Answer :
To determine how much you need to invest today to have [tex]$20,000 in 5 years with an average annual return of 6%, we use the formula for finding the present value of a future amount. The formula can be stated as:
\[ \text{Future Value} = \text{Present Value} \times (1 + r)^n \]
Here:
- Future Value (FV): The amount of money you want in the future, which is $[/tex]20,000.
- Present Value (PV): The amount you need to invest today, which we are trying to find.
- [tex]\( r \)[/tex]: The annual interest rate as a decimal, which is 0.06 (6%).
- [tex]\( n \)[/tex]: The number of years the money will be invested, which is 5 years.
We rearrange the formula to solve for Present Value (PV):
[tex]\[ \text{Present Value} = \frac{\text{Future Value}}{(1 + r)^n} \][/tex]
Substituting the given values into the formula:
[tex]\[ \text{Present Value} = \frac{20000}{(1 + 0.06)^5} \][/tex]
From the given options, we need to find the equation that matches this rearranged formula:
- Option A: [tex]\(20000 = x(0.06)^5\)[/tex]
- Option B: [tex]\(20000 = \Phi(1.06)^5\)[/tex]
- Option C: [tex]\(y = 20000(1.06)^5\)[/tex]
- Option D: [tex]\(y = 20000(0.06)^6\)[/tex]
The correct option that matches our present value formula is:
Option B: [tex]\(20000 = \Phi(1.06)^5\)[/tex]
This equation correctly represents the relationship where the future value is divided by the growth factor [tex]\((1 + r)^n\)[/tex] to find the present value, [tex]\(\Phi\)[/tex].
\[ \text{Future Value} = \text{Present Value} \times (1 + r)^n \]
Here:
- Future Value (FV): The amount of money you want in the future, which is $[/tex]20,000.
- Present Value (PV): The amount you need to invest today, which we are trying to find.
- [tex]\( r \)[/tex]: The annual interest rate as a decimal, which is 0.06 (6%).
- [tex]\( n \)[/tex]: The number of years the money will be invested, which is 5 years.
We rearrange the formula to solve for Present Value (PV):
[tex]\[ \text{Present Value} = \frac{\text{Future Value}}{(1 + r)^n} \][/tex]
Substituting the given values into the formula:
[tex]\[ \text{Present Value} = \frac{20000}{(1 + 0.06)^5} \][/tex]
From the given options, we need to find the equation that matches this rearranged formula:
- Option A: [tex]\(20000 = x(0.06)^5\)[/tex]
- Option B: [tex]\(20000 = \Phi(1.06)^5\)[/tex]
- Option C: [tex]\(y = 20000(1.06)^5\)[/tex]
- Option D: [tex]\(y = 20000(0.06)^6\)[/tex]
The correct option that matches our present value formula is:
Option B: [tex]\(20000 = \Phi(1.06)^5\)[/tex]
This equation correctly represents the relationship where the future value is divided by the growth factor [tex]\((1 + r)^n\)[/tex] to find the present value, [tex]\(\Phi\)[/tex].
