You want to accumulate $1 million by your retirement date, which is 25 years from now. You will make 25 deposits in your bank, with the first occurring today. The bank pays 7% interest, compounded annually. You expect to receive annual raise of 3%, which will offset inflation, and you will let the amount you deposit each year also grow by 3% (i. e., your second deposit will be 3% greater than your first, the third will be 3% greater than the second, etc.). How much must your first deposit be if you are to meet your goal

First deposit will be $11,213.87

Explanation:

To derive how much the first deposit must be, the deposit can be derived by using payment formula for growing annuity

P = FV x (r - g) / [(1 + r)^n - (1 + g)^n]

When FV = $1,000,000

r = 7%

g = 3%

n = 25

Hence, First payment will be:

P = 1,000,000 * (7% - 3%) / (1.07^25 - 1.03^25)

P = 1,000,000 * 4% / 5.427433 - 2.093778

P = 40,000 / 3.333655

P = 11998.842

P = $11,998.84

However, this formula is applicable when the payments are made at the end of the year. In this case the payments are upfront, occurring today. We need to adjust this first payment to reflect the early payment.

Hence, first payment = $11,998.84  / (1 + 7%)

First payment =  $11,998.84  / (1 + 0.07)

First payment = $11,998.84  / 1.07

First payment = 11213.8691588785

First payment = $11,213.87

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