Mrs. Hart, at age 65, can expect to live for 25 years. If she can invest at 5% per annum compounded monthly, how much does she need now to guarantee herself $900 every month for the next 25 years?
math - Reiny, Friday, April 3, 2015 at 3:05pm
Present Value
= 900(1 - 1.0041666..)^-300)/.0041666...
= $153,954.05
math - Judy, Friday, April 3, 2015 at 3:33pm
Reiny, 153,954.05 is coming up wrong. any suggestions?
900 dollars every Month...
do you need to divide the 153,954 by the number of months in 25 years?
I am not sure what happened but I got the answer .. 153,954 is correct. thanks for the help.
To calculate the present value that Mrs. Hart needs to guarantee herself $900 every month for the next 25 years, we can use the formula for the present value of an ordinary annuity:
Present Value = Cash Flow / Discount Rate
Where:
- Cash Flow is the amount to be received per period ($900 per month in this case)
- Discount Rate is the interest rate per period (5% per annum compounded monthly)
- We need to adjust the discount rate to match the compounding period, which is monthly in this case
Now, let's calculate the present value step-by-step.
Step 1: Calculate the monthly interest rate.
The annual interest rate of 5% needs to be converted to a monthly interest rate.
Monthly interest rate = (1 + Annual interest rate)^ (1/12) - 1
Monthly interest rate = (1 + 0.05)^(1/12) - 1
Monthly interest rate = 0.00407489 (rounded to 8 decimal places)
Step 2: Calculate the number of compounding periods.
Since Mrs. Hart can expect to live for 25 years and we are compounding monthly, the total number of compounding periods would be 25 years * 12 months = 300 months.
Step 3: Calculate the present value.
Using the formula:
Present Value = Cash Flow / Discount Rate
Present Value = $900 / 0.00407489
Calculating the present value:
Present Value = $220,912.06 (rounded to the nearest cent)
Therefore, Mrs. Hart needs to have $220,912.06 (rounded to the nearest cent) now to guarantee herself $900 every month for the next 25 years.
To calculate the amount Mrs. Hart needs to invest now to guarantee herself $900 every month for the next 25 years, we can use the formula for the present value of an annuity. The formula is:
Present Value = Payment Amount * ((1 - (1 + Interest Rate)^(-Number of Payments)) / Interest Rate)
In this case, the Payment Amount is $900, the Interest Rate is 5% per annum (or 0.05), and the Number of Payments is 25 * 12 = 300 (since there are 12 months in a year).
Plugging in these values into the formula, we have:
Present Value = 900 * ((1 - (1 + 0.05) ^ (-300)) / 0.05)
Now we can simplify this expression:
Present Value = 900 * ((1 - 1.0041666..^(-300)) / 0.0041666..)
To evaluate the expression inside the brackets, you can use a scientific calculator or spreadsheet software. Raise 1.0041666.. to the power of -300 and subtract the result from 1. Divide this value by 0.0041666.. to get the final answer.
After evaluating the expression, the correct present value should be obtained.