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Bonds: income, yield changes and duration

A bond pays fixed coupons and returns its face value at maturity. Its price moves opposite to market yields: if yields rise, existing bonds with lower coupons become less attractive and their prices fall. A bond's return comes from several separate sources.

Income (carry)
Coupon interest earned as time passes, captured through accrued interest. For most bond funds this is the steadiest part of the return.
Roll-down
As a bond ages it's priced off a shorter part of the yield curve. If the curve slopes upward, its yield falls and its price rises a little, even if nothing in the market moves.
Yield curve effect
The price change from moves in government yields: parallel shifts, and changes in the curve's slope and shape.
Spread effect
The price change from moves in the extra yield investors demand for credit risk over governments. Widening spreads hurt; tightening helps.
Convexity
The curvature that makes price gains from falling yields larger than price losses from rising yields of the same size.
Defaults and other
Credit losses, plus smaller effects such as liquidity and pricing differences.

Clean price, dirty price, accrued interest

Bonds are quoted at a clean price, without accrued interest. A buyer actually pays the dirty price: clean price plus the coupon interest accrued since the last payment. Returns must use the dirty price (or clean price plus accrued interest). Otherwise each coupon date looks like a sudden price drop.

Duration and convexity

Modified duration tells you how much a bond's price changes, in percent, for a 1% change in yield. A duration of 7 means a 1% rise in yields cuts the price by about 7%. For larger moves the estimate drifts, and convexity corrects it.

ΔPP ≈ −Dmod × Δy + ½ × Convexity × (Δy)²
In plain termsDuration is like a see-saw's length: the longer it is, the more the far end moves when the middle tilts. Long-dated bonds have long see-saws, so a small change in interest rates swings their price a lot.

Price vs yield

An annual-coupon bond priced at its starting yield. Shift yields and compare the exact price change with the duration estimates.

Exact price changeDuration onlyDuration + convexity
Modified duration
Convexity
Exact price change
Duration estimate
With convexity

Fixed income attribution

Bond portfolios are rarely explained with sectors alone. Instead, the return is split by source, in the spirit of the Campisi model: income, then the effect of government yield changes, then the effect of spread changes. Against a benchmark, the same idea becomes duration and curve positioning (were you longer or shorter than the index, and where on the curve?), sector and credit allocation, and issue selection.

Where did the bond return come from?

A corporate bond fund over a holding period. Set the moves in government yields and credit spreads.

For practitionersThis is a simplified Campisi decomposition: income as yield × time, treasury effect as −duration × Δgovernment yield, spread effect as −spread duration × Δspread, and convexity as a separate term; the residual in real data absorbs roll-down, curve reshaping and pricing noise. Production models use key-rate durations to split curve shifts into parallel, slope and curvature, option-adjusted spread and spread duration for callables and MBS, and separate inflation accrual for linkers. Income should come from accrued interest on the dirty price. Portfolio and benchmark must be priced from the same source and time; a few cents of bid-mid difference across a whole index is a visible return break.

All numbers are illustrative. The models are simplified for teaching: annual coupons, Black-Scholes options, simplified fee mechanics and no intra-period trading. Real systems add transaction-based returns, daily valuation, tax and corporate action processing, and reconciliation to the official TWR.