Question 4 (1 point) You have an investment that will mature in 20 months with the val \( \$ 2500.00 \). You need some quick cash and decide to sell it today discount rate of \( 10 \% \) compounded quarterly. What is the cash val \( \$ 2219.63 \)

To find the present value of the investment, we can use the formula for present value (PV) when interest is compounded quarterly. The formula is: \[ PV = \frac{FV}{(1 + r/n)^{nt}} \] Where: - \( PV \) = Present Value - \( FV \) = Future Value (the amount the investment will mature to, which is \( \$2500.00 \)) - \( r \) = annual interest rate (as a decimal, so \( 10\% = 0.10 \)) - \( n \) = number of compounding periods per year (quarterly means \( n = 4 \)) - \( t \) = time in years (20 months = \( \frac{20}{12} \) years) Now, let's calculate the present value step by step. 1. Convert the time from months to years: \[ t = \frac{20}{12} \approx 1.6667 \text{ years} \] 2. Substitute the values into the formula: \[ PV = \frac{2500}{(1 + 0.10/4)^{4 \cdot 1.6667}} \] 3. Calculate \( r/n \): \[ r/n = \frac{0.10}{4} = 0.025 \] 4. Calculate \( nt \): \[ nt = 4 \cdot 1.6667 \approx 6.6668 \] 5. Now substitute these values into the formula: \[ PV = \frac{2500}{(1 + 0.025)^{6.6668}} \] 6. Calculate \( (1 + 0.025)^{6.6668} \): \[ (1.025)^{6.6668} \] 7. Finally, calculate the present value \( PV \). Let's perform the calculations. Calculate the value by following steps: - step0: Calculate: \(\frac{2500}{1.025^{6.6668}}\) - step1: Convert the expressions: \(\frac{2500}{\left(\frac{41}{40}\right)^{\frac{16667}{2500}}}\) - step2: Simplify the expression: \(\frac{2500}{\frac{41^{\frac{16667}{2500}}}{40^{\frac{16667}{2500}}}}\) - step3: Rewrite the expression: \(\frac{2500\times 40^{\frac{16667}{2500}}}{41^{\frac{16667}{2500}}}\) - step4: Simplify: \(\frac{2500\times 40^{6}\sqrt[2500]{40^{1667}}}{41^{\frac{16667}{2500}}}\) - step5: Simplify: \(\frac{2500\times 40^{6}\sqrt[2500]{40^{1667}}}{41^{6}\sqrt[2500]{41^{1667}}}\) - step6: Multiply by the Conjugate: \(\frac{2500\times 40^{6}\sqrt[2500]{40^{1667}}\times \sqrt[2500]{41^{833}}}{41^{6}\sqrt[2500]{41^{1667}}\times \sqrt[2500]{41^{833}}}\) - step7: Multiply the numbers: \(\frac{2500\times 40^{6}\sqrt[2500]{40^{1667}\times 41^{833}}}{41^{6}\sqrt[2500]{41^{1667}}\times \sqrt[2500]{41^{833}}}\) - step8: Multiply the numbers: \(\frac{2500\times 40^{6}\sqrt[2500]{40^{1667}\times 41^{833}}}{41^{7}}\) After performing the calculations, we find that the present value (PV) is approximately: \[ PV \approx 2120.54 \] This means that if you sell the investment today, you would receive approximately \( \$2120.54 \). However, you mentioned a cash value of \( \$2219.63 \). It seems there might be a discrepancy. Based on the calculations, the present value of the investment at a \( 10\% \) discount rate compounded quarterly for \( 20 \) months is approximately \( \$2120.54 \). If you have any further questions or need additional clarification, feel free to ask!