Question

Your father is 50 years old and will retire in 10 years. He expects to live for 25 years after he retires, until he is 85. He wants a fixed retirement income that has the same purchasing power at the time he retires as $60,000 has today. (The real value of his retirement income will decline annually after he retires.) His retirement income will begin the day he retires, 10 years from today, at which time he will receive 24 additional annual payments. Annual inflation is expected to be 4%. He currently has $160,000 saved, and he expects to earn 8% annually on his savings. How much must he save during each of the next 10 years (end-of-year deposits) to meet his retirement goal? Round your answer to the nearest cent.

Answer 1

The required annuity payments:

Retirement income today	$ 60,000
Years to Retirement	10
Years of retirment	25
Inflation rate	4%
Savings	$ 160,000
Rate of return	8 %

Step1: Calculating value of saving in 10 years.We have,

Value of saving in 10 years = Retirement income today (1 + Inflation rate)^t

Value of savings in 10 years = 60,000 ( 1.04)¹⁰ = 60,000 x 1.4802 = $ 88,815

Value of savings in 10 years = $ 88,815

Hence, the amount $ 88,815 must be needed to match purchasing power of $ 60,000 in 10 years.

Step2: Calculating value of 25 begenning of year retirement payment at t = 10.We have,

The formula of present value of annuity = A [ 1 - ( 1 / (1+r)ⁿ ] / r

Where,

A = Constant periodic flow = $ 88,815

r = Discount rate = 8 %

n = number of periods = 25

By substituting these value in the above formula.We have,

Present value of annuity =  88,815 [ 1 - (1 / (1.08)²⁵ ] / 0.08

Present value of annuity = 88,815 [ 1 - 0.1460 ] / 0.08

Present value of annuity = $ 948,080

Hence, amount $ 948,080.25 he will need at the age of 60.

Step3: Calculating fixed retirement income in 10 years.We have,

Fixed retirement income in 10 years = Current Savings x (1+r)ⁿ

Fixed retirement income in 10 years = 160,000 ( 1.08)¹⁰ = 160,000 x 2.1589 = $ 345,424

Step4: Calculating net amount needed at the year 10.We have,

Value of Retirement Payment	$ 948,080
Less: Value of savings	$ 345,424
Net Amount needed	$ 602,656

Step5: Calculating annual saving needed for next 10 years.We have,

The formula for annual saving needed for retirement = A x - r / (1 - (1+r)ⁿ )

Annual saving needed for retirement = 602,656 x - 0.08 / ( 1 - (1.08)¹⁰ )

Annual saving needed for retirement = 602,656 x - 0.08 / ( 1 - 2.1589)

Annual saving needed for retirement = 602,656 x - 0.08 / - 1.1589

Annual saving needed for retirement = $ 41,602

Hence, he must save annually $ 41,602 during each of the next 10 year to meet his retirement goal.