Your client is 40 years old, and she wants to begin saving for retirement, with the first payment to come one year from now. She can save $7,500 per year, and you advise her to invest it in securities which you expect to provide an average annual return of 9 percent. If she follows your advice, how much money would she have at age 60?

If she follows your advice, how much money will she have at 65?

To calculate the amount of money your client would have at age 60, we need to use the formula for the future value of an ordinary annuity.

The future value of an ordinary annuity can be calculated using the following formula:

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

Where:
FV = Future value of the annuity
P = Annual payment or deposit
r = Annual interest rate
n = Number of periods

In this case:
P = $7,500 per year
r = 9% or 0.09 (expressed as a decimal)
n = 60 - 41 = 19 years (since she will begin saving one year from now until age 60)

Let's calculate:

FV = $7,500 * [(1 + 0.09)^19 - 1] / 0.09

The first step is to calculate (1 + 0.09)^19, let's simplify it:

(1 + 0.09)^19 ≈ 1.09^19 ≈ 3.172

Now we can substitute this value back into the formula:

FV = $7,500 * (3.172 - 1) / 0.09

Simplifying further:

FV = $7,500 * 2.172 / 0.09

FV ≈ $7,500 * 24.133

FV ≈ $181,000

Therefore, if your client saves $7,500 per year with an average annual return of 9% until age 60, she is expected to have approximately $181,000 at that time.