Effective duration and effective convexity

Why does yield duration break for bonds with options?

Modified duration assumes the promised cash flows will arrive on schedule, so there is one yield-to-maturity to nudge up and down. An embedded option removes that certainty. A callable bond lets the issuer repay early, which it will do when rates fall. A putable bond lets the investor demand early repayment, which they will do when rates rise. Either way the timing of the cash flows depends on future rates, so a callable or putable bond has no single, well-defined yield-to-maturity. Even the yield-to-worst is only one of several possible outcomes.

So Macaulay and modified duration are the wrong tools. The right tool shifts the market, not the bond’s own yield. It moves a benchmark yield curve (for example the government par curve) and asks what happens to the bond’s price once the model lets the cash flows react. That is effective duration , and because it is defined by a move in the curve it is a curve duration , not a yield duration.

Interest rate risk measuresa duration and a convexity for each family

• Yield-based Memorize shift the bond’s own yield-to-maturity
  • Macaulay and modified duration option-free bonds, fixed cash flows
  • Approximate duration and convexity three prices, yield bumped ± Δy
• Curve-based Memorize shift the benchmark yield curve
  • Effective duration and convexity Memorize embedded options, uncertain cash flows
  • Key rate duration one maturity at a time, for shape changes
• Empirical Know the idea estimated from historical prices

Effective duration also works for option-free bonds. There it is close to modified duration but not identical: a parallel shift in the par curve is not a parallel shift in spot rates, so the bond’s price moves a little differently from a change in its own yield. The two match exactly only on a flat curve, and the gap narrows when the curve is flatter, maturity shorter and the price closer to par.

How do you calculate effective duration and convexity?

Memorize

The formulas are the approximate duration and convexity you already know, with two changes: Δy becomes ΔCurve, and the present values (PV) after the curve shifts, PV₋ and PV₊, come from an option pricing model rather than from the bond’s own yield. Lower the benchmark curve to get PV₋, raise it by the same amount to get PV₊, and keep PV₀ as today’s full price.

$$\text{EffDur} = \frac{PV_- - PV_+}{2 \times \Delta Curve \times PV_0}$$

Effective convexity measures the second-order effect of the same parallel shift:

$$\text{EffCon} = \frac{PV_- + PV_+ - 2 \times PV_0}{(\Delta Curve)^2 \times PV_0}$$

Duration has a 2 and a single ΔCurve in the denominator. Convexity has no 2 and ΔCurve squared. Enter ΔCurve as a decimal (25 bp = 0.0025).

Exam tip The numerator of the convexity formula is a difference of large, nearly equal numbers. Keep every decimal of every price or the sign can flip.

Go deeper The five inputs to the option model, and which one you shift

The model that produces PV₋ and PV₊ needs (1) the length of the call protection period, (2) the schedule of call prices and dates, (3) an assumption about the credit spread over the benchmark (including any liquidity spread), (4) an assumption about future interest rate volatility and (5) the level of market rates, such as the government par curve. The analyst holds the first four constant and moves only the fifth, up and then down in parallel.

Go deeper Why a smaller shift is not always more accurate here

With a plain bond, a smaller yield bump gives a better duration estimate. For complex securities the model also predicts how issuers or borrowers behave. A company’s decision to call depends partly on its own credit spread, and a mortgage borrower’s decision to refinance depends on the value of the home. The model’s behavior changes with the size of the move, so a tiny shift is not automatically better.

Go deeper A real-world note: liabilities too

Curve-based measures are used for financial liabilities as well as bonds. A pension obligation whose payments depend on rates or inflation has an effective duration too, and the same repricing method measures it.

How do callable and putable bonds behave?

A callable bond is worth less than an otherwise identical straight bond; the gap is the value of the call, which the issuer owns. A putable bond is worth more than its straight twin by the value of the put, which the investor owns. Each option matters most where it is likely to be used.

	Callable bond	Putable bond
Who owns the option	Issuer	Investor
Price vs straight bond	Lower (straight − call value)	Higher (straight + put value)
When the option bites	Rates fall: price capped near the call price	Rates rise: price floored near the put price
Effective duration	Shorter than the straight bond, especially when rates are low	Shorter than the straight bond, especially when rates are high
Effective convexity	Can turn negative at low rates	Stays positive

Embedded optionits value is added to or taken from a straight bond

• Call (issuer owns it) Memorize price = straight bond − call value
  • Bites when rates fall price capped near the call price
  • Effective duration shorter most of all when rates are low
  • Effective convexity can turn negative Memorize at low rates
• Put (investor owns it) Memorize price = straight bond + put value
  • Bites when rates rise price floored near the put price
  • Effective duration shorter most of all when rates are high
  • Effective convexity stays positive Memorize always

When rates are high compared with the coupon, a call is far out of the money, so the callable and straight bonds behave almost the same, and both have positive convexity. As rates fall the call gains value, the callable’s price curve flattens against the call price and its slope drops. Past some point the curve bends the wrong way, and the effective convexity is negative convexity . The investor bears call risk: a called bond must be replaced at a lower rate. The put works the mirror way when rates rise, but a put never bends the curve downward, so convexity stays positive.

Both options shorten effective duration. Read a shorter effective duration as a shorter expected life: the call may end the bond early, and the put may let the investor leave early. Mortgage-backed securities behave like callable bonds, because borrowers refinance when rates fall.

Exam tip The call caps your upside when rates fall. The put caps your downside when rates rise. Duration shortens either way; only the callable can go negatively convex.