Question

What is the value today of a money machine that will pay $1,224.00 per year for 13.00 years? Assume the first payment is made 5.00 years from today and the interest rate is 7.00%.

Answer 1

The cash flow diagram for this problem looks like this:

$1,224 $1,224 $1,224 $1,224 CD 4 17 13 payments

The first step is to calculate the present value of 13 payments starting in year 5. The formula to calculate present value of an annuity is:

$PV=C\left [ \frac{1-\left ( 1+r \right )^{-n}}{r} \right ]$

Where

• C is the annuity
• r is the discount rate
• n is the period/no. of payments

Substituting the values, we get

$PV=1,224\left [ \frac{1-\left ( 1+0.07 \right )^{-13}}{0.07} \right ]=10,229.76$

but this value is currently at period 4. (The annuity formula calculates the present value for the period that is one period before the first cash flow)

So, in order to calculate its value at period 0, we need to discount this value using the formula

$PV=\frac{FV}{\left ( 1+r \right )^{n}}$

Where

• FV is the future value
• r is the discount rate
• n is the period

$PV=\frac{10,229.76}{\left ( 1+0.07 \right )^{4}}=7,804.238$

So, the value of this money machine is $7,804.238.