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How much money should be deposited today in an account that earns $3\%$ compounded semiannually so that it will accumulate to $\$12,000$ in three years? $\newline$ The amount of money that should be deposited is $\newline$ $\$\square$ . $\newline$ (Round up to the nearest cent.)

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Find probabilities using the binomial distribution

Full solution

Q. How much money should be deposited today in an account that earns

3\%

compounded semiannually so that it will accumulate to

\$12,000

in three years?

\newline

The amount of money that should be deposited is

\newline

\$\square

.

\newline

(Round up to the nearest cent.)

Identify Formula: Identify the formula to calculate the present value of an investment that will yield a future value at a given interest rate compounded semiannually. $\newline$ The formula for the present value (PV) is: $\newline$ $PV = \frac{FV}{(1 + \frac{r}{n})^{nt}}$ $\newline$ Where: $\newline$ - $PV$ = present value (the amount of money to be deposited today) $\newline$ - $FV$ = future value (the amount of money we want in the future, which is $\$12,000$ ) $\newline$ - $r$ = annual interest rate (which is $3\%$ or $0.03$ as a decimal) $\newline$ - $n$ = number of times the interest is compounded per year (which is $2$ for semiannual) $\newline$ - $t$ = number of years the money is invested (which is $PV$ $0$ years)
Plug Values: Plug the given values into the formula to calculate the present value. $\newline$ $FV = \$12,000$ $\newline$ $r = 0.03$ ( $3$ \%) $\newline$ $n = 2$ (semiannual compounding) $\newline$ $t = 3$ years $\newline$ $PV = \frac{12000}{(1 + 0.03/2)^{(2*3)}}$
Calculate PV: Calculate the present value. $\newline$ PV = $\frac{12000}{(1 + 0.015)^{6}}$ $\newline$ PV = $\frac{12000}{(1.015)^6}$ $\newline$ PV = $\frac{12000}{(1.093443)}$ $\newline$ PV = $\frac{12000}{1.093443}$ $\newline$ PV = $10976.372$
Round Up: Round up to the nearest cent. $\newline$ The present value needs to be rounded up to the nearest cent. $\newline$ $PV = \$10,976.37$

More problems from Find probabilities using the binomial distribution

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\newline

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\newline

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