. On the day that you were born, your grandfather opened a savings account in your name. At that time, he deposited a certain amount of money into the account, and has deposited the same amount on each of your first 20 birthdays. Today is your 21st birthday. Your grandfather has decided not to make another deposit but rather to give you a special gift. If the account has paid a consistent interest rate of 4% compounded semi-annually and has a current balance of $100,000, how much money did your grandfather deposit on each occasion?

Question

Reiny · Answer

In order for the ordinary formulas to work, the interest period must match the payment period.
In this question, the payment period is annually, but the interest period is semiannually.
So we must first find what annual interest rate is equivalent to 4% per annum compounded semi-annually
let that rate be i
1+i = 1.02^2 = 1.0404
so i = .0404

secondly, we have to be careful not to make a mistake in counting the number of payments, let's look at the pattern
at birth -- 1st payment
1st birthday -- 2nd payment
2nd birthday -- 3rd payment
...
21st birthday -- 22nd payment

so P(1.0404)^22 - 1)/.0404 = 100,000
P(34.407256) = 100,000
P = 2906.36