100,000 for 20 years compounded at 4 percent annually results in a rate per period of
$100,000 for 20 years compounded at 4% annually results in a rate per period of:
To calculate the rate per period, we need to use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment ($100,000)
P = the principal amount of the investment ($100,000)
r = the annual interest rate (4%)
n = the number of times that interest is compounded per year (we are not given this value, so we will assume it is compounded annually)
t = the number of years (20)
Let's substitute the values into the formula and solve for r:
100,000 = 100,000(1 + r/1)^(1 * 20)
Divide both sides by 100,000:
1 = (1 + r)^20
Take the 20th root of both sides:
(1)^(1/20) = 1 + r
Simplify:
1 = 1 + r
Subtract 1 from both sides to isolate r:
r = 0
Therefore, the rate per period is 0% since there is no growth or compounding.
To find the rate per period when compounding annually, you first need to know the number of periods in 20 years.
Since compounding is done annually, there is one compounding period per year. So the number of periods in 20 years is 20.
Now, let's calculate the rate per period. We can use the compound interest formula:
Future Value = Present Value * (1 + r/n)^(n*t),
where:
- Future Value is the final amount after compounding,
- Present Value is the initial amount,
- r is the annual interest rate,
- n is the number of compounding periods per year, and
- t is the number of years.
In this case:
- Present Value (PV) is $100,000,
- Future Value (FV) is unknown,
- r is 4% (or 0.04),
- n is 1 (since compounding is done annually), and
- t is 20 years.
Using the compound interest formula, we can isolate the rate per period (r/n):
FV/PV = (1 + r/n)^(n*t)
FV/PV = (1 + r/1)^(1*20)
FV/PV = (1 + r)^20
Now, substitute the given values:
100,000/(100,000) = (1 + r)^20
Simplifying the equation:
1 = (1 + r)^20
To isolate (1 + r), take the 20th root of both sides:
(1 + r) = 1^(1/20)
(1 + r) = 1
Finally, subtract 1 from both sides to get the rate per period:
r = 1 - 1
r = 0
Therefore, the rate per period when compounding annually is 0 (or 0%).