Annual par yields are 3.00% at one year and 3.50% at two years, with annual coupons on par 100. What is the two year zero rate?

Annual par yields are 3.00% at one year and 3.50% at two years, with annual coupons on par 100. What is the two year zero rate?

Approach: A par bond prices at 100 by definition. Discount the first coupon at the one year zero rate, then solve the remaining cash flow for the two year rate.

3.5088%. The two year par bond pays 3.5 after one year and 103.5 after two and prices at 100. The one year zero rate equals the one year par yield of 3.00%, so the first coupon is worth 3.5/1.03 = 3.39806 and the final flow must supply 100 - 3.39806 = 96.60194. Then (1 + z2)^2 = 103.5/96.60194 = 1.071407, giving z2 = 3.5088%. The zero rate sits above the par yield because the curve slopes upward and a par yield is a cash flow weighted average of the zero rates along it. Bootstrapping proceeds one maturity at a time with every earlier zero rate held fixed, so an error at the front end propagates into every point beyond it.

Follow-up: What is the three year zero rate if the three year par yield is 3.80%?

Key concepts: bootstrapping, zero rate, par yield, coupon.