BLSH Model - Half-a-Loaf Case

The graph shown here are for the Half-a-Loaf case. The colored lines are the four simulated results for SPY, total assets, risky capital, and cash reserve values.

In this case, there were more cash than risky asset at the beginning (2:1 ratio). The are negative values for the risky asset (meaning short positions are required for this simulation case). It is very clear a simple short strategy might not work well in actual operations. In real operations, taking short positions in a bull market needs a lot of confidences. So we feel we might not be confident enough to do so in actual mangement of portfolios.

However, we can gain insights from this case. Some of the current understandings are the followings.

(1) The ratio between cash and risky asset is very critical when setting up the initial portfolio positions. When there is too much cash relative to risky asset, we can easily run into a situation when we do not have enough shares to sell when the market continues going up as shown in this half-a-loaf approach.

(2) The total assets curve is very statble over the entire simulation period. It is more or less close to a straight line. This means the volatility for this case is very low. We can carefully device a balanced portfolio with very little volatility for the most conservative investors based on this case with improved parameters to control shares investing in the risky asset.

(3) Based on the previous three simulation models, we become more confident that we can optimize model parameters to improve the overall model results. The appropriate ranges for the basic parameters should be within the range already used in the three models.

BLSH Model - the Duality Case

When we let cash and risky assets change at the similar ways, we can observe that the cash asset (blue line) and risky asset (red line) move up at the similar rates. We call this case as the duality case because the dual (the cash and the risky asset) are moving together.

As we compared the total assets curve (green line) with the SPY (brown line) curve, we can observe the difference between them. The difference is expanding during the bull market; the difference is shrinking during the bear market. The two curves almost converge together during the worst time in the market from mid-2002 to early 2003.

Now the question is whether the two curves will converge again in the future. If they will converge, are there still incentives to hold the risky asset alone? I can not be fully confident to predict such a convergence. But I do understand the following insights based on this result.

(1) Holding risky asset is taking risks. At the same time, holding cash along is taking another kind of risks. It might be the two extremes in the case of duality can compensate each other in the long run with the minimum oveall risk.

(2) Market timing will add tremendous values if one can really do it correctly. If one can sell the risky asset when the price is high and buy it back when the price is low, he is becoming the guru of investment vey soon.

(3) If one realizes that he can not do market timing well, the duality case might be a balanced way to invest for the long run. Once in while (might be as long as 10 years), you will catch up with the guys who invest only in risky assets.

BLSH Model - the Base Case

Actually this is the base case for our BLSH model. The previous post was the result of the buy/sell factor of 1.07. Our base case was defined with the buy/sell factor of 1.06. However, the little change in buy/sell factor has made some difference in the model result. Based on this, we can state that the buy/sell factor is a sensitive parameter in the model.

One very critical simulation result is that the cash reserve is negative during the worst time of the bear market. That means we need to use margin to finance new purhases of risky asset shares. In real operations, we might not be able to do this due to many factors such as manager's confidence in the overall bear market conditions.

BLSH Model - B Case with BSF 1.07

The base case result from our BLSH Model is very promising. The brown colored line is based on the SPY closing price. The green line is our model result. Apparently the green line is less volatile than the brown line. Which line do we like? The brown line or the green line?

This was actually an alternative to the base case. The result shown here had an alternative buy/sell factor (BSF) of 1.07. We had defined our base case with the BSF = 1.06.

The base case result is posted on the following blog.

Visions Needed for the Poor Investors Only

The investment philosophy might be different for investors with different perspectives. If investors consider themselves to be poor, the main goal of investing might be for growth. Conversely, if investors consider themselves to be rich, the main goal of investing might be for not loosing money. The absolute amount of money they have might not play an important role in this.

Now the question is how these investors can achieve their respective goals. I was thinking about this question. I feel I have some clues about this question.

Now let’s first define the poor investors. The poor investors are those investors who think that the amount of money they have are too limited in their views. Their absolute amount of money does not play an important role here. We further assume that their goals of investing are for rapid growth of capital.

My view is that there are not poor investors in the world. Therefore, no investors should need to have rapid growth for their capital. In the following paragraphs, I am trying to prove my point of view.

There are about 60 trillion dollars of wealth in the United States. There are about 300 million people live in this country. The average wealth per person is approximately 200,000 dollars. Let’s further assume that the average number of people in a typical household is 4 people. Then the typical family wealth is about 800,000 dollars. Let’s assume the typical family has about 200,000 dollars of mortgage or other types of loans. Then the net wealth is about $600,000

Further let’s assume that only one person in this household makes all the investment decisions about what to invest in daily operations of the investment portfolio. This assumption is very critical because that two people might have opposite views about investing. If opposite views about investing are moderated into an actual investing strategy, all other assumptions about the poor investors might not be valid.

Let’s assume that the return for the risk-less investment is 6%. One choice for the poor investors is to invest 100% of their net wealth in such risk-free investments. They can get about 36,000 dollars per year. One has to notice that this $36,000 is very close to the average family income of the United States.

Now let’s assume that a family with an income of 100,000 dollars will have a nice living standard in a very typical suburban neighborhood. So apparently most people in the United States feel they are not wealthy. In order to obtain a risk-free income over 100,000 dollars, the typical family might need approximately 2 million dollars (3 x 600,000 = 1,800,000).

How can the poor investors grow their capital three times from $600,000 (a typical family) to $1.8 million in a short time? Now let’s calculate the required rate of returns for this growth.

If the short time is five years, the required rate of return is 25%. This rate can be achievable if one can invest like Warren Buffett. How can any poor investors reach such a high rate of returns? The only way to reach this high rate of returns consecutively for five years might be the investors who have visions.

If the short time is 10 years, the required rate of return is 12%. This rate is achievable if the poor investors invest 100% of their money in the stock market. In many 10-year periods, the stock market return is close to 12% annually. As long as the investors choose the right index funds, they are very likely to achieve their goals.

If the short time is 20 years, the required rate of return is 6%. This rate can be achieved as long as the poor investors invest wisely in combinations of risky and risk-free investments. One can see how easily the long-term views are critical in managing money. I would think the most conservative investors with the right balanced funds could achieve this goal easily. One minor note here is that the risk-free investments might not serve this purpose because the inflation might reduce the available inflation-adjusted capital.

If the short time is 40 years, the required rate of return is 3%. This rate can be achieved as long as the poor investors invest only risk-free investments. This is why the rich is getting richer and the poor is getting poorer. The rich does not need to do anything to transfer their money from generations to generations because they can always invest their money in the US treasuries to obtain the required rate of returns. If the poor investors take such a long time view about investing their money, I think the poor investors will not feel poor for too long.

In summary, I do not think there should be any poor investors in the world. If you want to reach your goal in five years, you might need visions. If you want to get there in ten years, you can rely on the index funds to move you there. If you can wait for twenty years to reach there, you can easily achieve your goal by conservatively investing your money in any balanced funds. If you have an investment horizon of forty years, it is very hard for you not to reach your goal.

Conversely, if the poor investors do not think they have visions about the future, they just need to extend their investment horizon a little bit longer, say from five years to ten years.

道德经 第十章 玄德

载

营魄抱一 能无离乎
专气致柔 能婴儿乎
涤除玄览 能无疵乎
爱民治国 能无为乎
天门开阖 能无雌乎
明白四达 能无知乎

生之畜之

生而不有
为而不恃
长而不宰

是谓玄德


[译]

Simultaneously,

Can you control your greed and fear to unity and never let them be separated?
Can you exercise your body to be as soft and flexible as a baby?
Can you clean your heart so deeply so that it becomes spotless?
Can you love people and govern a country without actions?
Can you open your mind so completely so that it is without feminine?
Can your understandings reach all over the world without knowledge?

It is produced this way and it should be nurtured in another way.

To produce without possession
To act without expectation
To control without dominance

These are the mysteries to all powers.

[注]

This chapter is one of the most difficult one to understand at the first few readings. I feel that this chapter is highly condensed in many ways to guide us throughout the entire life. It lays out the basic steps for people to educate themselves, to nurture themselves, and to finally master the mystic powers.

At first, from controlling greed and fear to unity, to exercising our bodies to be soft and flexible, to clearing our desires to the level of spotless, to loving people and governing countries without actions, to opening our minds completely with balance, and to knowing the world without knowledge, this chapter tells us what to do in each of these steps. It stresses that we have to given in when the nature takes its course.

When we have educated ourselves along these steps and finally reach at the level of finishing the production process. We have come to a new level of nurturing process. The nurturing process is to produce without passion, to act without expectation, and to control without dominance.

When we have finished the production and nurturing processes, we can finally reach at the level of mystic powers.

Basic Power Numbers

I am thinking of a series of number with a basic power of numbers: 1, 4, 27, 256, 3125, 46656… This series can expand very fast. This way the world can be easily quantified by the basic power numbers since the scaling factors are huge between the two numbers in sequence. I call them basic power numbers because the base and power are the same. The basic equation is the following simple equation.

N = B ^ P

In the above equation, N is the basic Power Number, B is the base, and P is the power. When B equals to P, we get the basic power number N.

I have thought about the potential applications for such basic power numbers. To think about the envelope numbers in the world is kind of very interesting to me.

Social-Political Layers of World Population

World population is a linear number. This number is changed by one at a time when there is one birth or death. The current world population is 6,576,494,555. This is a hard to remember. We can convert this number as a basic power number of 10 (or 9.873). Then it can be easily remembered. The U.S. population is 301,178,149. It basic number is approximately 9 (or 8.921). The basic numbers are so small and they are related to our daily life experience.

The basic number can be thought as the number of layers in the social-political systems. In the world population case, the basic power number is 10 for the world population, which is somehow related to the layers of social-political system of the world. I can classify the world social-political systems into 10 layers.

1. The World, such as the United Nations
2. Regions, such as the North America
3. Country, such as the USA
4. States, such as Texas
5. Counties, such as Collin County
6. Cities, such as Plano
7. Communities, such as the East Plano and West Plano
8. Neighborhoods, such as the Willow Bend
9. Families, such as the Smith Family
0. Individuals, such as John Smith.

The Internet

The Internet is another example of potential applications for the basic power numbers. The number of computers connected to the Internet is expanding very fast. If we use the basic power number to describe it, the number would not change very fast.

I do not know exactly how many computers are connected to the Internet right now. One good indication for this number is the close depletion of the available Internet addresses. The Internet address is 256.256.256.256. The total number of available addresses is approximately 4 billion addresses. Since the number of Internet addresses is close to depletion, the total number of connected computers must be close to 4 billion.

The basic power number for the Internet is 10. When I studied the Internet 15 years ago, I knew that the Internet was divided into seven layers. Now many years has passed. Does the Internet have 10 layers now?

The Human Genome

The human genome contains 3.1647 billion chemical nucleotide bases (A, C, T, and G). The total number of genes is estimated at 30,000. So the number of chemical nucleotide bases has a basic power number of 9.65 or approximately 10. The number of genes has a basic power number of 5.841 or approximately 6.

It is conceivable that the basic power numbers for the world population, the Internet, and the human genome are in the same order, approximately 10 in terms of basic power number. It is very interesting to me these seemingly unrelated numbers are in the same orders.

There is a saying that the computer power in the network is proportional to the square of the number of connected computers. We can think along the same line that the societal power is proportional to the number of people in the society. So the potential intelligence among these seemingly unrelated aggregated systems should be in the same order (about 10). So the societal power, the human brain power, and the Internet power are similar in this way if their potential powers are fully developed.

So now we have three aggregated systems, the human society, the Internet, and the human body, have the same number of subunits. Are intelligence coming from the random connections among its members. When the random connections are repeated billions of times, some beautiful pattern will form. These beautiful forms of repeated patterns might be what we call the Intelligence.

The US Wealth

The total wealth of the United States approaches to $65 trillion dollars. What is the basic power number for the wealth? Based on my estimate, the basic power is 12.57. In this case, what does this mean?

It can be a price. The $12.57 is a good price for a nice lunch. It is about the price to make a trade at most brokerage houses. One may find the cheapest price for the Internet connection at $12.57 per month. It is the price for a nice bottle of wine. So the basic power number in wealth can be thought as some basic price for some consumer goods or services.

It can be a nice annual return on investments. The 12.57% can be thought as the average return for the S&P 500 Index. It is indeed very close to average return of the S&P 500 index. I would be very happy if I could make 12.57% on my investments every year.

The 12.57 can be thought as some basic price. Now when the total wealth expands, do we get just more expensive lunches? Or we get something more substantial than that.

The 12.57 can be thought as some growth rate. Should our growth rate be increased when the total wealth expands. That the rich is getting richer seems approves this point.

Diverging and Converging Investments

Recently the Stock-Poll group is discussing about the investment styles. Basically there are two major opinions. One is about investing in S&P 500 will produce excellent results. This way of investing will outperform most of the mutual fund investors.

Another way of thinking is about market timing. One can achieve the above market return if one can predict the short time trend of the market. Buy when the market is low while sell when the market is high.

So I think that investing philosophy is discussed here between Yong and Alan. Basically, there are two believe in investing. One is about diverging while another one is about converging. Interestingly, diverging is often applied by technical analysts while converging is the basic assumption for fundamental analysts.

In the converging market, whatever is deviated from its long-term trend will converge back. That is why investing in S&P 500 can be an excellent choice for long term investment.

In the diverging market, the current trend (up or down) can persistently continue for an extended period of time. And that period is long enough for investors to take advantage of the market. So these investors can achieve the above market return.

The converging investors will have the long-term return similar to the market return. The diverging investors overall will have the long-term return also similar to the market return. The only difference is that about 50% of the diverging investors will have returns above the market return while another 50% of the diverging investors will have returns below the market return. This result is governed by the laws of mathematics,

Numerically Quantified Diversification II

Yesterday we reported our first real case study result using Vanguard 500 Index Fund as the base fund. That was based on the assumption that some of the growth-orientated investors like to have this fund as their anchor fund.

For some other investors, they may like to have growth and income at the same time. We assume that these investors may use Vanguard Wellesley Income Fund as the anchor fund. For these investors, their diversification needs are different. The results posted here today are for the estimated ranking based our modeling efforts.

The ranking of diversification benefits of the 50 funds based on the Vanguard Wellesley Income Fund are listed below. We list them from the least diversification fund to the highest diversification fund. The difference between the two rankings (today versus yesterday) is clear.

Life Strategy Income Fund
Life Strategy Conservative Growth Fund
Long-Term Investment-Grade Fund
Long-Term Bond Index Fund
Long-Term Treasury Fund
Wellington Fund
Intermediate-Term Bond Index Fund
Intermediate-Term Investment-Grade Fund
Total Bond Market Index Fund
STAR Fund
GNMA Fund
Intermediate-Term Treasury Fund
Short-Term Investment-Grade Fund
Short-Term Bond Index Fund
Balanced Index Fund
Life Strategy Moderate Growth
High-Yield Corporate Fund
Asset Allocation Fund
Equity Income Fund
Windsor II Fund
REIT Index Fund
Life Strategy Growth Fund
Selected Value Fund
Global Equity Fund
Health Care Fund
Dividend Growth Fund
Convertible Securities Fund
Windsor Fund
Value Index Fund
International Value Fund
Strategic Equity Fund
Small-Cap Index Fund
Total Stock Market Index Fund
European Stock Index Fund
International Growth Fund
Growth and Income Fund
500 Index Fund
Total International Stock Index Fund
Extended Market Index Fund
Morgan Growth Fund
Explorer Fund
Energy Fund
Growth Index Fund
PRIMECAP Fund
US Growth Fund
Precious Metals and Mining Fund
Emerging Markets Stock Index Fund
Growth Equity Fund
Capital Opportunity Fund
International Explorer Fund

Numerically Quantified Diversification I

We are reporting our first real case study result today. Diversification is very critical in portfolio management. However, there are no quantitative values for the diversification benefits. We are the first to quantify the diversification based on numerical values.

We used Vanguard family of funds as our first real case study. We selected 52 funds with data records over the last 10 years for this case study. We thought that the Vanguard 500 Index Fund as the base fund. The ranking of diversification benefits of the 50 funds based on the Vanguard 500 Index fund are listed below. We list them from the least diversification fund to the highest diversification fund.

Growth and Income Fund
Total Stock Market Index Fund
Growth Index Fund
Morgan Growth Fund
Value Index Fund
Life Strategy Growth Fund
Asset Allocation Fund
Equity Income Fund
Balanced Index Fund
PRIMECAP Fund
Life Strategy Moderate Growth
US Growth Fund
Windsor Fund
Windsor II Fund
STAR Fund
Strategic Equity Fund
Extended Market Index Fund
Wellington Fund
Dividend Growth Fund
Life Strategy Conservative Growth Fund
Growth Equity Fund
Small-Cap Index Fund
Health Care Fund
Global Equity Fund
Explorer Fund
Life Strategy Income Fund
Convertible Securities Fund
Wellesley Income Fund
European Stock Index Fund
International Growth Fund
Selected Value Fund
International Value Fund
Total International Stock Index Fund
Capital Opportunity Fund
Short-Term Investment-Grade Fund
High-Yield Corporate Fund
GNMA Fund
Short-Term Bond Index Fund
Energy Fund
Total Bond Market Index Fund
Intermediate-Term Investment-Grade Fund
Intermediate-Term Treasury Fund
Intermediate-Term Bond Index Fund
REIT Index Fund
Long-Term Investment-Grade Fund
Long-Term Bond Index Fund
Long-Term Treasury Fund
Emerging Markets Stock Index Fund
International Explorer Fund
Precious Metals and Mining Fund

Birds and the Wisdom of the Crowds

As a kid I used to blow soapy water to make bubbles into the air and then I watched them flying in the air and finally it broke. Recently I bought special bubble water from a local toy store for George. I taught him how to blow bubbles. The bubbles George made seemed much bigger than the ones I used to make. No matter how careful we are about bubbles, they will break eventually.

Another natural phenomenon like bubbles is flying-birds in a dense group. I have seen hundreds or maybe thousands of birds flying in a large group. The boundary of the flying birds group is very smooth and continuous. It looks like a giant bubble in a way. It quickly goes up and down in the air as the birds are coordinated together making the beautiful changing shapes.

Another bubble which investors are often referring to is the stock market bubble. When more and more investors are coming to the market to do similar investments, the market price goes up and all the way to the roof. Then the market price goes down quickly. Some investors loose their investments. The most recent case was the year 2000 U.S. stock market bubble.

When I put together the soapy water bubbles, the flying-birds bubbles, and the stock market bubbles, I tend to see the similarities among them. I can see clearly how the soapy water bubbles breaking into clear water vapors. I can also physically observe the flying-birds bubble landing on the trees when the perceived flying-birds bubble suddenly disappearing. I could not physically see how the crowds of investors coming to the stock market. But I can imagine how they can become coordinated in moving the market into the bubble zone.

In the soapy water bubble, the water molecules are connected together via capillary force to form the bubble. When the air is steady and mild, the bubble can stay live for quite a while. When the air is very turbulent and windy, the bubble can break into pieces in a short time. When the bubble touches some non-air boundary, especially the sharper edges, the bubble disappears very soon. The capillary force which connecting the water molecule is very weak. Any strong imbalance in the pressures inside the bubble and outside the bubble will cause the bubble to break. Although it is intuitively easy to feel the process for the bubble to break, it is very difficult to model the exact movement of the bubble and the exact time when the bubble will break.

In the flying-birds bubble, I do not know the exact mechanism for the birds to coordinate with each other to form the smoothly round shapes when they are flying together. I can be confident to know that it seams easy for them to coordinate with each other when they are flying. However, the smoothly round shape can not be maintained when they are landing on trees. When they are landing on the trees, their coordinated efforts become vanished. The coordination force is replaced by the needs of the individual birds to land accurately on the trees.

In the stock market bubble, the mechanism is the coordinated investment decisions made the participating investors. Although there is no real coordinator in the operation, the wisdom of the crowds is almost evenly distributed among most investors. If most of the participating investors share the common wisdom of the crowds, then the coordination is almost automatic.

I have seen this kind of automatic response when I feed fish with bread pieces in a pond. When I through the bread pieces onto the pond, I can feel almost all fish are jumping at the bread pieces at the same time. A water splash is caused by such sudden actions of the fish in the pond.

Actually, the shared wisdom of the crowds is conditional. The condition is that every investor can make money. If that condition is changed, this wisdom of the crowds can vanish quickly. It just as it depicted in the year 2000 market crash. This condition is very similar to the capillary force in the water bubbles.

Golf Scores

Golf is probably the only game with the lowest scores as winners. I remembered that Warren Buffett said something similar to this for investors. If every investor was given a punch card with a maximum of 20 punches, the investor would be very careful with each punch. Every time you invest in one stock, you loose one punch. If Buffett is correct, then investing is another game like playing golf.

In playing golf, there are three factors can have impacts on my scores. One is related to my internal skill level to play. Another is the physical environmental conditions of the golf course. The third one is my assessment of my skill level relative to the environmental conditions. Each one of these factors can have a dramatic impact to my final scores.

First factor is my internal skill level. This is related to my mental and physical strength to strike the golf ball to where I want it to land. It is related to how well I strike the ball accurately. This skill level can be improved by practice a lot on the driving range. If my maximum potential skill level is exhibited during my performance, I can play at par. But this is the maximum likelihood. The probability of reaching that level of performance may have a certain probability distribution.

The second factor is the golf course. The golf course conditions are more or less fixed. However, the landing position of my golf ball can have millions of possibilities. So the dynamic golf course conditions are not fixed at all. The golf course can exhibit millions of perspectives depending on where my golf ball is landed after each strike.

The third factor is my assessment of the golf course conditions looking from the point where my golf ball is landed or the starting point at the Tees. Since the staring Tees and the flag positions on the greens are changing every day, the Tees and flags themselves have many changes already. The landing positions of my golf ball in each game can be very different. Another factor is my selection of the golf club. I may use a wrong club for a strike even if the golf ball landing position is excellent.

Each of the above three factors are inherently dynamic. However, playing golf is also a self-correcting process. If my long clubs are not playing well, I need to play more long clubs in actual plays. If my putts are bad, I have to play more putts in order to putt the ball to the hole. So real golf game is actually a right process to improve my skills.

Investing is like playing golf. Each bad investment is like the missed strike in golf. The more misses we have, the worst investing results we get.

Snowballing from Golf to Swensen

George did not go to school today because it was snowing this morning. After lunch, George wanted me to play with him outside with the snow. He wanted to make a snowman. I first taught him to snowballing. He asked “how did you learn about snowballing?” I told him that it was snowing almost every year when I was a kid in China and I knew it then.

We both started with making a small snowball (the size of a golf ball) made of clean snow on the top of a bench in our backyard. I taught him the technique of snowballing at the bench. When the two snowballs become the size of eggs, we took the snowballs at the ground. We started rolling and rolling. George did very well. After a while he realized that his snowball was smaller than mine. He was a little bit unhappy about that. I told him to roll the snowball along the thick snow on the ground. He wanted me to roll behind him so that he could roll along the thick snows. I was rolling behind him and purposely along the shallower snow. Very soon his snowball was becoming the same size as mine. By then we both were feeling tired and our hands were very cold. We went back inside the house to rest for about 10 minutes.

Then we went to finish the rest of the project of making a snowman. Since the snowballs were very heavy. George could not lift the snowballs. Jie came to help us. Jie and I lifted the snowball and put it at the top of mine. I then added two orange-colored billboard balls as eyes, one red carrot as nose, and a red half-circle shaped toy as mouth. George was just laughing harder and harder after each piece was added into the snowman. Then Jie took several pictures as George was standing besides the newly made snowman.

Yesterday I went to the Beginner Golf class. We did not go out since it was snowing outside. He did coaching in the class about putting and holding clubs. The most important thing about holding a club is that the two Vs formed by two hands should be pointed at the right shoulder. I did not know this before the class.

The Portfolio Management class was fine. The most interesting part was about the fund of hedge funds. He believed that the fund of hedge funds did not add much value to the investors. He would rather like the pension funds to build this capability in-house. He thought that the Yale Endowment Fund manager David Swensen might be the first manager realized this importance. This was one of the main reasons why the Yale Endowment portfolio had the highest level of exposure to the hedge funds. I had researched about Swensen for his portfolio structure. But I did not know Swensen was the pioneer in investing in hedge funds.

As an after thought, I feel that snowballing is very much like investing. Starting with a very small ball in the size of a golf ball, one can roll the snowball to a much bigger one. The snowball is getting bigger as long as you roll along the fresh snow at the ground. And the snowball is becoming too big to roll easily. This is kind of strange - everything is connected and related to investing, even on a snowballing day without playing golf.

Distance between Two Funds

In our first project, we studied all major traditional asset classes, such as the style box funds, the sector funds, bond funds, and REIT funds. We have defined a new concept in asset management – Distance. The distance concept is a quantitative representation of diversification. Before us, the diversification is just a qualitative concept. Now we can give a number for the level of diversification when two funds are considered in a portfolio. At a later stage, we will try to determine the multi-dimensional distance concepts for a portfolio with more than two funds.

When the large cap fund is the center, the five funds with longest distance are Precious Metals funds, Technology funds, REIT funds, Bond Funds, and Energy Funds.

When the small cap fund is the center, the five funds with largest distance are Technology funds, Precious Metals funds, Utilities Funds, Energy funds, REIT funds.

When the energy fund is the center, the five funds with biggest distance are Technology funds, Precious funds, Small Cap Growth funds, REIT funds, and Consumer funds.

When the utilities fund is the center, the five funds with the best distance are Precious Metals funds, Technology funds, REIT funds, Consumer funds, and Energy funds.

When the bond fund is the center, the five funds with the farthest distance are Precious Metals funds, Small Cap Growth funds, Mid Cap Growth funds, Consumer funds, and Industrial funds.

When the REIT fund is the center, the five funds with the best diversification are Technology funds, Precious Metals funds, Large Cap Growth funds, Small/Mid Cap Growth funds, and Utilities Funds.

Now we have concluded the one-dimensional study for the general styles and sectors funds. We are satisfied with the current results. Now we are prepared to study one family of funds. Now the Vanguard family is the largest one in the United States. So we are ready to study the Vanguard funds.

In the first round of study, we have selected 52 Vanguard funds with at least 10 years of records. We are looking for the best pairs of funds within this family of funds.