Question

Ryan is 50 years old and will retire in 15 years. He expects to live for 25 years after he retires, until he is 90. He wants a fixed retirement income that has the same purchasing power at the time he retires as $40,000 has today. (The real value of his retirement income will decline annually after he retires.) His retirement income will begin the day he retires, 15 years from today, at which time he will receive 24 additional annual payments. Annual inflation is expected to be 3%. He currently has $100,000 saved, and he expects to earn 8% annually on his savings. How much must he save during each of the next 15 years (end-of-year deposits) to meet his retirement goal? Answer with 2 decimals (ex. $1,000.00).

Answer 1

First retirement income to have the same purchasing power as $40,000 = 40000*1.03^15

= $ 62,318.70

Now Ryan Wants to receive this amount(i.e $ 62,318.70)  constantly for 25 year starting from his date of retirement which is 15 years from now

so, we need to calculate the PV of these annuaties as on that date:

Where C = $ 62,318.70, i = 8% n =24

=62318.70*((1-((1+0.08)^-24))/0.08) + 62318.70

= $718,457.23

Since the First annuity is received on the date of retirement therefore Present value of 24 annuties is added to $62,318.70 thereby making it 25 annuties in total.

Therefore funds needed on retirement $ 718,457.23

Now The $ 100,000 already saved will grow to @ 8% For 15 years

PV	1,00,000.00
i	8%
t	15
n	1
Future Value is calculated using the formula given below
FV = PV * [ 1 + ( i / n ) ] (n * t)
FV = 100000 * [ 1 + (8% / 1 ) ] (1 *15)
Future Value	$317,216.91

Funds Needed on retirement (a) $718,457.23

Funds Available through Investment (b) $317,216.91

Remaining amount (a-b) $401,240.32

Therefore the amount need to be saved through annuties is $401,240.32

Annual savings required :

Present Value = Annuity / PVIFA(8%,15)

= $401,240.32/8.5594

= $46,876.21

Note: PVIFA is present value annuity factor @ 8% for 15 year

Hence Ryan will need to save $46,876.21 Annually for Next 15 years.