Question

1) A bond will mature in 20 years. It has a 5% coupon rate and will pay annual coupons. If the bond has a face value of $1,000 and a 4% yield to maturity, what should be the price of the bond today? What if YTM goes up to 5%? What if YTM goes up to 6%?

(2) What would be the price of the bond above in (1) if the coupons were paid semiannually?

(3) What is the relationship between the coupon rate of a bond and the yield to maturity in terms of the bond’s price. In particular, when do we have a discount bond, a par bond, and a premium bond? Please explain the rational behind the relationship as if you are explaining this relationship to a novice of Finance.

(4) Both Bond Tom and Bond Jerry have 4% coupons, make semiannual payments, and have YTM of 3%. Bond Tom has two years to maturity, whereas Bond Jerry has 30 years to maturity. If the interest rates (YTM) suddenly rise by 2 percent point to 5%, what is the percentage change in the price of Bond Tom and Bond Jerry? Based on your answer, what can you say about the relationship between the interest rate risk and the time to maturity?

Answer 1

Since, multiple questions have been posted and Question 1) and Question 2) are parts of each other and both questions have multiple subparts, I have answered the first two questions.

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Question 1:

a)

Price of Bond Today

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 = 4%, Nper = 20, PMT = 1,000*5% = $50 and FV = $1,000

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

Bond Price Today = PV(4%,20,50,1000) = $1,135.90

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b)

Price of Bond if YTM Goes Up to 5%

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 = 5%, Nper = 20, PMT = 1,000*5% = $50 and FV = $1,000

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

Bond Price if YTM Goes Up to 5% = PV(5%,20,50,1000) = $1,000

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c)

Price of Bond if YTM Goes Up to 6%

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 = 6%, Nper = 20, PMT = 1,000*5% = $50 and FV = $1,000

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

Bond Price if YTM Goes Up to 6% = PV(6%,20,50,1000) = $885.30

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Question 2:

a)

Price of Bond Today

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 = 4%/2 = 2% , Nper = 20*2 = 40, PMT = 1,000*5%*1/2 = $25 and FV = $1,000 [we use 2 since the bond is semi-annual]

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

Bond Price Today = PV(2%,40,25,1000) = $1,136.78

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b)

Price of Bond if YTM Goes Up to 5%

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 = 5%/2 = 2.5%, Nper = 20*2 = 40, PMT = 1,000*5%*1/2 = $25 and FV = $1,000 [we use 2 since the bond is semi-annual]

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

Bond Price if YTM Goes Up to 5% = PV(2.5%,40,25,1000) = $1,000

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c)

Price of Bond if YTM Goes Up to 6%

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 = 6%/2 = 3%, Nper = 20*2 = 40, PMT = 1,000*5%*1/2 = $25 and FV = $1,000 [we use 2 since the bond is semi-annual]

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

Bond Price if YTM Goes Up to 6% = PV(3%,40,25,1000) = $884.43