Example Question - present value formula
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Comparing Present Value of $8000 in 10 Years with $6433 Now
The question asks you to find the present value of $8000 in 10 years given an interest rate of 2.2% compounded quarterly. To decide which is larger, $6433 now or $8000 in 10 years, we need to calculate the present value of the $8000 using the present value formula for compound interest. The present value formula is given by: \[ PV = \frac{FV}{(1 + \frac{r}{n})^{n \times t}} \] where: PV = present value FV = future value ($8000) r = annual interest rate (2.2% or 0.022) n = number of times the interest is compounded per year (quarterly, so 4 times a year) t = number of years (10) Now let's calculate PV: \[ PV = \frac{8000}{(1 + \frac{0.022}{4})^{4 \times 10}} \] \[ PV = \frac{8000}{(1 + 0.0055)^{40}} \] \[ PV = \frac{8000}{(1.0055)^{40}} \] Using a calculator: \[ PV \approx \frac{8000}{(1.0055)^{40}} \approx \frac{8000}{2.48832} \approx 3215.77 \] Therefore, the present value of $8000 in 10 years at an interest rate of 2.2% compounded quarterly is approximately $3215.77. Now, comparing the present value of $8000 in 10 years ($3215.77) with $6433 now, it is clear that $6433 now is larger. Please note: The final answer has been rounded to the nearest cent as requested.
Calculating Present Value for Compound Interest Investment
To determine the present value required for an investment to grow to a future amount, we will use the present value formula for compound interest: \( PV = \frac{FV}{{(1 + r/n)}^{(nt)}} \) where: - PV = present value (the amount to be invested now) - FV = future value (the desired accumulated amount) - r = annual interest rate (as a decimal) - n = number of times the interest is compounded per year - t = number of years the money is invested Given information: - FV = $110,000 - r = 2% or 0.02 (as a decimal) - n = 4 (since the interest is compounded quarterly) - t = 4 years Plugging these values into the formula gives us the present value: \( PV = \frac{110,000}{{(1 + 0.02/4)}^{(4*4)}} \) \( PV = \frac{110,000}{{(1 + 0.005)}^{16}} \) \( PV = \frac{110,000}{{1.005}^{16}} \) Now we will calculate \(1.005^{16}\) and then divide 110,000 by this result: \( 1.005^{16} = 1.082856 \) (rounded to six decimal places) So, \( PV = \frac{110,000}{1.082856} \) \( PV ≈ 101,576.65 \) Therefore, the amount to be invested now, or the present value needed, is approximately $101,576.65.
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