Question

Consider a 4-years bond with a 8% annual coupon rate and semi-annual payments. Let us suppose that the zero coupon curve rate today with annual compounding is given by the one in Table 1 . (a) Calculate the discount factors for all the previous maturities and then the bond price. (b) Calculate the equivalent continuous compounding rates. What do you expect as result for the bond price with these rates? Should it be lower, higher or equal to the one in part (a)? Why? The equivalent continuous compounding rates should be lower than the annual rates. Why?

Answer 1

To calculate the present value of a two-year bond with an 8% coupon rate, we discount the future cash flows at the given discount rate. If rates rise to 11%, the present value is reduced due to increased discounting reflecting interest rate risk.

Step-by-step explanation:

To calculate the present value of a bond, we analyze the future cash flows it provides and discount them back to their present value using a discount rate. With a two-year bond issued for $3,000 at an 8% interest rate, the bond pays $240 in interest each year. The future cash flows over two years are $240 in the first year and $3,240 ($240 interest + $3,000 principal) at the end of the second year.

Using the initial discount rate of 8%:

• Year 1 PV = $240 / (1 + 0.08) = $222.22
• Year 2 PV = $3,240 / (1 + 0.08)2 = $2,777.78
• Total PV = $222.22 + $2,777.78 = $3,000

If the interest rates rise and the new discount rate is 11%, the calculations are:

• Year 1 PV = $240 / (1 + 0.11) = $216.22
• Year 2 PV = $3,240 / (1 + 0.11)2 = $2,625.23
• Total PV = $216.22 + $2,625.23 = $2,841.45

Thus, the present value of the bond decreases when the discount rate increases from 8% to 11% because the future cash flows are discounted more heavily. This reflects the interest rate risk inherent in bonds.