Question

A 5% 20-year semi-annual bond offers an YTM of 5%. If the market interest rate will increase to 6.5% in one year (there will 19 years left to maturity then), what is the change in price that the bond will experience in dollars during the next year?

Answer 1

We will calculate bond price today, then one year after when the interest rate has increased to measure the price changes.

Bond price when YTM is 5%

Price of the bond could be calculated using below formula.

P = C/ 2 [1 - {(1 + YTM/2) ^2*n}/ (YTM/2)] + [F/ (1 + YTM/2) ^2*n]

Where,

                Face value (F) = $1000

                Coupon rate = 5%

                YTM or Required rate = 5%

                Time to maturity (n) = 20 years

                Annual coupon C = $50

Let's put all the values in the formula to find the bond current value

P = 50/ 2 [{1 - (1 + 0.05/2) ^-2*20}/ (0.05/ 2)] + [1000/ (1 + 0.05/2) ^2*20]

    = 25 [{1 - (1 + 0.025) ^ -40}/ (0.025)] + [1000/ (1 + 0.025) ^40]

    = 25 [{1 - (1.025) ^ -40}/ (0.025)] + [1000/ (1.025) ^40]

    = 25 [{1 - 0.37243}/ (0.025)] + [1000/ 2.68506]

    = 25 [0.62757/ 0.025] + [372.43116]

    = 25 [25.1028] + [372.43116]

    = 627.57 + 372.43116

    = 1000.00116

So price of the bond is $1000

Bond price when YTM is 6.5%

                Face value (F) = $1000

                Coupon rate = 5%

                YTM or Required rate = 6.5%

                Time to maturity (n) = 20 years

                Annual coupon C = $50

Let's put all the values in the formula to find the bond current value

P = 50/ 2 [{1 - (1 + 0.065/2) ^-2*20}/ (0.065/ 2)] + [1000/ (1 + 0.065/2) ^2*20]

    = 25 [{1 - (1 + 0.0325) ^ -40}/ (0.0325)] + [1000/ (1 + 0.0325) ^40]

    = 25 [{1 - (1.0325) ^ -40}/ (0.0325)] + [1000/ (1.0325) ^40]

    = 25 [{1 - 0.27823}/ (0.0325)] + [1000/ 3.5942]

    = 25 [0.72177/ 0.0325] + [278.22603]

    = 25 [22.20831] + [278.22603]

    = 555.20775 + 278.22603

    = 833.43378

So price of the bond is $833.43

$ Change in price of the bond = 1000 – 833.43 = 166.57

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