Answer:
<em>The sum of the present value of the revenues from buying a bicycle is the closest to the price of a bicycle when we use an interest rate of:</em>
<em>d. 8 percent</em>.
Explanation:
<em>Making use of the present value concept is a useful way to compare cases where money is to be received in the future</em>. The higher the present value, the better.
- In this case, <em>we don't know the sum of the </em><em>present value</em><em> of the revenues from buying one bicycle</em>, so <em>we have to determine it by using different interest rates (a. 5%, b. 6%, c. 7%, and d. 8%</em><em>).</em>
- <em>Each time we determine a present value, we will compare it with the price of a bicycle (</em><em>$249.66</em><em>), </em>to see at <em>which interest rate is the present value closest to the price of the bicycle.</em>
- <em>Since we can sell the used bike for </em><em>$25</em><em> at the end of the second year, </em><em>we can add this money to the revenues of the second year</em><em> (</em><em>$115</em><em>), t</em>o simplify calculations. <em>We should then get a total of </em><em>$115+$25=$140</em><em> by the end of the second year</em>.
- The following formula is the one that we are going to use to determine our present value for each interest rate:
Where P: <em>Present value</em> (what we are going to determine),
C: <em>Cash flow at a given period of time</em> (what we are going to receive by the end of a given year; in this case, $140 for the first year, C₁, and another $140 for the second year, C₂),
r:<em> Interest rate</em> (we will use the different mentioned options of 5%, 6%, 7%, and 8%, or 0.05, 0.06, 0.07, and 0.08, respectively, until we get a present value that is the closest possible to the price of a bicycle), and
n: <em>Number of periods of time</em> (in this case, n=1 for the first year, and n=2 for the second year).
Now,<em> since we are getting money twice </em>(each year for two years), we can add the term for the second year to the formula, and we will get something like <em>this</em>:
We could further simplify our formula but it's better not complicate ourselves so much.
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Let's now start determining our present value of the revenues with the interest rate of 5%, or 0.05:
So <em>we got a present value of </em><em>P=$260.32 while using an interest rate of 5%</em><em>, which is pretty close to the price of $249.66, but maybe not close enough</em>. We should determine the difference between the present value and the price like this:
<em>The less difference there is, the closer we get to the price of the bicycle</em>. Let's use the following interest rate now.
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6%, or 0.06:
In the 6% interest rate case, our present value is of <em>P=$256.68</em>, which got us closer to the price of $249.66. We should calculate its difference with the price:
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7%, or 0.07:
The 7% case gave us a present value of <em>P=$253.12</em>. Its difference with the price is:
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8%, or 0.08:
This time, by using an interest rate of 8%, we got to the present value of <em>P=$249.66</em>, <em>which is exactly the price of a bicycle</em>. The difference in this case is therefore 0.
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<em>So the sum of the present value of the revenues from buying a bicycle is the closest to the price of a bicycle when we use an interest rate of</em>:
<em>d. 8 percent</em>.