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Fixed Income

Difficulty intermediate

Overview

Fixed income securities are debt instruments that pay regular interest and return principal at maturity. The bond market is larger than the stock market.

Bond Types

Type Issuer Risk Yield
Treasury Government Lowest Lowest
Agency Government-sponsored Very low Low
Municipal State/local gov't Low-Medium Tax-advantaged
Corporate Companies Medium-High Medium-High
High Yield Lower-rated companies High High
Emerging Market Foreign governments High High

Bond Pricing

Price = Σ (Coupon / (1+r)^t) + Face / (1+r)^n

where r = yield to maturity

where: Coupon periodic coupon payment · Face face (par) value paid at maturity · r yield to maturity (the discount rate that makes the PV equal the market price) · t period index from 1 to n · n total number of coupon periods until maturity. does: the price equals the present value of all future cash flows (coupons + face) discounted at YTM. Solving the equation for r (given price) yields YTM via numerical root-finding.

Price-Yield Relationship

Inverse relationship: When yields rise, prices fall
Longer duration = more sensitive to rate changes

Duration and Convexity

Duration

Modified Duration = -1/P × dP/dy

Approximate price change for 1% yield change:
ΔP/P ≈ -Duration × Δy

where: P bond price · dP/dy first derivative of price with respect to yield · Δy change in yield (in decimal — e.g. 0.01 for 1%). does: linear sensitivity of price to a small yield change. Modified duration of 7 means a 1% rate rise drops the bond ~7% in price. The headline interest-rate risk measure.

Convexity

Convexity = 1/P × d²P/dy²

Better approximation:
ΔP/P ≈ -Duration × Δy + ½ × Convexity × (Δy)²

where: d²P/dy² second derivative of price w.r.t. yield · the convexity term is always positive for plain bonds (price is convex in yield). does: the quadratic correction to duration. For large yield moves, duration alone underestimates the price gain on rate drops and overestimates the price loss on rate rises — convexity captures that asymmetry.

Yield Curve

yield % 5 3 1 3m 2y 5y 10y 20y 30y normal flat inverted
normal · long > short, typical expansion · inverted · short > long, historic recession leading indicator · flat · transitional
Normal: Short rates < Long rates (upward sloping)
Inverted: Short rates > Long rates (recession signal)
Flat: Short rates ≈ Long rates

Trading the curve:
Steepener: Long short-end, short long-end
Flattener: Short short-end, long long-end

Trading Fixed Income

Strategies

Strategy Description Risk
Buy and Hold Hold to maturity Interest rate
Yield Curve Trade curve shape Curve risk
Credit Spread Trade credit quality changes Default risk
Duration Bet on rate direction Rate risk
Roll-Down Capture yield curve roll Curve risk

Key Metrics

Metric Formula Use
Yield to Maturity IRR of cash flows Total return if held
Current Yield Annual Coupon / Price Income measure
Spread Bond yield - Treasury yield Credit risk premium
OAS Option-adjusted spread Spread with embedded options

Interest Rate Risk

Rates Rise → Bond Prices Fall
Rates Fall → Bond Prices Rise

Duration measures sensitivity:
10-year duration bond: 1% rate rise → ~10% price fall
2-year duration bond: 1% rate rise → ~2% price fall

Practical Guidelines

  1. Understand Duration — Key to rate risk
  2. Credit Quality Matters — Higher yield = higher risk
  3. Liquidity Varies — Treasuries liquid, corporates less so
  4. Inflation Risk — Real returns matter
  5. Call Risk — Issuers can call bonds when rates fall
  6. Tax Considerations — Munis offer tax advantages
  7. Yield Curve Signals — Inversion = recession warning

Next Steps