Question

Assume that the ‘expectations theory’ fully explains the Treasury Bond Yield Curve. The 10-year zero-coupon bond maturing in 2030 has a yield-to-maturity (YTM) of 3.5%. Assume that instead of simply buying the 10 year, you decide you will: • buy a 5-year bond today, then, when that bond matures in 2025, • then, you will buy a new 5-year bond that starts in 2025 and will mature in 2030.

Scenario A – Upward sloping yield curve. a) if today’s 5 year bond has a YTM of 1%, what would the second bond need for YTM for you to get a terminal value that would match simply buying the 10 year bond today?

Scenario B – Downward sloping yield curve b) if today’s 5 year bond has a YTM of 5%, what would the second bond need for YTM for you to get a terminal value that would match simply buying the 10 year bond today?

Bond investors love bad news and dropping interest rates are usually a sign of economic weakness or even recessions. c) Which scenario indicates the bond market foresees a bad economy (A or B)?

Answer 1

Scenario  A:Future Value of $1 after ten years of a 10 year bond with yield to maturity of 3.5% (0.035) =(1+0.035)^10=1.410599

Future value $1 after 5 years of a 5 year bond with YTM of 1% (0.01)=1.01^5=1.05101

Assume YTM of the second 5 year bond =R1

Future Value of $1.05101 after another five years invested in the second bond with YTM of R1=1.05101*((1+R1)^5)

As per expectation theory, the Future Value after 10 years will be same

1.05101*((1_R1)^5)=1.410599

(1+R1)^5=1.410599/1.05101=1.342136

1+R1=1.342136^(1/5)=1.060619

R1=0.0606(rounding to four decimal places)

YTM of the second 5 year bond=R1=0.0606=6.06%

Scenario B:Future Value of $1 after ten years of a 10 year bond with yield to maturity of 3.5% (0.035) =1.410599

Future value $1 after 5 years of a 5 year bond with YTM of 5% (0.05)=1.05^5=1.276282

Assume YTM of the second 5 year bond =R2

Future Value of $1.276282 after another five years invested in the second bond with YTM of R2=1.276282*((1+R2)^5)

As per expectation theory, the Future Value after 10 years will be same

1.276282*((1_R2)^5)=1.410599

(1+R2)^5=1.410599/1.276282=1.105241

1+R2=1.105241^(1/5)=1.020214

R2=0.0202(rounding to four decimal places)

YTM of the second 5 year bond=R2=0.0202=2.02%

Scenario B foresees reduction in interest rate , hence bad economy