### Problem

An individual retirement account is opened with a deposit of 5,000 and since opening, 200 is deposited into the account at the end of every week. The individual retirement account pays annual interest of 8.1% compounded every week with each deposit. How much will be in the account after 10 years?

### Solution

The balance on the individual retirement account is made up of the value of the initial deposit after 10 years and the value of the weekly annuity payments for 10 years.

This problem is solved in two steps.

- Calculate the future value of the initial deposit.
- Calculate the future value of the annuity payments.

##### 1. Calculate the Future Value of the Initial Deposit

The initial deposit is a lump sum of 5,000 (PV), and its future value (FV) is calculated using the future value of a lump sum formula.

As compounding is weekly, one period is one week, and the term of the individual retirement account is 10 years or 520 weeks (n). The interest rate has been quoted as an annual rate so the periodic (weekly) rate is 8.1%/52 per week (i).

The future value of the initial deposit is calculated as follows:

FV deposit = PV x (1 + i)^{n}PV = initial deposit = 5,000 n = number of weeks = 10 x 52 = 520 weeks i = nominal rate = 8.1%/52 per month FV deposit = 5000 x (1 + 8.1%/52)^{520}FV deposit = 11,232.46

After 10 years the initial deposit will be worth 11,232.46.

##### 2. Calculate the Future Value of the Annuity Payments

The weekly payments of 200 (Pmt) are an annuity, and their future value (FV) is calculated using the future value of an annuity formula.

As compounding is weekly, one period is one week, and the payments continue for the term of the individual retirement account of 10 years or 520 weeks (n). The interest rate has been quoted as an annual rate so the periodic (weekly) rate is 8.1%/52 per week (i).

The future value of the annuity payments is calculated as follows:

FV annuity = Pmt x ( (1 + i)^{n}- 1 ) / i Pmt = weekly payment = 200 n = number of weeks = 10 x 52 = 520 weeks i = nominal rate = 8.1%/52 per month FV annuity = 200 x ( (1 + 8.1%/52)^{520}- 1 ) / (8.1%/52) FV annuity = 160,043.39

The future value of the annuity payments is 160,043.39.

Finally, to find the balance on the individual retirement account, we simply add together the future value of the initial deposit and the annuity payments.

Individual retirement account balance = FV deposit + FV annuity Individual retirement account balance = 11,232.46 + 160,043.39 Individual retirement account balance = 171,275.85

After 10 years the balance on the individual retirement account will be 171,275.85.

### Explanation

Both the initial deposit and the weekly annuity payments combine to form the balance on the individual retirement account. The cash flows can be separated and the future value of each calculated independently. The initial deposit is a lump sum and the future value is given by the future value of a lump sum formula, and the weekly payments are an annuity whose future value is given by the future value of an annuity formula.

In this example it should be noted that the annuity payments are made weekly and the compounding period is also weekly, if this is not the case then an effective interest rate needs to be calculated to bring compounding period into line with the payment period.

### Individual Retirement Account Examples

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