• Global Frameworks: CFA Level 1 & 2 (Fixed Income), FIAT (Fixed Income Analyst Training standards).

1. Arbitrage-Free Bond Pricing
A bond’s price is determined by discounting each individual future cash flow by the specific spot interest rate that matches its payment date, rather than using a single average discount rate:
P_bond = Σ_{t=1..n} (PMT / (1 + z_t)^t) + (Principal / (1 + z_n)^n)
Where
z_t is the spot rate (yield on a zero-coupon government bond) for maturity period t.
 
2. Yield to Maturity (YTM) and Yield Curve Dynamics
Yield to Maturity (YTM) is the single internal rate of return that equates the present value of a bond’s promised future cash flows to its current market price. It operates on two key assumptions: the bond is held to maturity, and all intermediate coupon payments are reinvested at that exact same YTM rate.
The relationship between YTM and maturity dates across identical risk profiles is mapped via the Yield Curve:
Yield %
  ▲
  │       ┌─────────────────── Normal (Upward Sloping - Economic Growth)
  │     ┌─┘
  │    ─┼───────────────────── Flat (Uncertainty Transition)
  │     └─┐
  │       └─────────────────── Inverted (Downward Sloping - Recession Signal)
  └──────────────────────────────────────────────────► Time to Maturity

3. Interest Rate Risk: Duration and Convexity Mechanics
  • Macaulay Duration: The weighted average time until an investor receives all of a bond’s cash flows, measured in years.
  • Modified Duration: Measures a bond’s price sensitivity to interest rate shifts. It provides a direct estimate of the percentage change in a bond’s price for a 1% change in yield:

    Modified Duration = Macaulay Duration / (1 + YTM/m)


  • %ΔP_bond ≈ − Modified Duration × Δy

 

  • Convexity Adjustment: Because the relationship between a bond’s price and interest rates is curved (convex) rather than linear, duration estimates lose accuracy during large interest rate shifts. Convexity (C) measures this curvature, providing a more accurate price adjustment:

    %ΔP_bond ≈ (−Modified Duration × Δy) + ( (1/2) C × (Δy)^2 )

 
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