Questions: If you put up 39,000 today in exchange for a 6.5 percent, 16-year annuity, what will the annual cash flow be? Note: Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.

If you put up 39,000 today in exchange for a 6.5 percent, 16-year annuity, what will the annual cash flow be? Note: Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.
Transcript text: If you put up $\$ 39,000$ today in exchange for a 6.5 percent, 16 -year annuity, what will the annual cash flow be? Note: Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.
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Solution

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Solution Steps

To find the annual cash flow of an annuity given the present value, interest rate, and number of periods, we can use the formula for the present value of an annuity. The formula is:

\[ PV = C \times \left(1 - (1 + r)^{-n}\right) / r \]

where:

  • \( PV \) is the present value of the annuity,
  • \( C \) is the annual cash flow,
  • \( r \) is the interest rate per period,
  • \( n \) is the number of periods.

We need to solve for \( C \), the annual cash flow.

Solution Approach
  1. Rearrange the formula to solve for \( C \).
  2. Substitute the given values into the formula: \( PV = 39000 \), \( r = 0.065 \), and \( n = 16 \).
  3. Calculate the annual cash flow using Python.
Step 1: Identify the Given Values

We are given the following values:

  • Present Value (\( PV \)): \( 39000 \)
  • Interest Rate (\( r \)): \( 0.065 \)
  • Number of Periods (\( n \)): \( 16 \)
Step 2: Use the Present Value of Annuity Formula

The formula for the present value of an annuity is:

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

We need to rearrange this formula to solve for the annual cash flow (\( C \)):

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

Step 3: Substitute the Values

Substituting the given values into the rearranged formula:

\[ C = 39000 \times \frac{0.065}{1 - (1 + 0.065)^{-16}} \]

Step 4: Calculate the Annual Cash Flow

Calculating the value of \( C \):

\[ C \approx 3992.72538422617 \]

Rounding to two decimal places, we find:

\[ C \approx 3992.73 \]

Final Answer

The annual cash flow is \\(\boxed{3992.73}\\).

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