Duration and convexity

Duration is the single most-used number in fixed income after the yield itself. It answers the question every bond trader and portfolio manager cares about every minute the market is open: if yields move by some amount, how much does my bond's price move? Duration is the price sensitivity to yield, and although the formal derivation comes out of present-value calculus, the working intuition is much simpler — it is the weighted average time you have to wait, in years, to receive a bond's cash flows. Bonds that pay you sooner have low duration and small price reactions to yield changes; bonds that pay you later have high duration and large price reactions. The whole subject is variations on that one idea.

There are three flavours of duration that you will encounter — Macaulay, modified, and effective — plus a closely related quantity called DV01 that traders use on the desk. They look different but they all answer the same underlying question; the differences are in compounding conventions and in how they handle bonds with embedded options. We will walk through each one with the full variable list spelled out, so that when you meet 'modified duration of 7.2' in a research note you know exactly what was computed, what was assumed, and what to do with the number.

Macaulay duration — the original definition

Frederick Macaulay introduced his weighted-average-maturity concept in 1938 in a study of US bond markets, and it remains the cleanest entry point. The idea is straightforward: every coupon and principal payment a bond promises has a date and a present value. If we weight each date by the present value of the cash flow that arrives that day (relative to total bond price), and average, we get the bond's economic 'centre of mass' on the time line. A zero-coupon bond has all of its weight at maturity, so its Macaulay duration equals its maturity. A coupon bond has weight spread out, so its Macaulay duration is always less than maturity.

                  N
                ─────       t · CF_t / (1 + y)^t
  Macaulay D  =   ∑       ─────────────────────────
                  t = 1              P

where:
  N      = total number of cash-flow dates remaining on the bond
  t      = time index, in years from today, when the t-th cash flow arrives
  CF_t   = the cash flow received at time t (a coupon, or the final coupon + principal)
  y      = the bond's yield to maturity, in decimal form (e.g. 0.06 for 6%)
  (1+y)^t = the discount factor that brings cash flow at time t back to today
  CF_t / (1+y)^t = the present value of the cash flow at time t
  P      = the current full (dirty) price of the bond — the sum of all PVs

Read the formula slowly because every symbol earns its place. The factor t out front is the maturity (in years) of each cash flow, which is what makes duration a weighted average of times. The fraction CF_t / (1+y)^t × 1/P is the share of the bond's total present value attributable to that one cash flow — it acts as the weight. Multiply each maturity by its weight, sum across all the cash flows, and the answer comes out in units of years. A 10-year 6% coupon bond at par typically has a Macaulay duration of about 7.8 years, which says the average dollar of the bond's value comes back to you a bit before year 8.

Variable glossary, in plain language

• N — the number of remaining cash-flow dates. Higher N usually means higher duration, but only because more cash flows extend the time horizon; doubling N does not double duration if the new cash flows arrive sooner.
• t — time to each cash flow, expressed in years. A bond paying semi-annually has t = 0.5, 1.0, 1.5, … For Macaulay, t is what makes the weighted average sit in the units of time.
• CF_t — the actual cash you receive at time t. For a typical fixed-rate coupon bond this is the coupon at every date except the last, and coupon-plus-face-value at the last. For floaters, TIPS, or sinking funds, CF_t becomes more interesting.
• y — yield to maturity. This is the single rate that, when used to discount every cash flow, reproduces the bond's market price. A higher y means heavier discounting, which pulls more weight forward in time and shortens duration.
• (1+y)^t — the discount factor. Tells you how much weight a dollar arriving t years from now carries today. A dollar twenty years out is worth roughly half of a dollar arriving today at a 4% yield, less than a third at 6%.
• P — the price of the bond expressed as the sum of all discounted cash flows. Acts as the normaliser so that the duration weights sum to one and the answer is genuinely an average rather than just a sum.

Modified duration — what desks actually use

Macaulay duration measures time. To get the percentage price change for a yield move, we need a small adjustment that converts the time-weighted average into an elasticity. That adjustment is modified duration. It is what every Bloomberg terminal, every fixed-income textbook, and every portfolio-management system reports when the word 'duration' is used without qualification.

                          Macaulay duration
  Modified duration  =  ───────────────────────
                              1 + y / m

where:
  Macaulay duration  =  the weighted-average time computed above, in years
  y                  =  the bond's yield to maturity, decimal form
  m                  =  the number of coupon payments per year — 2 for US Treasuries,
                        1 for most European/African annual coupons, 12 for monthly-pay MBS
  y / m              =  the per-period yield used in compounding the discount factor

Once you have modified duration, the everyday workhorse formula falls out:

  %ΔPrice  ≈  − Modified duration  ×  Δy

where:
  %ΔPrice  =  the approximate percentage change in the bond's full price
              for a small change in yield
  Modified duration  =  the number you just computed, in years
  Δy       =  the change in yield, expressed in decimal form
              e.g. a 25-basis-point increase is Δy = +0.0025
  The minus sign is what encodes the inverse price-yield relationship —
  yields up, prices down.

DV01 — the trader's currency version

Modified duration is a percentage. Trading desks live in dollars (or shillings, or whatever the book is denominated in). To bridge the two, traders use DV01, which simply asks: how many dollars do I gain or lose for a one-basis-point fall in yield, on this exact position?

  DV01  =  Modified duration  ×  Price  ×  0.0001

where:
  Modified duration  =  in years, computed above
  Price              =  the full market value of the position you hold,
                        in the currency of the book
  0.0001             =  one basis point expressed in decimal form
                        (1 bp = 0.01% = 0.0001)

Example: a $50 million position in a bond with modified duration 8
  DV01 = 8 × $50,000,000 × 0.0001 = $40,000 per basis point

Reading: if yields fall 1bp, the position gains $40k; if yields rise 1bp,
it loses $40k. Every fixed-income desk has its DV01 limit; every trader
knows their book's DV01 at every moment of the trading day.

Convexity — why the linear formula isn't enough

Duration is the slope of the price-yield curve at the current yield. The full price-yield relationship, however, is not a line; it is a curve. As yields move further from today's level, the linear approximation drifts away from the true price by an amount that grows with the square of the yield change. The curvature of the price-yield relationship is captured by a second number called convexity, and the corrected approximation comes from a second-order Taylor expansion of the bond-pricing function.

  %ΔPrice  ≈  − Modified duration · Δy  +  ½ · Convexity · (Δy)²

where:
  %ΔPrice            =  approximate percentage price change for any size yield move
  Modified duration  =  the first-derivative slope, computed above (years)
  Δy                 =  yield change in decimal form
  Convexity          =  the second derivative of price with respect to yield,
                        divided by price — a number measuring curvature, with
                        no clean physical-units interpretation but typically
                        between 30 and 300 for plain bonds
  ½                  =  the factor that emerges from the second-order Taylor term;
                        always exactly one half, never something to adjust
  (Δy)²              =  the yield change squared — always positive whether yields
                        rise or fall, which is what makes the convexity term
                        a one-sided gift to bondholders

For option-free bonds, convexity is always positive. That has a meaningful real-world consequence: on a yield drop of 100 basis points, the bond gains slightly more than the linear approximation predicts; on a yield rise of 100 basis points, it loses slightly less. The bondholder gets an asymmetric kicker — better than expected when yields fall, less bad than expected when they rise. This is why portfolio managers consciously seek convexity, particularly in regimes where they expect large yield moves in either direction.

Effective duration for option-embedded bonds

Macaulay and modified duration both assume the cash flows are fixed regardless of where yields go. For a plain Treasury or a plain corporate bond, that is fine. For a callable bond, a putable bond, or a mortgage-backed security, the cash flows themselves depend on the yield path: callable bonds get called when rates fall, mortgage borrowers prepay when refinancing rates drop, putable bondholders exercise when rates rise. Plugging the static-cash-flow duration formula into one of these instruments gives an answer that is mechanically defensible but economically misleading.

The solution is effective duration, computed numerically using a pricing model that captures the option behaviour. The procedure is to bump the entire yield curve by a small amount up and down, re-price the bond under each scenario with the option behaviour active, and divide the resulting price change by the original price and twice the yield bump.

                              P(y − Δy)  −  P(y + Δy)
  Effective duration  =  ──────────────────────────────────
                                  2 · P(y) · Δy

where:
  P(y)        =  the bond's current price at today's yield curve
  P(y − Δy)   =  the model-implied price if the entire yield curve drops by Δy
  P(y + Δy)   =  the model-implied price if the entire yield curve rises by Δy
  Δy          =  the size of the parallel shift used in the bump (typically 10-25 bp)
  2 · Δy      =  the total range of the bump, since we are using a symmetric two-sided
                  finite difference around the current yield

Effective duration is what every trading desk uses for any bond with embedded optionality. A callable corporate bond might have a Macaulay duration of 8 but an effective duration of 4, because the call option caps the upside on a yield drop. A mortgage-backed pass-through with a stated 30-year maturity might have an effective duration that swings between 3 and 7 as prepayment expectations shift. The numerical effective-duration calculation captures all of that; the static formulas don't.

Duration matching and immunisation

Pension funds and life insurers face long-dated liabilities — pensions paid out over decades, life-insurance death benefits payable far in the future. The asset-liability problem they face every quarter is: how do we invest so the assets and liabilities move together as yields shift? The answer, since the 1950s, has been duration matching. If a pension fund's average liability is 14 years away and the matched-duration bond portfolio also has a duration of 14, then a parallel rise in yields of 100 basis points reduces the present value of both the assets and the liabilities by roughly 14%, leaving the funding ratio essentially unchanged.

The technique is called immunisation because it 'immunises' the funded status against parallel yield-curve shifts. It is the foundation of liability-driven investment, the dominant pension-fund paradigm globally since the early 2000s. The practical implementation often involves leveraged interest-rate swaps overlaying a smaller bond portfolio — and that is exactly the structure that broke in the September 2022 UK gilt crisis, when forced unwinds of the LDI overlays drove gilt yields higher in a destabilising spiral and the Bank of England intervened with emergency purchases.

Putting it all together — a worked example with every step shown

A 10-year US Treasury note trades at par (price = 100, face value = 100), with a 4% annual coupon (m=1 here for simplicity) and yield to maturity of 4%. Cash flows are 4 per year for years 1-9, then 104 in year 10. To compute Macaulay duration, take each cash flow's present value, divide by the current price (100), multiply by its year-index, and sum:

Year  CF      PV at 4%       Weight = PV/100      Year × Weight
────────────────────────────────────────────────────────────────
  1    4       3.846          0.03846              0.0385
  2    4       3.698          0.03698              0.0740
  3    4       3.556          0.03556              0.1067
  4    4       3.419          0.03419              0.1368
  5    4       3.288          0.03288              0.1644
  6    4       3.162          0.03162              0.1897
  7    4       3.040          0.03040              0.2128
  8    4       2.923          0.02923              0.2339
  9    4       2.811          0.02811              0.2530
 10  104      70.258          0.70258              7.0258
────────────────────────────────────────────────────────────────
                Sum            1.00000              8.4356

Macaulay duration   =  8.44 years
Modified duration   =  8.44 / (1 + 0.04/1)  =  8.11 years
DV01 on $10m       =  8.11 × $10,000,000 × 0.0001  =  $8,110 per bp

Now apply the duration approximation: if yields rise by 50 basis points (Δy = +0.005), the bond price should fall by approximately 8.11 × 0.5% ≈ 4.06%. The actual price (computed exactly by discounting the cash flows at 4.5%) is 96.04, a fall of 3.96%. The duration estimate over-states the loss slightly because it ignores convexity, which would add roughly 0.10% back in. The whole exercise — Macaulay → modified → DV01 → approximate price change → convexity correction — is what a desk runs through mentally before quoting any bond.