Economics 434 – Financial Market Theory Tuesday, August 25, 2009 Tuesday, August 24, 2010Tuesday, September 21, 2010Thursday, October 7, 2010 Economics 434 Theory of Financial Markets Professor Edwin T Burton Economics Department The University of Virginia

Economics 434 – Financial Market Theory Tuesday, August 25, 2009 Tuesday, August 24, 2010Tuesday, September 21, 2010Thursday, October 7, 2010 Administrative Exam LNEC Note-Taker

SECTION II MODERN PORTFOLIO THEORY

Economics 434 – Financial Market Theory Tuesday, August 25, 2009 Tuesday, August 24, 2010Tuesday, September 21, 2010Thursday, October 7, 2010 Introduction Modern Portfolio Theory To start, what is portfolio theory? It is the theory of how investors choose what assets to purchase out of the universe of assets available. In other words, it aims to explain how rational investors make the resource allocation decisions that they do.

Economics 434 – Financial Market Theory Tuesday, August 25, 2009 Tuesday, August 24, 2010Tuesday, September 21, 2010Thursday, October 7, 2010 Introduction Here is a hypothetical allocation of an investor’s financial resources: This raises the question – what is driving this investor to allocate his or her funds in this exact manner?

Economics 434 – Financial Market Theory Tuesday, August 25, 2009 Tuesday, August 24, 2010Tuesday, September 21, 2010Thursday, October 7, 2010 Introduction Secondly, here is a trend one might observe by tracking an investor over time: Age 50Age 60Age 70

Economics 434 – Financial Market Theory Tuesday, August 25, 2009 Tuesday, August 24, 2010Tuesday, September 21, 2010Thursday, October 7, 2010 Introduction How exactly is the investor re-optimizing the allocation of his or her portfolio over time? And what about investment in other assets, like cars, real estate, commodities, etc.? Portfolio Theory aims to answer these kinds of questions. But, why is it called “Modern” Portfolio Theory?

Economics 434 – Financial Market Theory Tuesday, August 25, 2009 Tuesday, August 24, 2010Tuesday, September 21, 2010Thursday, October 7, 2010 Evolution - Modern Portfolio Theory Portfolio theory has undergone a big evolution over the past years. Pre-1950s - Assets were segmented into two classes: Real assets – assets with intrinsic value (land, housing, etc.) Financial assets – assets purchased for investment (stocks, bonds, etc.) Stocks were viewed via a present value model of future dividends (John Burr Williams).

Economics 434 – Financial Market Theory Tuesday, August 25, 2009 Tuesday, August 24, 2010Tuesday, September 21, 2010Thursday, October 7, 2010 Evolution - Modern Portfolio Theory Harry Markowitz (1952) Realized the present value model ignored risk Developed the mean-variance view of assets Joined all assets together under one umbrella Introduced the concept of an “Efficient Portfolio” James Tobin (1958) Added a risk-free asset to the Markowitz model Found all investors will purchase a combination of the risk-free asset and one specific risky asset

Economics 434 – Financial Market Theory Tuesday, August 25, 2009 Tuesday, August 24, 2010Tuesday, September 21, 2010Thursday, October 7, 2010 Evolution - Modern Portfolio Theory Bill Sharpe (1964) Applied an equilibrium concept to Tobin’s model Created the Capital Asset Pricing Model (CAPM) Demonstrated the important of covariance (with the market) over variance All three models depend on an understanding of statistics and utility maximization

Economics 434 – Financial Market Theory Tuesday, August 25, 2009 Tuesday, August 24, 2010Tuesday, September 21, 2010Thursday, October 7, 2010 Review - Statistics Random Variable Variable whose value will be realized at a future point in time but is unknown now However, some information is known – the random variable’s “distribution” A Simple Example: Consider a game that pays $100 if a (fair) coin lands heads but costs $60 if the coin lands tails. What would you make on average?

Economics 434 – Financial Market Theory Tuesday, August 25, 2009 Tuesday, August 24, 2010Tuesday, September 21, 2010Thursday, October 7, 2010 Review - Statistics Intuitively, if you play the game twice, on average you’ll win $100 once and lose $60 once – so you’d win $40 on net for two plays. Hence, the average winnings per play should be $20. A summary of the game structure: EventOutcomeProbability Heads$100½ Tails-$60½ This is called the payoff’s “distribution.”

Economics 434 – Financial Market Theory Tuesday, August 25, 2009 Tuesday, August 24, 2010Tuesday, September 21, 2010Thursday, October 7, 2010 Review - Statistics From this, we can see a way to calculate the average payoff: Average = $100 ¢ ½ + (-$60) ¢ ½ = $50 - $30 = $20 EventOutcomeProbability Heads$100½ Tails-$60½ This is called the distribution’s “mean,” also known as its “expected value.”

Economics 434 – Financial Market Theory Tuesday, August 25, 2009 Tuesday, August 24, 2010Tuesday, September 21, 2010Thursday, October 7, 2010 Review - Statistics What if the coin is weighted so that heads only comes up 30% of the time? Mean = $100 ¢ 30% + (-$60) ¢ 70% = $30 - $42 = (-$12) The mean payoff is now (-$12) – i.e., now on average we lose $12. EventOutcomeProbability Heads$10030% Tails-$6070%

Economics 434 – Financial Market Theory Tuesday, August 25, 2009 Tuesday, August 24, 2010Tuesday, September 21, 2010Thursday, October 7, 2010 Review - Statistics From this, we can write a general equation for how to calculate the mean. For a random variable with: n different potential outcomes, x 1, x 2, x 3, …, x n-1, x n, Each with associated probabilities p 1, p 2, p 3, …, p n-1, p n ;  p i = 1 The mean of the random variable is: In other words, you simply multiply each outcome with its probability and sum up the results. ¹ =  p i ¢ x i i = 1 n

Economics 434 – Financial Market Theory Tuesday, August 25, 2009 Tuesday, August 24, 2010Tuesday, September 21, 2010Thursday, October 7, 2010 Review - Statistics Compare our first version of the game with one in which you make $20 with certainty; i.e.: This also has a mean payoff of $20. However, this game is very different – the distribution of the original version of the game is much more spread out. We capture this with a metric called “standard deviation.” EventOutcomeProbability Heads$20½ Tails$20½

Economics 434 – Financial Market Theory Tuesday, August 25, 2009 Tuesday, August 24, 2010Tuesday, September 21, 2010Thursday, October 7, 2010 Review - Statistics From the examples, we can write a general equation for how to calculate the standard deviation. For a random variable with: n different potential outcomes, x 1, x 2, x 3, …, x n-1, x n, Each with associated probabilities p 1, p 2, p 3, …, p n-1, p n ;  p i = 1 The variance of the random variable is: where ¾ is the distribution’s standard deviation and ¹ x is its mean. ¾ 2 =  p i ¢ [x i - ¹ x ] 2 i = 1 n

Economics 434 – Financial Market Theory Tuesday, August 25, 2009 Tuesday, August 24, 2010Tuesday, September 21, 2010Thursday, October 7, 2010 Review - Statistics Consider another distribution: What are the mean and standard deviation? Using the formulas introduced earlier, we see this distribution has a mean of 20 and a standard deviation of 10. ProbabilityOutcome 10%40 20%30 30%20 40%10

Economics 434 – Financial Market Theory Tuesday, August 25, 2009 Tuesday, August 24, 2010Tuesday, September 21, 2010Thursday, October 7, 2010 Review - Statistics So far, every distribution we’ve looked at so far has been “discrete.” That is, there have been a finite number of outcomes each with positive probability measure. However, distributions do not have to be discrete; they can be “continuous” instead.

Economics 434 – Financial Market Theory Tuesday, August 25, 2009 Tuesday, August 24, 2010Tuesday, September 21, 2010Thursday, October 7, 2010 Review - Statistics Example – the Normal Distribution f(x) x p(x i ) f(x) ¹ =  p i ¢ x i ¹ = s x ¢ f(x) dx ¹x¹x ¾ 2 =  p i ¢ [ x i - ¹ x ] 2 ¾ 2 = s [ x - ¹ x ] 2 ¢ f(x) dx ¾x¾x ¾x¾x

Economics 434 – Financial Market Theory Tuesday, August 25, 2009 Tuesday, August 24, 2010Tuesday, September 21, 2010Thursday, October 7, 2010 Review - Statistics Just as with discrete distributions, for continuous distributions, the mean represents the average and the standard deviation represents how spread out the distribution is. Changes in the mean shift the distribution horizontally, whereas changes in the standard deviation vary its spread.

Economics 434 – Financial Market Theory Tuesday, August 25, 2009 Tuesday, August 24, 2010Tuesday, September 21, 2010Thursday, October 7, 2010 Review - Statistics Increasing the mean shifts the distribution rightwards; increasing the st. dev. makes it more spread out… f(x) x Higher mean & higher standard deviation

Economics 434 – Financial Market Theory Tuesday, August 25, 2009 Tuesday, August 24, 2010Tuesday, September 21, 2010Thursday, October 7, 2010 Review - Statistics …whereas decreasing the mean and st. dev. does the opposite. f(x) x Higher mean & higher standard deviation Lower mean & lower standard deviation

Economics 434 – Financial Market Theory Tuesday, August 25, 2009 Tuesday, August 24, 2010Tuesday, September 21, 2010Thursday, October 7, 2010 Review - Statistics Why did we spend all this time on distributions? This was Harry Markowitz’s view of what defines an asset – the probability distribution describing its return. Markowitz characterized each asset’s probability distribution with two metrics representing its return: Its mean, representing its expected return, and Its standard deviation, representing the spread of the asset’s return, or its risk.

Economics 434 – Financial Market Theory Tuesday, August 25, 2009 Tuesday, August 24, 2010Tuesday, September 21, 2010Thursday, October 7, 2010 Review - Statistics In other words, to Markowitz, the entire universe of assets could be represented as follows: ¹x¹x ¾x¾x

Economics 434 – Financial Market Theory Tuesday, August 25, 2009 Tuesday, August 24, 2010Tuesday, September 21, 2010Thursday, October 7, 2010 Review - Statistics Every available asset is represented by a single point on this diagram. ¹x¹x ¾x¾x Asset 1 Asset 3 Asset 2 Asset 4 Asset 6 Asset 7 Asset 5 Asset 8 Asset 9 Asset 10

Economics 434 – Financial Market Theory Tuesday, August 25, 2009 Tuesday, August 24, 2010Tuesday, September 21, 2010Thursday, October 7, 2010 Review - Statistics Which asset on the previous diagram would an investor most likely purchase? Asset 7 has the highest return and lowest standard deviation – Markowitz says the investor would purchase that one over the others. What if Asset 7 did not exist? This is where utility maximization comes into play.