a zero - coupon bond is a bond that is sold now at a discount and pay its face value at the time when it…

a zero - coupon bond is a bond that is sold now at a discount and 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 the following returns? (a) 12% compounded monthly (b) 12% compounded continuously. (a) use the present value formula, $p = acdot(1+\frac{r}{n})^{-nt}$. make the necessary substitutions and solve for p.

a zero - coupon bond is a bond that is sold now at a discount and 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 the following returns? (a) 12% compounded monthly (b) 12% compounded continuously. (a) use the present value formula, $p = acdot(1+\frac{r}{n})^{-nt}$. make the necessary substitutions and solve for p.

Answer

Explanation:

Step1: Identify formula for compound - interest

The present - value formula for compound interest is $P = A(1+\frac{r}{n})^{-nt}$, where $A$ is the future value, $r$ is the annual interest rate (in decimal), $n$ is the number of times interest is compounded per year, and $t$ is the number of years. For part (a), $A = 10000$, $r=0.12$, $n = 12$ (compounded monthly), and $t = 20$.

Step2: Calculate present value for part (a)

Substitute the values into the formula: [ \begin{align*} P&=10000\left(1+\frac{0.12}{12}\right)^{-12\times20}\ &=10000(1 + 0.01)^{-240}\ &=10000\times(1.01)^{-240}\ \end{align*} ] Using a calculator, $(1.01)^{-240}\approx0.090757$, so $P\approx10000\times0.090757 = 907.57$.

Step3: Identify formula for continuous compounding

The present - value formula for continuous compounding is $P=Ae^{-rt}$, where $A$ is the future value, $r$ is the annual interest rate (in decimal), and $t$ is the number of years. For part (b), $A = 10000$, $r = 0.12$, and $t=20$.

Step4: Calculate present value for part (b)

Substitute the values into the formula: [ \begin{align*} P&=10000e^{-0.12\times20}\ &=10000e^{-2.4}\ \end{align*} ] Using a calculator, $e^{-2.4}\approx0.090718$, so $P\approx10000\times0.090718=907.18$.

Answer:

(a) $$907.57$ (b) $$907.18$