**Present value of an annuity**   Consider the following case.  ​(Click on the icon here

in order to copy the contents of the data table below into a​ spreadsheet.)

**Amount of annuity**

**Interest rate**

**Period**​ (years)

​$13 comma 00013,000

99​%

55

**a.**  Calculate the present value of the annuity assuming that it is

​(1)  An ordinary annuity.

​(2)  An annuity due.

**b.**  Compare your findings in parts

**a**​(1)

and

**a**​(2).

All else being​ identical, which type of

annuitylong dash—ordinary

or annuity

duelong dash—is

​preferable? Explain why.

Question content area bottom

Part 1

The present value of the ordinary annuity is

​$enter your response here.

​(Round to the nearest​ cent.) 13,000	9	5

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.

Present value of an annuity
Consider the following case.
(Click on the icon here in order to copy the contents of the data table below into a​ spreadsheet.)
Amount of annuity
Interest rate
Period​ (years)
$13,000
9%
5

a. Calculate the present value of the annuity assuming that it is
(1) An ordinary annuity.
(2) An annuity due.

b. Compare your findings in parts a(1) and a(2). All else being identical, which type of annuity—ordinary or annuity due—is preferable? Explain why.

The present value of the ordinary annuity is​ $enter your response here.

Learn how to calculate the present value of an ordinary annuity and an annuity due with a $13,000 payment, a 9% interest rate, and a 5-year period. This finance problem involves key concepts of annuity, present value, and time value of money, making it suitable for college-level students.

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