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

Tutorial Notes :

1. Due to lack of further information it is most rational to assume that no interest rate will change during the 15 years of employment and also 25 years after retirement.

2. PVAF = Present Value Annuity Factor

3. This problem will be solved in 4 steps. Please follow the explanation given at the begining of each step for clear understanding.

4. All the decimals are approximated upto two decimal numbers.

Step-1 : Calculation of Purchasing power equivalent to $40,000 after 15 years

Explanation :
Ryan needs a purchasing power of $40,000 after 15 years from today. Since there would be inflation during these 15 years period, the amount Ryan will need annually for a purchasing power of $40,000 would be much higher. So we need to calculate future value of $40,000 after 15 years under given inflation rate of 3%.

Given data :
Rate of Inflation (Inf) = 3% or 0.03
Present Value of the Amount (PV) = $40,000
Number of years (n) = 15 years

We know, Future Value (FV) = PV * (1 + Inf)^n
Or, FV = $40,000 * (1 + 0.03)^15 = $62,400 (approx)

Therefore, Ryan would need fixed income of $62,400 annually during his 24years of his retirement life; which would give him the same purchasing power of $40,000 as on today.

Step-2 : Computation of lumpsum amount required at the retirement date

Explanation :
Ryan needs $62,400 annually for 24 years.
By problem, the interest rate of savings as on today (r) = 8%
Hence, Ryan needs a lumpsum as on the retirement date. And that amount required as on retirement date would be the present value on that date whose annuity is $62,400 for 24 years at the interest rate of 8%

Given data :
Annuity = $62,400
Rate of Interest (r) = 8% or 0.08
Number of years (n) = 24 years

We know, PV of Annuity = Annuity * PVAF

Hence in the given case, PV of Annuity = $62,400 * PVAF (for 24years @ 8%)
= $62,400 * 10.53 = $ 657,072

Step-3 : Future Value of $100,000 after 15 years

Explanation :
As on today Ryan has a saving of $100,000, which will grow at 8% rate till retirement. We need to compute that future value.

Given data :
PV = $100,000
Rate of interest (r) = 8% or 0.08
Number of years (n) = 15 years

Hence, FV = PV * (1 + r)^n
Or, FV = $100,000 * (1 + 0.08)^15 = $317,000 (approx)

Step-4 : Computation of required Savings for the next 15 years till retirement

Explanation :
In order to meet the retirement plan Ryan needs to save a certain amount annually to get a lumpsum of $657,072 after 15 years i.e., on the date of retirement. Hence, an annual saving to be done by him for the next 15 years when rate of interest on saving is 8%.
However, Ryan already has a saving of $100,000, which will create a sum of $317,000 after 15 years. Hence, Ryan's annual saving should be only for a sum that can create a future value of $340,072 [i.e., $657,072 - $317,000]

Given data :
Rate of Interest (r) = 8% or 0.08
Number of years (n) = 15 years
FV of Annuity = $340,072

We know, FV of Annuity = Annuity * [ { (1 + r)^n - 1} / r ]
Or, $340,072 = Annuity * [ { (1 + 0.08)^15 - 1} / 0.08 ]

Or, Annuity = $12,537.22 (approx)

Hence, to meet the retirement plan Ryan should save $12,537.22 till his retirement for next 15 years.