What is the value today of a money machine that will pay $1,000.00 per year for 30.00 years? Assume the first payment is made 5.00 years from today and the interest rate is 7.00%.

Answer 1

To find the present value (PV) of the payment stream from the money machine that pays $1,000.00 per year for 30 years starting 5 years from today, we will break the problem into two parts.

1. Calculate the present value of the annuity (payments) at the time the annuity starts (after 5 years).
2. Discount that total back to today.

Step 1: Present Value of the Annuity at Year 5

We need to calculate the present value of an annuity that pays $1,000 per year for 30 years at a discount rate of 7%. The formula for the present value of an ordinary annuity is:

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

where:

• \( P \) = payment per period ($1,000)
• \( r \) = interest rate (0.07)
• \( n \) = number of payments (30)

Plugging in the values:

\[ PV = 1000 \times \left(1 - (1 + 0.07)^{-30}\right) / 0.07 \]

Calculating \( (1 + 0.07)^{-30} \):

\[ (1 + 0.07)^{-30} ≈ (1.07)^{-30} ≈ 0.131366 \]

Now substitute this value back into the annuity formula:

\[ PV = 1000 \times \left(1 - 0.131366\right) / 0.07 \]

\[ PV = 1000 \times 0.868634 / 0.07 \]

\[ PV ≈ 1000 \times 12.40906 \]

\[ PV ≈ 12409.06 \]

This is the present value of the annuity at time 5 years from today.

Step 2: Discounting the Present Value Back to Today

Now we need to discount this amount back to today (3 years before the annuity starts, since it starts in 5 years).

Using the present value formula:

\[ PV = FV / (1 + r)^n \]

Where:

• \( FV \) = future value which we calculated as \( 12409.06 \)
• \( r \) = interest rate (0.07)
• \( n \) = number of years until the first payment (5)

So, we calculate:

\[ PV = 12409.06 / (1 + 0.07)^5 \]

Calculating \( (1 + 0.07)^5 \):

\[ (1 + 0.07)^5 ≈ 1.402552 \]

Now substitute this back into the PV formula:

\[ PV ≈ 12409.06 / 1.402552 \]

\[ PV ≈ 8843.55 \]

Final Result

The value today of the money machine that will pay $1,000.00 per year for 30 years, starting in 5 years, at an interest rate of 7% is approximately:

\[ \text{$8,843.55} \]

Answer 2

what values do I enter for N, i/y, PV, PMT, and FV to solve on a finance calculator?

Answer 3

To solve for the present value of the cash flows using a finance calculator, you can use the following inputs based on the scenario described. Here's a breakdown of the values for each variable:

• N: This represents the total number of periods. In your case, since the payments occur for 30 years starting 5 years from today, you don't directly input 30 years here. Instead, you think of the payment stream starting in year 5, which means you'll have 30 total payments after that.

  • For the present value calculation at the time of the first cash flow (after 5 years): Enter 30 (for the total number of payments).

• i/y: This is the interest rate per period. In your case, it's 7%.

  • Enter 7.

• PV: This represents the present value of the cash flows at the time just before the first payment occurs (i.e., after 5 years). Since we are calculating it separately, you can leave this as 0 for now because you're solving for the present value of the annuity first.

  • Enter 0.

• PMT: This is the payment amount per period. You are receiving $1,000.00 each year.

  • Enter 1000.

• FV: This represents the future value, which is not applicable in this case since this function is primarily for calculating present value and we are interested in cash flows as they come in regularly.

  • Enter 0.

After you've entered these values, you would typically calculate for PV:

1. Set N = 30
2. Set I/Y = 7
3. Set PMT = 1000
4. Set FV = 0
5. Set PV = 0 (if your calculator requires an initial value to be input, but typically it's not needed for this calculation).

When you compute the present value using these inputs, you would be calculating the present value of the annuity that starts at year 5. After you have that amount, you would subsequently discount it back to today to find the present value as we calculated earlier.

Final Note

Remember that you are aiming to find the value at Year 0 (today), so after calculating the present value of the annuity as specified, you will need to use the present value function again to discount back to the present (Year 0) using another calculation if your calculator doesn't account for that directly.

So in summary, you will first solve for the present value of the annuity from Year 5 to Year 35 and then discount that amount back to Year 0.