Investment Choices
By Karl Preissner, Xavier University
Business Stats 500 Project
Winter 2011

Problem Description

This paper concerns a simple question, “What’s the best way to invest my money when presented with 13 alternative opportunities?” In this case, we’ll assume our investment goal is to maximize return while simultaneously minimizing the risk of loss. Since we cannot invest backwards in time, we need to develop an expectation for the future performance of our opportunities in order to make our choice. This leads to our two fundamental questions: 1. If investment skill exists, how can we find it? 2. Based on the above answer, what investment opportunity is most likely to maximize return while minimizing risk?

Problem Discussion and Historical Review
When did it start?
The problem of appropriate expectations has been with us as long as financial markets themselves. For much of our history, adaptive expectations theory was commonly used to guide investment decisions. Adaptive expectations suggest that past data can be used to predict future expectations. More recently, and accelerating after WWI, the theory of rational expectations has come to dominate the conversation. This theory suggests that expectations are simply our best guess of the future using all available information. Two key implications of this theory are that arbitrage opportunities are quickly taken advantage of, and that as a result, we cannot predict the future ahead of time (based on passed data or otherwise). When applied to financial markets, we call this the Efficient Markets Hypothesis or EMH (Fox, P164).
Many historical figures have contributed to the discussion over time. Adam Smith (1723-1790) introduced the “invisible hand” metaphor, perhaps the most well know argument for the rationality or “wisdom” of a market. Five Centuries earlier, while medieval scholars were arguing that governments should mandate market prices to ensure fair outcomes, St. Thomas Aquinas had already considered that fair prices were best set by the market (or at least that better alternatives were unavailable). (Fox, P xiii)
Alternative viewpoints have developed in parallel. One of the more notable these days was developed in “Security Analysis, 1939” and “the Intelligent Investor, 1949” by Benjamin Graham and David Dodd. They argued that intelligent selection is possible when a thorough analysis is completed and the “intrinsic value” is understood. Warren Buffett has offered his strong support for the philosophy outlined in these texts.
Why is it a problem?
The problem is that the Intelligent Selection approach has turned out to be neither obvious, nor unprovable. And what appears random to the untrained eye could actually be recognizable pattern to a knowledgeable party. Many other fields of study have proceeded along similar lines (weather patterns are now highly predictable).
If the Efficient Market Hypothesis is to be believed, then all unexploited opportunities will be eliminated and the possibility of “beating the market” is left to pure chance. One is best to give up trying to out-guess the market and instead to pursue strategies that minimize transaction costs (i.e. buy and hold) and eliminate non-systemic risk (i.e. portfolio diversification).
On the other hand, if intelligent selection is possible, as Graham and Dodd suggest, then accepting market returns would be foolish. We should seek strategies that produce higher returns at equal or lower risks.

How have others approached it?
Below are a few strategies for how people have attempted to demonstrate the existence (or non-existence) of skill.
Evidence in Favor of Market Efficiency (Miskin, P 171-176) * “Dartboard Approach” - Compare results of buy and sell recommendations for a sample of investment advisors to the results of the market as a whole * Variation: Select advisors who have a track record of past success and see if they can repeat the success at a higher rate than the market. * Event Analysis - The EMH suggests that stock prices reflect all publicly available information, thus company announcements should not impact stock prices if that information was already publicly available. This can be investigated using historical data such as the impact of stock splits on share prices. * Technical Analysis - Looks for patterns in historical stock prices or in publicly available information (money supply, interest rates, etc.) that reliable predict future performance. The existence of a reliable pattern is by definition a refutation of the EMH.
Evidence in favor of Intelligent Investing * Small Firm Effect - Empirical studies have demonstrated that small firms have delivered superior value over time (even accounting for their higher risk). * January Effect - Over long periods of time, predictable price rises were noted in January. * Value investing - A “mean reversion” phenomenon has been observed that has worked in favor of stocks with recent poor performance. As the stock price reverts to its mean, dependable returns in excess of market returns have been possible.
Related to the above debate, William Sharpe, John Litner, and Jack Treynor developed a useful tool called the Capital Asset Pricing Model or CAPM (Mishkin, P91-93). It took market efficiency as one of their assumptions, and used it to derive some useful tools for pricing risk. It also helped to formalize some of the benefits of diversification in a portfolio. One point it helped establish was that the marginal contribution of an asset to the risk of a portfolio depends not on the risk of the asset in isolation, but rather on the sensitivity of that asset’s return to changes in the value of the portfolio. When the portfolio is the market in general, the marginal contribution becomes the “Beta” of the asset. From this it was shown that the risk of a well-diversified portfolio depends on the systemic risk of the assets plus “alpha”.
E(RETi) = ∝i + βiE(RETm)
They extended this work further to derive the efficient portfolio frontier, which helped establish the market price of risk as the expected return of the market minus the expected return of a risk free investment (like a loan at current interest rates) all divided by the standard deviation of the market returns [E(Retmarket) – Retf)]/σmarket . We care about this because it helps us to evaluate whether a portfolio is efficiently developing return for the risk it is taking. Beta gives us a measure of relative risk and alpha provides us a measure of skill (or lack thereof) once the contribution of the market has been accounted for. Good management should result in a positive alpha while Beta can be selected based on an investors risk tolerance.
Description of the Data Set
For the purposes of this exercise, we will imagine that we have quarterly returns for 13 alternative investment opportunities (labeled A to M) available for selection as well as a benchmark for reference only. Our analysis includes 68 quarters of historical data from 1st Quarter 1994 to 4th Quarter 2010.

Methods of Analysis
As stated in the introduction, this paper seeks to accomplish two things: identify investment skill (or the absence thereof) and based on that result, recommend an appropriate selection from 13 investment opportunities. We’ll use the following approach.
To begin with, we will use graphical techniques of histograms and box plots to look for patterns that will help our analysis (the absence of outliers, the appropriateness of normality assumptions, etc).
Then we will seek to identify skill using two approaches. The first approach is to look at the matched pair returns between each investment strategy and the benchmark and look for statistical significance at 90% confidence. The second is to look at the frequency with which an opportunity beat the market (on a quarterly basis) and consider how likely it is that this was chance. We’ll compare this to a binomial experiment of a coin flip to answer this question. Here, we will look for success probabilities less than 10% as evidence of skill.
Should skill be discovered in all of our opportunities, we’ll use the following additional criteria to help with selection: * Was the return efficient? * Which opportunity yielded the highest return as observed by measures of central tendency (mean, median)? * Which opportunity offers the highest Sharpe Ratio? * Do we see an Alpha? Is the Return greater than the expected return as predicted by CAPM and the Security Market Line? * Was the investment consistent? * Who had the highest geometric mean? * Who had the lowest standard deviation? * Do some investments offer higher upside or lower downside? * Box Plots * Histograms
Note: Even if skill cannot be determined in our first analysis, we will use the above criteria to make a recommendation. Even if the Efficient Market Hypothesis is true, and any choice is as good as another, there can be no harm in making a choice. However, if skill does exist, and we are simply encountering Type II error, than the above criteria can help us to be confident in our selection.

Analysis
Histograms

A few things are determined via the histograms.
First, we can feel confident our analysis techniques are appropriate. These charts all are reasonably normally distributed. We also do not see any extreme values or outliers which could skew our data.
Second, we do see some interesting trends. G has a surprisingly tight distribution. This could be attractive if the returns are also good. L, J, and H also seem to have greater spread in their returns than the benchmark. This might imply greater risk. B, C, D, E, and F all seem to have a slight skew left. Not sure if this means anything, but we’ll see if it becomes a problem for returns.

Box Plots

Box plots suggest that each of these data sets will have similar central tendencies (median values are fairly similar).
Once again, G stands out once again for it’s very tight range of returns.
We also see high Max values for (H, J, and L) and lower Max values for (B, D, and F). This may suggest upside potential for opportunities with high max values. However, it also seems that H, J, and L have wider boxes and more of their box in negative territory than the other opportunities.
A couple other patterns stand out though it is unclear what they may mean: * A-F have higher minimum values than H-M. Perhaps these are related investment strategies? * B, D, F have three quarters of their values in positive return territory

Matched Pair Data

We first examine the likelihood that the mean value of the opportunity diverged from the mean value of the benchmark. | | | | | Table A | | | | | | | | | | | | | Matched Pair Summary for P-Value | | | | | | | | | | | | | | | | Attribute | A | B | C | D | E | F | G | H | I | J | K | L | M | Mean | -0.08% | 0.14% | -0.05% | 0.13% | -0.13% | -0.09% | -0.80% | 0.56% | 0.71% | -0.09% | 0.56% | 0.38% | 0.70% | sample StDev | 3.34% | 3.22% | 3.30% | 3.09% | 3.17% | 2.87% | 9.27% | 6.36% | 4.51% | 6.43% | 6.04% | 6.70% | 5.16% | P Value | 57.89% | 35.89% | 55.30% | 36.88% | 63.37% | 60.07% | 75.97% | 23.50% | 10.00% | 54.39% | 22.52% | 32.01% | 13.40% | Here we see that only one opportunity has met our standard for skill. With a p-value of 10%, we can say with 90% confidence that the return on Opportunity I is significantly different than the return on the benchmark! Of the rest, Opportunity M comes the closest to meeting our standard with a P-value of 13.4%. The other opportunities cannot be distinguished from the benchmark, thus we cannot conclude that skill was used based on this evidence.

Frequency to “Beat the Market”
Binomial Experiment

Table B | Frequency to Beat Benchmark |

Attribute | A | B | C | D | E | F | G | H | I | J | K | L | M | Freq out perform | 35 | 35 | 34 | 35 | 35 | 30 | 27 | 41 | 39 | 35 | 32 | 33 | 36 | Prob of Successif p=0.5 (values in %) | 45.18 | 45.18 | 54.82 | 45.18 | 45.18 | 86.25 | 96.59 | 5.71 | 13.75 | 45.18 | 72.77 | 64.18 | 35.82 | If beating the market were truly random, out of 68 trials we would expect 34 successes. The data above indicates that only opportunity H & G have met our standard for evidence. There is only a 5.71% chance that opportunity H would outperform the market 41 times if its performance were truly random. By symmetry, we can also say that there is only a 5.71% chance that opportunity G would fail to beat the market 41 times. Though outside our standard for success, opportunity I was the next closest, with a 13.75% chance that they could have beat the market if it were random. The other opportunities all appear closer to 50/50, or random chance.

Additional Criteria for Choice
Measures of Central Tendency Table C contains data of mean, median, and mode (top values are in blue and bottom values are in red). The top returns are highlighted in blue while the bottom returns are highlighted in red. We can immediately see that Opportunity I has the highest return by all three measures. We also see in table D that M & K consistently appear in the top 4 spots.
Interestingly, G and J are the only two to underperform the market on all measures. Table C | | Central Tendency (all values in %) |

Attribute | Benchmark | A | B | C | D | E | F | G | H | I | J | K | L | M | Mean | 2.33 | 2.25 | 2.47 | 2.27 | 2.45 | 2.20 | 2.24 | 1.53 | 2.89 | 3.04 | 2.24 | 2.88 | 2.71 | 3.03 | Median | 2.97 | 3.71 | 3.06 | 3.88 | 2.97 | 3.09 | 2.71 | 1.52 | 3.44 | 4.28 | 2.05 | 3.36 | 3.14 | 3.70 | Geometric Mean | 1.95 | 1.71 | 2.10 | 1.75 | 2.09 | 1.71 | 1.88 | 1.51 | 2.07 | 2.62 | 1.42 | 2.40 | 1.87 | 2.59 | Table D | | Rank of Top 4 Opportunities | Attribute | 1 | 2 | 3 | 4 | Mean | I | M | H | K | Median | I | C | A | M | Geometric Mean | I | M | K | B |

Sharpe Ratio Table E contains information on the Sharpe Ratio, the coefficient of variation (the inverse of Sharpe ratio), and the correlation of returns with the benchmark. We prefer an “efficient return”, defined for our purposes as one that delivers the lowest risk for a given return. We can identify this by a high Sharpe ratio (or low coefficient of Variation). It’s clear from the data that G is the most efficient, delivering 0.77 units of return for every unit of risk. Opportunities I, K, and M are next in line. Correlation data is also included in the table below. Interestingly, opportunity G is negatively correlated with the market benchmark (-0.19). Perhaps this is some kind of hedge. Table E | Measures of Deviation from Benchmark | Attribute | Benchmark | A | B | C | D | E | F | G | H | I | J | K | L | M | Sharpe Ratio (avg return/stdev) | 0.27 | 0.22 | 0.29 | 0.22 | 0.29 | 0.22 | 0.26 | 0.77 | 0.23 | 0.33 | 0.17 | 0.30 | 0.21 | 0.32 | Coefficient of Variation | 3.73 | 4.61 | 3.46 | 4.50 | 3.46 | 4.48 | 3.77 | 1.29 | 4.42 | 3.00 | 5.72 | 3.39 | 4.81 | 3.10 | Correlation | 1.00 | 0.95 | 0.93 | 0.95 | 0.94 | 0.95 | 0.94 | (0.19) | 0.89 | 0.87 | 0.89 | 0.79 | 0.88 | 0.84 | Table F | | Rank of Top 4 Opportunities | Attribute | 1 | 2 | 3 | 4 | Sharpe Ratio (avg return/stdev) | G | I | M | K | Coefficient of Variation | G | I | M | K |

Alpha and CAPM Table G provides several interesting pieces of data. We have calculated the beta here: β=correlation*(SDySDx) In this equation, correlation is between the opportunity and the benchmark, SDy is the standard deviation of the opportunity, and SDx is the standard deviation of the benchmark. This is equivalent to the slope of a regression line or the marginal contribution of risk this opportunity would bring to a well diversified portfolio. The alpha is also calculated: ∝ =y- β*x Here, y is the mean return of the opportunity and x is the mean return of the benchmark. It represents the intercept of a regression line and is a measure of the return after removing market risk related return. We have also calculated an “Expected Return by CAPM Security Market Line”. This was derived from the capital asset pricing relationship and is calculated: E(RETi) = E(RETf) + β* [E(RETm) – E(RETf)] Here, E(RETi) is the expected return of opportunity I, E(RETm) is the expected return of the market, and E(RETf) is the expected return of a risk free investment. As is shown in the appendix, we used an average of 7 mortgage rates to provide a quick estimate of E(RETf) (see APPENDIX D). E(RETf) = 4.79% If we compare E(RETi) to the mean return, we can derive a value for “excess return”. We can use this value to assess which opportunities have the greatest potential to improve the return of a well diversified portfolio.

As we examine Beta, we see further confirmation that G does not follow the general market closely. It has a negative Beta of -0.043. We also see several other opportunities that are contributing greater risk than the market (A, C, E, H, J, and L) and several opportunities that contribute less risk (B, D, F, I, K, and M).
As we examine Alpha, four investments stand out for having high values. G is the highest with 1.63% followed by M, I and K.
Examining Excess Return, we see opportunities H, L, J, and I showing the highest values with B, D, F, and G actually having negative values. Table G | Regression Values & Expected Returns | Attribute | A | B | C | D | E | F | G | H | I | J | K | L | M | Beta | 1.135 | 0.913 | 1.121 | 0.915 | 1.074 | 0.918 | (0.043) | 1.311 | 0.913 | 1.315 | 0.891 | 1.326 | 0.905 | Alpha | -0.40% | 0.34% | -0.34% | 0.32% | -0.30% | 0.10% | 1.63% | -0.16% | 0.91% | -0.82% | 0.81% | -0.38% | 0.92% | Expected Return by CAPM Security Market Line | 1.99% | 2.54% | 2.03% | 2.54% | 2.15% | 2.53% | 4.89% | 1.56% | 2.54% | 1.55% | 2.60% | 1.52% | 2.56% | Mean Return | 2.25% | 2.47% | 2.27% | 2.45% | 2.20% | 2.24% | 1.53% | 2.89% | 3.04% | 2.24% | 2.88% | 2.71% | 3.03% | Excess Return | 0.25% | -0.07% | 0.24% | -0.08% | 0.05% | -0.29% | -3.36% | 1.33% | 0.49% | 0.69% | 0.29% | 1.18% | 0.47% | Table H | | Rank of Top 4 Opportunities | Attribute | 1 | 2 | 3 | 4 | Beta | L | J | H | A | Alpha | G | M | I | K | Excess Return | H | L | J | I |

Consistency of Investment Table I looks at measures of the consistency of the return. We care about standard deviation because a smaller but consistent return can often outperform and larger return if the larger return is interrupted by large decreases as well. Slow and steady wins the race. Also, for our purposes, we will prefer an investment with predictable returns so that we can plan our retirement with greater confidence. Large swings in returns might mean we are unexpectedly surprised by a bad year close to our retirement which may require us to work longer. We wish to avoid this. Another way to measure this directly is to look at the geometric mean, which calculates the compounding rate of return. From the tables below, we see that G has the lowest absolute variation. This was expected from the box plot data. However, it also has the lowest geometric mean at 1.51% so we pay dearly for its consistency. L is the least consistent investment at 13.03%. We also see that once again that opportunity I delivers superior returns at 2.62% Geo Mean. Table I | | Measures of Consistency(values in %) |

Attribute | Benchmark | A | B | C | D | E | F | G | H | I | J | K | L | M | Standard Deviation | 8.69 | 10.34 | 8.53 | 10.23 | 8.50 | 9.83 | 8.45 | 1.98 | 12.76 | 9.10 | 12.82 | 9.77 | 13.03 | 9.37 | Geometric Mean | 1.95 | 1.71 | 2.10 | 1.75 | 2.09 | 1.71 | 1.88 | 1.51 | 2.07 | 2.62 | 1.42 | 2.40 | 1.87 | 2.59 | Table H | | Rank of Top 4 Opportunities | Attribute | 1 | 2 | 3 | 4 | Standard Deviation | G | D | B | F | Geometric Mean | I | M | K | B |

Conclusions This paper started with two challenges: 1. Identify skill in our investment opportunities 2. Based on the answer, select an opportunity that maximizes returns while minimizing risk We can now do both. Our recommendation is to invest in Opportunity I. * Opportunity I - Opportunity I represents the best balance between return and risk, while also providing some assurances of skill and consistency. * Good Return - I has the highest return by all three measures (Table D: mean, median, and mode). Also, the geometric mean was highest meaning that we have a nice consistent growth pattern. * Appropriate Risk - Next to G, I has the 2nd highest Sharpe ratio meaning we get good return for the risk we take (Table E). It also has an alpha in the top 3 and an excess return in the top 4 (Table H). Both these measures suggest the strategy employed is adding value beyond the market returns. * Presence of Skill - Mean paired data demonstrates the presence of skill (Table A) and the opportunity is also most likely to outperform the benchmark on a quarterly basis (Table B). Other opportunities that were considered but discarded: * Opportunity G - This opportunity has appropriate risk with the highest Sharpe Ratio, and we can also detect skill in Table A. However, the skill appears to be in consistently underperforming the market. We would be better off buying an index fund that tracked the market average. We might feel differently if our time horizon was much shorter (i.e. we are nearing retirement). In this case, the low risk might outweigh the low return. Alternatively, if we wanted to hedge our portfolio against the market, the negative correlation suggests this could be useful. However, neither of these are currently the case so the opportunity is not attractive. * Opportunity M - M has good returns that are comparable with I. It also has a comparable Sharpe ratio. However, we had difficulty detecting the presence of skill with M. It is possible that these returns are due to random luck. Given that I outperformed M and had the benefit of skill, we should not select M.
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APPENDIX A
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Investment Returns
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1Q1994 through 4Q2010
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| Benchmark | A | B | C | D | E | F | G | H | I | J | K | L | M | 1Q1994 | -3.81% | -4.38% | -3.29% | -4.41% | -3.51% | -4.98% | -3.87% | -2.88% | -3.10% | -2.83% | -4.08% | -1.37% | -2.94% | -1.58% | 2Q1994 | 0.41% | -1.49% | 0.39% | -1.04% | 0.63% | 0.52% | 1.13% | -1.03% | -4.40% | -0.26% | -6.30% | -1.79% | -6.47% | -0.95% | 3Q1994 | 4.92% | 7.85% | 2.78% | 7.70% | 2.57% | 7.95% | 1.48% | 0.61% | 7.10% | 4.48% | 9.34% | 4.73% | 9.63% | 4.82% | 4Q1994 | -0.03% | 0.60% | -1.73% | 0.73% | -1.56% | 1.69% | -0.58% | 0.38% | -1.39% | -3.35% | -0.72% | -2.95% | -0.81% | -3.41% | 1Q1995 | 9.74% | 9.12% | 8.89% | 9.52% | 9.52% | 8.98% | 9.21% | 5.04% | 10.81% | 10.05% | 5.48% | 3.70% | 8.46% | 6.26% | 2Q1995 | 9.49% | 9.84% | 8.94% | 9.83% | 8.95% | 10.61% | 9.12% | 6.09% | 8.03% | 8.67% | 9.92% | 8.77% | 8.92% | 8.89% | 3Q1995 | 7.95% | 9.31% | 8.70% | 9.08% | 8.74% | 8.73% | 9.14% | 1.97% | 9.85% | 7.93% | 11.37% | 8.32% | 11.11% | 7.99% | 4Q1995 | 5.96% | 4.24% | 6.27% | 4.55% | 6.64% | 5.79% | 7.67% | 4.26% | 1.88% | 4.54% | 1.48% | 2.91% | 1.75% | 3.84% | 1Q1996 | 5.44% | 5.41% | 5.54% | 5.37% | 5.66% | 4.89% | 5.69% | -1.78% | 6.45% | 5.59% | 5.74% | 4.39% | 7.00% | 4.52% | 2Q1996 | 4.51% | 6.30% | 1.94% | 6.36% | 1.72% | 7.53% | 1.63% | 0.57% | 3.74% | 1.92% | 5.84% | 4.09% | 5.11% | 3.07% | 3Q1996 | 3.06% | 3.14% | 2.76% | 3.60% | 2.91% | 3.69% | 2.92% | 1.84% | 3.40% | 2.88% | -0.85% | 1.49% | 1.37% | 3.10% | 4Q1996 | 8.37% | 5.46% | 9.99% | 6.04% | 9.98% | 7.36% | 10.63% | 3.00% | 2.89% | 8.62% | 0.26% | 10.06% | 0.93% | 10.03% | 1Q1997 | 2.62% | -0.54% | 2.27% | 0.54% | 2.56% | 2.19% | 2.97% | -0.56% | -3.65% | 1.70% | -10.49% | -0.25% | -8.00% | 0.81% | 2Q1997 | 17.49% | 18.79% | 14.78% | 18.91% | 14.74% | 20.48% | 15.74% | 3.68% | 14.72% | 12.60% | 17.55% | 15.09% | 16.25% | 14.17% | 3Q1997 | 7.52% | 8.42% | 10.25% | 7.52% | 9.96% | 5.40% | 8.59% | 3.32% | 13.99% | 12.75% | 16.92% | 12.89% | 16.64% | 12.45% | 4Q1997 | 2.87% | 0.49% | 4.19% | 1.52% | 4.47% | 3.05% | 4.68% | 2.95% | -2.75% | 4.07% | -8.20% | 1.68% | -8.01% | 2.83% | 1Q1998 | 13.95% | 14.83% | 11.33% | 15.15% | 11.66% | 16.23% | 12.50% | 1.54% | 11.94% | 9.99% | 11.88% | 8.35% | 11.15% | 9.54% | 2Q1998 | 3.30% | 3.57% | 0.06% | 4.54% | 0.45% | 6.01% | 1.94% | 2.34% | -0.06% | -2.57% | -5.74% | -3.61% | -4.88% | -3.58% | 3Q1998 | -9.94% | -10.26% | -12.15% | -9.08% | -11.58% | -7.12% | -10.50% | 4.23% | -16.69% | -13.66% | -22.36% | -17.88% | -22.21% | -16.47% | 4Q1998 | 21.29% | 26.49% | 15.97% | 26.74% | 16.60% | 26.80% | 18.13% | 0.34% | 26.47% | 13.57% | 23.64% | 9.07% | 25.35% | 11.17% | 1Q1999 | 4.98% | 5.76% | 0.56% | 6.36% | 1.43% | 7.04% | 3.65% | -0.49% | 3.42% | -3.11% | -1.68% | -9.69% | -1.05% | -8.00% | 2Q1999 | 7.05% | 4.60% | 11.65% | 3.85% | 11.28% | 2.41% | 11.31% | -0.88% | 10.42% | 11.18% | 14.75% | 16.56% | 16.81% | 16.02% | 3Q1999 | -6.25% | -3.76% | -9.65% | -3.66% | -9.80% | -3.40% | -9.44% | 0.68% | -5.01% | -10.64% | -4.92% | -7.82% | -3.47% | -8.64% | 4Q1999 | 14.88% | 25.70% | 5.14% | 25.14% | 5.44% | 22.47% | 6.18% | -0.12% | 39.47% | 3.77% | 33.39% | 1.53% | 39.36% | 4.08% | 1Q2000 | 2.30% | 7.29% | 0.71% | 7.13% | 0.48% | 4.18% | 0.26% | 0.94% | 21.12% | 1.01% | 9.28% | 3.82% | 15.15% | 4.67% | 2Q2000 | -2.66% | -3.03% | -4.24% | -2.70% | -4.69% | -1.52% | -5.96% | 3.01% | -7.41% | -1.68% | -7.37% | 1.95% | -6.91% | -0.59% | 3Q2000 | -0.97% | -5.28% | 7.83% | -5.38% | 7.86% | -6.96% | 7.15% | 3.02% | 2.52% | 9.65% | -3.97% | 7.34% | -2.94% | 6.89% | 4Q2000 | -7.82% | -21.27% | 3.89% | -21.35% | 3.60% | -20.93% | 1.28% | 4.21% | -23.25% | 9.44% | -20.20% | 8.11% | -19.35% | 8.62% | 1Q2001 | -11.86% | -20.50% | -5.40% | -20.90% | -5.86% | -20.00% | -6.86% | 3.03% | -25.08% | -3.53% | -15.20% | 0.97% | -19.92% | -1.97% | 2Q2001 | 5.86% | 9.13% | 5.35% | 8.42% | 4.88% | 6.85% | 3.94% | 0.56% | 16.18% | 7.05% | 17.97% | 11.64% | 21.27% | 9.98% | 3Q2001 | -14.68% | -20.02% | -11.12% | -19.41% | -10.95% | -17.58% | -10.71% | 4.61% | -27.80% | -11.55% | -28.08% | -13.34% | -27.07% | -12.45% | 4Q2001 | 10.69% | 15.83% | 8.00% | 15.14% | 7.37% | 12.85% | 5.52% | 0.05% | 27.06% | 12.03% | 26.17% | 16.72% | 25.90% | 16.27% |

| Benchmark | A | B | C | D | E | F | G | H | I | J | K | L | M | 1Q2002 | 0.27% | -2.54% | 4.49% | -2.59% | 4.09% | -2.77% | 2.50% | 0.10% | -1.77% | 7.90% | -1.96% | 9.58% | -2.95% | 8.51% | 2Q2002 | -13.39% | -18.47% | -8.03% | -18.67% | -8.52% | -18.76% | -10.21% | 3.70% | -18.26% | -4.67% | -15.70% | -2.12% | -16.63% | -3.49% | 3Q2002 | -17.28% | -15.49% | -18.96% | -15.05% | -18.77% | -14.51% | -19.11% | 4.58% | -17.18% | -17.95% | -21.52% | -21.29% | -19.06% | -18.38% | 4Q2002 | 8.43% | 7.17% | 8.91% | 7.15% | 9.22% | 6.66% | 10.11% | 1.57% | 9.16% | 7.07% | 7.51% | 4.92% | 8.28% | 5.44% | 1Q2003 | -3.15% | -1.25% | -4.88% | -1.07% | -4.86% | -1.33% | -5.19% | 1.39% | -0.02% | -4.06% | -3.88% | -5.08% | -3.20% | -4.77% | 2Q2003 | 15.40% | 14.92% | 17.65% | 14.31% | 17.27% | 13.17% | 17.02% | 2.50% | 18.76% | 17.89% | 24.15% | 22.72% | 22.72% | 21.28% | 3Q2003 | 2.64% | 4.37% | 2.47% | 3.91% | 2.06% | 3.12% | 0.45% | -0.15% | 7.16% | 5.94% | 10.47% | 7.72% | 9.97% | 7.77% | 4Q2003 | 12.18% | 10.58% | 14.35% | 10.41% | 14.19% | 9.97% | 13.74% | 0.32% | 12.16% | 15.22% | 12.68% | 16.37% | 12.00% | 16.43% | 1Q2004 | 1.70% | 1.14% | 3.33% | 0.79% | 3.03% | -0.26% | 1.99% | 2.66% | 4.83% | 5.35% | 5.58% | 6.92% | 5.50% | 6.18% | 2Q2004 | 1.71% | 1.79% | 0.88% | 1.94% | 0.88% | 2.19% | 0.48% | -2.44% | 1.05% | 1.73% | 0.09% | 0.85% | 0.13% | 0.51% | 3Q2004 | -1.87% | -5.29% | 1.42% | -5.23% | 1.54% | -5.49% | 1.46% | 3.20% | -4.33% | 1.73% | -6.01% | 0.15% | -5.61% | 0.25% | 4Q2004 | 9.23% | 9.66% | 10.62% | 9.17% | 10.38% | 7.70% | 9.01% | 0.95% | 13.94% | 13.46% | 15.08% | 13.20% | 14.91% | 13.64% | 1Q2005 | -2.15% | -4.32% | -0.26% | -4.09% | 0.09% | -4.90% | -0.23% | -0.48% | -1.67% | 0.78% | -6.83% | -3.98% | -4.33% | -2.12% | 2Q2005 | 1.37% | 2.55% | 1.96% | 2.46% | 1.67% | 2.10% | 0.30% | 3.01% | 3.43% | 4.70% | 3.48% | 5.08% | 3.56% | 5.33% | 3Q2005 | 3.61% | 4.21% | 3.81% | 4.01% | 3.88% | 3.06% | 3.24% | -0.67% | 6.55% | 5.35% | 6.32% | 3.09% | 6.29% | 3.55% | 4Q2005 | 2.08% | 2.86% | 1.21% | 2.98% | 1.27% | 2.80% | 1.24% | 0.59% | 3.44% | 1.34% | 1.61% | 0.66% | 2.72% | 0.92% | 1Q2006 | 4.20% | 4.07% | 6.59% | 3.10% | 5.93% | 1.35% | 5.21% | -0.65% | 7.61% | 7.62% | 14.36% | 13.51% | 11.83% | 10.47% | 2Q2006 | -1.44% | -4.22% | 0.28% | -3.90% | 0.59% | -3.58% | 1.09% | -0.08% | -4.69% | -0.56% | -7.25% | -2.70% | -6.13% | -2.53% | 3Q2006 | 5.67% | 3.44% | 5.88% | 3.94% | 6.22% | 5.31% | 7.26% | 3.81% | 0.89% | 3.53% | -1.76% | 2.55% | -1.20% | 2.27% | 4Q2006 | 6.70% | 6.16% | 8.10% | 5.93% | 8.00% | 5.48% | 7.81% | 1.24% | 6.95% | 8.50% | 8.77% | 9.03% | 8.24% | 9.14% | 1Q2007 | 0.64% | 1.29% | 1.26% | 1.19% | 1.24% | -0.02% | -0.07% | 1.50% | 3.96% | 4.86% | 2.48% | 1.46% | 4.03% | 3.08% | 2Q2007 | 6.28% | 6.84% | 4.69% | 6.86% | 4.93% | 6.93% | 5.41% | -0.52% | 6.74% | 3.65% | 6.69% | 2.30% | 6.98% | 2.91% | 3Q2007 | 2.03% | 3.85% | -0.75% | 4.21% | -0.24% | 5.24% | 0.96% | 2.84% | 2.15% | -3.55% | 0.02% | -6.26% | 0.66% | -6.04% | 4Q2007 | -3.33% | -0.88% | -5.91% | -0.77% | -5.80% | -0.32% | -5.74% | 3.00% | -1.70% | -5.97% | -2.10% | -7.28% | -2.09% | -6.97% | 1Q2008 | -9.44% | -10.39% | -8.55% | -10.18% | -8.72% | -9.83% | -8.74% | 2.17% | -10.95% | -8.64% | -12.83% | -6.53% | -11.08% | -7.22% | 2Q2008 | -2.73% | 1.51% | -5.17% | 1.25% | -5.32% | -0.28% | -7.13% | -1.02% | 4.65% | 0.07% | 4.47% | -3.55% | 3.62% | -1.24% | 3Q2008 | -8.37% | -11.93% | -5.26% | -12.33% | -6.11% | -10.00% | -5.54% | -0.49% | -17.75% | -7.52% | -6.99% | 4.96% | -12.09% | -1.22% | 4Q2008 | -21.94% | -23.15% | -22.41% | -22.79% | -22.18% | -20.99% | -20.17% | 4.58% | -27.36% | -27.19% | -27.45% | -24.89% | -27.77% | -24.86% | 1Q2009 | -10.93% | -4.54% | -17.00% | -4.12% | -16.77% | -4.39% | -17.51% | 0.12% | -3.36% | -14.67% | -9.74% | -19.64% | -5.97% | -16.32% | 2Q2009 | 15.93% | 16.82% | 16.81% | 16.32% | 16.70% | 14.73% | 15.14% | 1.78% | 20.67% | 20.94% | 23.38% | 18.00% | 21.79% | 18.76% | 3Q2009 | 15.61% | 14.12% | 18.59% | 13.97% | 18.24% | 12.70% | 16.22% | 3.74% | 17.58% | 23.62% | 15.95% | 22.70% | 17.17% | 22.77% | 4Q2009 | 5.93% | 7.65% | 4.17% | 7.94% | 4.22% | 8.40% | 3.81% | 0.20% | 6.69% | 5.21% | 4.14% | 3.63% | 5.57% | 4.65% | 1Q2010 | 5.39% | 4.87% | 7.05% | 4.65% | 6.78% | 3.54% | 5.63% | 1.78% | 7.67% | 9.61% | 7.61% | 10.02% | 8.81% | 9.57% | 2Q2010 | -11.43% | -11.55% | -11.09% | -11.75% | -11.15% | -12.33% | -11.79% | 3.49% | -10.20% | -9.57% | -9.22% | -10.60% | -9.77% | -10.16% | 3Q2010 | 11.29% | 12.98% | 10.10% | 13.00% | 10.13% | 12.37% | 9.23% | 2.48% | 14.65% | 12.13% | 12.83% | 9.72% | 13.15% | 11.39% | 4Q2010 | 10.76% | 12.26% | 10.92% | 11.83% | 10.54% | 10.99% | 9.75% | -1.30% | 14.01% | 12.24% | 17.11% | 15.36% | 16.00% | 13.84% |

APPENDIX B Matched Pair Data
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1Q1994 through 4Q2010

APPENDIX C Five Number Summary & Location Information
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1Q1994 through 4Q2010 Attribute | Benchmark | A | B | C | D | E | F | G | H | I | J | K | L | M | Min | -21.94% | -23.15% | -22.41% | -22.79% | -22.18% | -20.99% | -20.17% | -2.88% | -27.80% | -27.19% | -28.08% | -24.89% | -27.77% | -24.86% | 1st Quartile | -2.28% | -3.21% | -1.00% | -2.94% | -0.57% | -2.93% | -0.32% | 0.02% | -3.17% | -2.64% | -6.08% | -2.27% | -5.06% | -2.01% | Median | 2.97% | 3.71% | 3.06% | 3.88% | 2.97% | 3.09% | 2.71% | 1.52% | 3.44% | 4.28% | 2.05% | 3.36% | 3.14% | 3.70% | 3rd Quartile | 7.63% | 7.99% | 8.25% | 7.76% | 8.19% | 7.57% | 8.01% | 3.01% | 9.99% | 8.86% | 10.70% | 9.04% | 11.12% | 9.24% | Max | 21.29% | 26.49% | 18.59% | 26.74% | 18.24% | 26.80% | 18.13% | 6.09% | 39.47% | 23.62% | 33.39% | 22.72% | 39.36% | 22.77% | APPENDIX D 30 yr Fixed Interest Rate
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4/26/11 AimLoan.com | 4.50% | First Choice Bank | 4.75% | First Internet Bank | 4.75% | BNC National Bank | 4.88% | First Preferred Mortgage | 4.88% | Cap West Mortgage Corp. | 5.00% | Quicken Loans | 4.75% | Average | 4.79% |

Works Cited

1. Fox, Justin.2009. The Myth of the Rational Market. Harper Collins.

2. Mishkin, Frederic, and Stanley Eakins.1998. “Financial Markets and Institutions.” Second Edition. Addison Wesley.