You borrowed $50,000 which you must repay in 10 years.You plan to make an initial deposit today,then make 9 more deposits at the beginning of each the next 9 years,but with the deposits increasing at the inflation rate.You expect to earn 5% on your funds,and you expect a 3% inflation rate.To the nearest dollar,how large must your initial deposit be to enable you to reach your $50,000 target?

Question

Quizplus · Accepted Answer

We can set up a table with an arbitrary initial deposit that grows at the inflation rate and is then compounded at the nominal rate for (N F1F1F1S1F1F1F10 t)years.The sum of the compounded amounts should total to $50,000.With an arbitrary initial amount the ending amount is not likely to be $50,000,so we use goal seek as shown in the completed dialog box to find the correct initial deposit.Inputs:
 Years&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp10
Amount Needed (FV)&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp$ \$ 50.000 $
Nominal rate earned on account&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp5.00%
Inflation&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp$ 3.00 \% $
 1. Garal Seek appraach:
Use Gaal Seek in the following table to determine the iritial deposit. Start with $5,000 as the initial payment but this will change to $\$ 3,341.94$.
$\begin{array}{crr}
\text { Period }(\mathrm{t}) & \begin{array}{r}
\text { BOY Payment } \
\text { Initial }(1+I)^{\mathrm{t}}
\end{array} & \begin{array}{r}
\text { Computed } \
\text { Value }
\end{array} \
\hline 0 & \$ 3,341.94 & \$ 5,443.67 \
1 & \$ 3,442.20 & \$ 5,339.98 \
2 & \$ 3,545.47 & \$ 5,238.27 \
3 & \$ 3,651.83 & \$ 5,138.49 \
4 & \$ 3,761.39 & \$ 5,040.62 \
5 & \$ 3,874.23 & \$ 4,944.61 \
6 & \$ 3,990.45 & \$ 4,850.42 \
7 & \$ 4,110.17 & \$ 4,758.03 \
8 & \$ 4,233.47 & \$ 4,667.40 \
9 & \$ 4,360.48 & \$ 4,578.50 \
10 & \$ 0.00 & \$ 50,000.00=\text { Sum of the } 10 \text { compounded values }
\end{array}$
2. Farmula apprarch:
Find the real rate
$1.941749 \% = ( 1 + I 1572 ) / ( 1 + I 1573 ) - 1$
Find the PV of the required Future mamus discounted at the infletion rate. This is aur canstant dollar Future taret.
$$\text { 197,204.7] } = 11571 / ( 1 + I 1573 ) ^ { ( 1157 ) }$$
Use a calculatar or Eacel ta find the iritial payment.
$\mathrm { N } = 10 , \mathrm { I } / \mathrm { YR } = 1.941748 , \quad \mathrm {~V} = \mathbb { 0 } , \mathrm { FV } = - 37204.70$, and $5 \mathrm { et }$ to $\mathrm { BEGIN }$ made.
This is consistent with the Garl Seek salution.
$3,341.94 =$ PMT $( H 1593 , I 1570,0 , - H 1595,1 )$

You Borrowed $50,000 Which You Must Repay in 10 Years