Calculus Problem#1

[dcc3038026]

Can anybody please help me with this problem......
Find the lump sum deposited today that will yield the same total amount as this yearly payment (made at the end of each year for 20 years at the given interest rate, compounded annually. $3500 at 4%

 

[soroban]

Hello, dcc3038026!

Find the lump sum deposited today that will yield the same total amount as a yearly
payment of $3500 made at the end of each year for 20 years at 4% compounded annually.

I must assume you are familiar with the necessary formulas.

\(\displaystyle \text{Annuity: }\;A \;=\;D\cdot\frac{(1+i)^n-1}{i}\)

. . \(\displaystyle \text{where: }\;\begin{Bmatrix} D &=& \text{periodic deposit} \\ i &=& \text{periodic interest rate} \\ n &=& \text{number of periods} \\ A &=& \text{Final balance} \end{Bmatrix}\)


\(\displaystyle \text{We have: }\;D = 3500,\;i = 0.04,\;n = 20\)

\(\displaystyle \text{Then: }\;A \;=\;3500\cdot\frac{1.04^{20}-1}{0.04} \;\approx\;\$104.223.28\)

Depositing $3500 annually for 20 years at 4% interest, we will have that final amount.



\(\displaystyle \text{To have same final balance with a single deposit }P\text{ at 4 percent compounded annualy}\)

. . \(\displaystyle \text{we have: }\;P(1.04)^{20} \;=\;104,223.78\)


. . \(\displaystyle P \;=\;\frac{104,223.28}{(104)^{20}} \;= \;47,566.14448\)

\(\displaystyle \text{The lump sum deposit is: }\:\$47,566.14\)

 

[dcc3038026]

Thank you so much..........
dcc3038026

Hello, dcc3038026!

Find the lump sum deposited today that will yield the same total amount as a yearly
payment of $3500 made at the end of each year for 20 years at 4% compounded annually.

I must assume you are familiar with the necessary formulas.

\(\displaystyle \text{Annuity: }\;A \;=\;D\cdot\frac{(1+i)^n-1}{i}\)

. . \(\displaystyle \text{where: }\;\begin{Bmatrix} D &=& \text{periodic deposit} \\ i &=& \text{periodic interest rate} \\ n &=& \text{number of periods} \\ A &=& \text{Final balance} \end{Bmatrix}\)


\(\displaystyle \text{We have: }\;D = 3500,\;i = 0.04,\;n = 20\)

\(\displaystyle \text{Then: }\;A \;=\;3500\cdot\frac{1.04^{20}-1}{0.04} \;\approx\;\$104.223.28\)

Depositing $3500 annually for 20 years at 4% interest, we will have that final amount.



\(\displaystyle \text{To have same final balance with a single deposit }P\text{ at 4 percent compounded annualy}\)

. . \(\displaystyle \text{we have: }\;P(1.04)^{20} \;=\;104,223.78\)


. . \(\displaystyle P \;=\;\frac{104,223.28}{(104)^{20}} \;= \;47,566.14448\)

\(\displaystyle \text{The lump sum deposit is: }\:\$47,566.14\)