Answer:
The five deposits will be for $ 1,263.360
Explanation:
at the given rate of 4%
we are going to do 5 deposits.
Then, after 5 years we subtract 3,000 dollar during 3 years:
First, we solve for the PV of tat annuity:
[tex]C \times \frac{1-(1+r)^{-time} }{rate} = PV\\[/tex]
C 3,000.00
time 3
rate 0.04
[tex]3000 \times \frac{1-(1+0.04)^{-3} }{0.04} = PV\\[/tex]
PV $8,325.2731
Then, we discount for the 5 years waiting period:
[tex]\frac{Maturity}{(1 + rate)^{time} } = PV[/tex]
Maturity $8,325.2731
time 5.00
rate 0.04000
[tex]\frac{8325.27309968139}{(1 + 0.04)^{5} } = PV[/tex]
PV 6,842.7676
Now, we solve for which annuity of 5 deposits generates that amount:
[tex]FV \div \frac{(1+r)^{time} -1}{rate} = C\\[/tex]
FV 6,843
time 5
rate 0.04
[tex]6842.76763180258 \div \frac{(1+0.04)^{5} -1 }{0.04} = C\\[/tex]
C $ 1,263.360
The amount that should be deposited each year for five years is $ 1,263.36.
It is calculated by applying the annuity formula.
Annuity refers to a series of payments that are made at equal intervals, and are withdrawn after a specific time at a specified rate of interest. The formula to calculate the present value of annuity is:
[tex]\rm PV = P\times \dfrac{1-(1+r)^{-n}}{r}[/tex] , where PV is the present value, P is the principal contributed each year, r is the rate of interest and n is the number of periods.
Given:
Amount to be withdrawn each year is $3,000
Rate of interest is 4%
Number of period is 3 years
The present value of annuity is:
[tex]\rm PV = 3,000\times \dfrac{1-(1+0.04)^{-3}}{0.04}\\\\\rm PV = 3,000\times \dfrac{1-(1.04)^{-3}}{0.04}\\\\\rm PV = 3,000\times \dfrac{1-(1.04)^{-3}}{0.04}\\\\\rm PV = \$8,325.2731[/tex]
Therefore the present value of amount withdrawn is $8,325.27
To calculate the PV for the same after 5 years, the amount is to be discounted for 5 years at the rate 4%:
[tex]\rm PV = \dfrac{8,325.2731}{(1+0.04)^5}\\\\\rm PV = \dfrac{8,325.2731}{1.2167}\\\\\rm PV = \$6,842.7676[/tex]
The present value after 5 years is $6,842.77.
The contribution per year will be calculated using the annuity formula:
[tex]\begin{aligned} \rm 6,842.77 &=\rm P\times \dfrac{1-(1+0.04)^{-5}}{0.04}\\\\\rm 6,842.77 &=\rm P\times \dfrac{1-(1.04)^{-5}}{0.04}\\\\\rm 6,842.77 &=\rm P\times \dfrac{1-(1.04)^{-5}}{0.04}\\\\\rm P &= \$1,263.360 \end[/tex]
Therefore the annual deposit should be $1,263.36 per year for 5 years.
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