Question 21.80

image 50. We continue the theme of Exercises 47–49 by comparing in algebraic terms three kinds of investments for retirement: an ordinary after-tax investment, a tax-deferred investment (such as a tax-deferred annuity or an IRA), and a Roth IRA. Let an investment earn interest at a steady annual yield and let your income (in whatever year you receive it) be taxed at rate .

  1. Ordinary after-tax investment: Explain why if you earn , pay taxes on it, let what remains earn interest, and pay tax each year on that year’s interest, the grows after years to .
  2. Ordinary IRA: Explain why if you earn $E, defer taxes on it, let it earn interest, and defer taxes on all the interest, then the grows after years to .
  3. Roth IRA: Explain why if you earn , pay taxes on it, let what remains earn interest, and pay no taxes on any of the interest, the grows after years to .
  4. Which investment gives the best return after years?