Underhedging

A duration-based hedge is computed from today's numbers: today's portfolio value, today's duration, today's futures price. It is exact for a small move and it degrades for a large one.

The reason is that duration is a tangent. It describes the slope of the price-yield curve at one point, and the curve bends away from that line in both directions. Work a large move through a straight line and the true sensitivity has changed underneath you — so the number of lots calculated at the old yield is no longer the number the position needs.

The workbook states the outcome without hedging its words: "the limitation of employing a duration-based hedging strategy has much to do with the fact that duration measures are only accurate for small changes in yield. For large changes in yield, the price/yield relationship is not linear but is actually convex. Thus, using the strategy in the face of large moves in yield will result in underhedging."

Two mechanisms pull in the same direction.

The market value moves. The hedge ratio has the portfolio's market value in its numerator. When yields fall the portfolio is worth more, so more futures are needed — but the position was sized on the old, smaller value.

Duration itself moves. Modified duration is inversely related to yield: as yields fall, duration rises. So the numerator grows on both counts at once, while the hedge stands still.

The result is a hedge that was exactly right at inception and is short by the time the move is over. And because convexity is what caused it, the shortfall grows with the square of the yield move — negligible at 10 basis points, material at 150.

The workbook pairs underhedging with a second, distinct limitation in the same paragraph: "the price of portfolio & futures contract may not move in tandem leads to basis risk." These are different failures. Underhedging is a convexity problem and would occur even if the portfolio and the futures were written on the same bond. Basis risk is a mismatch problem and would occur even if the price-yield relationship were perfectly straight.

The remedy is not a better formula but a practice: rebalance as yields move, and add the convexity correction where the moves are large.

Lots at inception  = (MD_port × Value_port) ÷ (MD_fut × Price_fut ÷ PAR × 2,000)

After a yield move, BOTH inputs to the numerator have changed:

   Value_port  rises when yields fall
   MD_port     rises when yields fall  (duration is inversely related to YTM)

   →  Required lots > lots held  →  UNDERHEDGED

Shortfall grows with (Δy)², because it is a convexity effect.

The workbook's portfolio, taken through a large move. A Rs 26 crore GOI bond portfolio, duration 6.1, hedged with futures of duration 4.7 at Rs 98.50:

Lots at inception = (26,00,00,000 × 6.1) ÷ (98.50 × 4.7 × 2,000) = 1,713 lots, sold

A small move — the hedge works. Yields rise 50 bp:

A large move — the hedge does not. Yields fall 150 bp. Approximately:

Portfolio value → 26 × (1 + 6.1 × 1.5%) ≈ Rs 28.38 crore
Futures price   → 98.50 × (1 + 4.7 × 1.5%) ≈ Rs 105.44

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

Held: 1,713.  Short by 33 lots — 1.9% underhedged.

And that understates it, because the portfolio's duration has risen above 6.1 as its yield fell.

What the gap is worth. From the workbook's convexity table, a 200 bp fall in yields moves the bond +14.0846% in reality against the +13.2209% duration predicted — a shortfall of 0.86 percentage points. On Rs 26 crore:

0.8637% × Rs 26,00,00,000 = Rs 22,45,620

Rs 22 lakh of unhedged gain and loss on a book the desk believed was duration-neutral. In a rising-rate move the same arithmetic runs the other way — the portfolio falls 11.8543% rather than the predicted 13.2209%, so the short futures overshoot.

The discipline that fixes it: recompute the ratio as yields move and top the hedge up. A hedge set once and left alone is a hedge with a shelf life.

Chapter 5, section 5.2.2 (Portfolio Based Hedging), names underhedging directly as the limitation of duration-based hedging in large yield moves, and pairs it with basis risk from portfolio and futures prices moving out of tandem. Chapter 1, section 1.12.8 (Convexity Measure), supplies the mechanism and the error table, and section 1.12.6 states that modified duration follows a linear relationship that works well only for small rate changes.

Questions ask what happens to a duration-based hedge when yields move a long way, and why — the expected answer being underhedging, caused by the convexity of the price-yield relationship.

• Underhedging is a convexity problem, not a basis risk problem. The workbook lists them as two separate limitations in the same paragraph, and they have different causes and different fixes.
• It is caused by the size of the move, not by a mistake in the formula. The ratio was correct when it was computed; the inputs moved.
• Duration rises as yields fall. The numerator of the hedge ratio grows for two reasons at once, which is why the shortfall compounds.
• The shortfall grows with the square of the yield change. Doubling the move roughly quadruples the gap, which is the signature of a second-order effect.
• It is not fixed by selling more futures once. The remedy is periodic rebalancing, because the ratio keeps moving.
• Do not read underhedging as under-confidence. It is a measurable shortfall in contracts held, computable at any moment by re-running the hedge ratio at current levels.