Financial Management

35. You are saving for retirement. To live comfortably, you decide you will need to save $2 million by the time you are 65. Today is your 30th birthday, and you decide, starting today and continuing on every birthday up to and including your 65th birthday, that you will put the same amount into a savings account. If the interest rate is 5%, how much must you set aside each year to make sure that you will have $2 million in the account on your 65th birthday?

First we create a timeline to illustrate the contribution (C) that will be deposited into the savings account.

Timeline:

30 31 32 33 65

0 1 2 3 35

C C C C C

This timeline illustrates the initial contribution represented by 0 and the 35th contribution occurring on the 65th birthday. Each contribution will be of equal value and will incur a 5% annual interest rate.

The future value (FV) of annuity is $2 million dollars, therefore:

FV = $2 million

The PV of the cash flows must equal the PV of $2 million in 35 years. The cash flows consist of a 35-year annuity, plus the contribution today, so the PV is:

The PV of $2 million in 35 years is

Therefore, this equates to:

You would have to contribute $20,868.91 beginning today and ending on your 65th birthday with a savings interest rate of 5% per year to obtain $2 million dollars.

36. You realize that the plan in problem 35 has a flaw. Because your income will increase over your lifetime, it would be more realistic to save less now and more later. Instead of putting the same amount aside each year, you decide to let the amount that you set aside grow by 3% per year. Under this plan, how much will you put into the account today? (Recall that you are planning to make the first contribution to the account today).

This timeline illustrates the initial contribution represented by 0 and the 35th contribution occurring on the 65th birthday. Each contribution will increase in...