First Simple Bank pays 6 percent simple interest on its investment accounts. First Complex Bank pays interest on its accounts compounded annually.
Required:
What rate should the bank set if it wants to match First Simple Bank over an investment horizon of 10 years?
Assume an investment of $1.00
First Simple Bank:
amount after 6 years = 1 + PRT
= 1 + 1(.06)(10)
= 1.60
Complex Bank:
1(1+i)^10 = 1.6
take 10th root of both sides
( (1+i)^10)((1/10) = 1.6^(1/10)
1+i = 1.048122..
i = .048122..
or
appr 4.81%
Well, if First Simple Bank is paying 6 percent simple interest, First Complex Bank will need to put on its thinking cap and calculate the compounded interest rate to stay in the game.
Let me put on my clown glasses and do the math for you.
To match First Simple Bank's rate over 10 years, First Complex Bank needs to find an interest rate that will yield the same amount of money. Now, I promise you this won't be as complicated as finding your way out of a clown maze.
Let's break it down like a clown juggling balls. The formula for compound interest is A = P(1+r/n)^(nt), where:
A = the amount of money accumulated after the interest is compounded
P = the principal amount (the initial amount of money)
r = the annual interest rate (in decimal form)
n = the number of times that interest is compounded per year (annually in this case)
t = the number of years
We want to find the rate (r) that will make A equal to what First Simple Bank is offering. So, if First Simple Bank is paying 6 percent, we need to determine the value that will make A equal 6 percent over 10 years.
Now, let's not clown around and get straight to the point. First Complex Bank needs to solve for r in the equation 1.06 = (1+r)^10.
To make things simpler, let's use a calculator that isn't shaped like a rubber chicken. Using some advanced clown magic, I can tell you that the rate First Complex Bank needs to set to match First Simple Bank over 10 years is approximately 5.93 percent.
So, there you have it! First Complex Bank needs to set an interest rate of around 5.93 percent over 10 years to keep up with First Simple Bank. Now, let the circus of banking begin!
To calculate the interest rate that First Complex Bank should set in order to match First Simple Bank over an investment horizon of 10 years, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the initial principal amount
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, we want to find the interest rate (r) that would result in the same final amount (A) after 10 years as First Simple Bank's 6 percent simple interest.
Let's assume that First Simple Bank invests $1 initially, so P = $1.
For First Simple Bank:
A = 1 + (1 * 0.06 * 10) = 1.6
For First Complex Bank:
A = 1 * (1 + r/1)^(1 * 10) = 1 * (1 + r)^10
Since we want to find the interest rate (r) that results in A = 1.6 after 10 years, we can set up the equation:
1 * (1 + r)^10 = 1.6
Now we can solve for r:
(1 + r)^10 = 1.6
Take the 10th root of both sides:
1 + r = 1.6^(1/10)
r = 1.6^(1/10) - 1
Calculating the value of r gives us:
r ≈ 0.0545
Therefore, First Complex Bank should set an interest rate of approximately 5.45 percent (compounded annually) to match First Simple Bank's 6 percent simple interest over an investment horizon of 10 years.
To determine the interest rate that First Complex Bank should set to match First Simple Bank over a 10-year investment horizon, we need to find the equivalent compounded annual interest rate. The formula to convert a simple interest rate to a compounded interest rate is:
\( \text{Compounded Rate} = (1 + \text{Simple Rate})^n - 1 \)
Where:
- "Simple Rate" is the interest rate offered by First Simple Bank (6% or 0.06 as a decimal),
- "n" is the number of compounding periods over the investment horizon (10 years in this case).
Using this formula, we can calculate the compounded rate:
\( \text{Compounded Rate} = (1 + 0.06)^{10} - 1 \)
Let's calculate it step by step:
1. First, add 1 to the simple interest rate: \( 1 + 0.06 = 1.06 \).
2. Raise this sum to the power of the number of compounding periods: \( 1.06^{10} \).
3. Finally, subtract 1 from the result to get the compounded rate: \( 1.06^{10} - 1 \).
Using a calculator or spreadsheet, we can find that the compounded rate is approximately 0.7908, or 79.08% as a percentage.
Therefore, First Complex Bank should set the interest rate at approximately 79.08% compounded annually to match First Simple Bank over a 10-year investment horizon.