Question

In order to make CDs look more attractive as an investment than they really are, some banks advertise that their rates are higher than their competitors’ rates; however, the fine print says that the rate is based on simple interest. If you were to deposit $10,000 at 10% per year simple interest in a CD, what compound interest rate would yield the same amount of money in 3 years? Solve by formula and write the spreadsheet function to display the i value

Answer 1

To achieve the same amount of money in 3 years with a $10,000 deposit at a 10% per year simple interest rate, the equivalent compound interest rate would be approximately 10.34%.

Step-by-step explanation:

To determine the compound interest rate equivalent to a simple interest rate, we can use the formula for compound interest:

`\[A = P \left(1 + (r)/(n)\right)^(nt)\]`

Where:

-
`\(A\)` is the future value of the investment,

-
`\(P\)` is the principal amount (initial deposit),

-
`\(r\)` is the annual interest rate (as a decimal),

-
`\(n\)` is the number of times interest is compounded per year, and

-
`\(t\)` is the number of years.

Given
`\(P = $10,000\), \(r = 0.10\), \(n = 1\)` (as simple interest is compounded once a year), and
`\(t = 3\)` years, we rearrange the formula to solve for
`\(r\)`:

`\[r = n \left(\left((A)/(P)\right)^{(1)/(nt)} - 1\right)\]`

Substituting the values, we get:

`\[r \approx 1 \left(\left((P * (1 + rt))/(P)\right)^{(1)/(1 * 3)} - 1\right)\]`

Solving this gives
`\(r \approx 0.1034\)`, or approximately
`\(10.34%\)`. Thus, the compound interest rate required to yield the same amount of money in 3 years as a 10% simple interest rate is approximately
`\(10.34%\)`. To display this in a spreadsheet, you can use the formula
``=RATE(3,0,-10000,0,1)``, where 3 is the number of periods, 0 is the payment per period, -10000 is the present value (negative because it's an investment), 0 is the future value, and 1 represents payments at the beginning of the period.