Taxes

After Tax Value of Investment

Returning to the issue of what constitutes a good investment, we consider Warren Buffett’s advice that Stocks will beat Gold and Bonds. He reported in an article and speech to shareholders that from 1965 to 2012 the S&P 500 grew significantly more in the S&P than in Treasurey bills or gold. To consider this problem, we need to consider the payment of taxes that accompanies T - bills, but also be able to adjust for inflation.

For the first situation, we investigate the T-Bills, where any gains are included in your income that year. Accordingly, you have to pay a tax on that amount earned. Suppose you invest P dollars at i yearly interest, paying an annual interest rate of r. We would then have the following recursive situation.

Amount to Invest Next Year = Amount Invested Last Year + Interest Earned - Taxes Paid

\[a_{n + 1} = a_n + a_n \times i - r \times a_n \times i \] This is the same as the closed formula

\[ S_n = P(1 + i(1-r))^n \]

Accounting for Inflation

Despite having taken taxes into consideration, we have not considered inflation in the investment. Note, that whatever we earn, we want to discount by inflation. This amounts to multiplying by (1 - I) where I is the rate of inflation. Thus, if we start with \(P\) dollars, we earn \(iP\) interest, hence we have earned:

\[a_{n+1} = (a_n + a_n \times i )(1 - I)\]

In closed form we see this as:

\[P(1 + i - I)^n\]

We can put these together to account for both taxes and inflation as follows:

\[S_{tI} = P(1 + i(1 - r) - I)^n\]