Explain how the yield curve's shape, spot and forward rates, yield to maturity, and expected and realized bond returns relate to one another
The idea
Spot rates price single cash flows, forward rates are the rates for future periods locked in by today's spot curve, and yield to maturity is one blended rate that equates a bond's price with all its cash flows. They link through compounding: (1 + S₂)² = (1 + S₁) × (1 + 1y1y). A bond's yield equals its expected return only if it is held to maturity, every coupon is reinvested at that same yield and no default occurs. Realised return differs when coupons are reinvested at rates other than the yield. When the spot curve slopes up, forward rates sit above spot rates and par yields sit slightly below spot rates. The common slip is to average spot rates instead of compounding them.
Formulas
Forward rate between any two dates
(1 + long spot)^(B) = (1 + short spot)^(A) × (1 + forward)^(B − A)
- long spot
- the spot rate for B years
- short spot
- the spot rate for A years
- forward
- the annual rate for the B − A years starting in year A
- A
- the earlier date, in years
- B
- the later date, in years
Investing for B years at once must return the same as investing for A years and rolling into the forward, or there is an arbitrage. Solve for (1 + forward) and take the (B − A)th root before subtracting one.
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