Calculate the future value of an ordinary annuity consisting of monthly payments of $\$470$ for five years. The rate of return was $9.3\%$ compounded monthly for the first two years, and will be $7.2\%$ compounded monthly for the last three years. (Do not round intermediate calculations and round your final answer to $2$ decimal places.) Future value

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Grade 8

Compound interest

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Q. Calculate the future value of an ordinary annuity consisting of monthly payments of

\$470

for five years. The rate of return was

9.3\%

compounded monthly for the first two years, and will be

7.2\%

compounded monthly for the last three years. (Do not round intermediate calculations and round your final answer to

2

decimal places.) Future value

Calculate FV first two years: Calculate the future value of the annuity for the first two years with a $9$ . $3$ % annual interest rate compounded monthly. $\newline$ We have: $\newline$ Monthly payment (PMT) = $\$470$ $\newline$ Number of payments for the first two years (n) = $2$ years $* 12$ months/year = $24$ months $\newline$ Monthly interest rate (i) = $9.3\%$ annual rate / $12$ months = $0.093 / 12$ $\newline$ Future value of an annuity formula: $FV = PMT \times (((1 + i)^n - 1) / i)$ $\newline$ Let's calculate the future value for the first two years: $\newline$ $FV\_{first\_two\_years} = 470 \times (((1 + 0.093/12)^{24} - 1) / (0.093/12))$
Accumulated value first two years: Calculate the accumulated value of the annuity for the first two years. $\newline$ $FV_{\text{first two years}} = 470 \times \left(\left(1 + \frac{0.093}{12}\right)^{24} - 1\right) / \left(\frac{0.093}{12}\right)$ $\newline$ $FV_{\text{first two years}} \approx 470 \times \left(\left(1 + 0.00775\right)^{24} - 1\right) / 0.00775$ $\newline$ $FV_{\text{first two years}} \approx 470 \times \left(1.19985 - 1\right) / 0.00775$ $\newline$ $FV_{\text{first two years}} \approx 470 \times \left(0.19985 / 0.00775\right)$ $\newline$ $FV_{\text{first two years}} \approx 470 \times 25.7871$ $\newline$ $FV_{\text{first two years}} \approx \$12,119.94$
Calculate FV last three years: Calculate the future value of the annuity for the last three years with a $7.2\%$ annual interest rate compounded monthly. $\newline$ We have: $\newline$ Monthly payment (PMT) = $\$470$ $\newline$ Number of payments for the last three years (n) = $3$ years $* 12$ months/year = $36$ months $\newline$ Monthly interest rate (i) = $7.2\%$ annual rate / $12$ months = $0.072 / 12$ $\newline$ Future value of an annuity formula: $FV = PMT \times (((1 + i)^n - 1) / i)$ $\newline$ Let's calculate the future value for the last three years: $\newline$ $FV\_{last\_three\_years} = 470 \times (((1 + 0.072/12)^{36} - 1) / (0.072/12))$
Accumulated value last three years: Calculate the accumulated value of the annuity for the last three years. $\newline$ $FV\_{\text{last three years}} = 470 \times \left(\left(1 + \frac{0.072}{12}\right)^{36} - 1\right) / \left(\frac{0.072}{12}\right)$ $\newline$ $FV\_{\text{last three years}} \approx 470 \times \left(\left(1 + 0.006\right)^{36} - 1\right) / 0.006$ $\newline$ $FV\_{\text{last three years}} \approx 470 \times \left(1.243449 - 1\right) / 0.006$ $\newline$ $FV\_{\text{last three years}} \approx 470 \times \left(0.243449 / 0.006\right)$ $\newline$ $FV\_{\text{last three years}} \approx 470 \times 40.57483$ $\newline$ $FV\_{\text{last three years}} \approx \$\$$ $19$ , $070$ . $07$ \)
Calculate total FV: Calculate the total future value of the annuity by adding the future value of the first two years to the future value of the last three years. $\newline$ We have: $\newline$ $FV_{\text{first two years}} \approx \$12,119.94$ $\newline$ $FV_{\text{last three years}} \approx \$19,070.07$ $\newline$ Total future value ( $FV_{\text{total}}$ ) = $FV_{\text{first two years}} + FV_{\text{last three years}}$ $\newline$ $FV_{\text{total}} \approx \$12,119.94 + \$19,070.07$
Final total FV: Calculate the final total future value of the annuity. $\newline$ $FV_{\text{total}} \approx \$(12,119.94) + \$(19,070.07)$ $\newline$ $FV_{\text{total}} \approx \$(31,190.01)$