Multi-Factor Models – Applying the Lessons Learned from the Numbers

In Finance 1 last year, we were introduced to the idea of multi-factor models (MFM) originally explained by Fama and French as an alternative to the traditional Capital Asset Pricing Model (CAPM) for assessing systematic risk. Additional factors include small versus big (SML) and value versus growth (HML).

In our Business Analysis and Valuation class, we discussed a merger case in which a large company acquired a smaller company. We talked about what would be the best way to approximate beta. The method I used (which was the best method I could conceive, I’d be happy to hear criticism or suggestions otherwise) was to weight the betas by market cap and take an average.

However, there was some discussion about the fact that one entity was much smaller than the other. While we were having a conversation of what that would actually mean, I would suggest a mathematical method for expressing the quantitative effect of size, using FF’s MFM.

1. 1. Express both company’s re as a three factor MFM
   Re1 – RFR = beta1 (Rm – RFR) + betas1 (SMB) + betav1 (HML)
   Re2 – RFR = beta2 (Rm – RFR) + betas2 (SMB) + betav2 (HML)
2. Take the larger company’s size beta and apply it to the smaller entity
   betasnew = betas2
3. Recalculate re for both entities
   Re1new – RFR = beta1 (Rm – RFR) + betasnew (SMB) + betav1 (HML)
   Re2new – RFR = beta2 (Rm – RFR) + betasnew (SMB) + betav2 (HML)
4. Take a weighted average (by market cap) as the expected return of the combined entity

By taking the larger company’s size beta for both, what you are saying is that you expect the smaller company to have the size “characteristics” of the larger entity. I might even be more appropriate to add the size factors (Would that be appropriate? As the MFM is a linear regression, is it appropriate to add these factors?) and use that for new return on equity for each entity as it relates to the combined entity.

betasnew = betas1 + betas2

While there are some significant assumptions which are required for this to work, it is the best solution I can conjure based on information given. I would really appreciate any additional ideas for creating a more robust model.

Flight of Fancy: What If...? A Market for Bid Points

One common theme I've heard is that MBA's are often upset when they don't get all the elective courses that they want. While I certainly can't complain, it brings up an interesting question: "What if someone like me was able to sell their bid points? What would I get for them? And how would you value them?"

For example, my course choices weren’t very restrictive, I got 500 points to bid on four courses, most of which I could have gotten with a zero bid. Whereas, Mr(s). Ambitious was trying to take TMP and Value Investing while going on Exchange (physically impossible, Value Investing is a year long course and Exchange means you are physically gone). If there existed a mechanism (and therefore a market) for me to transfer my points for a price, what would I get for them? What should they be worth? Clearly, there is currently some "market inefficiency" as we are both unsatisfied: Mr(s). Ambitious because they didn't get all the courses they wanted [net deficiency] and me because I didn't realize the full value of my bid points because I had more than I could use - [net surplus].

Well let’s make some assumptions:

• Rotman tuition is C$35k per year (let’s not include first year as it’s common, or you can adjust the value of points accordingly if you feel second year courses are more / less important)
• You take 10 elective courses in your second year
• You are given 1000 points with which to bid

A “book value” of the points would simply be C$35k / 1000 points or about $35 per point.

But keep in mind that when something is inherently useful, especially in a scenario where a few points margin can mean the difference between getting the course you really want versus having to settle for a less popular course, there can potentially be bidding wars from “oversubscription” (points trade at a multiple above their book value) especially if they were in limited supply.

While people are paying C$70+k to go to school, for a marginal $35 x 100 points (a rough approximation of the average points allocated per student / course) or $3500 you can get any course you want (including the highly coveted TMP and Value Investing – which includes a trip to visit Warren Buffet – one of the reasons why this course is so wildly popular).

If you could some how do it, you could see how much additional probability you have of getting into the classes you wanted and put a dollar value on how badly you wanted to be in that class (regression analysis), you can determine a price you’d be willing to pay to attend that class. For example: Would there be a correlation between the number of points you consumed to get into classes of your choice against your overall earning power once out of university (thinking along the lines of DCF to value bid points like common shares).

And also imagine if this market had a “market maker”. For example, the PSO will (create and) sell you points for a certain value (either regulated and pre-determined or floating with the market). Students could liquidate their points at market value and get money back or buy points of the market to be more competitive for course selection and the school could potentially get revenue from selling points.

And since you have a market with underlying assets, imagine if you created financial instruments for those assets (shorts, puts, and calls for bid points, futures).

And imagine if other schools had market systems (I’m told that bidding systems are not uncommon at other MBA schools), you could trade between these. Or even other programs!

Of course, these points would inherently have an “expiry” as to their value (you wouldn’t want to be holding (take delivery of) 5000 MIT Engineering points if you were going to Stanford Law School).

There are some interesting implications. For instance, a new ranking system for schools where the relative value of a course is determined by the market value (determined by students taking courses there) in real time with comparisons to year over year values. Example: Would an engineering calculus class go for more at Waterloo or Toronto? Could you couple this with flexibility between schools (accreditation programs) which allow students to take equivalent courses at other schools and what do you get?

It would be a more sophisticated and real-time version of tuition regulated by the market. Taken to the extreme, here is another idea: drop the original tuition completely and have students buy bid points for classes. And then what if you were able to connect this market to actual financial markets? An S&P Index of Undergraduate studies to benchmark the valuation of your individual class’ performance.

Another thought: If the value of courses in a particular faculty started to "overheat" would that be a leading indicator of oversupply of labour in a particular industry in 4 years time?

Bidding Strategy - The Mechanics

So I've been lucky enough to receive all the classes I want in all the sections I want and it turns out that LBS doesn't use a "bidding" system per say (classes awarded based on listed "preference" - an ordinal system).

A few people were asking about how my bidding formula works and while it's hardly perfect, I figured I'd put up some of the details just for laughs (or a least as building blocks for someone who plans on taking this model to the next level). It uses only public information available to all students at the time of bidding.

In this model, each course bid is determined by three factors. The first is the inital base and most people will choose one of two initial bases: Last year's minimum bid or last year's median bid (depending on how competitive the class is).

After determining the appropriate bases for your five courses, the remaining points (“the Remainder”) can be divided amongst your courses to make your bids more competitive. But like all dilemmas in bidding, you want to assign just enough points so that you get the courses you want, but not so much that you jeopardize your chances of getting the other courses. So how do you do it?

I propose that the two major factors you should look at are what I call:

1. The Ballot factor (anticipated) (x% of the Remainder, or “X-Factor”)
2. The Historic factor (backward-looking) ([100% - x%] of the Remainder, or the “Y-Factor”)

Where x% is the weight of value of your Ballot factor versus your Historical factor (In other words: how much you believe your Ballot Factor represents real bidding behaviour versus historical).

Ballot Factor:

This factor accounts for the number of people who say they will take the course. A few notes:

• People don’t always bid for the courses they ballot for
• Use the numbers as guidance to see if the course is oversubscribed
• Calculate the expected utilization capacity = total number of students balloting for any course in that section / total class capacity
• Square the utilization capacity to create an “intensity factor”
• Total all the factors and express each factor as a percentage of the total
• Multiply the percentages by the X-Factor
• The result is each individual courses’ Ballot Factor offset

Example:

• 2 classes have a capacity of 40 people each
• You have 200 points allocated to Ballot Factor
• 20 people bid on Class A (fairly certain everyone who bids will get in… There is even a chance that a 0 point bid could win) has utilization 50% and Ballot “intensity factor” of .25
• Class B has 60 bidders has utilization 150% (red flag: guarantee that not everyone will get in) and it’s “intensity factor” is 2.25.
• Class A’s weight is .25/(.25+2.25) = 10%
• Class B’s weight is 2.25 /(.25+2.25) = 90%
• Class A’s Ballot factor offset is 10% * 200 points = 20 points (a non-zero bid with decent margin, you'll probably get in)
• Class B’s Ballot factor offset is 90% * 200 points = 180 points (a strong bid, considering an average of 100)

This model tries to account for the fact that only very high bids will win the competative class, but you also don't want to low ball Class A incase a few stray bids appear from people who take the class last minute (obviously, the less people who originally bid on the class, the less you have to worry about dark horse bidders).

Note that it is 9x because at least 20 people are guaranteed to not get in the class. Classes that are oversubscribed will have intensity factors much higher than 1 with much heavier weights and undersubscribed much lower than 1 with much lower weights. This accounts for the premium on variation and intensity due to the number of bids in a competitive environment. Note that in this pure form, this is a best effort bidding mechanism with the scaling of points to consume all remaining points.

Historical Factor:

Another way to try to guess what the bidding will look like is to use the historical bidding as guidance for the variation of bids (were the bids tight or across a broad range?) One indicator of that is the minimum and median bid. If you make some HUGE assumptions, you can use these two points to create a normal curve with standard deviations. Since the mechanics of this are taught in stats in first quarter, I won’t bore my readers with a poor facsimile of Prof. Krass’ lecture.

Even if you don’t technically know the actual distribution of the curve, you can also use Chebyshev's inequality to position yourself within a certain percentile (also looking at the expected capacity utilization of the class based on your previous calculations). How? Here’s a hint (shown above): the bidding percentiles (% of students bidding that are not successful being admitted into the class) should be the same as the bid oversubscription capacity (again, huge assumptions) to provide the number of standard deviations. Combine this fact with the distance from the median to the minimum should provide a clue as to size of a standard deviation. Note that using this method, you may not (probably won't) have enough points to guarantee getting into the courses you want (unless like me, you probably have a surplus of points or are taking unpopular courses), but it is probably one of the best mechanical methods for balancing aggresive bidding with conserving points as well as building a view for what the bidding landscape looks like. In practical terms, at this point you can use a best effort model similar to the one shown above using the Y-Factor.

Also, I’ve deliberately left out methodology for mechanically scaling up courses based on your individual preferences (ie rating courses from 1 to 10 and incorporating that into your bidding strategy). Also, there are huge economic implications for bidding strategy considering that the involved parties do communicate with each other and affect the bidding levels of courses (ie Friends talk to each other about how they plan to bid). Signalling, game theory and strategy all come into play.

While not perfect, this model will give you some perspective into what a reasonable, very mechanically inclined bid would be. Admittedly, while I built this model, I did do some “emotional” adjustments to my bids (there was one course where I wanted to work with my friends on their team, so I wanted to be CERTAIN that I got the course). Like anything done on a computer, it’s just a tool.

Disclaimer: Like anything on this blog, this model does not guarantee any degree of success. This post is intended as a conversation / pensive reflection piece only. It is possible for you to use this model and not get ANY courses you want. For instance, it is physically impossible to get both Top Management Perspective AND Value Investing because both courses usually require exceptionally high bids. Note that by definition, there will be some people who don't get the courses they want. The more you want to be certain that you are in one course, the less certain that you will be in another (almost like the Heisenberg uncertainty principle). For better or worse, it is a zero-sum game.

Also, more importantly, I've been told that it's all a wash and at the end of the day, after the drop and add periods are over, most people get the courses they want anyways.

Finance Industry Day - Group 1's first small victory

First some background:
Because I was very busy yesterday, I didn't get a chance to blog about our little finance prep team. We've split the team into two groups, the "advanced group" (Group 2: people with experience who just want to talk about the markets and current events) and the "beginner group" (Group 1: people with little or no experience).

Yesterday, we also had an intense speaker, CEO of Claymore Investments, Som Seif, give a fantastic talk, describing how correlations between different asset classes are increasing (even between international and geographic regions) because of increased globalization. He also described how asset allocation plays a major role in determining return and managing risk. One of his more poignant points was that weak fund managers who can't justify their performance fees in down markets will be quickly replaced by low cost ETFs (the philosophy that management fees are not necessarily supposed to provide amplification on the upside, but rather a sense of expertise and awareness to protect from the downside).

His talk was fairly technical at points, and I began to wonder about our "Group 1" members in the audience. I asked them after the talk, "Did you feel intimidated? Don't worry, the purpose of the group is that a someday (soon) you can have an intelligent conversation with Som with the same vocabulary and technical proficiency".

For Group 1, I've set a schedule of topics to supplement what we'll be learning in finance class, and began reviewing basics like Time Value of Money (TVM), NPV, IRR etc. It was a whirlwind class, but we took them from zero to being able to have the vocabulary and math to describe when perpetuity formulas fail, to how to select between multiple investment vehicles. We'll even be asking Group 2 members to lecture on topics for the Group 1 team as practice. While I am very happy with the result at the end of yesterday, it is also what happened today which made me smile:

Today was the Rotman Finance Industry day and we had a host of speakers in the afternoon from industry (recruiters and alumni) as well as current students and the CCC. So far there wasn't anything earth shatteringly new (mostly hearing the same message from different people: Finance is hard to get into, this is how you should prepare: read the Vault Guides, read the newspaper, be intense... etc)

My proudest moment was during the presentation from one of the industry speakers when he mentioned that you have to calculate IRR's for different projects as part of your job. Immediately, I saw a few heads turn around and catch eyes with me with a knowing smile on their face. If there was anyone who was living the message of "MBA school is great, but a large component of your success is what you do and who you are" it was those who turned around and knew they were doing the right thing by preparing early. These were the people who, less than 24 hours before, were intimidated by Finance TLAs (Three letter acronyms) were now getting a sense of confidence.

Word has spread also. While Group 2 membership is deliberately limited (not a "snobby thing" but rather a purely logistical issue - I've been helping other people put together their own "Group 2's"), the word has spread and Group 1 has picked up a couple more members.

Again, I don't know if people remember that I had promised in my GBC first year rep speech to help all the Rotman students "reach their maximum potential" and build more equity into the degree (and network) we all paid so much to have the opportunity to earn, but I hope I'm beginning to show active signs outside of the standard "job requirements" that there is a lot I can offer and that we can do together.

Multiple Regression Model - High Correlations like Dividing by Zero

In multiple regression models, statisticians will create formulas which incorporate different factors in order to predict a value. The example used in class is the popular model used by real estate agents to do market valuation of houses. A typical model will incorporate things like number of bed rooms, if there is a basement, square footage, home assessment value etc. However, you can very quickly identify that some of these factors are 'related' (a house with many bed rooms, a basement or high home assessment value will probably have more square footage).

However, we were shown a formula which only used AREA to predict value and one formula that used ASSESSMENT (home tax assessment value). Intuitively, you can tell that they are positively correlated to some degree and this was reflected in the formula.

MARKET1 = X + a AREA + e
MARKET2 = Y + b AREA + c ASSESS + e

Where 'e' is the error.

However, in the two formulas, b was less than a. What does that mean? Some of the 'correlated' value between AREA and ASSESS is encompassed both in 'a' and 'c'.

But what if they are VERY highly correlated (or if you deliberately choose one factor which was a linear construction of another factor for a correlation of 1), you can see that it is impossible to create a 'factor' as the two items will move in perfect harmony. Imagine AREA is perfectly correlated to ASSESS or

ASSESS = d AREA

then

MARKET1 = X + a AREA + e
MARKET2 = Y + b AREA + c ASSESS + e

However, if ASSESS = d AREA (perfectly linearly correlated)
then
MARKET2 = Y + b AREA + cd AREA + e
MARKET2 = Y + (b + cd) AREA + e
Since there are only one 'real' factor, this would mean that:

• MARKET1 = MARKET2, the second model would be identical to the first model.
• 'a' would be expressed as (b + cd)
• X = Y

Another problem is that you can allocate b and d in any proportion you want to create MARKET2 (there is no meaningful way to allocate the weight of the factors b and cd). It's a very awkward solution and reminds me of divide by zero when it comes to math problems, that there is a problem when you take the limit of a factor to asymptotically approach a value. Mathematically, I believe that it would be described by these factors having no orthogonal components which causes a unique case.

Another interesting result is that if 'b' is not significantly different from 'c', it shows that correlation is probably low.

Also, if any of the 'weight' letters (except for 'e') is close to zero, it implies that these prediction factors don't actually have any bearing on model, but high numbers don't necessary imply importance (it really matters what the factor variables are measured in as it has a relative impact on the final prediction).

Fixing "Problems" We Don't "Understand"

We were discussing statistics in our class and how to understand numbers you are presented with respect to their distribution. Particularly, we were focusing on prediction metics and describing how a residual value (calculated as actual minus prediction) is defined as biased if the mean value is not equal to zero (or FAR from zero, our heuristic is if the mean is approximately 20% of the standard deviation).

I proposed a solution that we could "correct" the bias (even if we didn't understand it) by making a permanent subtraction from our prediction model by the mean of the residual.

My teammate, Matt Clare, and the professor further refined this model to say it would work IF the residual pattern exhibits constant variability (the variation and volitility isn't changing).

While I am loath to use solutions which insufficently describe the underlying causes, this seems to be one of the FEW examples where math can actually be used to solve a problem without a keen knowledge of what is happening (assuming an intimate knowledge of the mechanics of why patterns occur, rather just a knowledge that they exist and are stable).

For residuals that increase with the size of the actual, I think that considering errors as a fraction of residual over actual (in the same way we normalize variation for residual based on actual size) can understand variation in the same way.

Population Distributions

In our statistics class, we have begun to move into the realm of distribution parameters and describing data for management purposes. We were discussing how standard deviations could be used to approximate distributions and probabilities of occurrences with appropriate assumptions. CFA candidates will immediately perk up and remember concepts like Chebyshev's inequality, which describes the percentage of a population which lies within k standard deviations regardless of the underlying population by the formula:

(1 - k^-2) * 100%

k = 2 --> 75%
k = 3 --> 89%

Or more common normal distributions where:
1 σ = 68%
2 σ = 95%
3 σ = 99%
(values shown above are 'expected knowledge' for MBAs and slightly oversimplified)

However, like the CFA exam, I anticipate that the trick in demonstrating an understanding of this concept will be more related to picking appropriate bounds rather than the pure memorization of the percentages for different values of sigma. For instance, in our stats class, we had a problem where we needed to know the probability that our value would not fall below 1 σ. As a clue, that indicates that we need to do a 'one-tailed' test. It is important not to fall into the trap of saying "I see 1 σ, therefore the answer is 68%". In this case, the answer is:
100% - ((100% - 68%) / 2)
100% - (32% / 2)
100% - 16%
= 84%
because it is only a one tailed test.

I have always loved stats. Not just for the pure math value, but more importantly, when trying to prove when math (or anything else for that matter) works or doesn't work, stats is the first place people should go to to determine correlations as a precursor to causality.

Diversification and Correlation - Understanding Risk and Reward

Often you'll hear people quoting investment cliches: "Don't put all your eggs in one basket" "Diversify, diversify, diversify" but not really understand what they mean. Most will understand that if they only invest in one stock and it tanks that they can lose everything straight away (or hit it big). However, few understand the risk and benefits of diversification.

The CFA Level II material begins to go into the idea of correlation, one of my favourite topics in math. While I am often criticized for loving math a little too much (one colleague went so far as to say that I think math can "solve all the world's problems" whereas I would prefer to think of it as "math can describe most of the world's patterns"). I even said that "there is math to describe when math fails" and that in my opinion is statistics.

A Quick Primer on Correlation
Correlation is the idea of how closely to items move together (in finance, the most notable example is stock prices) and the strength of their linear relationship. Relationships measured in correlation can have a value between 1 (perfectly linearly correlated) and -1 (perfectly negatively linearly correlated). What does this mean in layman's terms?

With a correlation of 1, two stocks will move in perfect harmony. If one stock rises, the other stock will rise proportionally. With a correlation of -1, if one stock rises, the other stock will fall proportionally. A correlation of 0 implies no linear relationship (strictly speaking not independent, but independent variables will have a correlation of 0).

Correlations of less than 1 mean that they move in the same direction, but do not have a perfectly linear relationship (most stocks in the stock market) and do not move proportionally (sometimes one will move faster or slower than the other). I would propose that the only way to find a perfect correlation is to buy more of the stock (or short it for a -1 correlation). Obviously, correlation is a bit more complicated that this but this will do for now.

Risk and Return of a Portfolio
Now that we have a basic understanding of correlation, how can that help us understand diversification, risk and reward? Let's look at two stocks A and B with expected returns 15% and 10% and a correlation of .5. Let's say the stocks have std dev of 9% and 6% respectively and the risk free rate is 4% (therefore the Sharpe ratio is 1.22 and 1 respectively). A is riskier, but offers more marginal return per unit of investment risk.

There are four possible actions:

1. Long (buy) A - Correlation to Long A: +1
   
2. Short (sell) A- Correlation to Long A: -1
   
3. Long (buy) B - Correlation to Long A: +0.5
   
4. Short (sell) B- Correlation to Long A: -0.5
   

Note that if you only care about maximum returns you will allocate all your capital to action 1: Long (buy) A. It has an expected return of 15% so it has the highest growth potential. But note that it also has the highest risk profile (largest standard deviation). If you were more moderate, you would Long (buy) a combination of A and B (with an expected return of between 10 to 15% depending on allocation and a standard deviation between 6 to 9%).

The lower risk portfolio construction would be from some combination of stocks with negative correlation (example Long A, Short B or Short A, Long B) because if one ever went down, the negative correlation will imply that the other will go up (possibly by more, possibly by less). However, also note that if their movements are counter each other as is usually the case in a negative correlation, your profit potential becomes much less.

Diversified Portfolio
In this over simplified scenario, assume that a portfolio, evenly weighted between a Long A position and a Long A and Long B. If the both hit their growth targets their combined return is 12.5% (equally weighted average between 10 and 15% and std dev between 6 and 9%). This is less reward than just buying A, but also less risk.

Assume another evenly weighted portfolio between a Long A and Short B position has it's Long A hit +15% and it's counterpart, the Short B hits -10%. The portfolio only gains 5%. Conversely, if the Long A drops to -15% and the Short B rises to 10%, the portfolio only loses 5%. Whereas the movement in the individual stocks is much more pronounced, the portfolio is dampened from extreme gains and losses.

Implication
There are times to over diversify and there are times to cherry pick. Arguably, in this recovering economy, it's easy to pick "sprouts in scorched earth". That is to say, most stocks are undervalued so it's not hard to pick "winners". This is a decent time to over diversify, because the general trend is to go up in value.

The worst time to over diversify is at the peak of the market, when most stocks are over valued. In this case, it is better to be very specific about your investments and be extra diligent in your homework (or find another asset class like fixed income - deleverage).

Weighted Averages

Another recurring mathematical theme in the CFA is the weighted average (probably because it is so universally useful). It appears in portfolio management, expected returns, WACC, indexing etc. It essentially takes the components of a group, takes the proportional weight of each and determines the value of the aggregate. Example:

S = (wα x vα) + (wε x vε) + (wρ x vρ) + ...
Where w is the weight of of each component expressed as a percentage of the whole and v is the value of each component.

For portfolio management, each component is an individual security's expected rate of return.

[Example] A portfolio is make of three stocks A, B and C. A has an expected return of 8% and makes up 20% of the portfolio. B has an expected return of 10% and makes up half of the portfolio. Finally C has a return of 12%. What is the expected return of the portfolio?

[Solution] E (Rp) = wA x E (RA) + wB x E (RB) + wC x E (RC)
= 8% x 20% + 10% + 50% + 12% x 30%
= 1.6% + 5% + 3.6
= 10.2%

For weighted average cost of capital (WACC), each component (debt, mezz and equity financing) of cost is weighted by proportion again:
[Example] A company issues bonds with at a cost of 4% which accounts for half of their capital. The required rate of return for their projects is 9% and they have an issue of preferred shares out for of 6%. They have twice as many common shares issued as preferred. If their tax rate is 30%, what is their WACC? Assuming that preferred shares are treated as debt, what is their total financial leverage ratio?

[Solution] Note in this case: we = 2wp and wd = 50%
wd = 50% = 100% - wp - we
wp = 16.7%
we = 33.3%

WACC = wd x kd x (1 - tax rate) + wp x kp + we x ke
= 50% x 4% x (1 - 30%) + 16.7% x 6% + 33.3% x 9%
= 1.2 + 1% + 3%
= 5.2%

Financial Leverage of Assets (FLA) = A / E
On a percentage basis:
A = wd + wp + we = 100%
FLA = 100% / 33.3%
= 3

[Example] An market weighted index is composed of three stocks A, B and C. A is worth $50 and composes 50% of the index. B is worth $10 and is 30% of the index. If C increases in value by 15%, what is the increase in the index?

[Solution] Initial Index = 100%
Final Index = 50% + 30% + 20% x (1.15) = 103%

The index increases by 3%

Why Bernoulli based models are rubbish - and why they might work

I've recently read from a brochure from a highly respected Canadian big bank about Behavioural Finance, particularly, the Gambler's Fallacy - The idea that future random events depend on past random events. It proposes the idea that if a stock rises five consecutive days in a row, that if you assumed it's less likely to rise on the sixth day, then you are a victim of a gambler's fallacy. Although I can see their rational, I strongly disagree.

If the analogy was taken into a casino, where future random events *are* independent of past events, then I would agree. However, I think that this model oversimplifies the stock market. And the principles on which this type of modeling is built, Bernoulli trials, should not hold under the Efficient Market Hypothesis.

The reason I'm so confident in making this statement is that I would like to believe that there are *some* rational value based traders out there who, when the stock price varies too far from its intrinsic value, will buy or sell accordingly. In this way, those traders who are doing their homework are essentially profiting off of those who are trading as if each day's outcome is random (or even if you model your risk management as if these outcomes are random). Prices *should* gravitate towards some number, and therefore each day's rise or fall *is* dependent on the day before it. A prime example is the price escalation we recently saw. Bernoulli trials would have simply labeled them as "unlikely" but I'd like to think that a value based trader would see a bubble forming.

Having said that, with all the emotional and fear based trading going on, this lack of faith causes high volatility in prices and using a Bernoulli type model might have some value as a risk management tool based on certain similarities. However, trying to get actionable value from it would be like attempting to profit from chaos which is risky and difficult (like creating a Garbage In Gold Out model).

However, those of you that can profit from this volatility will actually be providing a service to the economy in the form of marginal price stabilization. Anyone who is able to profiteer from the mood swings by reducing the spreads will bring us closer to something resembling that should happen under EMH.

But real stability won't happen until there is enough stable capital in the market that doesn't exhibit flight at the first hint of trouble. And that won't happen until there is broad based confidence in the fundamentals of the economy.

Sunday Reflection: A Theoretical Model for the Value of Derivatives

I was developing a model for understanding prices of options based on the betting system at a race track.

Consider European options (because they are easier conceptually and can only be exercised at expiration) on a stock selling at $50 with an excise price of $50. Assume a call is $5 and a put option is $2.

Also assume that the standard deviation for the stock is $2.

Race tracks can guarantee their profits because they will adjust the ratios in such a way that no matter who wins the race, the race track will take profits on all bets. But for simplicity sake, let's say that we can analyze this system as a snap shot in time first (an assumption we can remove later to strengthen our system).

First, let's say that for every call option you sell a put option (another assumption we can relax later).

So you sell 1 call for $5 and 1 put for $2 for a total premium of $7.

Note that with a standard deviation of 2, the following probabilities are likely (majorly oversimplified, but if you want to crack open a spreadsheet to do this properly, be my guest). Each set is one interval or between standard deviations:

E1 - P(X>56) = Negligible
E2 - P(56>X>54) = 2%
E3 - P(54>X>52) = 13.5%
E4 - P(52>X>50) = 34%
E5 - P(50>X>48) = 34%
E6 - P(48>X>46) = 13.5%
E7 - P(46>X>44) = 2%
E8 - P(X>44) = Negligible

However, because of the options, the relative values of each event are:

E2 - Ct = 50 - 56 = -6, Pt = 0, prem = +7, VE2 = -6 + 0 + 7 = +1
E3 - Ct = 50 - 54 = -4, Pt = 0, prem = +7, VE3 = -6 + 0 + 7 = +3
E4 - Ct = 50 - 52 = -2, Pt = 0, prem = +7, VE4 = -6 + 0 + 7 = +5
E5 - Ct = 0, Pt = 0, prem = +7, VE5 = +7
E6 - Ct = 0, Pt = 48 - 50 = -2, prem = +7, VE6 = -2 + 0 + 7 = +5
E7 - Ct = 0, Pt = 46 - 50 = -4, prem = +7, VE7 = -4 + 0 + 7 = +3

So the total expected value is:
E1 = 0
E2 = +1 x 2% = 0.02
E3 = +3 x 13.5% = 0.405
E4 = +5 x 34% = 1.7
E5 = +7 x 34% = 2.38
E6 = +5 x 13.5% = 0.675
E7 = +3 x 2% = 0.06
E8 = 0

E(R) = SUM(E1:E8) = 5.24

Notice a few key things... There are no events with a negative return (you are guaranteed to make profits) and for selling two options, you are expected to make $5.24 per pair.

Now, let's slowly start to remove some key assumptions and point out a few key points.

1. You can't actually sell (a large volume) of options (in pairs) to make this kind of profit. Usually, call options will be MUCH more popular than puts.

I simulated the above results and instead of having them in pairs, I tried saying that if a call is $5 and a put is $2, then a call is 2.5x more popular than puts ($5/$2). A HUGE assumption, but it turns out the expected value is still positive (although lower... Where as you are expected to profit $5.24 per $7 pair of premiums you sell, I worked it out to about E(R) of $2.10 per $20 of premiums you sell - in other words, still profitable, but less so... which makes sense). It doesn't really much ratio of sales of Call to Put you use as long as you calculate the Ct and Pt properly to understand your potential loses in all scenarios. But, obviously, as you start relaxing your assumptions to reflect "real life" your margins get smaller to reflect investor behaviour. GIGO.

Also to note, in the model where you sell more of one option than another and the underlying asset of that option suffers from extreme deviations from the expected mean, you can lose a LOT of money because the premiums from one type of option don't nearly cover the losses from the other.

2. For a stock trading at $50 (or there abouts) with an excise price of $50 (or there abouts) the Call will cost $5 and the Put will cost $2. Going back to Put Call parity, reducing arbitrage and finding the intrinsic / time value of options, you can adjust these values accordingly, however, they will directly affect your expected value. As you can see, the premiums carry directly to the "bottom line" (expected returns) proportional to the number of options sold.

3. You don't have to know the exact price of the stock nor be certain if it will be above or below that price any degree of certainty. You only need to know approximately where it will be (know the standard deviation).

4. You can make even more money if you take the premium and invest it in at the risk free rate (reinvestment in a T-bill) over the holding period.

5. You didn't actually invest any money. As long as you have money to cover "potential losses" you are fairly safe.

6. Every time you sell an option, you can raise the price to increase the spread to reflect demand. Of course, if the price rises too fast, you'll limit the multiplication factor to leverage your expected return.