A zerocoupon bond is a bond that is sold now at a discount a

A zero-coupon bond is a bond that is sold now at a discount and will pay its face value at the time when it matures; no interest payments are made. A zero-coupon bond can be redeemed in 20 years for $10,000.How much should you be willing to pay for it now if you want a return of:

(a) 11% compounded daily you hould pau?

(b) 11% compounded continuously you should pay?

Solution

(a). The formula for compound interest is F = P(1+r/n)^nt ,where P is the principal/initial amount, r is the rate of interest in decimals, t is the number of years ,n is the number of times the interest is compounded in an year, and F is the future value.

Here, F = $10000, r = (11/100)*1/365 , n = 365 and t = 20 so that 10000= P(1+11/36500)^365*20 = P(36511/36500)⁷³⁰⁰ = P*9.022022544. Then, P = 10000/9.022022544 = $1108.40 (on rounding off to the nearest cent). Thus, one should be willing to pay $1108.40 for the zero-coupon bond if a return of 11% compounded daily is desired.

(b). The formula for continuous compounding is F = Pe^rt, where P is the principal/initial amount, r is the rate of interest in decimals, t is the number of years and F is the future value.

Here, r = 11/100 = 0.11, t =20 and F =$10000 so that 10000=Pe^0.11*20. Hence, P= 10000/e^2.2 =10000/ 9.0250135 = $1108.03(on rounding off to the nearest cent). Thus, one should be willing to pay $1108.03 for the zero-coupon bond if a return of 11% compounded continuously is desired.