Investment income can be taxed in several ways by the government. In the United States, tax law generally chooses for each class of investment whether to tax growth in the value of the investment on an annual basis (even if the investor does not withdraw any funds) or defer taxation until a gain is "realized" by the investor, usually through withdrawing funds from the investment. This latter tax is often referred to as a "capital gains tax" and is generally applied to the difference between the value of the investment at the time of "realization" and the "basis" in the investment, generally the amount of money actually put into the investment over time.[more]
The choice of taxation timing significantly affects the share of total gains from the investment received by the individual investor and the share received by the government.
This Demonstration uses coupled recurrence equations to show how those two shares are affected by four parameters:
1. the gross interest rate received by the investor
2. the tax rate applied to the annual growth in the investment value
3. the capital gains tax rate
4. the "horizon", the length of time the investment was held
The "shares" are the wealth of the parties, assuming they could invest at the same rate after payment and receipt of all taxes.[less]
The formula for the share of the investor is:
where is the tax rate on annual investment growth, is the tax rate on capital gains, is the interest rate and is the time horizon. The government's share is one minus the investor's share.
This formula is derived as follows:
Step 1: Let be the amount invested by the taxpayer each year. Assume the tax is paid at the end of the year. Solve the dual recurrence equations and , where represents the taxpayer's account and represents the government's account after payment of income taxes but before consideration of any capital gains tax. Both accounts are set to zero at time 0. \:201a and . The solutions to these equations, which can be found using the Wolfram Language RSolveValue function, are and .
Step 2: Now recognize that the capital gains payable at time is the capital gains tax rate multiplied by the difference between and , the latter of which is the amount of money paid in by the taxpayer (the so-called "basis").
Step 3: Subtract the capital gains tax from the taxpayer's account and call it ; add the capital gains tax to the government's account and call it . Some algebra shows the results are and .
Step 4. Now divide the taxpayer's post-capital gains account by the total to get the taxpayer's share; divide the government's post-capital gains account by the total to get the government's share. The 's in both equations cancel out and the results are and , respectively. One can check that the results are plausible by adding the two values together; the result is 1.
Snapshot 1: reducing the time horizon reduces the government's share
Snapshot 2: lowering the interest rate reduces the government's share
Snapshot 3: treating the investment growth as "ordinary income" results in a greater government share than deferral resulting from treatment as a "capital gain"
 A. Auerbach, "Retrospective Capital Gains Taxation," National Bureau of Economic Research, Working Paper No. 2792, 1988. (Aug 22, 2019) www.nber.org/papers/w2792.pdf.