Interest Rates & Bond Pricing

Medium4–5 hrsBondsDurationYield CurveConvexityDV01

Overview

Bond pricing is discounted cash flow in practice. The yield is the internal rate of return. Duration measures interest rate sensitivity — it's the bond's 'effective maturity'. DV01 converts that sensitivity to dollar terms. The key insight: higher duration = more volatile when rates move. Long-term bonds are riskier in a rising-rate environment. This is fundamental to fixed income trading.

How to Recognize

→Bond pricing questions: 'what is this bond worth?'
→'How does the bond price change if rates move 1bps?'
→Yield curve questions: normal/inverted/flat
→Duration and hedging interest rate risk

Step-by-Step Approach

1.Bond price = PV(coupons) + PV(face value) — discounted at yield y
2.Duration: % price change ≈ −Duration × Δy (Modified Duration)
3.DV01 = dollar value of 1 basis point change = Price × Mod.Duration / 10000
4.Convexity: bond price is convex in yield → you gain more from a fall than you lose from a rise

Key Formulas

P = \sum_{t=1}^T \frac{C}{(1+y)^t} + \frac{F}{(1+y)^T}

\text{Duration} = \frac{1}{P}\sum_{t=1}^T t \cdot \frac{CF_t}{(1+y)^t}

(Macaulay Duration)

\frac{dP}{P} \approx -D_{\text{mod}} \cdot dy

where

D_{\text{mod}} = D/(1+y)

DV01 = P \times D_{\text{mod}} / 10{,}000

Worked Examples

0Quick Example

10yr zero-coupon bond, y=5%. P = 100/1.05^10 ≈

61.39. Duration = 10 (zero coupon always has D=maturity). If y rises 1bp: ΔP ≈ −

0.0585.

Problem

A 5-year bond has face value $100, annual coupon 6%, yield = 8%. What is the price?

Solution

Price = PV(coupons) + PV(face).

= 6/1.08 + 6/1.08² + 6/1.08³ + 6/1.08⁴ + 6/1.08⁵ + 100/1.08⁵.

= 6 × [1−(1.08)^(−5)]/0.08 + 100/1.08⁵.

= 6 × [1−0.6806]/0.08 + 100×0.6806.

= 6 × 3.993 + 68.06 = 23.96 + 68.06 = $92.02.

Bond trades at discount ( $92 <$ 100) because coupon rate (6%) < yield (8%).

Answer

$P = \$ 92.02$. When coupon rate < yield: bond trades at discount. When coupon rate > yield: premium.

Common Mistakes

!
Confusing Macaulay Duration (in years) with Modified Duration (the % sensitivity). Always clarify which one you mean.
!
Bond price and yield move INVERSELY. Rates up → bond price down.
!
Zero-coupon bond duration = maturity. Coupon bond duration < maturity (intermediate cash flows reduce it).

Practice Problems

Click "Show Answer" to reveal

A bond has DV01 = $500. You want to hedge a$ 1M position. How many of a 10yr Treasury (DV01 = $80/contract) do you need to short?

Hint: Match total DV01: N ×

80 =

500.

Why does an inverted yield curve historically predict recessions?

Hint: Short rates > long rates: what does this imply about expected future rates?

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Volatility — Realized, Implied, & VIX