To calculate the present value (PV) of an annuity, we need to apply the following formulas:

1. Present Value of an Ordinary Annuity (payments are made at the end of each period): $PV_{\text{Ordinary}} = P \times \left(1 - (1 + r)^{-n}\right) \div r$ where:

   • $P$ = amount of each annuity payment ($13,000)
   • $r$ = interest rate per period (9% = 0.09)
   • $n$ = number of periods (5 years)

2. Present Value of an Annuity Due (payments are made at the beginning of each period): $PV_{\text{Annuity Due}} = PV_{\text{Ordinary}} \times (1 + r)$

Step 1: Calculate the Present Value of the Ordinary Annuity

Using the ordinary annuity formula:

$PV_{\text{Ordinary}} = 13,000 \times \left(1 - (1 + 0.09)^{-5}\right) \div 0.09$

Let me calculate that for you.The present value of the ordinary annuity is $50,565.47 (rounded to the nearest cent).

Step 2: Calculate the Present Value of the Annuity Due

Now, using the formula for the present value of an annuity due:

$PV_{\text{Annuity Due}} = PV_{\text{Ordinary}} \times (1 + r)$

Let's calculate that next.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can calculate the present value of the annuity due using the formula:

$PV_{\text{Annuity Due}} = PV_{\text{Ordinary}} \times (1 + r)$

With $PV_{\text{Ordinary}} = 50,565.47$ and $r = 0.09$, you can compute it as:

$PV_{\text{Annuity Due}} = 50,565.47 \times (1 + 0.09)$

Let me know if you have any questions, or if you'd like me to clarify!

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Related Questions:

1. How does the interest rate affect the present value of an annuity?
2. What is the difference between an ordinary annuity and an annuity due in terms of cash flow timing?
3. Can annuities be compounded monthly instead of annually, and how does that change the calculation?
4. How would the present value change if the interest rate were higher?
5. How does the length of the annuity term affect the present value?

Tip: For financial problems like this, rounding off values too soon can cause slight differences in final answers, so it’s good to keep precise values until the final step.