Bootstrapping Zero-Coupon Spot Rates from Par Bonds

Scenario

Fixed income desks trade par bonds (bonds trading at $P = 100$) of various maturities. Zero-coupon yields (spot rates) are not directly observable for all tenors and must be extracted inductively.

Your Task

1. Given annual par yields $y_1, y_2, \dots, y_N$ for bonds with maturities $1, 2, \dots, N$ years, derive the recursive formula to compute zero-coupon discount factors $d_k = (1 + z_k)^{-k}$ and annual spot rates $z_k$.
2. Explain why the spot rate curve is steeper than the par curve when the yield curve is upward sloping.
3. Bootstrap the 1-year, 2-year, and 3-year spot rates given par yields: $y_1 = 3.00\%$, $y_2 = 4.00\%$, and $y_3 = 5.00\%$.

Constraints

• Annual coupon payments, face value $M = 100$.
• Annual compounding of spot rates.

Examples For year 1, $z_1 = y_1 = 3.00\%$. For year 2, discounting coupon 1 at $3.00\%$ requires the remaining cash flow at year 2 to be discounted at a rate higher than $4\%$, so $z_2 > 4.00\%$.