Intro to Financial Theory: Part 1

I have an approximately 20,000 word work in progress ‘intro to financial theory for social scientists’ paper in progress. It is seeming like it will never be done though, so I decided to go ahead and start posting it in parts even if it is a little rough. It’s not as though I’m being graded.

 

Introduction

During the end of the great depression until the end of the 20th century financial regulation was not a hugely popular topic for social scientists to study. There were more important issues. Even during the dismantlement of welfare projects under Reagan and Clinton, when partisan politics began its infamous rise, financial regulation was not high up on the list. For social scientists today it is relevant to public discourse. And those that work in the finance industry should be able to articulate why their job is useful and important. With the financial world gaining larger scope and power in the world, understanding its foundation is now equally important to introductory courses in micro and macro economics. Even for political theorists or labor economists, current topics in finance are relevant. In these following posts I plan on teaching an introduction to financial markets. My main goal is to write palpable and intuitive descriptions that helps simplify how financial markets work. I will focus on the theory and not on the math. The math is not necessary for a basic theoretical understanding. In addition I will tie together these topics in investments with their counterparts in the financial industry, to provide a realistic and non-abstract lesson. My hope is that this be useful to students of sociology, political science, or other non-economic social sciences, as well as be useful to students of investments or economics that would like to learn financial theory in areas outside their expertise.

I will explain central theories and assumptions that will allow for an applied social scientists to have a strong theoretical understanding of the benefits of financial markets and how they function. Asset pricing and equity premiums are core areas of research in investments. However, it seems that often more academic and less (immediately) useful theory is ignored by those in the investment industry. This isn’t surprising. A firm being paid to invest money for a client is being paid to make them money. Whether or not the firm puts forward important theories to long-term academic questions is not as important as generating returns in the next year.. I remain concise and educational, rather than speculative. My primary goal will be to teach from John Cochrane’s seminal book on ‘Asset Pricing’ as well as pulling information from other key literature.

Introduction to Assets

First and foremost forget everything you have heard about stock picking and making easy money. While there are some particularly astute investors who might be able to identify the next ‘Apple,’ it is foolish to emulate them without understanding the basic theory. Once you understand asset pricing theory you will have a more refined and skeptical eye. For example, I often hear phrases such as “Apple is overvalued.” For Apple to be overvalued you should be able to express a reasonable theory of what price it should be, why you think it should be that price, and how you plan on making money off that difference. While often investing is said to have artistic properties, the art only can show its true grace after a strong foundation in empirical financial theory. The best gymnast in the world still needs to be in top physical condition to elegantly complete an artistic routine. A financial market is an aggregation of individual investors. As a result the most robust theory in finance ought to be micro-founded and can explain decisions at the level of an individual.

Every individual investor needs to make decisions on consumption, savings, and, assets held in his portfolio. At an investment equilibrium, the marginal utility loss of consuming less today, and instead investing into an asset, will equal the marginal utility gain of consuming more of the asset’s payoff in the future. That is to say, at a certain level an investor will be indifferent to additional short-run consumption today (such as an iPod) to instead enjoy extra consumption in the future once the asset has appreciated (such as an iPad). If the price and payoff of an asset change, the investor’s preferences should change as well. For example, if the future payoff drops by half, the investor will have less incentive to forgo current consumption for future consumption. It would be similar to the past scenario, only now in the future the iPad has less memory. This basic description explains how individual investors make decisions regarding current and future consumption based on the price of an asset and/or its expected payoff.

Now it is important to understand what drives the price of an asset, such as the price of a share of Apple. The fundamental theory is that an asset’s price ought to be equal to the expected discounted value of the asset’s payoff, using the investor’s marginal utility to discount the payoff. Put more simply, instead of imagining a single ‘asset’ selling for $100 imagine a series of future payoffs (I will be using the word payoff frequently, whether it is in the form of expected capital gains, dividends, or bond payments is not yet relevant). The current value of the asset is the discounted value of all the future expected payoffs. We will then use a certain discount value an investor holds to determine how much we should give up today for a certain amount of money in the future. In this case the act of buying one share of Apple would be equivalent to buying all the future payoffs of Apple, which would end up equalling the cost of one share of Apple.

To deconstruct the price of an asset to a very simplified state, imagine a single investor is all that exists and that there is no inflation or interest. Sitting across from him is a man that tells him “The only financial asset in the world is a single bond. This bond will pay you exactly $10 in one years from now” The man now wishes to calculate the maximum amount he would be willing to pay for the bond. If he pays $10 for the bond he will lose, because he will pay $10 now and then  not have access to his money for a full year with no gain. He now considers what might happen if he pays $9. He knows if he invests $9 into his personal business he will make a $1.50 profit and receive $10.50 back. However, he estimates that working for a year in his personal business would not be as fun as leisurely activities he might receive if he just bought a bond–and prices this personal cost at 0.51 cents, leading his end return from investing in his business at $9.99. As a result he decides he would rather invest $9 to buy this bond than invest it in his business. In this simplified market this man has just determined the price he would be willing to pay for an asset. However, this is a a micro-economic phenomenon. To examine where financial economics explains the price of an asset we must consider a discount factor.

In finance a discount factor is used to properly price an asset. The discount factor allows us to consider assets, investments, choices, prices, interest, inflation, time, and all other factors that would affect the price of an asset. If we examine the previous example, only the market has thousands of people, we would aggregate the price every individual would pay for the bond. The discount factor of the bond would be end up being the return it offers considering all individuals personal micro-economic utilty functions.

To start with the, the main discount factor used in asset pricing is based on investor consumption. Not surprisingly, it is called the consumption model. This model allows us to compare an investor’s future value of consumption to present value of consumption. Perhaps the most obvious, but theoretically important factor, is the idea that investors’ want to have consumption generally smooth throughout their life. We borrow when we are young and save as we grow older for retirement. On average we want to forgo consumption today to be sure we can continue to consume in the future. This is similar to the idea that most people would prefer to spend two days in a two and a half star hotel than one day in a five star hotel and one day on the street.This is known as an aversion to risk. We want to be sure that there is an extremely low probability our savings will be sufficient for retirement or that our home will foreclosed. This aversion to risk that appears at the level of an individual investor has profound implications for greater financial markets and the price of nearly all assets.

To properly conceptualize this idea of using a consumption model to price an asset on board financial markets consider the following: Each person has their own consumption based discount factor depending on the stage of their life, how much they have saved, how much they earn, when they expect to retire, if they plan on having kids, and every other variable that will affect consumption. Imagine two investors, one is about to go study an expensive masters program in London the following year and another is in a stable career at the age of 28. Each investor, in evaluating whether or not to buy an equity, will come to a different conclusion on what the price they would be willing to pay. The first investor knows he will have to consume a lot very shortly, so he does not want to invest money for future consumption but instead borrow. The second investor knows he will start a family in ten years and send his kids to college in 28 years, so he wants to save money for future consumption. Whether they buy or sell depends on if in using their own consumption based discount factor the price they value the equity at is below or above its market price.

A Primer on Efficiency: The law of one price and No Arbitrage.

As I have explained so far with the consumption model, a discount factor is a random variable that generates the price of an asset from the asset’s payoffs. The discount factor earlier was used to explain intertemporal substitution (substitution of current consumption for future consumption across time) in the consumption model. However, a discount factor can hold any variable that affects the price of an asset, such as risk of future payoffs failing to deliver due to bankruptcy or a global recession. Now that I have explained how a discount factor works and explained the primary consumption factor, it is important to understand how a discount factor can be extrapolated to other market theories. There are two conditions that must hold for discount factors to appropriately price assets: The Law of one Price and No Arbitrage. Knowing the individual discount factor for all assets for each investors is not possible. Other than consumption and risk there are infinitely many factors. If a single investor decides to only buy stocks that begin with the letter B that will impact the price of assets in the market and his or her personal discount factor. While this factor is likely not to be as popular as being averse to risk, it will still impact prices. I give this example to represent the absurdity of knowing each individual investors’ preference. However, while financial theorists attempt to identify the main factors, many are too small or niche to meaningfully prove their existence. The amount of variables and individuals that would need to be accounted for is absurd.

The first theory is the law of one price. The definition of the law of one price is that in an efficient market in equilibrium all identical assets (or payoffs) must have only one price. This means if two assets each have identical expected payoffs the price of the two assets must be identical. The reason the law of one price demands an equilibrium, is that if one of the two identical assets was cheaper we assume shrewd investors would immediately bid the price back up to identical. It is useful to now abstract away from reality (temporarily). Instead of thinking of microsoft stock and Treasury bonds, imagine a ‘payoff space.’ In this payoff space you have all the expected potential payoffs from every different stock, bond, future, option, or combination. The law of one price says that no matter how you create a certain expected ‘payoff’ it will cost the same amount. Intuitively this makes sense: If I were to create a portfolio ‘A’ by combining different assets that have a perfect correlation with the payoffs of portfolio ‘B’ they will be the same price. There should never be an asset or portfolio on the market that is cheaper than another asset or portfolio that offers the same payoff. To offer one last example consider Apple and Amazon. I should not be able to buy one share of each, package them together as a portfolio called “Apple Amazon portfolio,” and then sell it for anything greater than the combined share price. Even though it’s a new portfolio (and subsequently asset), it still holds the exact same payoffs as before.

Conversely, the existence of the law of one price allows us to derive the discount factor. To a certain theoretical degree this theoretical approach contains more substantial information. It is impossible to consider every single factor all individual investors might consider, which cause the price of a payoff to vary. Since we know the law of one price exists if the market is in equilibrium, we can instead examine payoffs. If we examine that all similar payoffs have similar prices this implies a rational and linear pricing structure, and a linear pricing function will imply that at least one discount factor exists. The reason this is true is that if all identical payoffs have identical prices, we know that there is at least one factor that all investors consider when buying and selling assets. This discount factor keeps is proven because in equilibrium identical assets have identical prices. If there was no discount factor, we would have no reason to expect identical assets to have identical prices.  It is likely that there are in fact multiple discount factors. To reiterate this means that there might be a discount factors to consider impatience, time, risk, and many other factors.

To add to my previous example, an example of market equilibrium means that If my special ‘Amazon Apple’ Portfolio were to trade at less than the combined value of Amazon and Apple (because someone just sold their holdings), another investor would instantly buy it and send it back to equilibrium. These profits are typically captured by high-frequency trading firms. A very simple program might logically be as follows: If portfolio ‘Amazon Apple’ is trading at a value less than the price of a single share of Amazon plus Apple instantly buy and resell at the net asset value (NAV).

From this explanation of the existence of a discount factor and the law of one price I have partially explained the no arbitrage condition. As expressed in the previous paragraph a market in perfect equilibrium will have no arbitrage opportunities (if any do exist they will near instantaneously be corrected). However, a temporary witness of a violation of the law of one price does not guarantee riskless profit. If an investor were to buy my ‘Apple Amazon’ portfolio at a discount and attempt to resell it there is a chance in that process he might have lost money if both assets crashed in price. A formal definition for arbitrage requires absolutely no risk with the potential for reward. No arbitrage means an investor, in a market in equilibrium, cannot receive an asset for free that would never have any associated cost and only potential gain. An example would be if Apple is trading at $500 on one stock exchange and $505 on another. An investor could buy one share for $500 while simultaneously selling the other share short for $505 (to sell a share ‘short’ means that you borrow a share and then sell it. Think of it as taking an inverse position where you make money when the stock declines). This guarantees a profit of $5 with no risk. Since even if the shares increase or decrease by $100 an investor is only betting that the identical sock will converge back to the same identical price. While differences in the same company on different exchanges do exist, they usually only exist in fractions of cents for fractions of a second (assuming no exchange has a different tax structure or other extra fees). This demands that any payoff must trade at a positive price and as a result the discount factor must always be positive (a negative discount factor is identical to ‘giving away money.’) This is obvious, however, we have now created a theoretical proof. As a brief example consider insurance. On average insurance investments have a negative expected value, after all insurance firms must make more money than they pay out. However, insurance products have a positive discount factor since we are willing to invest in an insurance that pays off greatly to offset a potential disaster. Conversely, selling insurance has a positive discount factor for an insurance firm, since they hedge their risks. So by focusing on a discount factor to price our costs and payoffs we create a more robust framework that considers all factors that might affect our valuation of an asset, rather than just focusing on absolute return.

So far we have related price to consumption and payoffs. It is important to note how academic views on assets begins with these models that originate from rational behavior of individuals and all advanced models (and most trading strategies) are built on top of these assumptions. A model for assets without a theoretical basis is questionable not only because it lacks the logical rigor expected from academia, but it also is not reliable for understanding future actions. The greatest leap is not to create a beautiful mathematical formula, but to let that mathematical formula express individual human actions in aggregate. If a model is not micro-founded, or, built on the rational actions of an individual, there is reason to be skeptical. This formal explanation of investments is necessary even for purely qualitative investors. Now that we have set up a way of viewing a rational investor, I will introduce variations on the theme.

I explained the discount factor earlier as a consumption model that encompasses various corrections for desires to avoid risk or smooth consumption over time for an individual investor. However, there are many other factors that must be considered when discounting the future value of an asset. Consider $10 that will be received in ten years with absolute certainty. To find the present value of that $10, in a situation with no uncertainty, we discount it by the risk free rate. The use of a risk free rate is the idea that if we invested in some perfectly riskless asset, how much money would we need to invest now to receive $10 in ten years from now. US debt has traditionally been used to define the risk free rate. Using the risk free rate would be inappropriate for an asset with risk.

To understand why this is required imagine discounting the expected payoff of Apple in 10 years by the risk-free rate and its risk-adjusted rate. The former would suggest a higher current stock value than the latter. This makes sense intuitively because if the gains of Apple had no risk you would be willing to pay more money since the payoff is certain. Market participants, in equilibrium, would bid the price of Apple up until it offered only the risk free rate. However, since Apple is risky, we demand a cheaper price to compensate us for the fact that despite Apple having done well so far, might fall out of fashion. A cheaper price for Apple would de facto result in a higher return. This is because you are able to pay less money for the same fraction of Apple’s cash-flow. The correlation between asset-specific payoff risks and the general investors discount factor generates asset-specific risk corrections.

 

Part Two:
Efficient Markets

Coming Soon

 

 

Thanks for reading!

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