Catherine Dohanyos plans to retire in 15 years. She will make 15 years of monthly contributions to her retirement account. One month after her last contribution, she will begin the first of 10 years of withdrawals. She wants to withdraw $3500 per month. How large must her monthly contributions be in order to accomplish her goal if the account earns interest of 7.9% compounded monthly for the duration of her contributions and the 120 months of withdrawals?
We can use the formula for the future value of an annuity to solve this problem. Let x be the monthly contribution amount. Then, the future value of the annuity after 15 years of contributions is:
FV = x * (((1 + 0.079/12)^180 - 1) / (0.079/12))
This formula calculates the value of the annuity after 15 years of contributions, assuming monthly compounding at 7.9% interest rate. We use 180 for the number of months because there are 15 years times 12 months per year.
Next, we need to calculate the present value of the withdrawals over 10 years. We can use the formula for the present value of an annuity for this:
PV = 3500 * (((1 - (1 + 0.079/12)^-120) /(0.079/12)))
This formula calculates the present value of the annuity that pays out $3500 per month for 10 years, assuming monthly compounding at 7.9% interest rate.
Now, we want the future value of the annuity to be equal to the present value of the withdrawals, so we set FV equal to PV and solve for x:
x * (((1 + 0.079/12)^180 - 1) / (0.079/12)) = 3500 * (((1 - (1 + 0.079/12)^-120) /(0.079/12)))
Simplifying this equation gives:
x = 3500 * (((1 - (1 + 0.079/12)^-120) /(0.079/12))) / (((1 + 0.079/12)^180 - 1) / (0.079/12))
Plugging in the numbers and calculating gives:
x = $519.51 (rounded to two decimal places)
Therefore, Catherine needs to contribute $519.51 per month in order to withdraw $3500 per month for 10 years after she retires.
At the average annual inflation rate of 7.6%, about how long would it take for the general level of prices in the economy to double?
To estimate how long it would take for the general level of prices in the economy to double, we can use the rule of 72, which says that the number of years it takes for an investment to double is approximately 72 divided by the annual growth rate. In this case, we can use the rule of 72 with an inflation rate of 7.6%:
Approximate number of years to double = 72 / 7.6% = 9.47 years
Therefore, at an average annual inflation rate of 7.6%, it would take about 9.47 years for the general level of prices in the economy to double. Note that this is an approximation and the actual time it takes for prices to double may be slightly different.
A building contractor gives a $15500 promissory note to a plumber who has loaned him $15500. The note is due in 9 months with interest at 6%. six months after the note is signed, the plumber sells it to a bank. If the bank gets a 7% return on its investment, how much will the plumber receive? Will it be enough to pay a bill for $16125?
To find out how much the plumber will receive by selling the promissory note to the bank, we need to calculate the present value of the note at the time of sale, which is 6 months after it was signed. We can use the formula for the present value of a promissory note:
PV = FV / (1 + r*t)
where PV is the present value, FV is the future value (or face value) of the note, r is the discount (or interest) rate, and t is the time in years.
In this case, the face value of the note is $15,500 and the time to maturity is 9 months, or 0.75 years. The interest rate is 6%. To calculate the present value of the note after 6 months, we need to adjust the time to maturity to 0.25 years (9 months minus 6 months):
PV = 15,500 / (1 + 0.06*0.25) = $14,745.28
This is the amount that the bank would pay to the plumber to purchase the note. However, we also need to calculate how much the bank would earn on this investment at a rate of 7% over the remaining 3 months until the note matures:
Interest earned by bank = 14,745.28 * 0.07 * (3/12) = $256.40
Therefore, the total amount that the plumber would receive from the bank is:
14,745.28 + 256.40 = $15,001.68
This is less than the amount needed to pay the bill of $16,125, so the plumber would not have enough funds to pay the bill. They would need to find additional sources of funds to cover the remaining balance.
A man deposits $15000 at the beginning of each year for 17 years in an account paying 8% compounded annually. He then puts the total amount on deposit in another account paying 9% compounded semiannually for another 11 years. Find the final amount on deposit after the entire 28-year period.
We can use the future value formula for annuities to find the total amount of money in the account after the first 17 years of deposits:
FV = A * ((1 + r)^n - 1)/r
where A is the annual payment, r is the annual interest rate, and n is the number of years.
Plugging in the numbers, we get:
FV = $15,000 * ((1 + 0.08)^17 - 1) / 0.08 = $537,308.41
So after 17 years, the account has $537,308.41.
Next, we can use the future value formula for compound interest to find the total amount of money in the account after the next 11 years:
FV = P * (1 + r/n)^(n*t)
where P is the principal, r is the annual interest rate, n is the number of compounding periods per year, and t is the number of years.
Plugging in the numbers, we get:
FV = $537,308.41 * (1 + 0.09/2)^(2*11) = $1,560,121.86
So after 28 years, the account has $1,560,121.86 on deposit.
Fritz Benjamin buys a car costing $21100. He agrees to make payments at the end of each monthly period for 8 years. He pays 9.6% interest, compounded monthly. What is the amount of each payment? Find the total amount of interest Fritz will pay.
We can use the formula for the present value of an annuity to find the amount of each payment:
PV = PMT * (1 - (1 + r/n)^(-n*t)) / (r/n)
where PV is the present value of the car, PMT is the monthly payment, r is the annual interest rate, n is the number of compounding periods per year, and t is the number of years.
Plugging in the numbers, we get:
PV = $21,100
r = 0.096/12 = 0.008
n = 12
t = 8
Solving for PMT, we get:
PMT = PV * (r/n) / (1 - (1 + r/n)^(-n*t)) = $279.58
So Fritz will need to make monthly payments of $279.58.
The total amount of interest Fritz will pay over the 8-year period is equal to the total amount of payments minus the principal amount. The total amount of payments is equal to the monthly payment times the number of months (12 months/year times 8 years = 96 months). Therefore:
Total interest = PMT * t * 12 - PV = $279.58 * 96 - $21,100 = $10,595.68
So Fritz will pay a total of $10,595.68 in interest over the 8-year period.
Find the final amount in the following retirement account, in which the rate of return on the account and the regular contribution change over time.
$381 per month invested at 5%, compounded monthly, for 7 years; then $565 per month invested at 7%, compounded monthly, for 7 years.