To solve this problem, we need to calculate the present value of the annuity using two different approaches: for an ordinary annuity and an annuity due.

Key information:

• Amount of annuity ($P$) = $13,000
• Interest rate ($r$) = 9% (or 0.09)
• Period (years) ($n$) = 5 years

Formulas:

1. Present Value of an Ordinary Annuity (PVOA): $PVOA = P \times \left(1 - \frac{1}{(1 + r)^n}\right) \div r$ Where:

   • $P$ = annuity payment
   • $r$ = interest rate per period
   • $n$ = number of periods

2. Present Value of an Annuity Due (PVAD): $PVAD = PVOA \times (1 + r)$ The difference between an ordinary annuity and an annuity due is that for an annuity due, payments are made at the beginning of the period, hence the present value will be slightly higher.

a. Present Value Calculations:

1. Ordinary Annuity (PVOA):

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

Let me calculate this for you.

2. Annuity Due (PVAD):

$PVAD = PVOA \times (1 + 0.09)$

Once we have the PVOA, we can compute the PVAD.

Let’s calculate these step by step.