Burton Bush wants to retire in Arizona when he is 80 years of age. Burton, who is now 55, believes he will need $400,000 to retire comfortably. To date, he has set aside no retirement money. If he gets an interest rate of 6% compounded annually, he will have to invest today (use the tables in the handbook):
A. $92,300
B. $69,900
C. $96,500
D. $93,200
To determine how much Burton Bush will have to invest today to retire comfortably, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the principal amount (the initial investment)
r = the annual interest rate (as a decimal)
n = the number of times that interest is compounded per year
t = the number of years
In this case, Burton is 55 years old and wants to retire at age 80, meaning he has 25 years until retirement. He believes he will need $400,000 at that time, and the interest rate is 6% compounded annually.
So, we need to find the principal amount (P) that Burton needs to invest today to achieve the desired future value (A). Rearranging the formula, we have:
P = A / (1 + r/n)^(nt)
Now let's calculate:
A = $400,000
r = 6% = 0.06
n = 1 (as the interest is compounded annually)
t = 25 years
P = 400,000 / (1 + 0.06/1)^(1*25)
P = 400,000 / (1 + 0.06)^(25)
P = 400,000 / (1.06)^25
P ≈ 69,867.76
Therefore, Burton will have to invest approximately $69,867.76 today to retire comfortably in Arizona at age 80.
The closest option to the calculated amount is option B: $69,900.
To calculate the amount that Burton will have to invest today, we can use the future value formula for compound interest:
\(FV = PV \times (1 + r)^t\)
Where:
FV = future value (amount needed at retirement)
PV = present value (amount that Burton needs to invest today)
r = interest rate
t = number of years
Given:
FV = $400,000
r = 6% (expressed as a decimal, 0.06)
t = (80 - 55) = 25 years
Let's plug in the values in the formula and solve for PV:
\(400,000 = PV \times (1 + 0.06)^{25}\)
Next, we can divide both sides of the equation by \((1 + 0.06)^{25}\):
\(\frac{400,000}{(1 + 0.06)^{25}} = PV\)
Calculating this using a calculator or spreadsheet will give us the answer.
\(PV \approx 92,300\)
Therefore, the amount Burton will have to invest today is approximately $92,300.
So, the correct answer is A. $92,300.
P*1.06^25 = 400000
Now just solve for P