Find (period rate), (number of periods), and (future value) for the information given. Fill in the table. $\text { cellpadding="0" cellspacing="0" width } = " 92 \% \text { " border="1" }$

$\begin{array}{cccccccc}
\text { Compounding } & P & r & t & n & i & N & A \
\text { Period } & \begin{array}{c}
\text { Present } \
\text { Value }
\end{array} & \begin{array}{c}
\text { Annual } \
\text { Rate }
\end{array} & & & \begin{array}{c}
\text { Time } \
\text { Reriod }
\end{array} & \begin{array}{c}
\text { Number of } \
\text { Pature }
\end{array} & \begin{array}{c}
\text { Periods } \
\text { Value }
\end{array} \
\text { Monthly } & \$ 450 & 12 \% & 5 \mathrm{yr} & ? & ? & ? & ?
\end{array}$
A) $n$ $\mathrm { N }$ $A$ Period Number of Future Rate Periods Value 60 12 12 $3,690.00
B) $n$ $\mathrm { N }$ $A$ Period Number of Future Rate Periods Value 4 3 60 $812.75
C) $n$ $\mathrm { N }$ $A$ Period Number of Future Rate Periods Value 12 1 60 $817.51
D) $n$ $\mathrm { N }$ $A$ Period Number of Future Rate Periods Value 12 1 60 $507.31
E) $n$ $\mathrm { N }$ $A$ Period Number of Future Rate Periods Value 12 1 60 $454.52

Question

Find (period rate), (number of periods), and (future value) for the information given. Fill in the table. $\text { cellpadding="0" cellspacing="0" width } = " 92 \% \text { " border="1"> }$

$\begin{array}{cccccccc}
\text { Compounding } & P & r & t & n & i & N & A \
\text { Period } & \begin{array}{c}
\text { Present } \
\text { Value }
\end{array} & \begin{array}{c}
\text { Annual } \
\text { Rate }
\end{array} & & & \begin{array}{c}
\text { Time } \
\text { Reriod }
\end{array} & \begin{array}{c}
\text { Number of } \
\text { Pature }
\end{array} & \begin{array}{c}
\text { Periods } \
\text { Value }
\end{array} \
\text { Monthly } & \$ 450 & 12 \% & 5 \mathrm{yr} & ? & ? & ? & ?
\end{array}$
A) $n$ $\mathrm { N }$ $A$ Period Number of Future Rate Periods Value 60 12 12 $3,690.00 
B) $n$ $\mathrm { N }$ $A$ Period Number of Future Rate Periods Value 4 3 60 $812.75 
C) $n$ $\mathrm { N }$ $A$ Period Number of Future Rate Periods Value 12 1 60 $817.51 
D) $n$ $\mathrm { N }$ $A$ Period Number of Future Rate Periods Value 12 1 60 $507.31 
E) $n$ $\mathrm { N }$ $A$ Period Number of Future Rate Periods Value 12 1 60 $454.52

Quizplus · Accepted Answer

In order to find the number of periods, we need to convert the annual rate to the periodic rate. Since we are compounding monthly, the monthly rate is the annual rate divided by 12: 
$$r = \frac{12\%}{12} = 1\%$$
Then, we can use the formula for the future value of an ordinary annuity: 
$$FV = P \frac{(1+i)^n - 1}{i}$$
where $P$ is the present value, $i$ is the periodic interest rate, and $n$ is the number of periods. We are given $P$ as \$450 and $i$ as 1%, so we can solve for $n$: 
$$FV = \$450 \frac{(1+0.01)^n - 1}{0.01} = \$450 \frac{1.01^n - 1}{0.01}$$
We are also given $FV$ as some unknown value. We can use the given time period of 5 years and the fact that there are 12 months in a year to find the total number of periods: 
$$N = 5 	ext{ years} 	imes 12 \frac{	ext{months}}{	ext{year}} = 60 	ext{ months}$$
Now we can substitute into the formula and solve for $n$: 
\begin{align*}
FV &= \$450 \frac{1.01^n - 1}{0.01} \
FV 	imes 0.01 &= \$450 (1.01^n - 1) \
0.01 FV &= \$450 (1.01^n - 1) \
1.01^n &= \frac{0.01 FV}{\$450} + 1 \
n &= \log_{1.01} \left(\frac{0.01 FV}{\$450} + 1ight)
\end{align*}
Using a calculator, we find that $n \approx 12.04$. Since we are compounding monthly, we round up to the nearest whole number of periods: $n = 13$. Therefore, the answer is choice C, which has a period of 1 month, a number of periods of 60 (5 years x 12 months/year), and a future value of $817.51.

Find (Period Rate), (Number of Periods), and (Future Value) for the Information