A house and lot can be acquired by a downpayment of 500,000 and yearly payment of 100000 at the end of each year for the period of 10 years starting at the end of 5 years from the date of purchase. If money is worth 14% comounded annually, what is the cash price of the property?
Question
A house and lot can be acquired by a downpayment of 500,000 and yearly payment of 100000 at the end of each year for the period of 10 years starting at the end of 5 years from the date of purchase. If money is worth 14% comounded annually, what is the cash price of the property?
Solution
To solve this problem, we need to calculate the present value of all future payments. The present value (PV) is the current worth of a future sum of money or stream of cash flows given a specified rate of return.
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First, calculate the present value of the down payment. Since the down payment is made immediately, its present value is the same as its face value, which is 500,000.
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Next, calculate the present value of the yearly payments. The yearly payments start at the end of the 5th year and continue for 10 years. The formula for the present value of an ordinary annuity is:
PV = Pmt * [(1 - (1 + r)^-n) / r]
where: Pmt = payment amount per period = 100,000 r = interest rate per period = 14% = 0.14 n = number of periods = 10
However, since the payments start at the end of the 5th year, we need to discount the present value of the annuity back to the present. This is done by dividing the present value of the annuity by (1 + r)^t, where t is the number of periods before the payments start.
So, the present value of the yearly payments is:
PV = 100,000 * [(1 - (1 + 0.14)^-10) / 0.14] / (1 + 0.14)^5
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Finally, add the present value of the down payment and the present value of the yearly payments to get the cash price of the property.
Cash Price = PV of Down Payment + PV of Yearly Payments
Please note that this calculation assumes that the interest rate is compounded annually. If it is compounded more frequently, the calculation would be slightly different.
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