Answer:
To calculate the present value of the payouts of the savings plan, we need to discount each future cash flow back to present value using the given effective interest rate of 2%. The formula for calculating the present value of a future cash flow is:
�
�
=
�
�
(
1
+
�
)
�
PV=
(1+r)
n
FV
Where:
�
�
PV = Present Value
�
�
FV = Future Value of the cash flow
�
r = Effective interest rate
�
n = Number of compounding periods
Let's break down the calculations step by step:
Calculate the present value of the annual investments for the next twenty years.
Each investment is $10,000, and it occurs at the start of each year.
�
�
annuities
=
∑
�
=
1
20
10000
(
1
+
0.02
)
�
PV
annuities
=∑
t=1
20
(1+0.02)
t
10000
Calculate the present value of the semi-annual payouts for the next forty years.
Each payout is $5,000, and they occur every 6 months.
�
�
semi-annual
=
∑
�
=
1
80
5000
(
1
+
0.02
)
2
�
PV
semi-annual
=∑
t=1
80
(1+0.02)
2t
5000
Add the present values of the annual investments and the semi-annual payouts to get the total present value.
�
�
total
=
�
�
annuities
+
�
�
semi-annual
PV
total
=PV
annuities
+PV
semi-annual
Let's calculate these values:
Annual Investments (Annuities):
�
�
annuities
=
∑
�
=
1
20
10000
(
1
+
0.02
)
�
PV
annuities
=∑
t=1
20
(1+0.02)
t
10000
�
�
annuities
=
10000
1.02
+
10000
(
1.02
)
2
+
…
+
10000
(
1.02
)
20
PV
annuities
=
1.02
10000
+
(1.02)
2
10000
+…+
(1.02)
20
10000
�
�
annuities
≈
10000
×
18.78349
PV
annuities
≈10000×18.78349
�
�
annuities
≈
187834.90
PV
annuities
≈187834.90
Semi-Annual Payouts:
�
�
semi-annual
=
∑
�
=
1
80
5000
(
1
+
0.02
)
2
�
PV
semi-annual
=∑
t=1
80
(1+0.02)
2t
5000
�
�
semi-annual
=
5000
(
1.02
)
2
+
5000
(
1.02
)
4
+
…
+
5000
(
1.02
)
160
PV
semi-annual
=
(1.02)
2
5000
+
(1.02)
4
5000
+…+
(1.02)
160
5000
�
�
semi-annual
≈
5000
×
50.74381
PV
semi-annual
≈5000×50.74381
�
�
semi-annual
≈
253719.05
PV
semi-annual
≈253719.05
Total Present Value:
�
�
total
=
�
�
annuities
+
�
�
semi-annual
PV
total
=PV
annuities
+PV
semi-annual
�
�
total
≈
187834.90
+
253719.05
PV
total
≈187834.90+253719.05
�
�
total
≈
441553.95
PV
total
≈441553.95
Based on the calculations, the closest option is O $29,739.01, which is the third option. However, this doesn't match the calculated value of approximately $441,553.95. It seems like there might be an error in the options provided or in the calculations themselves. Please double-check the calculations or the provided options.