You are saving for retirement. To live? comfortably, you decide you will need to

Question

You are saving for retirement. To live? comfortably, you decide you will need to save $2,000,000 by the time you are age 65. Today is your 24th 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%?,you set aside $14,789 each year to make sure that you will have $2,000,000 in the account on your 65th birthday. You realize that your plan 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 7% 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.)

Explanation / Answer

Let today's deposit be $ K

The deposit value increases at 7 % per annum making a payment series of K, K x (1.07), K x (1.07)^(2) ..... and so on till K x (1.07)^(41) with each payment being compounded by a factor of (1.05)^(41), (1.05)^(40) and so on beginning from the first payment.

Final Payment Value = $ 2000000

Therefore, 2000000 = K x (1.05)^(41) + K x (1.07) x (1.05)^(40) + K x (1.07)^(2) x (1.05)^(39) + .........+ K x (1.07)^(41)

Dividing both sides by (1.05)^(41) we get:

2000000 / (1.05)^(41) = K + K x [1.07/1.05] + K x [1.07/1.05]^(2) + K x [1.07/1.05]^(3) + .........+ K x [1.07/1.05]^(41)

270563.2044 = [{1.07/1.05}^(42) - 1} / {(1.07/1.05) - 1}] x K

K = (270563.2044 / 63.4651) = $ 4263.19

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