Question

you want an annuity that will pay $450 at the end of each month for 8 years. you have an account that is earning 6% interest per year compounded monthly. (3.1) (14 points) how much money must the annuity contain at the start, so that at the end of 8 years there is no money left? answer: (3.2) (14 points) how much of what you are paid over 8 years is from interest? answer: (3.3) (14 points) you would like to increase the amount you get each month and have it sent to you at the start of every month. the interest rate is still 6% interest per year compounded monthly. if you were able to start the annuity with $50,000, how much would your monthly payments be?

Answer 1

• Start with $34,243 to get $450 monthly for 8 years (ending with nothing left).
• You actually lose $8,957 to interest over 8 years (compounding works wonders!).
• To get $657 monthly upfront each month for 8 years, start with $50,000! Remember, interest plays tricks!

Annuity Calculations:

3.1 Initial Annuity Amount:

To find the initial amount needed for 8 years of $450 monthly payments with no remaining balance, we can use the formula for the present value of an annuity due:

Present Value (PV) = Payment / (Interest Rate * (1 - (1 + Interest Rate)^(Number of Payments)))

In this case:

• Payment = $450
• Interest Rate = 6% / 12 (monthly) = 0.5%
• Number of Payments = 8 years * 12 months/year = 96 months

Plugging in the values:

PV = 450 / (0.005 * (1 - (1 + 0.005)^(96))) ≈ $34,242.85

Therefore, you need approximately $34,242.85 at the start of the annuity.

3.2 Total Interest Earned:

The total interest earned over 8 years can be calculated as the difference between the total received payments and the initial investment:

Total Interest = Total Payments - Initial Investment

Total Interest = (450 * 96) - 34,242.85 ≈ $-8,957.15

Interestingly, you end up with negative interest because the 6% monthly compounding on the remaining balance outweighs the incoming payments, gradually depleting the account even before all payments are received.

3.3 Increased Monthly Payments with $50,000 Starting Balance:

With a higher initial investment of $50,000, you can increase the monthly payments while still ensuring the annuity lasts 8 years. Again, we can use the present value formula with the desired initial amount:

Payment = Interest Rate * Initial Investment / (1 - (1 + Interest Rate)^(Number of Payments))

Payment = 0.005 * 50,000 / (1 - (1 + 0.005)^(96)) ≈ $657.07

Therefore, you can receive approximately $657.07 per month at the start of each month for 8 years with a starting balance of $50,000.

Remember, these are approximations based on the provided information. Consulting a financial advisor or using specialized annuity calculators is recommended for personalized and accurate calculations.