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Sagot :
To determine how much more money is required to fund an ordinary annuity paying $340 per month for 25 years compared to 15 years when the money can earn 8.4% compounded monthly, follow these steps:
### Step 1: Understand the Formula for Present Value of An Annuity
The present value of an annuity (PVA) can be calculated using the formula:
[tex]\[ PV = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \][/tex]
where:
- \( P \) is the annuity payment
- \( r \) is the monthly interest rate
- \( n \) is the total number of payments
### Step 2: Convert the Annual Interest Rate to Monthly
Given the annual interest rate is 8.4% compounded monthly:
[tex]\[ r = \frac{8.4\%}{12} = \frac{0.084}{12} = 0.007 \][/tex]
### Step 3: Calculate the Present Value for 25 Years
For 25 years, the number of monthly payments (\( n_{25} \)):
[tex]\[ n_{25} = 25 \times 12 = 300 \][/tex]
Using the annuity payment of $340, the present value for 25 years:
[tex]\[ PV_{25} = 340 \times \left( \frac{1 - (1 + 0.007)^{-300}}{0.007} \right) \][/tex]
### Step 4: Calculate the Present Value for 15 Years
For 15 years, the number of monthly payments (\( n_{15} \)):
[tex]\[ n_{15} = 15 \times 12 = 180 \][/tex]
Using the annuity payment of $340, the present value for 15 years:
[tex]\[ PV_{15} = 340 \times \left( \frac{1 - (1 + 0.007)^{-180}}{0.007} \right) \][/tex]
### Step 5: Determine How Much More Money is Required
Next, calculate the difference between the present value required for 25 years and the present value required for 15 years:
[tex]\[ \text{More money required} = PV_{25} - PV_{15} \][/tex]
### Step 6: Interpret the Results
Given the computed values:
- The present value for 25 years \( PV_{25} \) is approximately $42,579.87
- The present value for 15 years \( PV_{15} \) is approximately $34,733.34
So, the more money required is:
[tex]\[ \text{More money required} = 42579.87 - 34733.34 = 7846.53 \][/tex]
### Conclusion
Therefore, [tex]$7,846.53 more is required to fund an ordinary annuity paying $[/tex]340 per month for 25 years than for 15 years.
### Step 1: Understand the Formula for Present Value of An Annuity
The present value of an annuity (PVA) can be calculated using the formula:
[tex]\[ PV = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \][/tex]
where:
- \( P \) is the annuity payment
- \( r \) is the monthly interest rate
- \( n \) is the total number of payments
### Step 2: Convert the Annual Interest Rate to Monthly
Given the annual interest rate is 8.4% compounded monthly:
[tex]\[ r = \frac{8.4\%}{12} = \frac{0.084}{12} = 0.007 \][/tex]
### Step 3: Calculate the Present Value for 25 Years
For 25 years, the number of monthly payments (\( n_{25} \)):
[tex]\[ n_{25} = 25 \times 12 = 300 \][/tex]
Using the annuity payment of $340, the present value for 25 years:
[tex]\[ PV_{25} = 340 \times \left( \frac{1 - (1 + 0.007)^{-300}}{0.007} \right) \][/tex]
### Step 4: Calculate the Present Value for 15 Years
For 15 years, the number of monthly payments (\( n_{15} \)):
[tex]\[ n_{15} = 15 \times 12 = 180 \][/tex]
Using the annuity payment of $340, the present value for 15 years:
[tex]\[ PV_{15} = 340 \times \left( \frac{1 - (1 + 0.007)^{-180}}{0.007} \right) \][/tex]
### Step 5: Determine How Much More Money is Required
Next, calculate the difference between the present value required for 25 years and the present value required for 15 years:
[tex]\[ \text{More money required} = PV_{25} - PV_{15} \][/tex]
### Step 6: Interpret the Results
Given the computed values:
- The present value for 25 years \( PV_{25} \) is approximately $42,579.87
- The present value for 15 years \( PV_{15} \) is approximately $34,733.34
So, the more money required is:
[tex]\[ \text{More money required} = 42579.87 - 34733.34 = 7846.53 \][/tex]
### Conclusion
Therefore, [tex]$7,846.53 more is required to fund an ordinary annuity paying $[/tex]340 per month for 25 years than for 15 years.
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