Bond Duration

Why one number for an entire bond?

Look at a 10-year corporate bond. Twenty semi-annual coupons are due, plus a face-value payment at the end. Each cash flow has its own date and its own present-value sensitivity to interest rates. If you wanted the most precise possible price-rate relationship, you would track all twenty-one of them. But traders, portfolio managers, and risk officers don't get to think about a vector of twenty-one numbers for every bond on the desk — they need a scalar that summarises rate risk well enough to compare bonds, hedge positions, and report to a risk committee.

Duration is that scalar. Frederick Macaulay defined it in 1938 in a study for the National Bureau of Economic Research as a way to characterise "the time-shape" of a bond's payments. The clever step is the weighting: each cash-flow time t_i is weighted by the present-value of the cash flow as a share of total price. A high-coupon bond gets most of its weight on early dates, so its duration is short. A zero-coupon bond's weight is concentrated entirely on the maturity date, so its duration equals its maturity. The single number captures, in years, when on average the bondholder is getting their money back.

The reason duration became universal isn't its interpretation as a weighted maturity — it's the discovery, formalised by John Hicks in 1939 and rediscovered by Lawrence Fisher and Roman Weil in the 1970s, that this same number is also the first derivative of price with respect to yield, modulo a small denominator correction. That gave fixed-income desks one number that did two jobs: described the bond's payment timing, and predicted what would happen to price if rates moved.

Macaulay duration — the weighted average time

For a bond with cash flows C_1, C_2, …, C_n at times t_1, t_2, …, t_n and yield y (per period), the present value of cash flow i is PV_i = C_i / (1+y)^t_i, and the bond price is P = Σ PV_i. Macaulay duration is

D_Mac = Σ ( t_i × PV_i ) / P
      = Σ t_i × w_i      where w_i = PV_i / P

The weights w_i sum to 1 and are the share of the bond's value attributable to each cash flow. D_Mac is just the weighted average of times — measured in the same time units as the discounting (years, if y is an annual yield).

Three immediate consequences fall out of the definition:

• Zero-coupon bonds. All weight is on the single maturity payment, so D_Mac = T.
• Coupon bonds with maturity T have D_Mac < T. The coupons pull the weighted average forward; the higher the coupon, the more it pulls.
• Perpetuities (a bond that pays forever) have D_Mac = (1 + y) / y, which converges as y → 0 to a large but finite number. The infinite-cash-flow stream has finite duration because distant cash flows are discounted to near-zero weight.

Modified duration — turning timing into sensitivity

Differentiate the bond price with respect to yield. The result, written as a percentage change, is

dP/dy = -Σ t_i × C_i / (1+y)^(t_i+1) = -(1/(1+y)) × Σ t_i × PV_i

(1/P) dP/dy = -(1/(1+y)) × D_Mac = -D_mod

Modified duration is just Macaulay duration divided by (1 + y):

D_mod = D_Mac / (1 + y)

The first-order Taylor expansion of price in yield then reads

ΔP / P ≈ -D_mod × Δy

That is the single most-used equation in fixed-income risk management. A bond with D_mod = 8 loses roughly 8% of its value if yields rise by 1 percentage point; gains roughly 8% if yields fall by 1 percentage point. The minus sign is the inverse price-yield relationship every bondholder learns first.

Two cosmetic conventions worth noting. Dollar duration (sometimes "DV01" when scaled to a 1-basis-point move) is the absolute price change, not the percentage change: $D = -D_mod × P × Δy. And "effective duration" is a numerical version that bumps yield up and down by some small Δy and measures the average price change — necessary for callable bonds, MBS, and other instruments where cash flows themselves depend on yield.

Convexity — the second-order correction

The price-yield curve isn't a line; it bends upward. Duration is the tangent at the current yield, which under-predicts price on big down-moves in rates and over-predicts loss on big up-moves. The second derivative captures the bend:

Convexity = (1/P) × d²P/dy²
          = (1/(1+y)²) × Σ t_i(t_i+1) × w_i

ΔP / P ≈ -D_mod × Δy  +  ½ × Convexity × (Δy)²

For small yield moves the quadratic term is negligible — at Δy = 0.0025 (25 bp) with Convexity = 90 (typical for a 10-year bond), the second-order term contributes about 0.03% to ΔP/P. For large moves it matters: at Δy = 0.03 (300 bp), the quadratic term is roughly 4%, a meaningful correction.

Convexity is a desirable feature: it makes losses smaller than duration predicts and gains larger. Two bonds with identical duration but different convexity are not identical assets — the higher-convexity bond is preferred, and the market typically prices it slightly richer. Negative convexity, where the curve bends the wrong way, occurs in callable bonds and mortgage-backed securities: when rates fall, prepayments accelerate and effective duration shortens, capping the upside.

Worked example — 10-year zero versus 10-year 5% coupon

Take two bonds, both yielding 4% semi-annually compounded, both with face value $100, both maturing in 10 years.

Zero-coupon bond. One cash flow of $100 at year 10. The bond price at 4% semi-annual yield (y/2 = 0.02 per period, 20 periods) is

P = 100 / (1.02)^20 ≈ $67.30

D_Mac is trivially 10 years. D_mod = 10 / (1.02) ≈ 9.80. The bond loses about 9.8% per 1-point yield rise.

5% coupon bond. Twenty semi-annual coupons of $2.50 plus $100 at maturity. Computing the PV-weighted average time of all 21 cash flows gives D_Mac ≈ 8.03 years; D_mod ≈ 7.87. The bond loses about 7.9% per 1-point yield rise.

The coupon bond has shorter duration despite identical maturity, because the coupons pull weight forward. A 30-year 5% bond by contrast has D_mod ≈ 15.5, while a 30-year zero has D_mod ≈ 29.4 — three times the rate-risk for the same nominal maturity. This is why pension funds buying long bonds for duration matching prefer STRIPS (zero-coupon Treasuries) when they want to maximise duration per dollar.

Bond	Maturity	Coupon	D_Mac (yr)	D_mod (yr)	Loss on +100 bp
1-yr T-bill	1 yr	0% (zero)	1.00	0.98	-0.98 %
2-yr Treasury	2 yr	4%	1.93	1.89	-1.89 %
5-yr Treasury	5 yr	4%	4.55	4.46	-4.46 %
10-yr 5% coupon	10 yr	5%	8.03	7.87	-7.87 %
10-yr zero	10 yr	0%	10.00	9.80	-9.80 %
30-yr 5% coupon	30 yr	5%	15.83	15.52	-15.52 %
30-yr zero (STRIP)	30 yr	0%	30.00	29.41	-29.41 %
Perpetuity at 4%	∞	$4/yr	26.00	25.49	-25.49 %

Immunization and asset-liability matching

The single most important institutional use of duration is matching the duration of an asset portfolio to the duration of a liability stream. If asset duration equals liability duration, then a parallel yield-curve move changes the present value of both by the same percentage, and the funded ratio is unchanged to first order. This is the foundation of liability-driven investing (LDI) in pensions, asset-liability management (ALM) in life insurance, and bank treasury management.

The discovery is due to Frank Redington (1952), who showed that if a portfolio's duration matches the liability's duration and its convexity weakly exceeds the liability's convexity, then the portfolio is "immunised" against small parallel shifts: any rate move produces at least a non-negative change in net worth.

The technique is simple to describe and operationally hard. The challenges that make ALM a profession in itself:

• Liability cash flows are uncertain. Pension benefits depend on mortality, salary growth, and member behaviour. Insurance claims depend on actuarial assumptions. The "liability duration" is itself a model output.
• The yield curve doesn't shift in parallel. Real moves are twists and bends; matching scalar duration leaves residual key-rate exposure (see below).
• Convexity matching costs money. Achieving positive convexity surplus typically requires holding longer or more-convex instruments, which may yield less.
• Re-balancing. Asset duration drifts as time passes and rates move. Without periodic re-balancing the match decays.

The 2022 UK gilt crisis was an LDI implementation failure: pension funds had used derivative leverage to extend asset duration up to liability duration, but the gilt sell-off triggered margin calls that forced them to dump the very gilts they were hedging with, creating a fire-sale loop the Bank of England had to break with an emergency buying programme. The strategy was correct; the operational design wasn't robust to the speed of the move.

Key rate duration — when one number isn't enough

Modified duration assumes a parallel shift: every point on the yield curve moves by the same Δy. In practice the curve twists. Key rate duration (Ho, 1992) decomposes total duration into sensitivities to a finite set of curve points — usually 2y, 5y, 10y, 30y — by bumping each in turn and re-pricing. The resulting vector of key-rate durations must sum (approximately) to the scalar modified duration.

Two bonds can share the same modified duration yet have very different key-rate profiles. A "bullet" bond concentrates duration at one point on the curve; a "barbell" portfolio combining short and long bonds spreads duration across two points. When the curve steepens (long rates rise more than short rates), the barbell loses more even though its scalar duration is identical to the bullet's. Professional fixed-income managers run their portfolios on the full key-rate vector, not just the scalar.

Empirical case — the 2022 bond rout

From January to October 2022 the U.S. 10-year Treasury yield rose roughly from 1.50% to 4.50% as the Federal Reserve front-loaded rate hikes against the post-pandemic inflation. The actual draw-down for representative bond portfolios:

Portfolio	Approx D_mod	Δy (rough)	Predicted ΔP/P	Actual 2022 return
Bloomberg US Aggregate (AGG)	~6.3	+2.4 pp	-15.1 %	-13.0 %
10-yr Treasury (IEF)	~7.8	+3.0 pp	-23.4 %	-16.6 %
20+ yr Treasury (TLT)	~17.5	+2.5 pp	-43.8 %	-31.2 %
Long zero-coupon (EDV/ZROZ)	~24	+2.5 pp	-60.0 %	-38 to -40 %

The first-order duration approximation systematically overstates the loss because it ignores positive convexity (which makes losses smaller than the linear prediction) and because real yield moves are paths, not a single shock — coupons reinvest at higher rates as the year progresses. But the order of magnitude is right, and the lesson for any holder of long-duration bonds is unforgettable: long Treasuries, traditionally treated as the safest possible asset, lost roughly a quarter of their value in a single calendar year because a single parameter — duration — was very large.

Empirical case — Silicon Valley Bank, March 2023

The collapse of Silicon Valley Bank on 9-10 March 2023 was the most expensive duration mismatch in modern banking history. The setup:

• Asset side. By Q4 2022, SVB's investment portfolio held roughly $91 billion of held-to-maturity (HTM) agency mortgage-backed securities and Treasury bonds at amortised cost. The realised market value was about $76 billion — an unrecognised loss of roughly $15 billion against $16 billion of tangible common equity. The portfolio's effective duration was roughly 6 years.
• Liability side. SVB's deposit base — concentrated in venture-backed technology firms — was 97% uninsured demand deposits. The behavioural duration of an uninsured, sophisticated, run-prone depositor base in 2023 was near zero.
• The trigger. On 8 March 2023 SVB announced a $1.8 billion realised loss on a $21 billion bond sale plus a planned $2.25 billion capital raise. Within 24 hours, depositors coordinated via Slack, WhatsApp and Twitter and requested $42 billion of withdrawals — roughly 25% of total deposits — in a single day.
• The math the depositors did. Total assets: $209 B. Mark-to-market HTM losses: ~$15 B. Tangible equity: $16 B. Marking the HTM portfolio to market reduced the equity cushion to roughly $1 B against $173 B of deposits. A run was rational.

The textbook lesson: if asset duration (years) exceeds liability duration (days, in a run) by orders of magnitude, the bank is a leveraged carry trade on the yield curve, not a maturity transformer. Every accounting trick that hides the mismatch — HTM accounting, the AFS / AOCI carve-out in regulatory capital, the assumption that "core deposits" are sticky — only hides the problem until depositors do the duration arithmetic themselves. Three banks (SVB, Signature, First Republic) failed in March-May 2023 with a combined $549 billion of assets, the second-largest banking failure episode in U.S. history.

Other duration concepts in practice

• Effective duration. A numerical estimate that bumps yield up and down and measures the average price change. Necessary for instruments where cash flows depend on yield (callable bonds, MBS, putable bonds). The standard formula is D_eff = (P_- - P_+) / (2 P_0 Δy).
• Option-adjusted duration (OAD). Effective duration computed inside an option-pricing model that prices the embedded option (call, put, prepayment). The standard reported duration for MBS.
• Spread duration. Sensitivity of price to a change in the credit spread over the Treasury curve, holding the Treasury curve fixed. The relevant duration for credit-risk hedging.
• Empirical duration. A regression of observed price changes on observed yield changes. Used when model assumptions about cash flows are suspect — e.g., for high-yield bonds whose cash flows are de facto contingent on the issuer's distance to default.
• Partial / key rate duration. The Ho (1992) decomposition by curve point.
• Duration of equity. The behavioural duration of bank deposits, used in ALM. Estimated from historical sensitivity of deposit balances to rates. Treating zero-duration deposits as having an "implied" duration of years is the central bank ALM assumption that 2022-23 stress-tested.

Common pitfalls

• Confusing maturity with duration. A 30-year 6% coupon bond has duration around 14 years, not 30. The bond market quotes maturity for reference but trades on duration.
• Using duration alone for big yield moves. For Δy > 100 bp, ignoring convexity introduces errors of 1 to 5 percent. For Δy > 300 bp (the 2022 environment) the error can exceed 10 percent.
• Forgetting that duration assumes parallel shifts. Curve steepening or flattening generates losses that scalar duration cannot capture. A duration-neutral portfolio can lose substantially on a twist.
• Applying Macaulay duration where modified is needed. Risk management uses D_mod; only investment-strategy descriptions and academic discussions need D_Mac.
• Holding-period mismatch. Duration measures sensitivity at a moment. Over a long enough holding period, higher yields are partly compensated by reinvestment income — the convexity of total return is more favourable than the convexity of price.
• Behavioural deposits. Bank treasurers assume non-maturity deposits have duration of years based on historical stickiness. In a fast run that assumption breaks; SVB found out how fast.