Long-term optimizing begins with Modern Portfolio Theory.

For a set of asset classes, an efficient frontier curve reveals the range of mixes that are best-diversified. These are the best portfolios to consider.

But for choosing among them for a long-term plan, this graph does not provide the right comparison. It compares the portfolios for only the single year, omitting long-term effects that reshape the comparison. For long-term portfolio selection, further analysis must be done.

To analyze long-term prospects for a plan, Monte Carlo simulation is applied.

Each simulation run shows a sample of what the future results may be, year by year through the length of the plan. From multiple simulation runs, investors can see that future results cannot be known -- but the outlook for result probabilities begins to take shape.

This is the very best use of data from history. Return-rate probabilities revealed by history are translated into scenarios of probability for the future.

By running ten thousand simulations of an investor’s long-term plan, and recording the long-term result for each, we develop the shape of the probability curve for the long-term result of the plan.

Investors can see the probabilities for their long-term results. At heights where the curve is wider, results are more likely.

On a probability graph for long-term results produced from Monte Carlo simulations, moving to various target heights makes it easy for investors to see what the graph reveals.

At each target height you move to, the area and number above the goal line, shown red, show the probability of meeting-or-beating that target. The area and number below, shown gray, show the risk of falling short.

But for finding the best long-term investment plans, Monte Carlo analysis is not by itself sufficient either. It requires thousands of simulation runs to assess one plan, and offers no system for efficiently sifting thousands of alternatives to narrow in on the best.

First Modern Portfolio Theory, to find the range of portfolios that are best-diversified . .   

The results are shown on Goal Frontier graphs.

Then Monte Carlo simulation, to compare the best-diversified in  probabilities for long-term results.

On a Goal Frontier graph, the curve shows the range of portfolios that are best-diversified -- and the axes compare them in measures of long-term prospects and risk.

Higher is higher expected result -- the best single measure of long-term prospects.

Further to right is greater probability of meeting the long-term goal -- a measure of safety. Further to left is more risk of falling short.

The point on the curve that is furthest to right, shown green, is safest in likelihood of meeting the long-term goal.

Points higher along the curve, shown purple, are competitive. They offer higher prospects with only a little less probability of meeting the goal.

All the portfolios along the curve below the safest, shown gray, are revealed to be inferior in both prospects and safety for the long-term plan.

By comparing positions of portfolio points on the graph, you can see and show how they compare for a long-term plan -- in long-term safety or risk along the horizontal axis, and in long-term prospects on the vertical axis.

With this graph, you can see which portfolios are best for long-term plans and goals.