Question

A 40-year maturity bond has a 9% coupon rate, paid annually. It sells today for $1,057.42. A 30-year maturity bond has a 8.5% coupon rate, also paid annually. It sells today for $1,069.5. A bond market analyst forecasts that in five years, 35-year maturity bonds will sell at yields to maturity of 10% and that 25-year maturity bonds will sell at yields of 9.5%. Because the yield curve is upward-sloping, the analyst believes that coupons will be invested in short-term securities at a rate of 8%.

Answer 1

Part 1)

Step 1: Calculate Price of Bond with 35 Years to Maturity

The price of the bond can be calculated with the use of PV (Present Value) function/formula of EXCEL/Financial Calculator. The function/formula for PV is PV(Rate,Nper,PMT,FV) where Rate = Interest Rate (here, Yield to Maturity), Nper = Period, PMT = Payment (here, Coupon Payment) and FV = Future Value (here, Face Value of Bonds).

Here, Rate = 10%, Nper = 35, PMT = 1,000*9% = $90 and FV = $1,000

Using these values in the above function/formula for PV, we get,

Price of Bond with 35 Years to Maturity = PV(10%,35,90,1000) = $903.56

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Step 2: Calculate Future Value of Coupon Payments at 8% Interest Rate for 5 Years

The future value of coupon payments can be calculated with the use of FV (Future Value) function/formula of EXCEL/Financial Calculator. The function/formula for FV is FV(Rate,Nper,PMT,PV) where Rate = Interest Rate, Nper = Period, PMT = Payment (here, Coupon Payment) and PV = Present Value.

Here, Rate = 8%, Nper = 5, PMT = 1,000*9% = $90 and PV = 0

Using these values in the above function/formula for FV, we get,

Future Value of Coupon Payments = FV(8%,5,90,0) = $527.99 or $528

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Step 3: Calculate Total Proceeds

The total proceeds from the bond is determined as below:

Total Proceeds = Price of Bond with 35 Years to Maturity + Future Value of Coupon Payments = 903.56 + 527.99 = $1,431.55

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Step 4: Calculate Expected Rate of Return of the 40-Year Bond

The expected rate of return of the 40-year bond is calculated as follows:

Expected Rate of Return of 40 Year Bonds = (1+Five-Year Rate of Return)^(1/Periods) - 1

Here, Five Year Rate of Return = (Total Proceeds/Current Selling Price) - 1 = (1,431.55/1,057.42) - 1 = 35.38% and Periods = 5

Substituting these values in the above formula, we get,

Expected Rate of Return of 40 Year Bonds = (1+35.38%)^(1/5) - 1 = 6.25% (answer for Part 1)

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Part 2)

Step 1: Calculate Price of Bond with 25 Years to Maturity

The price of the bond can be calculated with the use of PV (Present Value) function/formula of EXCEL/Financial Calculator. The function/formula for PV is PV(Rate,Nper,PMT,FV) where Rate = Interest Rate (here, Yield to Maturity), Nper = Period, PMT = Payment (here, Coupon Payment) and FV = Future Value (here, Face Value of Bonds).

Here, Rate = 9.5%, Nper = 25, PMT = 1,000*8.5% = $85 and FV = $1,000

Using these values in the above function/formula for PV, we get,

Price of Bond with 25 Years to Maturity = PV(9.5%,25,85,1000) = $905.62

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Step 2: Calculate Future Value of Coupon Payments at 8% Interest Rate for 5 Years

The future value of coupon payments can be calculated with the use of FV (Future Value) function/formula of EXCEL/Financial Calculator. The function/formula for FV is FV(Rate,Nper,PMT,PV) where Rate = Interest Rate, Nper = Period, PMT = Payment (here, Coupon Payment) and PV = Present Value.

Here, Rate = 8%, Nper = 5, PMT = 1,000*8.5% = $85 and PV = 0

Using these values in the above function/formula for FV, we get,

Future Value of Coupon Payments = FV(8%,5,85,0) = $498.66

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Step 3: Calculate Total Proceeds

The total proceeds from the bond is determined as below:

Total Proceeds = Price of Bond with 25 Years to Maturity + Future Value of Coupon Payments = 905.62 + 498.66 = $1,404.29

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Step 4: Calculate Expected Rate of Return of the 30-Year Bond

The expected rate of return of the 30-year bond is calculated as follows:

Expected Rate of Return of 30 Year Bonds = (1+Five-Year Rate of Return)^(1/Periods) - 1

Here, Five Year Rate of Return = (Total Proceeds/Current Selling Price) - 1 = (1,404.29/1,069.5) - 1 = 31.30% and Periods = 5

Substituting these values in the above formula, we get,

Expected Rate of Return of 30 Year Bonds = (1+31.30%)^(1/5) - 1 = 5.60% (answer for Part 2)