The time value of money calculation can be performed using formula equations or online calculators.
The correct responses are;
- 3) The difference in principal is approximately $8,000
- The difference in interest earned is approximately $2,977.87
- 4) It is better to invest more money at the beginning of the 30 years
Reasons:
Option 1: Present value = 0
Amount invested per month, A = $25/month
The Annual Percentage Rate, APR, r = 3.25%
Number of years = 30
The future value of an annuity is given by the formula;
[tex]\displaystyle FV_{A} = \mathbf{A \cdot \left (\frac{ \left(1 + \frac{r}{m} \right)^{m\cdot t} - 1}{\frac{r}{m} } \right)}[/tex]
In option 1, m = 12 periods per year
Therefore;
[tex]\displaystyle FV_{A} = 25 \times \left (\frac{ \left(1 + \frac{0.0325}{12} \right)^{12 \times 30} - 1}{\frac{0.0325}{12} } \right) \approx \mathbf{15,209.3}[/tex]
Contribution = $25 × 12 × 30 = $9,000
Total interest earned = $15,209.3 - $9,000 = $6,209.3
Final balance = $15,209.3
Option 2: Present value = 0
Amount, A = $75/quarter
m = 4 periods per year
The Annual Percentage Rate, APR = 4.00%
Therefore;
The effective interest rate is therefore;
[tex]\displaystyle r_{eff} = \left(1 + \frac{0.04}{4} \right)^4 - 1 \approx \mathbf{0.04060401}[/tex]
[tex]\displaystyle FV_{A} = 75 \times \left (\frac{ \left(1 + \frac{0.04060401}{4} \right)^{4 \times 30} - 1}{\frac{0.04060401}{4} } \right) \approx 17,437.7[/tex]
Using an online calculator, FV = $17,467.04
Contribution = $75 × 4 × 30 = $9,000
Total interest earned = $17,467.04 - $9,000 = $8,467.04
Final balance = $17,467.04
Option 3: Present value = $1,000
APR = 6.25%
m = 12 period per year
Number of years, t = 30 years
Therefore;
[tex]\displaystyle FV = \left (1 + \frac{0.0625}{12} \right)^{12 \times 30} \approx \mathbf{6,489.17}[/tex]
Contribution = $1,000
Total interest earned = $6,489.17 - $1,000 = $5,489.17
Final balance = $6,489.17
The table of values is therefore;
- [tex]\begin{tabular}{|c|c|c|c|}Option \# &Contribution &Total Interest Earned&Final Balance\\1&\$9,000&\$6,209.3 & \$15,209.3\\2&\$9,000&\$8,467.04 &\$17,467.04\\3&\$1,000&\$5,489.17&\$6,489.17\end{array}\right][/tex]
1) The option that has the least amount invested are option 3
Option 3 investment plan is a present value of $1,000, invested for 30 years at 6.25% APR compounded monthly.
2) Option 2 yielded the highest amount at the end of 30 years, given that the APR is higher than the APR for option 1, although the amount invested over the period are the same.
The basis of option 2 investment plan is $75 invested quarterly at 4.00% APR compounded monthly for 30 years.
3) The difference in the principal invested for the highest and lowest final balance is $9,000 - $1,000 = $8,000
The difference in the interest earned is; $8,467.04 - $5,489.17 = $2,977.87
4) In option 1 the present value is zero, therefore zero amount was invested at the beginning.
The interest to investment ration is 6,209.3:9,000 ≈ 0.7:1
In option 3, all the money was invested at the beginning.
The interest to investment ratio of option 3 is; 5,489.17:1,000 ≈ 5.5:1
Given that the interest to investment ratio, which is the return on investment is larger when more money is saved at the beginning as in option 3, it is better to invest more money at the beginning.
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