is which yields moved and where the portfolio was exposed.

The hedge that removes the total but keeps the trade

Now hold B and short $10 million of A, assuming the short can be arranged. Net node DV01 becomes [+$1,201.92, −$4,807.69, +$3,605.77]. The sum is zero. The steepening scenario still loses $115,079 because the five-year hedge does nothing when that rate is unchanged.

The linear estimate is −Σ(DV01ᵢ × shockᵢ in basis points). It predicts a $120,192 loss in that scenario, around $5,113 more than exact repricing. Even a correctly specified risk vector is a local approximation. FastBull’s bond convexity analysis explains the separate problem of nonlinear pricing errors.

The hedged book gains about $427 under the parallel rise and $358,460 under the third shock, before costs. Those figures are dollar profit and loss. Dividing them by zero initial net market value would create a meaningless return. Gross positions, collateral, financing, borrowing availability and liquidity still consume capital.

What would a genuinely better hedge change?

Choose instruments for their node exposures, not simply their labels. With several hedge instruments, arrange each instrument’s dollar sensitivities as a column and solve for positions that reduce the target vector. One five-year instrument generally cannot neutralize independent two- and ten-year exposures. If a perfect match is unavailable, state the residual vector and test the scenarios under which it hurts.

For assets backing liabilities, compare asset and liability sensitivities in money, not just their duration in years. Equal duration on unequal market values leaves dollar risk. Futures bring additional issues: delivery baskets, the cheapest-to-deliver bond, conversion factors and basis can change the effective hedge. A cash-bond example does not settle those risks.

A four-check routine before calling the book neutral

1. Fix the valuation time, currency, curve inputs, compounding and interpolation; calculate the same nodes for the position and hedge.
2. Reconcile the node sum with the independently calculated parallel DV01. Investigate large discrepancies before trading the result.
3. Show the remaining vector, then reprice parallel, steepening, flattening and middle-of-curve shocks. Include the relevant credit, volatility and basis scenarios separately.
4. Apply funding and execution costs, define tolerable losses in currency, and recalculate after market moves or position changes.

Key rate duration does not forecast the next curve move or assign it a probability. It makes a narrower, useful claim: if a particular part of this specified curve moves a little, this is where the book should react. That is the right level of precision for deciding whether a hedge removes risk or merely renames it.