A used piece of rental equipment has 3½ years of useful life remaining. When rented, the equipment brings in $300 per month (paid at the beginning of the month). If the equipment is sold now and money is worth 5%, compounded monthly, what must the selling price be to recoup the income that the rental company loses by selling the equipment "early"?
(a) Decide whether the problem relates to an ordinary annuity or an annuity due.
1
annuity due
ordinary annuity
.
(b) Solve the problem.
The selling price= the present value of
the annuity due.
A(42)=300*(1-v^42)/(1-v) where v=1/(1+i)
i=5%/12
To determine whether the problem relates to an ordinary annuity or an annuity due, we need to consider when the cash flows occur. In an ordinary annuity, the cash flows occur at the end of each period. In an annuity due, the cash flows occur at the beginning of each period.
In this problem, the equipment rental income is received at the beginning of each month. Therefore, the problem relates to an annuity due.
To solve the problem, we need to calculate the present value of the future rental income using the formula for the present value of an annuity due.
The rental income is $300 per month, and there are 3½ years of useful life remaining, which is equivalent to 42 months.
The interest rate is 5% compounded monthly. To convert the annual interest rate to a monthly rate, we divide it by 12: 5% / 12 = 0.4167% (approximately).
Now, we can calculate the present value of the rental income:
PV = PMT × [(1 - (1 + r)^(-n)) / r]
Where:
PV = Present value
PMT = Periodic payment (monthly rental income)
r = Interest rate per period (monthly interest rate)
n = Number of periods (months)
PV = $300 × [(1 - (1 + 0.004167)^(-42)) / 0.004167]
Using a calculator, we can evaluate the formula to find the present value of the rental income.
Once we have the present value, that will be the selling price needed to recoup the income that the rental company loses by selling the equipment "early".