Art is a self-employed installer of home entertainment systems, and he drives his car frequently to installation locations. Art
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Answer: <em>The lowest interest rate at which the present value of the second contract exceeds that of the first is </em><em>a. 7 percent</em><em>.</em>
Explanation:
<em>Calculating present values is a useful way to compare cases where money is to be received in the future. The higher the present value (when comparing cases where you get money), the better</em>. To calculate it, we make use of the next formula:
Where PV: Present value,
C: Cash flow at a given period,
r: Interest rate, and
n: Number of periods that will have passed (in this case, we are talking about years).
Now, since we are getting money twice in each case (the first payment one year from today, and the final payment two years from today), we can restructure our present value formula to include these two payments. We will get something like this:
<em>Notice how each fraction represents one of the payments received, with one having an 'n' of 1 year, and the other one having an 'n' of 2 years. C₁ and C₂ represent the first and the second payment, respectively.</em>
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Now that we have our completed formula, let's review each contract's present value (PV) with the lowest interest rate (7%), just to see how it turns out. <em>Remember that 7% equals 0.07 in any formula</em>:
<em>Contract A) This one gives her $100,000 one year from today and $100,000 two years from today</em><em>.</em>
So Contract A's present value at 7% interest rate would be equal to <em>$180801.817</em>.
<em>Contract B) The second one gives her $132,000 one year from today and $66,000 two years from today</em><em>.</em>
So Contract B's present value at 7% interest rate would be equal to <em>$181011.442, </em><em><u>which exceeds that of Contract A</u></em><em>.</em>
<em>Since among our options of interest rates, 7 percent is the lowest one, and, with this taken into account, the present value of the second contract (Contract B) exceeded that of the first (Contract A), </em><em>the answer is a. 7 percent</em><em>.</em>