You are trying to decide how much to save for retirement. Assume you plan to sav

Question

You are trying to decide how much to save for retirement. Assume you plan to save $5,000 per year with the first investment made one year from now. You think you can earn 10% per year on your investms and you plan to retire in 43 years, immediately after making your last $5,000 investment a. How much will you have in your retirement account on the day you retire? If, instead of investing $3,000 per year, you wanted to make one lump-sum investment today for your retirement that will result in the same retirement saving, how mach would tat mp sum need to be? If you hope to live for 20 years in retirement, how moch can you wihdraw every year in retirement (starting one year after retirement) so that you wil just exhaustyour savings with the 20th withdrawal (assume your savings will continue to earn 10% in retirement)? d. If, instcad, you decide to withdraw $300,000 per year in retirement (again w the first withdrawal one year after retiring), how many years will isake until you exhaust your savings? Assuming the most you can afford to save is $1,000 per year, but you want to retire with S1 million in your investment account, how high of a return do you need to earn on your investments? Annual saving Interest rate Years to retirement a. How much will you have in your retirement account on the day you retire? Future valuc If, instiead of investing $5,000 per year, you waned to make one lump-sum investment today for your retirement that will result in the same retirement saving, how mach would that ump sum need to be? Lump-sum investment If you hope to live for 20 years in retirement, how much can you withdraw every year in retirement (starting one year after retirement) so that you will just exhaust your savings with the 20th withdrawal (assume your savines will continue to earn 10% in fetretett ? Years of withdrawal Annual withdrawal If, instead, you decide to withdraw $300,000 per year in retirement (again w the first withdrawal one year afier retiring), how many years will itake until you cxhaust your savings? Annual withdrawal Number of periods Assuming the most you can afford to save is $1,000 per year, but you want to retire with $1 million in your investment account, bow high of a return do you need to carn on your invessments? Annual saving Future value Rate of return

Explanation / Answer

a. We can use following formula for Future value calculation of periodic savings

FV = PMT * [(1+i) ^n – 1] /i

Where FV = future value of savings at the time of retirement after 43 years =?

PMT or saving per year = $5,000 per annum

And i= I/Y = 10% is the interest rate per annum

The time period n = 43 years

Therefore,

FV = $5,000 * [(1+10%) ^43 -1]/ 10%

FV = $2,962,003.46

You will save $2,962,003.46 in your retirement account on the day of your retirement after 43 years.

b. To know the lump-sum amount of your investment you have to calculate the Present value (PV) of the Future value of your retirement account

PV = FV/ (1+i) ^n

Where,

Present Value PV =?

Future value FV=$2,962,003.46

Annual interest rate i = 10%

Time period n = 43 years

Therefore,

PV = $2,962,003.46/ (1+10 %) ^43

= $49,169.99

Therefore you should invest a lump-sum amount of $49,169.99 today.

c. We can use following Present Value of an Annuity formula to calculate the periodic annual withdrawals with 10% interest rate

PV of deposits = PMT * [1-(1+i) ^-n)]/i

Where,

Present value of loan (PV) = $2,962,003.46 (here the future value or saving of your retirement account will become present value for the withdrawals after retirement)

Annual withdrawal PMT =?

Number of annual payments n = 20

Annual interest rate I = 10%

Therefore

$2,962,003.46 = PMT * [1- (1+10%) ^-20]/10%

Or PMT = $347,915.81

Therefore you can withdraw $347,915.81 per year for 20 years after your retirement

d. Suppose you want to withdraw only $300,000 per year, then the time when your savings will exhaust

Again we can use same Present Value of an Annuity formula to calculate the time period of withdrawals with 10% interest rate

PV of deposits = PMT * [1-(1+i) ^-n)]/i

Where,

Present value of loan (PV) = $2,962,003.46

Annual withdrawal PMT =$300,000

Number of annual payments n =?

Annual interest rate I = 10%

Therefore

$2,962,003.46 = $300,000 * [1- (1+10%) ^-n]/10%

Or n = 45.84 years

Therefore you can withdraw $300,000 per year for 45.84 years after your retirement

e. We can use following Future Value of an Annuity formula to calculate the rate of return

FV = PMT * [(1+i) ^n – 1] /i

Where,

Future value of annuity (FV) = $1,000,000

Annual savings PMT =$1000

Number of annual payments n = 43 (for 43 years)

Annual rate of return i =?

Therefore

$1,000,000 = $1,000* [(1+i) ^43 – 1] /i

Or i= 11.74% per year

Therefore required annual rate of return is 11.74%.

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