Question

A zero coupon bond can be redeemed in 10 years for $2000.How much should you be willing to pay for it now withreturn a) 5% compounded monthly b) 4% compoundedcontinuously.

Answer 1

The price we would be willling to pay is equivalent to the present value (P) calculated using the discount rate for each case.

The future value (A) is equal to $2000 and the period is t = 10 years.

We can relate the present value with the future value with the formula:

`A=P(1+(r)/(m))^(t\cdot m)`

a) For this case we have:

r = 0.05

m = 12 (monthly compound)

t = 10 years

A = 2000

Then, we can calculate P as:

`\begin{gathered} P=(A)/((1+(r)/(m))^(t\cdot m)) \\ P=(2000)/((1+(0.05)/(12))^(10\cdot12)) \\ P=(2000)/((1.004167)^(120)) \\ P=(2000)/(1.647) \\ P=1214.32 \end{gathered}`

The price we would be willing to pay today at this discount rate is $1214.32.

b) In this case, r is r = 0.04 and it is compounded continously. In this case, we have to use another equation for continously compounded interest:

`A=P\cdot e^(r\cdot t)`

For this case, we have:

`\begin{gathered} 2000=P\cdot e^(0.04\cdot10) \\ 2000=P\cdot e^(0.4) \\ P=(2000)/(e^(0.4)) \\ P\approx(2000)/(1.4918) \\ P\approx1340.64 \end{gathered}`

The price we would be willing to pay today at this discount rate is $1340.64.

Answer:

a) $1214.32

b) $1340.64