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

Step-by-step 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+(r)/(365))^(t*365)`

This assumes that there are 365 days in a year.

We now have;

`10000=P(1+(0.08)/(365))^(20*365)`
`10000=P(1.00021917808)^(7300)`
`10000=P(4.95216415047)`

Now we divide both sides by 4.95216415047;

`P=(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*365)`
`10000=Pe^(29.2)`

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

`(10000)/(e^(29.2))=P`