Question

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 the following returns?(a) 8% compounded daily(b) 8% compounded continuously

Answer 1

EXPLANATION:

We are given a zero-coupon bond that will be worth $10,000 if redeemed in 20 years time at an annual rate of 8% compounded;

(a) Daily

(b) Continuously

The formula for compounding annually is given as follows;

$A=P(1+r)^t$

Here the variables are;

$\begin{gathered} P=initial\text{ investment} \\ A=Amount\text{ after the period given} \\ r=rate\text{ of interest} \\ t=time\text{ period \lparen in years\rparen} \end{gathered}$

Note that this zero-coupon bond will yield an amount of $10,000 after 20 years at the rate of 8%. This means we already have;

$\begin{gathered} A=10,000 \\ r=0.08 \\ t=20 \end{gathered}$

(a) For interest compounded daily, we would use the adjusted formula which is;

$A=P(1+\frac{r}{365})^{t\times365}$

This assumes that there are 365 days in a year.

We now have;

$10000=P(1+\frac{0.08}{365})^{20\times365}$$10000=P(1.00021917808)^{7300}$$10000=P(4.95216415047)$

Now we divide both sides by 4.95216415047;

$P=\frac{10000}{4.95216415047}$$P=2019.31916959$

We can round this to 2 decimal places and we'll have;

$P=2019.32$

(b) For interest compounded continuously, we would use the special formula which is;

$A=Pe^{rt}$

Note that the variable e is a mathematical constant whose value is approximately;

$e=2.7183\text{ \lparen to }4\text{ }decimal\text{ }places)$$10000=Pe^{0.08\times365}$$10000=Pe^{29.2}$

With the use of a calculator we have the following value;

$\frac{10000}{e^{29.2}}=P$