Question

You are 25 years old and decide to start saving for your retirement. You plan to save $3000 at the end of each year (so the first deposit will be one year from now), and will make the last deposit when you retire at age 65. Suppose you earn 10% per year on your retirement savings.

1. How much will you have saved for retirement?
2. How much will you have saved if you wait until age 35 to start saving (again, with your first deposit at the end of the year)?

Answer 1

Since, the first deposit will be made one year from now after attaining the age 25 and the last deposit will be made at the 65th year.
Hence, the total no. of years of savings = 40 years.

The savings amount at the end of each year = $3000

Interest rate on the savings account = 10%

The future value of the total retirement amount saved for 40 years @ 10% interest rate can be computed with the help of the following formula:-

where, A is the value of annuity

S_n represents the compounded sum of an annuity

CVIFA represents the appropriate factor for the sum of the annuity of $1.
CVIFA can also be defined as the Future Value Interest Factor for a one dollar annuity compounded at k% Interest rate for n Periods.

Symbolically,  CVIFA _k,n = [(1 + k)ⁿ - 1] / k

Since, it is not possible to calculate manually with the help of the above formula, the future value of an annuity compounded @ k% for relatively large no. of periods (when the value of n is large). Hence, annuity tables are used in the field of investment banking in order to guide the depositors as to what sum, an amount P deposited for n no.of years will accumulate to, at a stated rate of compound interest rate, say, k%.

From the above given formula we get the following data:-

A (the value of annuity) = $3000

n = 40 years

Interest rate = 10%

The appropriate factor for the sum of a 40 year annuity @ 10% is obtained from the table of Future Value Interest Factor for a one dollar annuity compounded at k% Interest rate for n Periods. It is given as 442.5926.

Therefore, the amount of retirement savings at the age of 65 is given by:

Or, S_n = 442.5926 * $3000 = $1,327,777.8

Therefore, the total amount of retirement savings = $1,327,777.8

If the savings is started at the end of 35 years of age.

n = 30 years, Interest rate = 10%

CVIFA _k,n i.e. CVIFA _{10, 30} as obtained from the table of Future Value Interest Factor for a one dollar annuity compounded at k% Interest rate for n Periods is = 164.4940.

Hence, the appropriate factor for the sum of a 30 year annuity @ 10% = 164.4940.

Therefore, the amount of retirement savings at the age of 65 is given by:

or, S_n = 164.4940 * $3000 = $493,482

Thus, the total amount of retirement savings, if the savings started at the age of 35 will be = $493,482.