Respuesta :
Given that a 40 years old woman wants to begin saving for retirement with
the first payment to come one year from now. she can save $5,000 per
year, and she invested it in the stock market, which she
expects to provide an average of 9% in the future.
Part A:
The value of her investment at the age of 65 (25 years) is given by:
[tex]FV=P \frac{(1+r)^n-1}{r} [/tex],
where FV is the value of the investment n years from now; P is the periodic payment = $5,000; r is the rate of return = 9% = 0.09 and n is the number of years = 25 years.
[tex]FV=5000\left( \frac{(1+0.09)^{25}-1}{0.09} \right) \\ \\ 5000\left( \frac{8.6231-1}{0.09} \right)=5000(84.70) \\ \\ =\$423,504.48[/tex]
Part B:
The value of her investment at the age of 70 (30 years) is given by:
[tex]FV=P \frac{(1+r)^n-1}{r} [/tex],
where FV is the value of the investment n years from now; P is the periodic payment = $5,000; r is the rate of return = 9% = 0.09 and n is the number of years = 30 years.
[tex]FV=5000\left( \frac{(1+0.09)^{30}-1}{0.09} \right) \\ \\ 5000\left( \frac{13.2677-1}{0.09} \right)=5000(136.31) \\ \\ =\$681,537.69[/tex]
Part C:
If she retires at the age of 65 and she expects to live additional 20 years after retirement, the present value her investment where an equal amount is withdrawn at equal interval of times is given by:
[tex]PV=P \frac{1-(1+r)^{-n}}{r} [/tex]
where PV is the present value = $423,504.48; P is the amount withdrawn at the end of each year; r is the rate of return = 9% = 0.09; n is the number of years = 20 years.
Thus,
[tex]423,504.48=P\left( \frac{1-(1+0.09)^{-20}}{0.09} \right) \\ \\ =P\left(\frac{1-0.1784}{0.09}\right)=9.1285P \\ \\ \Rightarrow P= \frac{423,504.48}{9.1285} =\$46,393.42[/tex]
Therefore, the amount she will be able to withdraw at the end of each year after retirement is $46,393.42
Part D:
If she retires at the age of 70 and she expects to live additional 15 years after retirement, the present value her investment where an equal amount is withdrawn at equal interval of times is given by:
[tex]PV=P \frac{1-(1+r)^{-n}}{r} [/tex]
where PV is the present value = $681,537.69; P is the amount withdrawn at the end of each year; r is the rate of return = 9% = 0.09; n is the number of years = 15 years.
Thus,
[tex]681,537.69=P\left( \frac{1-(1+0.09)^{-15}}{0.09} \right) \\ \\ =P\left(\frac{1-0.2745}{0.09}\right)=8.0607P \\ \\ \Rightarrow P= \frac{681,537.69}{8.0607} =\$84,550.80[/tex]
Therefore, the amount she will be able to withdraw at the end of each year after retirement is $84,550.80
Part A:
The value of her investment at the age of 65 (25 years) is given by:
[tex]FV=P \frac{(1+r)^n-1}{r} [/tex],
where FV is the value of the investment n years from now; P is the periodic payment = $5,000; r is the rate of return = 9% = 0.09 and n is the number of years = 25 years.
[tex]FV=5000\left( \frac{(1+0.09)^{25}-1}{0.09} \right) \\ \\ 5000\left( \frac{8.6231-1}{0.09} \right)=5000(84.70) \\ \\ =\$423,504.48[/tex]
Part B:
The value of her investment at the age of 70 (30 years) is given by:
[tex]FV=P \frac{(1+r)^n-1}{r} [/tex],
where FV is the value of the investment n years from now; P is the periodic payment = $5,000; r is the rate of return = 9% = 0.09 and n is the number of years = 30 years.
[tex]FV=5000\left( \frac{(1+0.09)^{30}-1}{0.09} \right) \\ \\ 5000\left( \frac{13.2677-1}{0.09} \right)=5000(136.31) \\ \\ =\$681,537.69[/tex]
Part C:
If she retires at the age of 65 and she expects to live additional 20 years after retirement, the present value her investment where an equal amount is withdrawn at equal interval of times is given by:
[tex]PV=P \frac{1-(1+r)^{-n}}{r} [/tex]
where PV is the present value = $423,504.48; P is the amount withdrawn at the end of each year; r is the rate of return = 9% = 0.09; n is the number of years = 20 years.
Thus,
[tex]423,504.48=P\left( \frac{1-(1+0.09)^{-20}}{0.09} \right) \\ \\ =P\left(\frac{1-0.1784}{0.09}\right)=9.1285P \\ \\ \Rightarrow P= \frac{423,504.48}{9.1285} =\$46,393.42[/tex]
Therefore, the amount she will be able to withdraw at the end of each year after retirement is $46,393.42
Part D:
If she retires at the age of 70 and she expects to live additional 15 years after retirement, the present value her investment where an equal amount is withdrawn at equal interval of times is given by:
[tex]PV=P \frac{1-(1+r)^{-n}}{r} [/tex]
where PV is the present value = $681,537.69; P is the amount withdrawn at the end of each year; r is the rate of return = 9% = 0.09; n is the number of years = 15 years.
Thus,
[tex]681,537.69=P\left( \frac{1-(1+0.09)^{-15}}{0.09} \right) \\ \\ =P\left(\frac{1-0.2745}{0.09}\right)=8.0607P \\ \\ \Rightarrow P= \frac{681,537.69}{8.0607} =\$84,550.80[/tex]
Therefore, the amount she will be able to withdraw at the end of each year after retirement is $84,550.80
