Duration and convexity — the love story between bond prices and interest rates

If bonds had emotions, duration would be their nervous system and convexity the part that makes them dramatic when rates move. Buckle up.

You already know how to read a term structure and spot curve (we built that toolkit in Position 3). You also know the nitty-gritty of yield measures and day count conventions (Position 2). Now we ask: given those yields and spot rates, how sensitive is a bond’s price to the cruel ballet of interest rates? Welcome to Duration and convexity — the quantitative romance novel of fixed income.

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What is Duration? (And why portfolio managers care)

Duration is a measure of a bond’s price sensitivity to changes in interest rates. In plain language: how much will the bond’s price swing when yields move? Think of duration as an investment’s steering wheel responsiveness. Higher duration = more twitchy.

Two common definitions:

• Macaulay duration: a weighted average time to cash flows (measured in years). Useful conceptually and for immunization.
• Modified duration: a price sensitivity measure — approximates the percentage change in price for a 1% (100 basis points) parallel shift in yield.

Formulae (clean and useful):

Macaulay Duration = Σ[t * (CF_t / (1+y)^t)] / Price
Modified Duration = Macaulay Duration / (1 + y)

Where CF_t = cash flow at time t, y = yield per period, Price = present value of cash flows.

Quick intuition: if Modified Duration = 5, then a 1% increase in yield implies roughly a 5% drop in price (and vice versa).

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What is Convexity? (The correction term that saves your approximation)

Duration is a linear approximation — great for small yield changes, but terrible for drama. Convexity captures the curvature of the price–yield relationship. It corrects the error from using duration alone and becomes especially important for large yield moves or long-term bonds.

Discrete-period convexity formula (practical):

Convexity = (1 / Price) * Σ[CF_t * t * (t+1) / (1+y)^(t+2)]

And the combined approximation for percent price change:

ΔP / P ≈ -Modified_Duration * Δy + 0.5 * Convexity * (Δy)^2

So convexity is the second-order term — the safety net for nonlinearity.

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How do these tie back to term structure and yield measures?

• Duration uses the yield (y) or discount rates in the denominator. If you computed price using the spot curve (from Position 3), you can compute cash-flow-weighted durations more precisely (especially when a bond’s cash flows are best discounted using spot rates rather than a flat YTM).
• Day count conventions (Position 2) affect accrual and exact timing of cash flows — that changes the time weights in Macaulay duration, so calendar precision matters.

In short: the better your yield inputs (spot rates, correct day counts), the more accurate your duration and convexity.

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Examples: small numeric walkthrough

Imagine a 3-year annual-pay 5% coupon bond, face = 100, YTM = 4%.

Step-by-step:

1. Compute cash flows: 5, 5, 105.
2. Discount using y = 4% and sum to get Price.
3. Compute Macaulay duration = Σ[t * PV(CF_t)] / Price.
4. Compute Modified duration = Macaulay/(1+y).
5. Compute convexity via the formula above.

I’ll spare the arithmetic heartbreak, but you’ll find Modified Duration ≈ 2.85 years and Convexity ≈ 8.6 (units: years^2). If yields rise 100 bps (Δy = 0.01):

ΔP/P ≈ -2.850.01 + 0.58.6*(0.01)^2 = -0.0285 + 0.00043 ≈ -2.42%.

Compare: using duration alone gives -2.85% — convexity correction reduces the error.

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Why does convexity matter in practice?

• Large rate moves: Duration is linear; convexity prevents systematic under/overestimation of price change for big moves.
• Long-dated bonds: More time → more curvature → more convexity effect.
• Portfolio construction: Two bonds with the same duration might have different convexities. You’d usually prefer higher convexity (all else equal) because it reduces downside and increases upside on rate moves.
• Hedging and immunization: Duration matches PV sensitivities; convexity explains residual risk when yields move non-parallel or non-small.

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Key-rate duration and the term structure nuance

Duration assumes a parallel shift in the yield curve. But you, wise reader, know from the term structure/spot curve module that yield changes are often non-parallel — the curve twists and bends.

• Key-rate durations decompose sensitivity across specific maturities (1y, 2y, 5y, 10y, ...). Useful when the spot curve moves unevenly.
• For accurate risk control, compute key-rate durations using the spot curve so each cash flow is discounted to the correct horizon.

If the yield curve were a sea, modified duration tells you how your boat heels with a gust; key-rate durations tell you how each wave (short, medium, long) slams you.

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Table: Quick comparison

Measure	Unit	What it tells you	Use case
Macaulay Duration	years	Average time to receive cash flows	Immunization, maturity intuition
Modified Duration	% price change per 1% yield	First-order price sensitivity	Quick P&L estimates, hedging
Convexity	years^2	Second-order curvature of price-yield	Improves estimates for large Δy; portfolio choice
Key-rate durations	% per 1% change at a specific tenor	Sensitivity to non-parallel curve moves	Managing term-structure risk

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Common mistakes (and how to stop doing them)

• Assuming duration gives exact P&L for big yield moves. (Nope — add convexity!)
• Using yield-to-maturity discounting for each cash flow when the spot curve is the correct input — this misstates price and duration if the curve is not flat.
• Confusing Macaulay with Modified — one measures time, the other sensitivity.
• Ignoring day count conventions — half-year vs actual/365 shifts those weights subtly but meaningfully for large portfolios.

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Practical questions (so you actually know how to use this)

• How to hedge duration? Short a position with offsetting duration or use interest rate swaps/futures to neutralize portfolio duration.
• Should I always pick higher convexity? Generally yes, but higher convexity often comes with lower yield or higher cost — it's a trade-off.
• How does this connect to Risk, Return, and Probability? Duration and convexity are the sensitivities you plug into probabilistic scenarios for rate changes. Combine them with your probability distribution for Δy (from your statistical model) to estimate expected losses or VaR.

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Final takeaways — make it stick

• Duration ≈ linear sensitivity (first derivative). Convexity ≈ curvature (second derivative).
• Use duration for quick estimates and hedging. Use convexity to correct and choose among bonds with similar durations.
• Always respect the spot curve and correct day counts when computing PVs — garbage in, useless duration out.

If duration is your bond’s heartbeat, convexity is its mood swings: both are required for a real diagnosis.

Now go calculate, compare a couple of bonds, and watch how convexity rescues your P/L estimate when the market does something dramatic. You’ll never look at a price-yield graph the same way again.

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Version note: this piece builds directly on your previous modules (term structure & spot curves; yield measures & day count) and the statistical foundations from Risk, Return, and Probability — use those concepts as inputs when you compute duration and convexity.