Comparing Present Values for Financial Decision
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 years
First, 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 quarters
Now we can use the present value formula for compound interest:
\[ PV = \frac{FV}{(1+ i)^n} \]
where
PV = 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.
