The man should seek a payment of approximately $434,047.20.

UpStudy Free Solution:

To determine the size of the insurance payment the man should seek, we need to calculate the present value of an annuity. The annuity will provide him with $2800 at the end of each month for 25 years, and the account pays 6% interest compounded monthly.

First, let's define the variables:

 \(PMT\) = $2800 (the monthly payment)

 \(r\) = 0.06 / 12 = 0.005 (the monthly interest rate)

 \(n\) = 25 * 12 = 300 (the total number of payments)

The present value \(PV\) of an annuity can be calculated using the formula:

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

Substituting in the values:\[PV = 2800 \times \left ( 1 - ( 1 + 0.005) ^ { - 300} \right ) / 0.005\]

First, calculate \(( 1 + r) ^ { - n} \):

\[( 1 + 0.005) ^ { - 300} = ( 1.005) ^ { - 300} \]

Using a calculator:

\[( 1.005) ^ { - 300} \approx 0.22313\]

Now, calculate the rest of the formula:

\[PV = 2800 \times \left ( 1 - 0.22313\right ) / 0.005\]

\[PV = 2800 \times \left ( 0.77687\right ) / 0.005\]

\[PV = 2800 \times 155.374\]

\[PV \approx 434,047.20\]

Thus, the man should seek a payment of approximately $434,047.20.

Supplemental Knowledge

In financial mathematics, an annuity is a series of equal payments made at regular intervals. To solve for the present value of an annuity, which is the amount needed today to provide a series of future payments, we use the Present Value of Annuity formula:

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

Where:

\(PV\) is the present value (the amount needed today).

\(PMT\) is the payment amount per period.

\(r\) is the interest rate per period.

\(n\) is the total number of periods.

For this problem:

The monthly payment (\(PMT\)) is $2800.

The annual interest rate is 6%, so the monthly interest rate (\(r\)) is \(0.06 / 12 = 0.005\).

The total number of periods (\(n\)) over 25 years with monthly payments is \(25 \times 12 = 300\).

Plugging these values into the formula:

\[PV = 2800 \times \left ( 1 - ( 1 + 0.005) ^ { - 300} \right ) / 0.005\]

This calculation will give us the present value, which represents how much money should be sought now to ensure $2800 monthly payments for 25 years at a compounded monthly interest rate of 6%.

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