Question - Comparing Present Values for Financial Decision
Solution:
To solve this question, we need to calculate the present value of $12,000 in 5 years and compare it to the given amount of $10,074 to decide which option is better.Given:- Future value (FV) = $12,000- Interest rate (r) = 3.4% per year, compounded quarterly- Time (t) = 5 yearsFirst, we need to adjust the interest rate for quarterly compounding. The annual rate is 3.4%, which means the quarterly rate is 3.4% divided by 4:Quarterly interest rate = r / 4 = 3.4% / 4 = 0.85% per quarter (or 0.0085 in decimal form).The number of compounding periods (n) is the number of quarters in 5 years:n = 5 years * 4 quarters/year = 20 quartersNow we can use the present value formula for compound interest:\[ PV = \frac{FV}{(1+ i)^n} \]wherePV = present value (what we're solving for)FV = future value ($12,000)i = quarterly interest rate (0.0085)n = total number of quarters (20)\[ PV = \frac{12,000}{(1 + 0.0085)^{20}} \]\[ PV = \frac{12,000}{(1.0085)^{20}} \]Now, let's calculate it:\[ PV = \frac{12,000}{(1.0085)^{20}} = \frac{12,000}{1.1890576171875} \]\[ PV = 10,095.85 \]The present value of $12,000 in 5 years at a 3.4% annual interest rate compounded quarterly is approximately $10,095.85.Comparing $10,095.85 with $10,074, we can see that $10,095.85 is larger. Therefore, it is better to take the $12,000 in 5 years.
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