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 $$ \$ 2219.63 $$

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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!
The present value of the investment is approximately \$2120.54.

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 \( \$ 2219.63 \)

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