Growth Functions - Analysis

[mexx]

Who can solve these exercises or at least try to explain me how to go about them..?

1. Suppose, you deposit $25,000 in an account to accrue interest for 40 years. The account pays 4% compounded annually. Assume that the income tax on the earned interest is 30%. Which of the following plans produces a larger balance after all income tax is paid?
a. Deferred: The income tax on the interest that is earned is paid one lump sum at the end of 40 years.
b. Not Deferred: The income tax on the interest that is earned each year is paid at the end of each year.

2. Which of the following would produce a larger balance? Explain!
a. 4.02% annual interest rate, compounded monthly.
b. 4% annual interest rate, compounded continously.

3. Suppose you deposit $1000 in each of two saving accounts. The interest for the accounts is paid according to the options described in Question 2. How long would it take for the balance in one of the accounts to exceed the balance in the other account by $100? By $100,000?

4. No income tax is due on the interest earned in some types of investment. Suppose you deposit $25,000 into an account. Which of the following plans is better? Explain!
a. Tax-Free: The account pays 5%, compounded annually. There is no income tax due on the earned interest.
b. Tax-Deferred: The account pays 7%, compounded annually. At maturity, the earned interest is taxable at a rate of 40%.

Thank you

 

[soroban]

Hello, mexx!

Here's #4 . . .
\(\displaystyle \;\;\)But there is no algebraic way to solve the final equation.

4. No income tax is due on the interest earned in some types of investment.
Suppose you deposit $25,000 into an account.
Which of the following plans is better? Explain!

A. Tax-Free: The account pays 5%, compounded annually.
\(\displaystyle \;\;\;\)There is no income tax due on the earned interest.

B. Tax-Deferred: The account pays 7%, compounded annually.
\(\displaystyle \;\;\;\)At maturity, the earned interest is taxable at a rate of 40%.

Of course, it depends on how long you make the investment.

Plan A
$25,000 invested at 5% compounded annually for \(\displaystyle n\) years will become: \(\displaystyle \,25,000(1.05)^n\) dollars.
\(\displaystyle \;\;\)The interest is: \(\displaystyle \,25,000(1.05)^n\,-\,25,000\;\;\Rightarrow\;\;I_a\:=\:25,000(1.05^n\,-\,1)\) dollars.

Plan B
$25,000 invested at 7% compounded annually for \(\displaystyle n\) years will become:\(\displaystyle \,25,000(1.07)^n\) dollars.
\(\displaystyle \;\;\)The interest is: \(\displaystyle 25,000(1.70)^n\,-\,25000\;=\;25,000(1.07^n\,-\,1)\) dollars.
But you will be taxed 40% of that interest; you will keep only 60% of it.
\(\displaystyle \;\;\)Your interest is: \(\displaystyle \,0.60\,\times\,25,000(1.07^n\,-\,1)\;\;\Rightarrow\;\;I_b\:=\:15,000(1.07^n\,-\,1)\)


When is Plan A better than Plan B?
\(\displaystyle \;\;\)That is, when is \(\displaystyle I_a\,>\.I_b\) ?

We have: \(\displaystyle \,25,000(1.05^n\,-\,1)\;>\;15,000(1.07^n\,-\,1)\)

Divide by 5,000: \(\displaystyle \,5(1.05^n\,-\,1)\;>\;3(1.07^n\,-\,1)\)

. . . . . . . . . . . . . \(\displaystyle \,5(1.05)^n \,-\,5\;>\;3(1.07)^n\,-\,3\)

. . . . . . . . . \(\displaystyle 5(1.05)^n\,-\,3(1.07)^n\;>\:2\)


I found no neat way to solve for \(\displaystyle n.\)
We can use a graphing utility or use trial-and-error.

I found that the inequality holds for \(\displaystyle \,n\:\geq\:16\)

If you plan to keep the investment for at least 16 years, Plan A is better.