Assume you want to accumulate $125,000 for a long-range goal in 40 years. You have two choices. Plan A is an account with annual compounding and an APR of 5%. Plan B is an account with continuous compounding and an APR of 4.5%. How much of an investment does each plan require to reach your goal?
math - bobpursley, Tuesday, May 14, 2013 at 7:41pm
125=M*e^(.045*40)
M=125K/6.04== 20.66236 K
125=M(1+.05)^40
log 125=logM + 40log1.05
M= = 17.75571 thousand
Assume you want to accumulate $250,000 for a long-range goal in 40 years. You have two choices. Plan A is an account with annual compounding and an APR of 3%. Plan B is an account with continuous compounding and an APR of 3.25%. How much of an investment does each plan require to reach your goal? -
To calculate the required investment for each plan, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = Accumulated amount
P = Initial investment (or principal)
r = Annual interest rate (in decimal form)
n = Number of times interest is compounded per year
t = Number of years
For Plan A:
r = 5% = 0.05 (APR)
n = 1 (annual compounding)
t = 40 years
125,000 = P(1 + 0.05/1)^(1 × 40)
Simplifying the equation:
125,000 = P(1.05)^40
To find the value of P, divide both sides by (1.05)^40:
P = 125,000 / (1.05)^40
P ≈ $36,740.74
Therefore, Plan A would require an initial investment of approximately $36,740.74 to accumulate $125,000 in 40 years.
For Plan B:
r = 4.5% = 0.045 (APR)
Since continuous compounding is used, n is not required.
t = 40 years
125,000 = P * e^(0.045 × 40)
To find the value of P, divide both sides by e^(0.045 × 40):
P = 125,000 / e^(0.045 × 40)
P ≈ $35,920.49
Therefore, Plan B would require an initial investment of approximately $35,920.49 to accumulate $125,000 in 40 years.
To find out how much of an investment each plan requires to reach your goal, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the final amount or goal
P = the principal or initial investment
r = the annual interest rate (in decimal form)
n = the number of times the interest is compounded per year
t = the number of years
Let's calculate the investment required for each plan:
Plan A:
A = $125,000 (goal)
r = 5% (APR) = 0.05
t = 40 years
Since Plan A mentions "annual compounding," the interest is compounded once per year (n = 1).
Using the formula:
$125,000 = P(1 + 0.05/1)^(1*40)
Simplifying:
$125,000 = P(1.05)^40
To solve for P (the investment required), divide both sides of the equation by (1.05)^40:
P = $125,000 / (1.05)^40
Now, let's calculate the investment required for Plan B:
Plan B:
A = $125,000 (goal)
r = 4.5% (APR) = 0.045
t = 40 years
Since Plan B mentions "continuous compounding," we can use another version of the formula:
A = Pe^(rt)
Using the formula:
$125,000 = Pe^(0.045*40)
Simplifying:
$125,000 = Pe^(1.8)
To solve for P, divide both sides of the equation by e^(1.8):
P = $125,000 / e^(1.8)
Now, you can use a calculator or software to evaluate the equations and find the investment required for each plan.