Question

The Wall Street Journal reports that the current rate on 10-tear Treasury bonds is 2.65 percent and the rate on 20-year Treasury bonds is 4.90 percent.Assume the maturity risk premium is zero. What is the expected rate on a 10-year Treasury bond purchased 10 years from today?

Answer 1

Current rate on 10 year treasury bond = YTM of 10 year Treasury bonds = 2.65%

As it is known that Treasury bond pays coupon semi annually, hence

Semi annual YTM of 10 year Treasury bond = YTM of 10 year Treasury bonds / 2 = 2.65% / 2 = 1.325%

No of half years to maturity of 10 year Treasury bond = 2 x no of years = 2 x 10 = 20

Current rate on 20 year Treasury bonds = YTM of 20 year Treasury bonds = 4.90%

Semi annual YTM of 20 year treasury bond = YTM of 20 year Treasury bonds / 2 = 4.90% / 2 = 2.45%

No of half years to maturity of 20 year Treasury bond = 2 x no of years = 2 x 20 = 40

We know that

Expected rate of return on 10 year Treasury bond 10 years from today = YTM of 10 year Treasury bond 10 years from today = 2 x Semi annual YTM of 10 year Treasury bond 10 years from today

No of half years in 10 year Treasury bond 10 years from today = 2 x no of years = 2 x 10 = 20

Now we have

(1+Semi annual YTM of 20 year treasury bond)⁴⁰ = (1+Semi annual YTM of 10 year Treasury bond)²⁰ (1+Semi annual YTM of 10 year Treasury bond 10 years from today)²⁰

(1+2.45%)⁴⁰ = (1+1.325%)²⁰ (1+Semi annual YTM of 10 year Treasury bond 10 years from today)²⁰

(1.0245)⁴⁰ = (1.01325)²⁰ (1+Semi annual YTM of 10 year Treasury bond 10 years from today)²⁰

2.633167 = (1.301166)(1+Semi annual YTM of 10 year Treasury bond 10 years from today)²⁰

Semi annual YTM of 10 year Treasury bond 10 years from today = (2.633167/1.301164)^1/20 - 1 = (2.023701)^1/20 - 1 = 1.035874 - 1 = 0.035874 = 3.5874%

Expected rate of return of 10 year bond 10 years from today = 2 x 3.5874% = 7.1748% = 7.17%

Note: As maturity risk premium is zero, so no adjustment in final YTM is required for maturity risk premium