Question

Assume that the City of Rockwall sold an issue of $1,000 maturity value, tax-exempt (municipal bond), zero coupon bonds 5 years ago. The bonds had a 25-year maturity when they were issued, and the interest rate built into the issue was a nominal 10 percent, but with semiannual compounding. The bonds are now callable at a premium of 10 percent over the accrued value. What effective annual rate of return would an investor who bought the bonds when they were issued and who still owns them earn if they were called today?

Answer 1

Computation of Issue Price

Face Value, FV = $ 1,000
Interest Rate, i = 10% Annually or 5% semi annually
Period,n = 25 years or 50 Semi Years

Face Value = Present Value*(1+Rate)ⁿ
1000 = PV*(1+5%)⁵⁰
1000 = PV*(1.05)⁵⁰
1000 = PV*11.4674
PV = 1000 /11.4674
PV = 87.20

Accrued Value after 5 years

Issue Price, PV = $ 87.20
Interest Rate, i = 10% Annually or 5% semi annually
Period,n = 5 years or 10 Semi Years

Accrued Value = Present Value*(1+Rate)ⁿ
Accrued Value = 87.20*(1+5%)¹⁰
Accrued Value = 87.20*(1.05)¹⁰
Accrued Value = 142.05

If called after 5 Years

Issue Value = $ 87.20
Current Callable Price (FV) = 142.05 with 10% premium = 156.25
Period,n = 5 years or 10 Semi Years

FV = Present Value*(1+Rate)ⁿ
156.25 = 87.20*(1+Rate)¹⁰
156.25 / 87.20 = (1+Rate)¹⁰
1.7918 = (1+Rate)¹⁰
(1.7918)^1/10 = (1+Rate)
1.0601 = (1+Rate)
1.0601 - 1 = Rate
Rate = 0.0601 semi annually

Effective Int Rate = (1+ Annual Rate/ No. of Compounding in Year)^{No. of Compounding in Year} - 1
Effective Int Rate = (1+ 0.0601)² - 1
Effective Int Rate = (1.0601)² - 1
Effective Int Rate = 1.1237 - 1
Effective Int Rate = 0.1237
Effective Int Rate = 12.37 %