Bond Duration and Convexity Approximation
You hold a fixed-rate bond with face value $F$, annual coupon rate $c$, maturity $T$ years, and yield-to-maturity $y$ (annual compounding).
- Write the bond price $P(y)$ as a discounted cash-flow sum.
- Define Macaulay duration $D_{\text{Mac}}$, modified duration $D_{\text{mod}}$, and convexity $C$ in terms of the derivatives $\frac{dP}{dy}$ and $\frac{d^{2}P}{dy^{2}}$.
- Derive the duration-convexity approximation for the relative price change $\frac{\Delta P}{P}$ under a small yield shock $\Delta y$. When do you expect this approximation to be accurate, and when does it break down or become biased?
Hints
- Start by writing out each cash flow and discounting it back at the yield -- the price is just a sum of present values.
- Duration and convexity are the first and second derivatives of the price function, normalized by $P$. Think Taylor expansion.
- For part (iii), expand $P(y + \Delta y)$ to second order and divide by $P$. The convexity term is always positive for vanilla bonds -- what does that tell you about the approximation's bias for large shocks?
Worked Solution
How to Think About It: This is the bread and butter of fixed-income risk. A bond is just a stream of known cash flows, so its price is a present-value sum. Duration tells you how sensitive that price is to yield changes -- it is the first-order (linear) effect. Convexity captures the curvature -- the second-order correction. Before writing any formulas, you should know: duration always makes the price move look worse than it really is for a plain vanilla bond, because convexity is positive and works in the bondholder's favor. A senior trader would say "duration is the hedge ratio, convexity is the P&L you pick up when yields move big."
Quick Sanity Checks: - A zero-coupon bond has Macaulay duration equal to $T$ (all cash flow arrives at maturity). - Modified duration is just Macaulay duration discounted by one period: $D_{\text{mod}} = D_{\text{Mac}} / (1 + y)$. - Convexity is always positive for a standard fixed-rate bond (the price-yield curve is convex downward toward the yield axis). - For small $\Delta y$, the duration term dominates. For large $\Delta y$, ignoring convexity leads to significant error.
Derivation:
(i) Bond price as a discounted cash-flow sum:
The bond pays coupon $cF$ at times $t = 1, 2, \ldots, T$ and returns the face $F$ at time $T$. Discounting each cash flow at yield $y$:
$$P(y) = \sum_{t=1}^{T} \frac{cF}{(1+y)^{t}} + \frac{F}{(1+y)^{T}}$$
This can also be written as:
$$P(y) = cF \cdot \frac{1 - (1+y)^{-T}}{y} + \frac{F}{(1+y)^{T}}$$
The first term is the annuity value of coupons; the second is the present value of the principal repayment.
(ii) Duration and convexity from derivatives of $P(y)$:
Take the first derivative:
$$\frac{dP}{dy} = -\sum_{t=1}^{T} \frac{t \cdot cF}{(1+y)^{t+1}} - \frac{T \cdot F}{(1+y)^{T+1}}$$
Macaulay duration is the weighted-average time to cash flows, with weights proportional to PV of each cash flow:
$$D_{\text{Mac}} = \frac{1}{P} \sum_{t=1}^{T} t \cdot \frac{CF_t}{(1+y)^{t}}$$
where $CF_t = cF$ for $t < T$ and $CF_T = cF + F$. This equals:
$$D_{\text{Mac}} = -\frac{(1+y)}{P} \frac{dP}{dy}$$
Modified duration removes the $(1+y)$ factor so it directly measures the percentage price sensitivity:
$$D_{\text{mod}} = -\frac{1}{P} \frac{dP}{dy} = \frac{D_{\text{Mac}}}{1+y}$$
So $D_{\text{mod}}$ tells you: if yield moves by $\Delta y$, the bond price changes by approximately $-D_{\text{mod}} \cdot \Delta y$ in percentage terms.
Convexity comes from the second derivative:
$$\frac{d^{2}P}{dy^{2}} = \sum_{t=1}^{T} \frac{t(t+1) \cdot CF_t}{(1+y)^{t+2}}$$
$$C = \frac{1}{P} \frac{d^{2}P}{dy^{2}}$$
Convexity is always positive for a standard fixed-coupon bond because every term in the sum is positive.
(iii) Duration-convexity approximation:
Taylor-expand $P(y + \Delta y)$ around $y$:
$$P(y + \Delta y) \approx P(y) + \frac{dP}{dy} \Delta y + \frac{1}{2} \frac{d^{2}P}{dy^{2}} (\Delta y)^{2}$$
Divide through by $P(y)$:
$$\frac{\Delta P}{P} \approx -D_{\text{mod}} \cdot \Delta y + \frac{1}{2} C \cdot (\Delta y)^{2}$$
This is the standard duration-convexity approximation.
When is it accurate? - For small yield shocks (say $|\Delta y| < 50$ bps), the approximation is very tight. - It works best for bonds with moderate duration and positive convexity.
When does it break down or become biased? - For large yield shocks ($|\Delta y| > 200$ bps), third- and higher-order terms become non-negligible and the approximation degrades. The direction of the bias depends on the sign of the move: the leading omitted term is $\frac{1}{6}\frac{d^{3}P}{dy^{3}}(\Delta y)^{3}$ with $\frac{d^{3}P}{dy^{3}} < 0$ for a vanilla bond, so for large yield *increases* the parabola overstates the true price, while for large yield *decreases* it understates the true price. (Check with $P(y) = (1+y)^{-10}$ at $y = 5\%$: for $\Delta y = +500$ bps the quadratic approximation gives $0.398$ vs true $0.386$ -- overstates; for $\Delta y = -500$ bps it gives $0.983$ vs true $1.000$ -- understates.) A common misstatement is that the approximation "always underestimates the true price": that is a property of the duration-only (linear) approximation, which positive convexity corrects, and it does not carry over to the duration-convexity (quadratic) approximation. - For bonds with embedded options (callables, putables), the price-yield relationship is not smoothly convex -- convexity can turn negative, and the Taylor expansion around the current yield can be a poor local approximation. - At very low yields near zero, a given basis-point shock is a larger *relative* move in yield, so higher-order terms matter more.
Practical Interpretation: On a trading desk, you hedge the duration risk (first order) and then think about convexity as the residual P&L driver. If you are long convexity, you benefit from large moves in either direction -- the convexity term $\frac{1}{2} C (\Delta y)^{2}$ is always positive. That is why traders say "long convexity = long gamma" -- it is the bond analogue of being long options.
Answer:
$$P(y) = \sum_{t=1}^{T} \frac{CF_t}{(1+y)^{t}}, \quad D_{\text{mod}} = -\frac{1}{P}\frac{dP}{dy}, \quad C = \frac{1}{P}\frac{d^{2}P}{dy^{2}}$$
$$\frac{\Delta P}{P} \approx -D_{\text{mod}} \cdot \Delta y + \frac{1}{2} C \cdot (\Delta y)^{2}$$
The approximation is accurate for small yield shocks and degrades on large moves as the omitted higher-order terms grow: on a plain vanilla bond the quadratic understates the true price for large yield falls and overstates it for large yield rises.
Intuition
Duration and convexity are really just the bond-world names for delta and gamma. A bond's price is a smooth, decreasing, convex function of yield. Duration captures the slope (how much price moves per unit yield change), and convexity captures the curvature (how that slope itself changes). Because the curve bows upward away from the tangent line, a bondholder always does slightly better than the linear (duration-only) approximation predicts -- yields go up and you lose less than expected, yields go down and you gain more than expected. This is the "convexity advantage."
In practice, traders hedge duration continuously but are willing to pay for convexity. A portfolio that is duration-neutral but long convexity makes money on big moves in either direction -- exactly like being long a straddle. The duration-convexity approximation is the fixed-income equivalent of delta-gamma hedging in options, and understanding when it breaks down (large moves, embedded optionality, low-rate environments) is essential for managing a bond book in stressed markets.