COMAP Problem 28 Page 828

Here is the answer to problem 28 on page 828 of our text. The problem compares three types of investment for retirement. All investments earn interest at the same annual yield r and all income is taxed at the same rate t. (The statement of the problem in the book contains a typo: the first i should be an r.) The crucial point is that if an investment of P grows to P(1+j) in one year it grows to P(1+j)n after n years.

  1. Ordinary after tax investment. In this kind of investment you earn E dollars, pay taxes on it leaving P=E(1-t) to invest. At the end of each year you pay taxes on the interest earned that year. If at the beginning of the year the account holds A dollars it earns rA in interest that year and you pay trA dollars on that interest. If you pay the taxes from the account the account grows from A to A+rA-trA=A(1+r-tr). This is the same as if the account paid annual interest at the rate of r-tr so, by the "crucial point" (reading r-tr for j) the size of the account after n years is

    A=P(1+r-tr)n=E(1-t)(1+r-tr)n.

  2. Ordinary IRA. In this investment you earn E and invest it without paying taxes. The interest again accumulates tax free, but when the money is withdrawn it is taxed. By the "crucial point" the account grows to E(1+r)n in n years. If all the account is withdrawn at this point, t times the account is deducted in taxes leaving

    A=E(1+r)n-tE(1+r)n=(1-t)E(1+r)n.

  3. Roth IRA. In this investment you earn E and pay taxes on it leaving P=(1-t)E to invest. The interest on this investment is tax free so after n years it grows to

    A=P(1+r)n=(1-t)E(1+r)n.

As stated the two kinds of IRA yield the same return, but in fact the ordinry IRA is probably better since the tax rate t is not constant; most people will have a lower income (and hence a lower tax rate) at retirement. The after tax investment is worse than the other two because (1+r-tr)<(1+r).