Exercise on Bond Valuation. An investor has two bonds in her portfolio that have

ID: 2717191 • Letter: E

Question

Exercise on Bond Valuation.

An investor has two bonds in her portfolio that have a face value of $1000 and pay a 10% annual coupon. Bond A matures in 15 years, while Bond B matures in 1 year.

What will the value of each bond be if the going interest rate is 5%, 8%, and 12%. ? Assume that only one more interest payment is to be made on Bond B at its maturity, and that 15 more payments are to be made on bond A.

Why does the longer-term bond price vary more than the price of the shorter-term bond when interest rates change ?

Please show your work solve problem in detail.

Explanation / Answer

Answer

YTM

Price of Bond A

Price of Bond B

1518.98

1047.62

1171.19

1018.52

12%

863.78

982.14

Prices of the longer term bonds varies more than the prices of shorter term bonds due to the discounting of coupon payments involved. The present value of coupon payments increases with the time to maturity.

Bond A

Face Value =$ 1000

Annual coupon rate = 10% pa

Annual coupon amount = 1000 * 10% = 100

Time to maturity = 15 years

If YTM is 5%, price of the bond is

Price = 100 * [(1-(1/(1+0.05)^15))/0.05] + 1000/((1+0.05)^15

Price = 100 * [(1-(1/(2.078928)/0.05] + 1000/2.078928

= 100 * [(1-0.481017)/0.05] + 1000 * 0.481017

= 100 * (0.518983/0.05) + 1000 * 0.481017

= 100 * 10.37966 + 1000 * 0.481017

= 1037.966 + 481.0171

= 1518.983 or 1518.98 (rounded off)

If YTM is 8%, price of the bond is

Price = 100 * [(1-(1/(1+0.08)^15))/0.08] + 1000/((1+0.08)^15

= 100 * [(1-(1/3.172169)/0.08] + 1000/3.172169

= 100 * (1-0.315242)/0.08 + 1000 * 0.315242

= 100 * (0.684758/0.08) + 1000 * 0.315242

= 100 * 8.559479 + 1000 * 0.315242

= 855.9479 + 315.2417

= 1171.19

If YTM is 12%, price of the bond is

Price = 100 * [(1-(1/(1+0.12)^15))/0.12] + 1000/((1+0.12)^15

= 100 * [(1-(1/5.473566)/0.12] + 1000/5.473566

= 100 * (1-0.182696)/0.12 + 1000 * 0.182696

= 100 * (0.817304/0.12) + 1000 * 0.182696

= 100 * 6.810864 + 1000 * 0.182696

= 681.0864 + 182.6963

= 863.7827 or 863.78 (rounded off)

Bond B

Face Value = $ 1000

Annual Coupon rate = 10%

Annual coupon amount = 100

Time to maturity = 1 year

If YTM = 5%, then price of the bond is

Price = 100/1.105 + 1000/1.05

= 95.238095 + 952.38095

= 1047.6190 or 1047.62 (rounded off)

If YTM = 8%, then price of the bond is

Price = 100/1.08 + 1000/1.08

= 92.59259 + 925.92593

= 1018.5185 or 1018.52 (rounded off)

If YTM = 12%, then price of the bond is

Price = 100/1.12 + 1000/1.12

= 89.2857 + 892.8571

= 982.1428 or 982.14 (rounded off)

YTM