Find the future value of the following annuities. The first payment in these annuities is made at the end of Year 1, so they are ordinary annuities. (Notes: If you are using a financial calculator, you can enter the known values and then press the appropriate key to find the unknown variable. Then, without clearing the TVM register, you can "override" the variable that changes by simply entering a new value for it and then pressing the key for the unknown variable to obtain the second answer. This procedure can be used in many situations, to see how changes in input variables affect the output variable. Also, note that you can leave values in the TVM register, switch to Begin Mode, press FV, and find the FV of the annuity due.) Do not round intermediate calculations. Round your answers to the nearest cent.

   a.  $600 per year for 10 years at 12%.

    

    b. $300 per year for 5 years at 6%.

    

    c. $600 per year for 5 years at 0%.

    

d. Now rework parts a, b, and c assuming that payments are made at the beginning of each year; that is, they are annuities due.

    Future value of $600 per year for 10 years at 12%  

    Future value of $300 per year for 5 years at 6%  

    Future value of $600 per year for 5 years at 0%

To find the **future value (FV)** of an ordinary annuity and an annuity due, we use the formulas for the future value of annuities.

### 1. **Future Value of an Ordinary Annuity (Payments at the End of Each Year)**:
The formula for the future value of an ordinary annuity is:

\[
FV = P \times \frac{{(1 + r)^n - 1}}{r}
\]

Where:
- $FV$ is the future value of the annuity.
- $P$ is the annual payment.
- $r$ is the interest rate per period (as a decimal).
- $n$ is the number of periods (years).

### 2. **Future Value of an Annuity Due (Payments at the Beginning of Each Year)**:
The formula for the future value of an annuity due is:

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

The difference between the two is that in an annuity due, each payment is shifted one period earlier, earning an extra period of interest.

---

### Part a: $600 per year for 10 years at 12%

- Payment ($P$) = \$600
- Interest rate ($r$) = 12% = 0.12
- Number of periods ($n$) = 10

#### Future Value of the Ordinary Annuity:
\[
FV = 600 \times \frac{{(1 + 0.12)^{10} - 1}}{0.12}
\]

#### Future Value of the Annuity Due:
\[
FV_{\text{due}} = 600 \times \left( \frac{{(1 + 0.12)^{10} - 1}}{0.12} \right) \times (1 + 0.12)
\]

---

### Part b: $300 per year for 5 years at 6%

- Payment ($P$) = \$300
- Interest rate ($r$) = 6% = 0.06
- Number of periods ($n$) = 5

#### Future Value of the Ordinary Annuity:
\[
FV = 300 \times \frac{{(1 + 0.06)^{5} - 1}}{0.06}
\]

#### Future Value of the Annuity Due:
\[
FV_{\text{due}} = 300 \times \left( \frac{{(1 + 0.06)^{5} - 1}}{0.06} \right) \times (1 + 0.06)
\]

---

### Part c: $600 per year for 5 years at 0%

- Payment ($P$) = \$600
- Interest rate ($r$) = 0% = 0
- Number of periods ($n$) = 5

When the interest rate is 0%, the future value is simply the sum of all the payments.

#### Future Value of the Ordinary Annuity:
\[
FV = 600 \times 5 = 3000
\]

#### Future Value of the Annuity Due:
Since there is no interest, the future value will also be the sum of the payments, but with one extra period due to the payment timing.

\[
FV_{\text{due}} = 600 \times 5 = 3000
\]

Now, let me calculate all the exact future values for you.Here are the future values for the annuities:

### Part a: $600 per year for 10 years at 12%
- Future Value (Ordinary Annuity): \$10,529.24
- Future Value (Annuity Due): \$11,792.75

### Part b: $300 per year for 5 years at 6%
- Future Value (Ordinary Annuity): \$1,691.13
- Future Value (Annuity Due): \$1,792.60

### Part c: $600 per year for 5 years at 0%
- Future Value (Ordinary Annuity): \$3,000
- Future Value (Annuity Due): \$3,000

If you have any questions or need further clarification, feel free to ask!

Here are some follow-up questions:
1. How would the future value change if the interest rate increased?
2. What happens to the future value if the number of periods is extended?
3. How does the payment amount affect the future value of an annuity?
4. Why is the future value of an annuity due always higher than that of an ordinary annuity?
5. How can you calculate the present value of an annuity given its future value?

**Tip**: An annuity due always earns more interest than an ordinary annuity because payments are made earlier, allowing them to compound longer.

Find the future value of the following annuities. The first payment in these annuities is made at the end of Year 1, so they are ordinary annuities. (Notes: If you are using a financial calculator, you can enter the known values and then press the appropriate key to find the unknown variable. Then, without clearing the TVM register, you can 'override' the variable that changes by simply entering a new value for it and then pressing the key for the unknown variable to obtain the second answer. This procedure can be used in many situations, to see how changes in input variables affect the output variable. Also, note that you can leave values in the TVM register, switch to Begin Mode, press FV, and find the FV of the annuity due.) Do not round intermediate calculations. Round your answers to the nearest cent.

 a. $600 per year for 10 years at 12%.

 b. $300 per year for 5 years at 6%.

 c. $600 per year for 5 years at 0%.

 d. Now rework parts a, b, and c assuming that payments are made at the beginning of each year; that is, they are annuities due.

Calculate the future value of ordinary annuities and annuities due for various payments and interest rates. This problem covers annuities with $600 per year for 10 years at 12%, $300 per year for 5 years at 6%, and $600 per year for 5 years at 0%. Step-by-step calculations and key formulas included.

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