Convexity

Plot a bond's price against its yield and you get a curve that bends upward. Modified duration is the tangent to that curve at today's yield — a straight line that touches it at one point and drifts away from it everywhere else.

For a small move the line and the curve are close enough. For a large move they are not, and the gap is always in the same direction: the real curve lies above the straight line on both sides.

Convexity measures how much the curve bends. Mathematically it is the second derivative of the bond pricing equation with respect to yield, where duration is the first.

The consequence of that upward bend is worth stating precisely, because the exam asks it in exactly this form:

• when yields rise, duration overestimates the loss — the bond falls by less than the straight line says
• when yields fall, duration underestimates the gain — the bond rises by more

So for a plain coupon bond the duration error always favours the holder. The workbook puts it as: "duration underestimates the price change in case of interest rate fall and over estimates the price change in case of an increase in interest rate."

And it compounds for the hedger. A portfolio hedged on a duration basis will underhedge when yields move a long way, because the position's true sensitivity has grown beyond what the straight line predicted. The workbook flags this as the central limitation of duration-based hedging.

The exact formula uses the bond price, yield, time to maturity and every discounted cash flow — tedious for a long bond. The approximation below needs only three prices.

                 P(yield falls) + P(yield rises) − 2 × P0
Convexity  ≈   ────────────────────────────────────────────
                          2 × P0 × (Δy)²

  P0 = current bond price
  Δy = the yield shift used, in decimal form

The exact measure is the second partial derivative of the pricing equation with respect to yield, summing CFt × t × (t+1) discounted over (1+y).

A semi-annual bond's convexity is divided by 4 (= 2²) to make it comparable with an annual bond's. The workbook's worked case — a 3-year bond, 10% coupon, semi-annual compounding, required yield 9%, price 102.58 — comes to 8.2135.

The workbook shocks one bond's yield and tabulates what modified duration predicted against what actually happened. Every figure is a percentage price change.

At 10 basis points the error is two-hundredths of a percent — noise. At 200 basis points it is 1.37 percentage points, roughly 70 times larger for a 20-fold bigger shock. The error grows with the square of the move, which is precisely what a second-order term does.

In money, on Rs 10 crore of that bond:

That is Rs 22 lakh of value, on a single Rs 10 crore position, invisible to a duration-only calculation. A desk that hedged this book purely on duration into a 200 bp selloff would have been short too few futures.

Chapter 1, section 1.12.8 (Convexity Measure), sets out the concept, the shock table above and the approximation formula, immediately after modified duration and PV01. Chapter 5 then returns to it as the reason duration-based hedge ratios underhedge: "using the strategy in the face of large moves in yield will result in underhedging."

Expect conceptual questions rather than the full convexity computation: which direction duration errs in, why the error grows with the size of the move, what convexity is the second derivative of, and why a semi-annual figure is divided by four.

• Convexity corrects duration, it does not replace it. The price estimate is the duration term plus the convexity adjustment; quoting convexity alone answers nothing.
• Get the direction right. Duration overestimates the loss when yields rise and underestimates the gain when they fall. For a plain bond the error is always in the holder's favour — never a nasty surprise, always a pleasant one.
• The error is second-order. Doubling the yield shock roughly quadruples the error, which is why nobody bothers with convexity for a 10 bp move and nobody ignores it for a 200 bp move.
• Modified duration acts on the dirty price, and so does the convexity correction built on top of it.
• Divide a semi-annual bond's convexity by 4, not by 2, to compare it with an annual bond — the adjustment is 2², matching the squared term.
• Convexity is why duration-based hedging underhedges, and it is a distinct source of hedge slippage from the amount and maturity mismatches that make up basis risk.