Question

Bond value and

time—Changing

required returns  Personal Finance Problem   Lynn Parsons is considering investing in either of two outstanding bonds. The bonds both have

​$1,000

par values and

13​%

coupon interest rates and pay annual interest. Bond A has exactly

10

years to​ maturity, and bond B has

20

years to maturity.  

a.  Calculate the present value of bond A if the required rate of return​ is: (1)

10​%,

​(2)

13​%,

and​ (3)

16​%.

b.  Calculate the present value of bond B if the required rate of return​ is: (1)

10​%,

​(2)

13​%,

and​ (3)

16​%.

c. From your findings in parts a and

b​,

discuss the relationship between time to maturity and changing required returns.

d.  If Lynn wanted to minimize interest rate​ risk, which bond should she​ purchase? ​ Why?

Answer 1

Par Value = 1000 | Coupon Rate = 13%

Bond A Time to maturity = 10 years | Bond B Time to maturity = 20 years

a) PV of Bond A

1) Required rate of return = 10%

Since coupon rate is 13%, hence, Annual Coupon payment = 13% * 1000 = 130

To find the Present Value of a bond, we need to consider the equal annual coupon payments as an annuity and calculate its PV accordingly and the PV of final principal repayment will be calculated by simple discounting.

Formula for PV of Bond = (Coupon / R) * [1 - (1+R)^-T] + Prinicipal (Par Value) / (1+R)^T

Inputting the values in the formula-

PV = (130 / 10%) * [1 - (1+10%)^-10 + 1000 / (1+10%)¹⁰

PV of Bond A at 10% return = 798.79 + 385.54 = $ 1,184.34

2) Required rate of return = 13%

Using the Formula for PV of bond and changing R to 13%, we can calculate the PV of the bond

PV = (130 / 13%) * [1 - (1+13%)^-10 + 1000 / (1+13%)¹⁰

PV of Bond A at 13% return = 705.41 + 294.59 = $ 1,000 (Since Required rate and Coupon Rate are same, hence, price of bond is same as Par value)

3) Required rate of return = 16%

Using the Formula for PV of bond and changing R to 16%, we can calculate the PV of the bond

PV = (130 / 16%) * [1 - (1+16%)^-10 + 1000 / (1+16%)¹⁰

PV of Bond A at 16% return = 628.32 + 226.68 = $ 855

a) PV of Bond B

1) Required rate of return = 10%

Similar to Bond A, we can use PV of Bond formula used earlier, however, the Time period for Bond B is 20 years, hence, we will change that.

PV = (130 / 10%) * [1 - (1+10%)^-20 + 1000 / (1+10%)²⁰

PV of Bond B at 10% return = 1,106.76 + 148.64 = $ 1,255.41

2) Required rate of return = 13%

Using the Formula for PV of bond and changing R to 13%, we can calculate the PV of the bond

PV = (130 / 13%) * [1 - (1+13%)^-20 + 1000 / (1+13%)²⁰

PV of Bond B at 13% return = 913.22 + 86.78 = $ 1,000 (Since Required rate and Coupon Rate are same, hence, price of bond is same as Par value)

3) Required rate of return = 16%

Using the Formula for PV of bond and changing R to 16%, we can calculate the PV of the bond

PV = (130 / 16%) * [1 - (1+16%)^-20 + 1000 / (1+16%)²⁰

PV of Bond B at 16% return = 770.75 + 51.39 = $ 822.13

c) If we compare both bond A and B's Present value calculated in Part (a) and (b), we will see that at a rate of 10% Bond A with 10 years maturity has PV of 1,184.34, whereas, Bond B with 20 years maturity has PV of 1,255.41. This means that a bond with longer maturity period sees a higher price fluctuations with change in Interest rate, compared to one with shorter maturity period. For example, Bond A which is at Par value of 1000 moved to 1,184.34 and 855 with decrease and increase in interest rate by 3%, respectively, whereas Bond B which is at Par value of 1000 moved to 1,255.41 and 822.13 with decrease and increase in interest rate by 3%. This is the effect of Time value of money, where longer maturity gets discounted more number of times than shorter maturity which reflects in their Present values. Hence, longer maturity faces higher interest rate risk.

d) If Lynn want to minimize the interest rate risk, then, he should buy Bond A which has maturity of 10 years. As discussed in part (c), Bond A showed lesser price fluctuation with change in interest rate compared to Bond B. The reason behind this was Bond B's longer maturity than Bond A which makes Bond B's cashflows more risky. Therefore, Bond A faces less interest rate risk than Bond B.