Frontiers vs. years

How frontier curves are changed
for longer time horizons.

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As number of investment years increases, the frontier curve of best-diversified portfolios changes dramatically, reshaping portfolio comparison. Portfolios with higher expected return rates and larger standard deviations become more attractive, not only in prospects but even in risk.

As a starting point for illustrating how increasing the number of years changes the frontier curve, a standard efficient frontier for the single year is shown at right.

Along the curve, points for three portfolios are shown. According to common labeling of this graph, with the vertical axis labeled “return” and the horizontal “risk”, portfolio A is shown to be “safest” with “risk” of 5% while C is shown to have much greater “risk” of 25%.

On this traditional graph, it appears the principal difference among the three portfolios is “risk”. It appears that by moving from portfolio C to A, an investor can reduce “risk” by 20% while reducing “return” by less than half that amount.

“Risk”

To show and compare frontiers for various numbers of investment years, the measures of expected result and risk must be redefined so they can measure results for any length of time.

On the graph below, the same frontier as on the graph above is shown -- but the axes measure expected results and risks in different ways, so the curve appears somewhat different. On the graph below, expected result and risk are measured in terms of dollars at the end -- total including initial investment. This graph shows end-results for investment of $1000 for one year

To measure risk for dollar end result, the horizontal axis measures safety vs. risk in terms of how large a minimum end result the investor can have 80% confidence of meeting-or-beating. Since larger minimums are further to right, further to right is safer according to this measure. Further to left means lower minimum result and therefore greater risk, as labeled beneath the graph.

On this graph the frontier curve is much smaller, and horizontally it goes the other way -- but otherwise it looks the same as the curve on the standard efficient frontier. The reason it’s smaller is that on the graph at right, total results including initial investment are shown, and on these scales the differences among the portfolios from one investment year are relatively small. The reason that horizontally the curve goes the other way is that on the graph at right, greater risk is further to left.

In sum, the graph at right show a different view of what the efficient frontier graph shows -- the range of best-diversified portfolios, shown in terms of expected result and risk for total dollar results at the end of one year instead of percentage rates of return.

With the frontier axes revised to measure expected results and risks in terms of dollar results at the end, a frontier curve can be drawn for any number of investment years.

On the graph below, frontier curves are shown for investment periods of 1, 5, 10, and 15 years.

As investment years increase, the frontier curve changes so dramatically that the graph must be drawn in a different scale, with the axes reaching larger amounts. The vertical axis for expected results must reach so much higher that, to keep the two axes in the same scale, the graph must be shown very tall and narrow.

On this graph, the single-year frontier, shown red, is so small it is barely visible.

As you move from 1 year to 5, to 10, to 15 years, the changes in the curve are stunning. It changes from tiny and nearly horizontal, to huge and almost vertical. The most important effect this shows is that of compounding. In expected result, on the vertical axis, for the single year the differences among the portfolios are tiny. But due to compounding, for longer terms the differences among the portfolios in expected result are huge.

Another change  that can be seen on this graph is that as investment years increase, the frontier curve slowly pivots, clockwise.

This shows that as investment years increase, portfolios above the bottom of the curve move further not only up to higher expected result, but also further to right to greater safety. Portfolios with higher expected return rates (and larger standard deviations) become more favorable in risk. 

To show how these dramatic changes in the frontier curve change portfolio comparison and selection for longer-term investment, on each curve on the graph below, points are shown for the same three portfolios shown on the standard efficient frontier: portfolios A, B, and C.

On the standard efficient frontier, it appears that compared to portfolio A, portfolio C offers only modestly higher “return” but far more “risk”. But that’s for the single year.

On this graph’s frontier curves, it’s clear that for longer investment plans and goals, the principal difference among the three portfolios is in expected result: moving from A  to B to C, the investor moves to vastly higher expected results.

Looking closely, one can also see that for longer investment periods, compared to A, portfolios B and C improve in relative favorability on the horizontal axis measure of risk.

More broadly, it is clear that compared to the great differences among the portfolios in expected result, the differences in risk are tiny. The overwhelming difference is the vast improvements in expected result as you move from A  to B to C.

On the graph below, from the points for portfolio C on the curves for 10 and 15 years, vertical dotted lines are drawn down to to horizontal axis. These dotted lines enable closer comparisons of the three portfolios in the horizontal-axis measure of risk.

The dotted lines show that for 10-year investment, according to the horizontal-axis measure of risk for dollar results, portfolio C is very slightly safer than A. Portfolio B, in the middle, is slightly safer than either of the others.

For 15 years, on the horizontal axis measure of long-term risk, C is slightly but clearly batter than A and essentially tied with B.

In a broader view, one might say that for 15 years, in the measure of long-term risk the three portfolios are too close to consider significantly different. The really significant difference is in their expected results. In going from A  to B to C, the long-term investor moves to prospects for much higher dollar results. For portfolio C, expected 15-year dollar results are nearly three times those for portflio A.

With a tool that shows the frontier for only the single year -- the traditional efficient frontier -- long-term investors and planners would be misled to think that compared to portfolio A, C offers only modestly higher “return” and far greater “risk”.

Portfolio PATHFINDER starts by producing and showing the standard single-year efficient frontier, but goes on to complete the analyses and graphs for optimizing for long-term plans and goals, including frontier curves of long-term dollar results like those on the graphs just above. These are one of PATHFINDER’s two kinds of unique Goal Frontier graphs, produced by combining powers of Modern Portfolio Theory and Monte Carlo simulation. Modern Portfolio Theory to identify the range of best-diversified portfolios, then Monte Carlo simulation to compare them in expected results and risks for investor’s long-term plans and goals.

On these PATHFINDER graphs, the planner can see and show investors how the frontier portfolios compare in prospects and risks for their long-term plans and goals.