We are asked to determine the value of a deferred annuity on the former student's 50th birthday. The investment will begin paying a 9% APR return starting when the student turns 65, and the student expects to receive these payments for 24 more years. However, the student wants a return of 8% on their investment.

To calculate the value today of this deferred annuity, we need to break the problem down into two parts:

1. Determine the value of the annuity payments at age 65.
2. Discount that value back to the present day (age 50).

Step 1: Value of the annuity at age 65

First, we calculate the annuity payment the student will receive once they turn 65. The program pays a 9% annual return on the initial investment, so for an investment of $9,400:

$\text{Annual Payment} = 9,400 \times 9\% = 9,400 \times 0.09 = 846 \text{ per year.}$

The payments will continue for 24 years after the student turns 65.

The present value of an ordinary annuity (because payments start immediately after the student turns 65) is given by the formula:

$PV_{\text{annuity}} = P \times \left[ \frac{1 - (1 + r)^{-n}}{r} \right]$

Where:

• $P = 846$ (the annual payment),
• $r = 8\% = 0.08$ (the desired return rate),
• $n = 24$ (the number of payments).

Substituting the values:

$PV_{\text{annuity}} = 846 \times \left[ \frac{1 - (1 + 0.08)^{-24}}{0.08} \right]$

Calculating this:

$PV_{\text{annuity}} = 846 \times \left[ \frac{1 - (1.08)^{-24}}{0.08} \right]$

$PV_{\text{annuity}} = 846 \times \left[ \frac{1 - 0.0952}{0.08} \right]$

$PV_{\text{annuity}} = 846 \times \left[ \frac{0.9048}{0.08} \right]$

$PV_{\text{annuity}} = 846 \times 11.31 \approx 9,579.86$

Thus, the present value of the annuity at age 65 is approximately $9,579.86.

Step 2: Discount this value back to age 50

Now, we need to discount the value of the annuity at age 65 back to the present day (age 50). Since the student is 50 and the payments will start at age 65, we need to discount for 15 years (65 - 50). The formula to discount a future value is:

$PV_{\text{today}} = \frac{FV}{(1 + r)^t}$

Where:

• $FV = 9,579.86$ (the future value at age 65),
• $r = 8\% = 0.08$ (the desired return rate),
• $t = 15$ (the number of years to discount).

Substituting the values:

$PV_{\text{today}} = \frac{9,579.86}{(1 + 0.08)^{15}}$

$PV_{\text{today}} = \frac{9,579.86}{(1.08)^{15}}$

$PV_{\text{today}} = \frac{9,579.86}{3.172} \approx 3,016.97$

Thus, the value of the deferred annuity today (on the student's 50th birthday) is approximately $3,016.97.

Conclusion:

The value of this deferred annuity today, considering the student’s desired return of 8%, is $3,016.97.

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Would you like more details on any part of the calculation? Let me know if you have further questions!

Follow-up questions:

1. How would the value change if the student wanted a higher rate of return, say 9%?
2. What is the effect of increasing the number of years the student expects to live after age 65?
3. How would the value of the annuity change if the student received payments for 30 years instead of 24?
4. What would the present value be if the student had invested a different amount, such as $10,000?
5. How does the choice of discount rate (8% vs 9%) affect the investment decision?

Tip:

When dealing with time value of money problems, always make sure to distinguish between the present value (today's value) and future value (the value at a specific future point), and adjust your calculations accordingly using the correct formulas.