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:

$A=P(1+\frac{r}{m})^{t\cdot m}$

a) For this case we have:

r = 0.05

m = 12 (monthly compound)

t = 10 years

A = 2000

Then, we can calculate P as:

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

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:

$\begin{gathered} 2000=P\cdot e^{0.04\cdot10} \\ 2000=P\cdot e^{0.4} \\ P=\frac{2000}{e^{0.4}} \\ P\approx\frac{2000}{1.4918} \\ P\approx1340.64 \end{gathered}$

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

Answer:

a) $1214.32

b) $1340.64