FINA 4321 Notes (1) PDF

Course Overview “Modern” Portfolio Theory ○ Mean-Variance Portfolio ○ CAPM ○ These should be a lot of review! Application; Relevant to Careers ○ Performance Evaluation ○ Investment Policy ○ Indexes ○ Personal Investing The Parad...

Course Overview “Modern” Portfolio Theory ○ Mean-Variance Portfolio ○ CAPM ○ These should be a lot of review! Application; Relevant to Careers ○ Performance Evaluation ○ Investment Policy ○ Indexes ○ Personal Investing The Paradox of Finance A large portion of this course will focus on the fact that markets are efficient However, markets are efficient largely because people are compensated for making them efficient If markets were efficient, people wouldn’t make money by making them efficient (they would already be efficient) Some of the unease with this material is natural! Statistics in Finance; Basics Investment returns are viewed as random, with variance in the average return ○ We have an expectation of what that return and variance is going to be, but we are not perfect at predicting the future Our job as investors is to choose the best portfolio ○ This does not necessarily mean the highest return! We shall use risk and return definitions to help us select securities and create the best portfolio Risk and Return Descriptive statistics usually refer to annualized values ○ We may collect daily, monthly, etc., returns and use those, too ○ Bond markets almost always annual yields Don’t get me started on bond people For securities, we try to estimate returns: ○ Expected return ○ Expected variance (standard deviation) ○ Expected correlation (covariance) This requires knowing returns and information about other securities! Sometimes, other expected values, too! Expected Values Expected values are the mean of a random variable that we are interested in ○ Return, variance These expected values typically are heavily based on historical returns  ○ Uh oh!! We are going to try to predict the future ○ Use a model which has some predictive power ○ Those are our expectations ○ Is the future hard to predict?......... accurately? Expected Value Example: 50% chance of $1, 50% chance of $0 What is the expected value of this event? ○ $1 * 0.50 + $0 * 0.50 = $0.50 + $0.00 = $0.50 Returns Expected returns are typically the arithmetic mean return Investors experience the geometric mean ○ Geometric Mean ≈ Arithmetic mean – ½ * variance This says that over time, variance eats away at your return. Can experience negative returns, even if expected mean is positive From a portfolio’s perspective, can imply that a negative expected return asset may increase portfolio returns, if it reduces variance enough! Returns! Determining performance is not as straight forward as it may seem. ○ Not always END / BEGINNING – 1 ○ Some real nerds use log returns (log(end/beginning)) Do we care about the absolute dollar amount gained, or how much each dollar itself could gain? ○ Time-weighted return ○ Dollar-weighted return Time-Weighted Return Reflects a ‘buy and hold’ strategy ○ How much does $1 invested at beginning of a period earn per period? Geometric mean ○ Important to use when invested amount changes due to cash distributions AND those cashflows are outside of your control ○ My personal favorite Calculation Steps Calculate the period returns each time a cashflow happens ○ This makes sure investment cashflows do not affect the return Annualize ○ This portion may be difficult, since cashflows may not come at ‘nice’ periods. ○ It may be easier to break down into days Dollar-Weighted Return Internal Rate of Return How much does each dollar earn, on average? Useful when we have control over cashflows, and understanding that further investments are necessary ○ Typically a good idea for levered investment funds, private equity, real estate Imagine $1 earning 100%, you investing an additional $10, but then only earning 2% on that additional amount.  ○ Dilution, etc. Calculating We must include all cashflows along the way Set up a timeline Use IRR ○ Difficult to calculate algebraically (Good luck, Seriously it’s hard to do) 10 annual periods Initial investment of $1000 Additional investment of $1000 in year 4 (end of 3) Determine the IRR for the given period ○ Assemble a timeline ○ Result is a dollar-weighted return (IRR) of positive 0.061% Currency Sometimes you will purchase securities denominated in a different currency There are two different types of returns you can have: hedged and unhedged Think about what happens if you buy a stock with euros: you have sold dollars, and bought euros, then used those euros to buy the stock. At the end of this, you will do the opposite. ○ You are implicitly long euros and short dollars. Exchange variation will have an effect on your returns. ○ If dollar appreciates, this makes you sad. More expensive to buy dollars back! Pre-tax and After-tax returns  Tax situations vary by investor, and it’s important to be able to compare returns! Imagine comparing a tax-exempt investment to a non-exempt one. You’d likely tolerate lower returns from the exempt one – up to the point that it is equal to other one after taxes! Typical Taxes Taxes on cashflows ○ Dividends and interest often are taxed. No choice on timing! Taxes on capital gains ○ These taxes happen after you sell the asset. Choice on timing! Notice: the cashflows result in taxes paid now! ○ Do you want to pay taxes now, or later? ○ Investments with same pre-tax return may be different post-tax! ○ Capital gains are preferred, since taxes are deferred Leveraged Returns Leverage is common in investing – borrow money to buy assets Returns are the calculated the same way ○ Need to remember to include cost of borrowing ○ And if that cost is tax-deductible Real Estate Just like Bonds or Stocks – there are two components to return: Cashflow and Capital Gains Assume you purchase an apartment for $250k to rent out. Use $50k cash to do this. 30y@7% ○ Cost ($250,000) ○ Loan $200,000 ○ Equity ($50,000) ○ Rate 7% ○ NPER 30 ○ PMT ($16,117) Estimate the Cashflows Just like stocks or bonds, we want to know the cashflows Use your financial modeling skills! Remember your accounting Total Return (Leveraged) Despite having negative cashflows, this is a positive return! Notice the change in value was only 2.7% on the apartment, and the cashflow was negative ○ But 11.0% return on our initial investment of $50,000 ○ You would want to create multi-year projection, do IRR Transacting in Financial Markets Through your broker, you may place orders to buy, sell, short sell, or buy to cover ○ Derivatives often have different terms Your broker will route your order to exchanges, or to market makers Your broker is responsible to make sure you get the ‘best price’  ○ Must be at least the National Best Bid Offer Regulations on how brokers handle orders Order Types Many ways to send an order to a broker, giving them instructions on how you would like the trade executed Popular: Market, Limit ○ Market orders are transacted immediately at the market price ○ Limit orders transact only if the price is no worse than the limit you set Bid/Ask = Bid/Offer ○ Price people are willing to buy/sell at. These are the ‘market prices’ Statistics and Measurements Measuring and Quantifying It is useful to have information and data about your investments! ○ We can quantify a lot of things which are useful for us, but we need to be able to measure these things consistently A lot of nuance in finance, and we can be misleading with just about any number. ○ We saw this with returns ○ How much money does that account have??? Not as simple as you might expect! Margin may cloud our figures. Cash or Credit? Brokerage firms typically offer two types of accounts: cash or margin (Reg T) With cash accounts, you must* have funds settled in the account to make a trade Margin accounts allow you to use the securities in your account as collateral on a loan ○ Much higher risk, and there are more requirements and regulations around margin accounts ○ This is called hypothecation Margin, Requirements Folks did not have a good time in the 1920s, so regulations have become tighter (by existing) Regulation T requires brokers to set margin limits, such as Initial Margin and Maintenance Margin Very sophisticated investors are able to do secondary calculations beyond this, which often lead to looser margin requirements ○ “Portfolio Margin” Margin Example You wish to buy 100 shares of NKE, but do not have $9,406.00 With margin, you only need 50% equity  ○ 9406.00 * 0.50 = $4703.00 in funds; $4703.00 borrowed from broker Always think of this from an asset and liability perspective. You have a liability of that loan amount. Rarely does that change. The asset, the stock, changes all the time! So, too, does the equity you have. Uh oh! What if the price goes down? Maintenance margin is 25% ○ Like initial margin, brokers may set that amount higher, and different securities may be higher or lower than that ○ Must always be above that amount. ○ Once you dip below it, must* reach the 50% mark again *Sometimes brokers allow you to just go to maintenance margin amount, instead of the larger initial margin. May depend on the security type Whoops Ok so the price of NKE went down It’s now $75.37. A loss of 19.9%. What about for you? Are we in trouble if our broker sets a 35% maintenance margin? 40%? 4703 liability – does not change! 7537 asset 2834 equity Top off the tank If we had a 40% maintenance margin, we would violate this Must now bring it up to 50% -- meaning we must have equity of 7537 * 0.50 = 3768.5 We currently have equity of 2834 Must find $934.50 ○ Can I just sell shares to do this? How many? Remember your accounting!!! Does the liability change? How else can I fulfill this? Portfolio Management Our jobs as portfolio managers is to fulfill specific client needs relating to risk, return, time, and other characteristics Understanding how markets have behaved in the past is often a good indicator of the future. ○ Not always! Build an intuition for market and asset class behaviors Stock Markets: Perspective Cumulative US Capitals Market Returns Basic Risk Measures We see that margin can cause big swings in portfolio value! Measuring the size of these swings, and putting in rules for our portfolio’s risk may be useful Risk measures could safeguard a portfolio from losing too much money, failing to meet investment goals. Variance and Standard Deviation Mathematically, it’s how much zigging and zagging a time series does (in relation to its mean). Variance is the square of Standard Deviation Standard Deviation is used more extensively in Finance. Lowercase sigma used. Covariance, Correlation Typically in finance, it’s how much zigging and zagging a time series does with another time series. Joint variability of two random variables. ***UNDERSTAND RELATIONSHIP BETWEEN COVARIANCE CORRELATION AND VARIANCE/STD Regressions Let’s draw the ‘best’ line through a bunch of dots Drawing lines on graphs is called Economics  Drawing lines through dots is called Research Beta We will revisit later in the class Beta = Covariance(Stock, Index) / Variance(Index) Of returns! ○ Over a specified interval, with specified periodicity E.g. 3 year, weekly ○ CHANGE IN VALUE AS PERCENT! Beta is simply a regression value! It is the slope of the company’s returns (y) vs. the index (x). Draw the line through the dots!!! Example Calculations Excel file on Canvas Calculate these descriptive statistics on MSFT, TGT, XEL, SPY ○ Standard Deviation, Correlations Can you calculate the Betas, too? ○ Remember our old friend, y=mx + b Beta is the m alpha is b The ‘real’ way to do this would involve excess returns Generalization about risk We say they have a standard deviation in the returns Stocks, other securities, move in random walk ○ They have a ‘drift’ higher (mean) and will move around randomly (variance) along the way We ‘know’ the destination, but not the path Most consider ‘risk’ as losing money ○ Note: variance does not care which direction from the mean Risk is just LOSING MONEY!!! ??? Securities will zig and zag, and we will discuss how they do not do that in equal amounts, nor in same behavior Can we measure standard deviation, but only for the bottom returns? Risk Measurement Understanding the risks of your portfolio is usually the #1 job as a portfolio manager You cannot simply use one or two risk measures and call it ‘good’ Know the weaknesses and strengths of each measurement! Know how to calculate it! Understand the market in which you are operating. Brownian Motion Now, some particle physics This type of motion was first described after observing pollen in water ○ Little bits of matter will randomly move around in a stochastic process ○ Asset prices behave very similarly! We can model this behavior to help quantify the risk of our portfolios ○ Einstein and other folks did some stuff on this in the 1900s Make your own! =norm.inv ○ Inputs include the probability/percentile (use rand()), mean, and standard deviation This is Brownian motion also known as “Random Walk” Drift Asset prices go up (we hope!) If we just modeled randomly going up or down, we shouldn’t expect a positive return on assets We add a drift, or direction, that biases the random walk to go higher This is the mean of returns Jumps Stocks do not behave exactly in the random walk way ○ We observe ‘jumps’. Big news information releases often lead to huge jumps in assets – much more than the random walk can predict. These jumps can be negative ○ The random walk implies a normal distribution, with no jumps  Assets do not behave this way, though it is useful to start from this foundation Kurtosis Not a disease Fancy statistics measure of how ‘normal’ a distribution is, or how fat the tails are (how likely events are to occur that are very far from the mean) We find that assets have excess kurtosis, meaning that seemingly rare events occur more often than expected in a normal distribution Skew Not only do assets show non-normal tails and distribution, we also tend to see that stocks exhibit skew ○ The tails are not equal! They are skewed. Stocks tend to have positive skew, meaning there are fat tails in the positive outcomes (+500%, +2000%, etc.) ○ *Can only lose -100% !!! Skewness is roughly defined with the location of the mean in relation to the median Why Do We Care Our analyses are pretty good, but miss actual real world complications Important to recognize our constraints and where our assumptions simplify too much Be aware of this throughout finance ○ Investment are rarely normally distributed. Kurtosis and skew Utility, Risk Aversion Quick Recap Understand different risk measures We face tradeoffs in investment ○ Higher returns imply higher risk Balance risk vs reward Try to estimate what that risk and reward is going to be; ex ante. ○ Predict the future! Risk vs Reward When we think about corporate finance, we often think about projects and whether or not they have positive NPV ○ If positive, do project This is a basic and useful strategy ○ Has weakness: assumes an infinite budget In reality, we face variance in the outcome, and could lose money, or ultimately go bankrupt due to chance.  ○ Discount rate attempts to fix some of this Efficient Frontier This should look familiar! We can graph our risk vs. return At any point along the curve, we have an ‘efficient’ portfolio ○ No portfolio achieves more return for that amount of risk ○ We prefer all portfolios directly to the left, or directly above ‘dominate’ This portfolio is typically constructed with all risky assets If we can determine the amount of risk we are willing to tolerate, we should know the appropriate portfolio Efficient Frontier This tradeoff means that the higher our risk, we should expect more return. We want the highest ratio of the two we can get. Wanna Play a Game? Fair coin: coin with a :) on one side, :( on the other side Stake starts at $2. You win what the stake is when the game ends. The game ends when a :( is flipped. The stake doubles when a :) is flipped, game continues. Expected Value of the game? ○ Infinity What would you pay to play the game? Creating a Portfolio of Bets Our job as a portfolio manager is to also maximize our geometric* return ○ While maintaining certain risk budgets We face tradeoffs when selecting securities ○ Risk vs reward We do not simply put all of our budget into a single project because its NPV is positive! ○ This is applicable beyond securities Asset Class Selection There are many risky assets which we can invest in, and all are available to us (in this class...) ○ Stocks, corporate bonds, mortgages, Pokemon cards, etc. Some combination of these assets will result in the best risk vs return profile for us ○ We may reduce risk by including risk-free assets in the mix ○ Something must be the best, and it might as well be your portfolio! Investing in assets which have lower expected returns may result in higher portfolio returns! Consider your asset class risk! Corporate bonds typically will only earn the interest payment. Nothing exciting! ○ The exciting part is when they don’t pay the principal. What does this frequency distribution look like? Most stocks lose money over their lifetime. ○ Some go up by 10,000% What does this frequency distribution look like? These are individual security distributions, does it look different when combining them into an asset class? Is the shape of a portfolio different than an individual asset? Stock Returns JP Morgan has a separate study, with the median stock earning -54% excess returns over its lifetime. 40% negative in absolute terms over lifetime. Tradeoffs We like return, but at a certain point, even huge returns cannot be justified if the risk is too great. We want to balance the return with commensurate risk. Understand that putting many different uncorrelated bets in a portfolio is likely the best portfolio ex-ante we can make ○ Ex-ante: before the fact ○ Ex-post: after the fact Non-satiation and Risk Aversion Investors are assumed to display two fundamental characteristics Non-satiation ○ Prefer more terminal wealth to less ○ Given portfolios with equal standard deviations*, will always select the one with the highest expected return Risk aversion ○ Will not accept a fair bet (e.g., flipping a coin to win or lose $1) ○ The satisfaction of earning an extra dollar is less than the dissatisfaction of losing an extra dollar ○ Given two portfolios with equal expected returns, will always select the one with the lowest standard deviation Utility Economics concept that gives those researchers something to do We can* calculate the amount of ‘utility’ someone receives for giving up something in return, like money or risk ○ *We can’t, but economists try For an individual person, we model how much return they need for the risk they take Utility of Wealth Utility = satisfaction We are interested in the utility associated with wealth A utility of wealth function expresses the investor’s level of satisfaction produced by various levels of wealth The shape of the utility of wealth function is determined by the characteristics of non-satiation and risk aversion ○ Because of the non-satiation characteristic, the function is upward sloping ○ Because of the risk averse characteristic, the function is concave (that is, bowed downward) But also because we don’t experience satisfaction in a linear fashion. Indifference Curve – Beer and Pizza Each combination of beer(in ounces!) and pizza rolls provides the same level of satisfaction. Bring curve down from upper right to bottom left (less utility) until it meets budget constraint. Upper right is more utility, bottom left is less. Loss Aversion, Risk Tolerance Humans are loss averse! We do not take losing well. ○ It’s why nobody will play board games with you anymore ○ Most call this risk aversion; studies show we aren’t averse to risk, just losing. Some people tend to tolerate or embrace risk more/better than others ○ Risk tolerance may be cumulative. Folks who take a lot of risk don’t feel it anymore. Dangerous! How much risk you should take is related to many things: mostly age, financial condition Utility  Utility in our case, means wealth at the end of our period. Investors want as much of the stuff as they can get! Thought process: You start with $500,000; does losing $250,000 hurt more than gaining? Do we have a linear relationship with wealth? Is $10 billion better than $9 billion by the same amount as $1 billion and $0 billion? Since we are decision making, we often focus on MARGINAL utility. ○ This is an important concept to remember in any financial decision! Use this everywhere. Risk Aversion, Utility Certainty Equivalents and Risk Premiums A risk averse investor will always choose a certain (guaranteed) outcome over an uncertain outcome with equal expected value  ○ The investor’s utility is greater for the guaranteed alternative than the risky alternative Put another way, a risk averse investor will accept a lower certain outcome instead of a risky outcome with the same expected value ○ The certain outcome that makes the investor indifferent between it and the risky outcome is the certainty equivalent ○ The amount by which the expected value of the risk outcome must exceed the certain outcome to make the investor indifferent is the risk premium Putting it together Certainty Equivalent of Wealth Evidence Investors Are Risk Averse Expected Return VS Standard Deviation Up and to the left is more utility Down and to the right is less utility Everywhere along the curve is the same utility Be careful assuming this direction, as we often rearrange the formula! This particular person’s utility graph slopes upwards. ○ What does this mean? ○ Indifference Curve How does the value of “A”, their Risk Aversion, affect the shape of this line? What would a downward sloping curve imply about the preferences of the individual? Do you think I’d ask these sorts of questions on a quiz or exam? Let’s do some Economics! As you may recall, drawing lines on graphs is called Economics We may construct many portfolios (Red Line) of differing risk Omg it touches, Nobel Prize pls Imagine different values of A That Green Circle ○ Seems pretty important ○ Maximizes our utility ○ Risk-reward payoff is best we can hope for – More risk results in less reward than we want, given risk – Less risk results in less reward than we want, given risk Diversification Real World Models Sophisticated quantitative models may predict the risk vs. return (expected values) of individual stocks, as well as their relationships (covariance) with others Using these basic models in the real world often leads to a bad time ○ Yada yada yada, linear algebra, multicollinearity, inversion matrix ○ The assumptions we make don’t hold in the real world! Our foundational learnings and intuition will still hold, and provide meaningful ways to think about the world ○ These are methods for how to think, not necessarily do US Capital Markets Returns 1926 Correlation Matrix  Diversification is important! Two Risky Assets Return of the portfolio: weighted average Risk (variance) of the portfolio: more math than that! ○ Consider the fact that the two assets may not move in same direction ○ If correlation is not equal to 1.0, there are diversification gains Variance will be less than weighted average! We are able to reduce risk without affecting return. This is the good stuff. Efficient Set Theorem An investor will select his or her portfolio from the set of portfolios that: ○ Offer maximum expected return for varying levels of risk AND ○ Offer minimum risk for varying levels of return The set of portfolios that meets these two conditions is called the efficient set (aka efficient frontier) Because investors exhibit non-satiation and risk aversion, they will want to be as far to the NW as possible ○ So portfolios will necessarily reside on the NW of the feasible set Efficient Set and the Feasible Optimal Portfolio and Risk Aversion Variance As Risk Variance (risk) in a portfolio can be reduced if the correlation is less than 1.  Better risk vs. return? Standard Deviation Portfolio Variance with 2 Assets Final term is covariance, which can also be expressed as: X represents the weight of the asset Bill Hwang Econ and MBA degrees from Carnegie Mellon Worked at a few hedge funds, did well. ○ “Tiger Cub” ○ Some of his former colleagues own NFL teams Started own fund for his wealth in early 2010s. ○ Archegos Diversification is boring though Bill does not like diversifying his portfolio Bill does not believe in risk reduction through non- correlated assets Bill does like leverage, though. Bill is known to be a successful investor and has banks which will give him leverage: ○ Credit Suisse, Nomura, Morgan Stanley, UBS, Mitsubishi, Goldman The Holdings/The Fraud Much of Bill’s wealth was in the stocks of consumer media and tech-related companies. In some cases, Bill Hwang’s trading activity (and related bank hedging) accounted for nearly all trades in some thinly-traded stocks. Banks gave Bill lots of leverage through asset swaps. ○ As Bill’s holdings got larger, it meant he wasn’t going to need to file with the SEC on his outsized holdings Bill lied to the banks about his holdings, borrowings, and collateral. ○ You’re not supposed to lie to banks Bill is going to jail. The Fallout The SEC wants a rule for funds which use asset swaps or other means for large holdings, to report their holdings. Prime brokers (those banks) will likely scrutinize their clients more carefully before exposing to large margin loans. Some of these banks lost billions of dollars. ○ Goldman Sachs, despite being one of the lenders, did not lose. ○ Archegos accidentally sent $470MM to Goldman, instead of asking for $470MM Bill is going to jail. Let’s Beat Bill at Portfolio Management Use the diversification_STOCKS.csv and calculate the standard deviation of the individual stocks. Also estimate the standard deviation of a portfolio of these stocks. Do the same thing with diversification.csv A portfolio with these two ETFs would be pretty good! N-risky Assets Ooo yeah math time ○ Double Sigma ○ Linear Algebra could be fun if you’re into it Can do some handwaving to separate >>>>>>>>> Linear algebra to the rescue! Two asset portfolio with linear algebra Diversification – The Math The total risk of a portfolio can be decomposed as follows: The systematic and unique risk can be further broken down into: Diversified Portfolio? When we have more and more assets in our portfolio, the complexity of our calculation increases exponentially Notice we must take covariance of each combination of assets ○ N2-N terms to calculate Feasible for 5000 asset portfolio? Diversification Let us assume an equal-weighted portfolio of N assets: weight = 1/N ○ This just makes math a little easier, doesn’t really affect much  Variance of portfolio: Diversification Wonderful part of investing! We show that if your portfolio has non-correlated assets, it will improve your risk-reward! This is most important concept in portfolio management! Consider what happens as number of assets gets really large? Types of Risk We can broadly classify risk into two types of categories: Systemic risk ○ Risk related to economic/business cycle. Everyone has some of this Idiosyncratic risk ○ Specific risks to the business itself What can we relate to our equation? Diversification – The Intuition The total risk (standard deviation) of a security can be separated into two parts: ○ Systematic (common, market, or non-diversifiable) risk ○ Unique (independent, unsystematic, specific, or diversifiable) risk When securities are combined into portfolios, the total risk of the portfolio can be similarly separated Adding more securities to the portfolio has little impact on the market risk of the portfolio ○ The factors impacting all securities will still impact the portfolio regardless of how many securities are held in it Adding more securities to the portfolio reduces unique risk ○ With enough securities, the factors impacting individual securities tend to offset each other Impact of Adding Securities on Risk As N Approaches Infinity... We find that only covariance should be the determinant of return for an asset, given all participants act rationally ○ Covariance is related to which other risk measure? With a little more imagination and handwaving (assumptions), everyone should hold ALL risky assets! ○ This portfolio will have the best risk-return profile It’s a Free Lunch If all market participants are rational, we would assume everyone wants to maximize their Mean-Variance tradeoff Investors are able to reduce their portfolio Variance by investing in assets which diversify their portfolio ○ This means buying all assets! Points of Clarification Buying all assets at their current values, with weights based on total value, is a ‘market portfolio’ We will discuss later in class, but this is very similar to the S&P 500 Index Risk is defined as our portfolio’s standard deviation (variance) ○ Individual stocks should not be measured in this way Many individual stocks held together achieve diversification and the portfolio’s risk goes down Only the covariance with the market matters! Globally Optimal Portfolio, Market Portfolio, and Capital Allocation Line Globally Optimal Portfolio (MVEP) All investors want to maximize their return for the risk they take ○ We assume they are savvy and can take advantage of mispricings This is mostly true, especially when you ‘dollar weight’ the investors! We are in equilibrium There must exist a portfolio of assets which gives us best risk-reward Equilibrium If an investment doesn’t have good risk-return profile (not enough return for its covariance), it will be shorted, price will fall, and expected return will increase until equilibrium This means that we assume the ‘market portfolio’ is the most efficient portfolio ○ Market participants have determined the equilibrium price Important to keep this in mind! Should We Give Up? If we are in equilibrium, there is no value in deviating from the market portfolio ○ No analysis will benefit the investor, as all prices are deemed fair ○ Deviating from the market portfolio will reduce the risk- return profile of the investment portfolio Are we in equilibrium? ○ If not, how far away? What Evidence? Tons of time ‘spent’ trying to show how the Efficient Markets Hypothesis is not correct, or has validity ○ I’ll leave this as an exercise for you to find your favorite bit of research Do you believe people act rationally? Do you believe incentives are large enough to take advantage of mispricings? Even if market isn’t perfect, is it ‘good enough’? Individuals vs. Market While individuals might not be rational, market participants as a whole likely are in the medium and long term (dollar-weighted basis) We know that individual stocks tend to have negative returns, with only a few superstars carrying the team ○ This is like Mike Trout and the LA Angels Holding the market portfolio is best way to ensure long-term success ○ How to find the market portfolio? Feasible Set of Risky Assets Portfolios from E to S are said to be efficient ○ No other portfolios are efficient Some call this curve the ‘Markowitz efficient frontier’ Mean Variance Efficient Portfolio (MVEP) We found the efficient frontier of all investment combinations The risk-free rate: this is 0 variance. Treasury. ○ Earn a rate with 100% certainty Draw a tangent line from that point to the efficient frontier ○ This is the MVEP ○ This portfolio has the highest Sharpe Ratio Sharpe Ratio  We can draw a tangent line from the risk free rate to the frontier of all possible combinations ○ Heck yeah, Economics The Star is the MVEP, the portfolio with the highest Sharpe Ratio ○ Market Portfolio Can choose an efficient portfolio with any amount of risk now along red line ○ Capital Allocation Line: The slope of this line is the Sharpe ratio This, but with Math While line drawing is good fun, it may not be practical. We can find this analytically! And we can eventually add in constraints, too. It is useful to think of problems like this from several perspectives, and how we might be able to use tools Nothing but the Best Our definition of the MVEP is that it has the highest Sharpe Ratio. Mathematically, what we are saying is that we want to maximize: ○ One other way to think of this, is the difference between the CML and the Efficient Frontier should equal 0 Set Up, Assumptions Have at it Other Ways to Find the MVEP We know that this point has the highest risk-reward profile to any other portfolio ○ Remember all that talk about markets being in equilibrium? The MVEP is the Market Portfolio! By definition, if we are in equilibrium, the best portfolio to have is the market portfolio All other combinations of assets are suboptimal, as they do not achieve risk-reward of market portfolio ○ Worse diversification Too Much Now That I Found You, I might not want to take this much risk ○ Could use a utility curve to determine how much risk Feels Right Holding anything other than MVEP results in worse risk-return ○ For Sure, we will hold MVEP, but how much? The MVEP will not be holding a Party For One Two Fund Separation Investors should hold two funds, with the weights depending on risk tolerance of the investor: ○ Risky fund ○ Risk-free fund The risky fund is the market portfolio of risky assets, the MVEP They should also hold risk-free assets, if they want to hold less risk than the MVEP CAL (Capital Allocation Line) 100% risk-free point, connected with the 100% risky assets, creates our Capital Allocation Line ○ Because the risk-free has no volatility, it is a straight line connecting the two. No fancy math. An investor’s risk-reward profile is the same, no matter their weights ○ No other combination earns better Borrowing (Margin) What if you want more risk than the market portfolio? ○ Theory says that we include all risk assets in there, which means a lot of loans! In Finance Land, we often like to wave a magic wand and say that you can borrow at the Risk Free Rate! Margin In reality, we cannot borrow at the risk-free rate ○ Because lending to us to buy risky assets is not risk-free! Borrowing at the higher rate means we get a worse Sharpe Ratio (slope of CAL) We also notice that we would invest in a different portfolio! SLOPE = SHARPE RATIO We’re Doing Lines! If we’re accepting no more risk than the MVEP, we will achieve the highest Sharpe Ratio, and we will use the MVEP and Risk-Free If we want lots of risk, we find a different efficient portfolio by drawing a line from the margin rate ○ We invest in that portfolio OVER 100% Between those two portfolios, we may choose along the frontier All Portfolios We now have every efficient portfolio we may want Any investor can determine their optimal portfolio along these lines/curve, given their risk tolerance We can superimpose utility curves on top of this now! Is It The Best? Consider all risky assets: if market participants believe there is a different combination of assets which maximize their risk-return, participants would instead hold a portfolio in that different combination Doing so would affect the prices until we reach equilibrium ○ Then the market portfolio necessarily becomes this new ‘efficient’ portfolio Issues with saying market portfolio is not best? Maybe you don’t like the idea that you can’t beat the market! ○ On average, over time You not being able to beat the market does not mean you are dumb. Active investing (deviating from the market) typically results in worse returns ○ Partially due to what we’ve discussed, but also due to fees! Gotta pay people big bucks to do this,Or your own time, but good luck Issues with all this theory Are investors rational? Frictions? Perfect Information? Arbitrage? Taxes exist  All assets available to buy? Liquidity Expected vs actual Returns not gaussian / normal Margin Underperformance of Active Management Consistently, active fund management underperforms the index In some areas, active isn’t as bad ○ Small cap international ○ Why might that be? Time is not on your side The longer the horizon, the worse off you are in active management. Is this the whole picture? I am showing you returns, is anything else missing from this to complete the whole analysis? Just mutual fund returns ○ Though hedge funds typically don’t perform any differently ○ Individual investors certainly don’t, either! The best case for active management is in esoteric or small markets, where there are unsophisticated investors with little knowledge, and where you can gain an edge CAPM and You! CAPM Capital Asset Pricing Model ○ Harry Markowitz again and others Basically everyone had the same idea in the 1960s Useful for us! ○ Expectations. We have a shiny model now that can give us the expected return of an asset. And we are assigning a risk to it, too! CAPM This is market risk premium Often denoted as 𝜇 I will try to confuse you here (expected market return, expected market risk premium) This Beta is the Beta of market returns ○ We will have other Betas ○ Beta is just the slope (coefficient) of a regressor variable If CAPM Holds... We should expect that the Beta of a stock will be the predictor of returns Securities should return what their risk is ○ In this case, risk is the market beta ○ Error term cancels to 0 in long run ○ Excess is considered alpha. Under CAPM, we say this should be 0 Any securities that do not ‘fall on the line’, are said to be in violation of CAPM ○ If we know this ahead of time, we can create arbitrage profits Asset Pricing Assumptions Assets are infinitely divisible No taxes or transactions costs Unlimited short sales and margin purchases at the riskfree rate Investors are price takers Investors are characterized by risk aversion and non-satiation Investors evaluate portfolios using expected returns and standard deviations All investors operate under the same conditions with respect to: ○ Time horizon ○ Risk free asset ○ Information ○ Expectations regarding expected returns, standard deviations, covariances (homogeneous expectations) “Perfect Markets” Separation Theorem All investors will hold the same tangency portfolio ○ Because all investors agree on estimates for securities’ expected returns, volatilities, and covariances along with a common riskfree asset, the linear efficient set must be the same for all investors ○ This implies one tangency point with the feasible set of risky assets Leads to the separation theorem ○ Regardless of investors risk-return preferences, the optimal risky portfolio will be the same for all investors Investors do differ in terms of their levels of risk aversion ○ Thus have different indifference curves ○ Results in different optimal combinations of the riskfree asset and the optimal risky portfolio One Tangency (Risky) Portfolio, Different Optimal The Market Portfolio The single tangency portfolio of risky assets held by all investors is referred to as the market portfolio ○ Consists of all assets (stocks, bonds, real estate, commodities, cars, boats, human capital, etc.) held in proportion to their relative market values ○ Relative market value of a security is the market value (price units outstanding) of that security divided by the sum of the market values of all securities Prices of securities will adjust so that in equilibrium all securities are held at amounts where the quantity demanded equals the quantity supplied The risk-free rate will adjust so that the amount of riskfree borrowing equals the amount of riskfree lending ○ Ehhhhhhhhhhhhhhhhhhhhhhhhh... sure................. Identifying the true market portfolio is impossible ○ Various proxies are used (S&P 500, Russell 3000, MSCI All-World, FTSE, etc. ○ This is a source of great controversy – more later Meet the New Line. Same as the Old Line. If we have a diversified portfolio, the only risk that matters is the market beta Our expectations is that all securities and their risk/returns will fall on that line ○ Otherwise, we have a violation of CAPM CML Example SML Compared to the CML Which Risk Free Rate? Great Question! Nobody will get too mad at you if you use a 3-month, 12-month, or 10-year Treasury rate! Because we typically are talking about stocks (long- term assets), and typically for longer time horizons, it’s probably slightly better to use 10-year rate Make sure it’s in the currency that you’re analyzing What is the Market Expected Return? Great Question!  When the market is doing well, and people are willing to take risk, we find that the market risk premium shrinks! ○ When people are scared, the risk premium increases! Historically, we see the market return about 10%, and the risk-free rate is about 3%. ○ Is history good? Model Requirements Mean Variance Analysis needs the expected return and standard deviation ○ Portfolio selection required knowing the covariance CAPM is an Equilibrium Model ○ Expectations on what the return should be, given that everyone is behaving rationally, constructing logical portfolios Individual participants are motivated and able to achieve equilibrium by acting rationally Assumptions We assume that investors are not rewarded for taking risk due to variance, but only covariance An individual stock’s movements due to its own unique risks are irrelevant for investment risk compensation ○ Investors are able to reduce portfolio risk by buying many assets which have low (not 1.0) correlation to each other! ○ We assume investors do this, because it’s in their best interests. Market risk is the only risk that should be rewarded, if everyone is acting rationally Risk Compensation This assumption that we only include covariance is what we just talked about! We only care about systematic risk – risk coming from the economy and business cycle Beta measures exactly that – the covariance ○ Beta equals Covariance / Variance (of market) ○ 1.0 being same risk as broad market Market Portfolio Everyone holds all risky assets in the proportion of their value (value-weighted index) ○ This is essentially how the S&P500 is constructed ○ Remember, the CAPM is saying it is all risky assets, not just S&P500: we just use S&P500 as a proxy to this ‘factor’ ○ We have the same assumption with Mean Variance. The market is all risky assets. Equilibrium Pricing Remember, our assumption is that everyone is acting rationally, and will work to find efficient prices of all assets ○ If too high, gets shorted. If too low, gets bought  Moves price to equilibrium If we assume everyone is acting rationally, everyone has constructed same portfolio, found efficient prices ○ This gives us the Mean Variance Efficient Portfolio This is the market! Risk Compensation Market risk is measured as the covariance of an asset with the market ○ We use the S&P500, but this should really be all risk assets For a given asset’s amount of market risk, we expect it to return a specific return over the risk-free rate ○ Security Market Line Equilibrium Condition Risk Reward Match We want more of good stuff, less of bad! Sell bad, buy good. As this happens, we should observe the securities come into equilibrium with their Sharpe Ratio: Rearrange (Light Math) Portfolio Beta Risk Examples Why are some stocks higher Beta than others? ○ Stocks with higher beta values tend to be more volatile, are often in riskier sectors, may have higher financial leverage, and are usually smaller companies that are more sensitive to market conditions. Does Beta change over time? ○ Yes, beta can change over time due to factors such as shifts in a company's business, market conditions, or the time period used in the calculation. How is Beta Calculated?  ○ Beta is calculated by dividing the covariance between a stock's return and the market return by the variance of the market return, typically using regression analysis. Differences in Industries Utilities and Consumer Staples tend to have Betas under 1.0 ○ Makes sense! These businesses don’t drastically change in recession As an owner of a portfolio, you also have personal risks – income! You may lose your job. ○ This is related to the economy, typically. You likely want to diversify these risks to smooth your wealth ○ Low covariance stocks are awesome for you! Awesome for everyone else, too. Because of this, people pay more for them, they have lower expected returns: Low Beta! Intuition High beta stocks are risky, and must therefore offer a higher return on average to compensate for the risk ○ Why are high beta stocks risky?... because they pay up just when you need the money least, i.e. when the overall market is doing well Thus if anyone is to hold this security, it must offer a higher expected return (i.e. have a low price). You only get compensation for covariance, NOT variance. ○ E.g. if an assets has huge variance but zero covariance with the market: CAPM says you get expected return = risk-free rate, i.e. no compensation for variance If you ONLY hold this security, you’re in trouble But investors hold diversified portfolios, so you don’t need to worry about the variance of this security: You don’t need compensation! Criticisms of the CAPM Unrealistic assumptions ○ Perfect markets, homogeneous expectations Inability to identify the market portfolio ○ Small changes in proxies for the market portfolio can have large impacts on relative return rankings Failure to recognize other sources of systematic risk ○ Perhaps factors like Value, Size, Momentum or others are priced... Empirical tests provide little support Portfolio managers rarely use it in its pure form ○ Probably the most condemning criticism  Yet the concept of beta and its relationship to expected returns are strong and enduring Applications The CAPM can be used to obtain one of the key inputs (expected returns) necessary to perform a mean-variance portfolio choice Also, it can be used to guide portfolio decisions: the CAPM provides a measure of the “fair price” of a security and thus we can use it find underpriced and overpriced securities in the market. Finally, it can be used to provide a benchmark for portfolio performance evaluation: Given the level of risk of a particular fund, is the fund doing better (i.e. obtaining higher expected returns) than the expected return predicted by the CAPM? CAPM Evidence, Other Models CAPM We attempt to model an individual asset’s returns by using a linear regression ○ We estimate Beta for the stock, and use the market’s returns as the multiple of that This is pretty simple! Also, somewhat effective! One-Factor Model Sample Regression Plot Key One-Factor Model Relationships General form of the factor model relationship Expected return  Variance Covariance Diversification One More Time Once again, we can partition the variance of a portfolio into diversifiable and non-diversifiable elements Systematic (factor) risk is averaged through diversification Unique (non-factor) risk becomes small through diversification Investors will not be compensated for taking on non-factor risk ○ It can cheaply be eliminated Only factor risk will be compensated Regression Recap and Deep Dive When we run a regression, the slope of our line is ‘beta’, and our fit is an R-squared value. ○ R-squared is correlation squared It’s best to use variables that you think have explanatory power! ○ Sometimes random variables look like they actually are very good predictors! ○ It is unlikely that these will be good variables outside of sample. Analytics! We want to find the linear estimator of our model: This works, in a way, which minimizes the distance of all points to the line we are regressing to. ○ Minimization and maximization are important mathematical concepts! Covariance divided by Variance!!! R-squared and Adjusted R-squared Each time you add a variable to a regression, you will increase its r-squared. ○ It does not matter what that variable is, or if it has any relationship whatsoever to the system you are trying to model. Predictive analytics provides us with a few tools to avoid this ‘overfitting’ of data: ○ Adjusted r-squared ‘adjusts’ the value to take into account chance. ○ Apply a penalty for the size of the variable: regularization Don’t Overfit! When we add more variables to the regression, we may run the risk of overfitting. We can tell the computer to settle down! Don't use variables if they don't have predictive power beyond basic random chance. ○ The equation needs to 'spend' money on the Beta. If it's not worth it, it won't use that variable. We can tell it how expensive it is. Optimize Regressions seek to find a minimum amount of error. We may impose costs on using variables, 'regularize', to reduce overfitting.  Lambda (λ) is a value used in regression to limit the influence of certain variables, helping to keep the model from being too complicated or overfitted. ○ Many flavors of this in math! Let’s try it out! Understanding Statistical Methods You will have the pleasure of coming across and using many different statistical methods in your lifetime! Make sure you understand the flaws of them, so you are not fooled! Data literacy is incredibly important in finance, and many folks in finance are illiterate. ○ You don’t have to know how to do all of the math, but you do need to understand it. Fama-French (Not the people who made math, but an individual who did math) What’s better than one X value to regress your line onto? ○ A BUNCH OF X VALUES TO REGRESS YOUR LINE ONTO ○ XTREME ECONOMICS When we calculate an asset’s beta, we also include ‘value’ and ‘size’ factors, which are purported to have explanatory power in returns Market Beta We are using returns on the market as a predictor! The returns that we gather here are typically the 'excess' returns - beyond the risk-free rate This is a value X (independent) for what we think predicts our value Y (dependent). We believe the move in market value will affect the move in our stock value. ‘Value’ Factor We can categorize stocks as ‘value’ if they have low price-to-book ratio ○ Be careful, some researchers switch this around and say ‘high book-to-price’ It seems logical that paying less for a company may result in higher returns! ○ Pay less, expect more Value companies may have excess returns not explained by market beta Explanatory Variables Our market risk variable (we assign as market beta) can explain a lot of an asset’s returns! ○ Wonderful! We may find that we can do a better job explaining the returns of an asset if we also include the ‘Value’ factor This acts as an additional variable in our regression Value factor sometimes called High Minus Low (HML) ○ High book/price minus Low book/price ‘Size’ Factor Market beta and value are not the only ways we can explain returns We may find that the size of the company has explanatory value in the return of the stock, too! ○ This factor is often called Small Minus Big (SMB) Small cap companies may have excess returns that are not explained by market beta Size Factor Smaller companies have fewer people looking at their performance ○ This implies the market may not incorporate information as well into the prices ○ Less liquidity and small tend to be riskier! Risk = returns? Fama French 3-Factor Model Use Market Beta, along with value and size factors, to explain security returns It was found that these 3 factors explained asset returns very well ○ They were required to continuously create research for their job, so they added a few extra factors over the years Ooo 5-factor model! 3 Factor Model Use the Factors for market return, size, and value ○ There just weren't enough dimensions! Results in an Alpha  ○ In theory, this alpha should be zero, because we are controlling for all factors of risk and return Could use other factors to regress on! ○ Best to use current estimates of future variables ○ Should make economic sense, like GDP. Explainability is important. Statistical Methods Does any of this actually work? How does this perform out of sample? Individual Estimates Estimates on single stocks tend to show that low beta stocks have higher performance than what is otherwise predicted And high beta have returns below what is expected! CAPM IS DEAD ○ WHY DID WE SPEND SO MUCH TIME ON IT ○ LONG LIVE CAPM Empirical Evidence on the CAPM STRATEGY 1: BETA strategy, Consider the following strategy: ○ At the end of each year, obtain the beta of each stock using a sample of the previous 5 years. ○ Then form 10 portfolios of these stocks based on beta deciles: Portfolio 1 contains the stocks that have betas in the lowest decile (bottom 10%) Portfolio 10 contains the stocks that have the highest betas (betas in the top 10%) Divide portfolio 1 and 10 in half (So in total we get 12 portfolios). ○ Keep these portfolios for a year and compute average annual returns of these portfolios ○ Repeat the process at the end of next year and so forth.  CONCLUSION: The betas of the different portfolios are very DIFFERENT (by construction), but the average returns of these portfolios are almost all the SAME! This is not consistent with the CAPM. Empirical Evidence on the CAPM STRATEGY 2: SIZE strategy ○ Exactly same procedure as before but instead of forming portfolios based on BETA, you form a portfolio based on the SIZE (MARKET CAPITALIZATION) of each stock: At the end of each year, get the closing stock price and calculate firm size (market capitalization) Then form 10 portfolios of these stocks based on size decile: Portfolio 1 contains the stocks that have market capitalization in the lowest decile (bottom 10%) Portfolio 10 contains the stocks that have the highest market capitalization (largest top 10% stocks) Divide only portfolio 1 and 10 in half (So in total we get 12 portfolios). Track returns over next year; repeat Calculate average monthly returns and betas CONCLUSION: Small stocks have higher average returns than large firms. Consistent with the CAPM small stocks have higher betas. BUT, for the very small stocks the average returns are much higher than predicted by its beta! Why do I say this? Let’s look at the alphas. Conclusion: Small stocks have higher CAPM alphas (and statistically significant) than large stocks. That, is even after adjusting for risk via the CAPM, small stocks have earned higher average returns than large stocks. Empirical Evidence on the CAPM STRATEGY 2: Size strategy (cont.) Just for discussion: Is this an anomaly (and so we can profit from it) or is this because we are not measuring risk correctly? The size effect disappeared after 1981! Was the pre-1981 size effect a market anomaly?  ○ Suppose it was. In 1981 investors realized that small stocks were on average underpriced (and hence provide higher returns). Then they bought a lot of small stocks, thereby driving their prices up and lowering their future returns. ○ Suppose it was not. Suppose small stocks were earning average returns pre-1981 because they were riskier. If this is true, what should happen when investors discover the size effect in 1981? Nothing! Small stocks should continue earning their higher average returns because that is a fair compensation for their risk ○ Does the data support the risk story? Not really, the size effect disappeared after it was discovered, but we can never know for sure... But here, the first story (anomaly) seems more plausible. Empirical Evidence on the CAPM STRATEGY 3: Value strategy Some researchers, starting with Graham and Dodd in the late 1930s, noticed that value stocks outperformed growth stocks A value stock is a stock with a low market price relative to the book value of assets ○ Some people believe these stocks are undervalued by the market and thus should present good investment opportunities A growth stock is a stock with a high market price relative to the book value of assets ○ Some people believe these stocks are “glamour” stocks that are overvalued by the market, and as such the expected returns from holding them will be poor. STRATEGY 3: Value strategy (cont.) Implementing the strategy: same procedure as before but now you form portfolios based on the book-to-market of each stock: ○ Look at the firm’s balance sheet and the closing stock price of the firm at the end of each year to calculate firm book-to-market ratio. ○ Form 10 portfolios based on book-to-market deciles ○ Divide portfolio 1 and 10 in half (So in total we get 12 portfolios). But the actual number is irrelevant. Using 10 , 12, 18, or 5 doesn’t matter much, ○ Track each portfolio’s returns over next year; repeat process ○ Calculate average monthly returns and betas of the portfolios What are the average (monthly) returns and the betas of this strategy? CONCLUSION: High Book-to-Market stocks (aka value stocks) have much higher average returns than Low Book-to-Market stocks (aka Growth stocks) but they have almost the same BETA! STRATEGY 3: Value strategy (cont.) What about their alphas? (UPDATED VALUES FOR VALUE STRATEGY 3) CONCLUSION: High Book-to-Market stocks (aka value stocks) have much higher average returns and CAPM alphas than Low Book-to-Market stocks (aka growth stocks). This is sometimes called “The value premium anomaly”  That is, even after adjusting for difference in risk as measured by the CAPM beta, value firms have higher average returns than growth firms. And note that value stocks (that have high returns) have even lower market beta than growth stocks (that have low returns)! => CAPM is rejected STRATEGY 3: Value strategy (cont.) Is the value effect due to risk or is it mispricing? ○ Its not market risk (because we just saw that the CAPM fails), but maybe there is some other risk(s) that we are missing.... Here is a potential risk explanation....: Distress risk: Value stocks tend to be stocks that have underperformed in the past. A lot of them are on the verge of bankruptcy and may be particularly risky. Investors may require an additional risk premium in order to hold them. That would explain why these stocks trade at such a low price relative to fundamentals (book value of assets). Now what if it is mispricing? Investors should profit from it. But the value effect is very strong during most of the 20th and the 21st century, as well as in many other countries. Why didn’t the markets correct this anomaly? Given this, a risk explanation (i.e. value firms are indeed riskier than growth firms) seems more likely since the fact remains even after it was found (early in the 20th century). But we need to model this risk! The CAPM market beta is not good. STRATEGY 2 + 3: Size and Value Effect Together Again, no relationship between average returns and market beta....This doesn’t look like a SML! STRATEGY 4: Momentum strategy Same procedure as before but now form portfolios based on the returns of each stock in the previous 12 months (to capture momentum) ○ At the end of each year, compute the accumulated returns of each stock during the past 12 months. ○ Form 10 portfolios based on these accumulated past returns. Portfolio 1 contains the “losing” stocks (i.e. the bottom 10% of stocks that had the lowest accumulated returns in the past 12 months) ○ Portfolio 10 contains the “winning” stocks (top 10%) Track each portfolio’s returns over next year; repeat process Calculate average monthly returns and betas of the 10 portfolios STRATEGY 4: Momentum strategy What are the average returns and the betas of this strategy? CONCLUSION: Loser stocks, i.e. stocks that had the lowest returns in the past 12 month, tend to also have low returns in the subsequent years (the opposite is true for winner stocks (momentum). But the have SAME beta! STRATEGY 5: Sales growth strategy What are the average returns and the betas of this strategy? CONCLUSION: A portfolio of stocks with low past sales growth has on average much higher returns than a portfolio of stocks with high past sales growth. But they have almost the SAME (or even smaller) BETA! Empirical Evidence on the CAPM: Other Anomalies There are many other anomalies: e.g. calendar anomalies CAPM does not say anything about returns being different for different regular time periods ○ Examples: January vs. other months of the year Monday vs. other days of the week ○ Most famous is January effect Look at average stock returns in January vs. other months of year We’re drawing a lot of lines! These are all linear* estimates of future returns ○ Keep in mind, we are typically using characteristics of the past to help project into the future The past is not the future! Do we find many linear relationships in investments? Are we using the ‘market’ portfolio for these tests? Are there dynamics which prevent arbitrage? ○ Short sales, liquidity, taxes P-hacking? Any regularization??? Value? Note: Value uses Book Value of the company Over the past many years, companies have less book value compared to their profits, revenues, etc. More intrinsic value to companies that are not recorded as book value Some sectors are going to be more book-value heavy than others ○ Banking, e.g. Beware Do you think these excess returns can be captured if everyone knows about them? ○ What assumptions are made in these models? Do those assumptions hold in the real world? Fact(or)s change: Regime Change If you torture data long enough, it will confess to anything Each variable you add to a regression will improve R-squared ○ No matter what Seriously any variable added will increase the R-squared Finance Gone Mild Jim Simons Math Guy! Formulas and whatever named after him. Worked with the NSA to break codes. Smart dude. ○ Didn’t like the Vietnam War, so they made him quit. Went on to teach math at MIT, but got bored. Renaissance Technologies Jim and a few of his smart friends decided to start a hedge fund. They wanted to use their skills in math, data analysis, and noise filtering to quantitatively trade. They hit a few speed bumps in the 1980s, but kept tuning their methods. One thing led to another, and Jim Simons became worth $30 billion. Medallion Fund One of the company’s funds, which has had annualized returns well over 50% since inception. – Pretty good. Very secretive! They are using proprietary models. Using huge amounts of information to determine if it can predict the movement in prices of financial instruments. Complex signal processing and noise filtering. This is quantitative finance. These people do not have finance backgrounds – they have math and physics backgrounds. Arbitrage (Pricing Theory) If we are able to create a portfolio which has no risk, but is able to achieve returns beyond the risk-free rate, we have arbitrage! We may find factors which are present in all assets in varying amounts, which correspond to their risk and return To take advantage of these, we may buy/short the market to offset risk to get to 0 Prior to 1981, we could have done this with Size! Example Where we have excess return associated with each factor ○ Return – risk free We may have specific factors that we are able to predict better, or may have better risk-adjusted returns than the market (provide alpha). Find securities with very high or low factors, relative to their other risks Optimize the portfolio Models CAPM considers the relationship between the asset and the market as one of the factors Fama and French propose several over time which explain asset returns  ○ Or at least did before they blabbed about it... We can find other good explanatory variables and use those as our ‘betas’ Typical Multifactor Approaches Each asset may have specific fundamental attributes, such as its price-to-book, revenue growth, or other characteristics which serve as the factor. Time series values may explain the returns of companies. Economic data is very popular here, as it is regular, relevant, and reliable. Proposals over time... Fama French and their original 3 factor model was a pioneer in this field ○ They had to obviously keep ‘working’ so they came up with a few more over the years Q Factor Model Uses a few Fama French portions (CAPM, size) Investment ○ How much a company changes its assets over time Is this a good predictor of future returns? ROE ○ Is this a good predictor of future returns? This model did a great job of getting the dots to sit on the line! Other May include GDP inputs, energy prices, unemployment, etc. ○ You are trying to break out what the components are to returns – what explains returns. Each of those factors will have a beta for a stock, just like our market beta has for stocks May be able to find specific combinations of stocks which result in alpha ○ Buy stocks which earn more than beta expects, short ones where expectation is lower Expected Risk shall equal Expected Return A priori, we need to understand that the risks we take in a well-diversified portfolio shall necessarily be commensurate with the returns we expect ○ If not, we are able to achieve risk-free profits We can create a zero-risk portfolio with long/short positions These “Arbitrage Price Theory” models are just trying to estimate risk and returns. ○ Can get betas to 0 to create a ‘pure factor’ portfolio of specific factors. Law of One Price We can extend our philosophy to fit with these models. If we believe returns can be explained by the factors, then we believe that no returns can be generated beyond the model.  Like with CAPM, we can use our model to estimate returns, or to hedge risks. ○ If we do analysis which shows estimated returns are different than the factor model, we may achieve excess returns beyond the risk we take. Portfolio Management! If you stray away from index investing, understanding the size of your positions and bets is incredibly important! We want to know how well our models perform. At least our bosses do! How good is our edge? Do we always win when we make our bets? What’s the variance? My Alpha is Bigger (Information Ratio) Investing has two paths: index, or active. When we choose active investing, we must prove that it was better than the alternative. Similar to the Sharpe Ratio, we want to see how much return we are getting, per unit of deviation from the index: Any issues using IR to determine which fund is best? My Crystal Ball is Cloudy Being able to predict the outcomes of random variables is a useful skill. We can measure it with Information Coefficient: ○ 𝐼𝐶=(2∗% 𝑐𝑜𝑟𝑟𝑒𝑐𝑡) −1 Where 1 is perfect prediction, -1 is perfectly bad, and 0 is random. A lot of time and effort goes into finding relationships between stock returns and factors. So much so, that it might take away from other important aspects of your job. Just Think There are going to be many factors which you can say explain market returns ○ Just by chance! Do these have economic intuition? Are these repeatable outside of your sample? Are there measurement errors? Do you have the ability to know that data at the time to make your investment decisions? Do restrictions exist which do not allow the market to adjust for these anomalies? Indexes, Financial Markets Which Pill do you Choose? You can invest in the market and earn the theoretical maximum risk-reward! You can choose assets on your own! It’s What the Cool Kids Do Lower fees and higher returns! ○ Seems like a good spot to be This trend has been going on for years, and is unlikely to reverse Index (Passive) A list that you just buy everything on it, in the quantity it says to S&P500, Dow Jones Industrial Average, NASDAQ 100, Russell 2000, Nikkei 225, FTSE 100, Bloomberg Aggregate, JUC0, etc., etc. Indexes are typically provided by financial information companies and maintained based upon specific criteria for inclusion, size, etc. ○ $$$ Pay for use! Why is passive better (on average?) You don’t have to pay a huge team of finance people! ○ Just buy the list On average, the index is able to also beat active management before fees ○ Theoretically, it’s due to better Sharpe ratio, diversification, more ‘lottery tickets’ Less Churn ○ Fewer transaction costs and taxes often result from passive investing Passive Management Managers of passive funds work to replicate (mirror) the index they are benchmarked to They would like their returns and price movements to match the index Some indexes are really easy to match! ○ Others have thousands of investments, many of which are hard to get your hands on What might cause deviations? Discuss. Benchmarks Benchmarks are collections of securities or factor sensitivities and associated weights designed to represent the persistent and prominent investment characteristics of an asset class or manager’s investment process Properties of valid benchmarks ○ Unambiguous ○ Investable ○ Measurable ○ Appropriate  ○ Specified in advance Types of Benchmarks Absolute – CPI + 3% Manager universes – All domestic equity managers Broad market indexes – S&P 500 Style indexes – Russell 1000 Growth Factor model based – Market model Returns based – Combinations of style indexes Custom security based – Manager’s “fishing hole” Example: Dow Jones Industrial Average Bad index construction ○ Uses share price as the weight Any issues with share price? Take all 30 companies and their share prices, add them up ○ Seriously that’s all you do Include a factor to divide by, and that’s the index value Continuity: Divisor! If a stock was to split, or pays a dividend, its share price will change ○ This has no economic value, so the index should not reflect this change We incorporate a ‘divisor’ to our calculation, so that the index appears to be unaffected by this change Example Index Calculation Index with XYZ $50b market cap $10/share and ABC $30b market cap $22/share If XYZ splits its stock, nothing happens to its market value, so a market-value index does not adjust A price-weighted index would create a divisor, such that the resulting index does not change. ○ E.g.: Index = $10 + $22 = 32.00 After = $20 + 22 = 42.00 Divisor = 1.3125 S&P500 How many stocks? Market value weighted ○ Add up the market caps Some little nuances with ‘float’ Include a factor to divide by, and that’s the index value No factor adjustment for typical dividends! ○ Cannot use over a time period to determine actual return Issues with many indexes What about cash payments??? ○ Dividends and interest are important parts of investment! ○ S&P500 has been about 2% in recent years Typically not included in the index value When we do our return comparison against the index, we need to make sure we are treating everyone fairly ○ Include all sources of return in index, and in our comparison portfolio ○ This is called total return (use total return index) That’s not quite right! When you go to your favorite website to compare returns, keep in mind that some are just doing ‘price’ returns! You want to compare to a Total Return Index Remember, there are two components to return: ○ Capital Gains ○ Cashflows (Dividends) Is it valuable to be in an index? Do you think securities have more value in them if they are part of an index? Why would someone prefer to own a security that is part of an index? “Fallen Angels” and “Rising Stars” Understanding index methodology is important in the financial markets Finance Gone Mild How Much Should I Charge? Imagine: You are a bank, and you want to know how much to charge on a loan. But it’s hard to know! You want to make money, but your customers are rate- sensitive. You also don’t want to lose money. It’s a tough world out there, and interest rates move around. How much do I charge my customers? That Sounds Lovely Your costs are based on your borrowing costs! It would be nice to just pass this borrowing cost onto the customer. You and your bank buddies propose setting up a little index, which measures how much it costs you as a bank to borrow money. You work in London, and this borrowing happens Interbank. Can tell what the Offered Rate is. LIBOR This becomes a popular idea. Loans are created which reference this Benchmark, plus some spread (so the bank makes money). It’s important to know the methodology of how this is collected! Define It The rate at which an individual Contributor Panel bank could borrow funds, were it to do so by asking for and then accepting inter-bank offers in reasonable market size, just prior to 11.00 London time. Cover multiple maturities, currencies. Technical Terms For the USD LIBOR, there were 18 banks which reported their estimate of borrowing costs. ○ Yes, this was an estimate. The top 4 and bottom 4 were thrown out. This creates a trimmed mean. This number is then later reported as the official LIBOR number. Any Issues With This? Could there be conflicts of interest with this methodology? Any controls in place to prevent malfeasance? Are people going to go to prison for this and have embarrassing chat logs released because of said malfeasance? Do People Know How to Count? Presumably, market participants can count to 500 That means we should not see an effect of inclusion/exclusion Index Sampling, Pricing Matrix Sometimes the Index is really hard to replicate ○ This is especially true in Fixed Income/Bonds Most popular index, ‘The Agg’, has over 13000 bonds Hard to find them all! Some people might not sell them to you. Index investors may be forced to do Stratified Sampling – which attempts to use multiple characteristics of the index and match those  Principal Component Analysis ○ You’ll never guess what type of math is used here Replication Strategy The asset class will dictate the types of stratifications necessary to replicate: ○ Equity may be most focused on industry weights, beta. Remember the factors of Fama-French! Typically it is easier for equity indexes to be fully replicated due to fewer constituents, lower trading costs ○ Bonds may be most focused on: Duration, convexity, asset type, credit risk, issuer Bonds tend to have much higher correlation to each other Whatever techniques used, the ultimate goal is to have performance characteristics match the index. Minimize tracking error. ○ Direction and magnitude ??? Hard to Replicate, Hard to Price ??? If some assets are hard to find, that is a pretty good indication that they are hard to price ○ Or that they do not have many transactions which indicate their price How do you price these assets? May not trade often! ○ Does Private Equity outperform Public Equities with less risk? Similar techniques to the replication strategy: stratification Beware of prices! Other Index Funds Uses, Markets Active funds may use ETFs which are passively investing in an index ○ Get some cash in today, put it to work today. Can also short sell these ETFs to achieve less risk, or capture alpha – returns not explained by risk ○ Bond funds, too! ○ Which measure of risk is brought down by short-selling the index? Derivatives uses are likely here Some regulations may prohibit extent of this ETF Market Exchange Traded Fund: They trade on exchanges Most are passive index investments, work similar to a mutual fund: portfolio of assets You may ‘deliver’ or ‘receive’ the ‘baskets’ of investments in return for shares of the ETF. ○ This pricing mechanism helps keep the funds near their intrinsic value / Net Asset Value May use this mechanism to acquire or even short sell assets which are hard to find Basket Arbitrage on Canvas, download the file ETF  Determine how you would create free money, given: ○ $500 fee for basket creation or redemption ○ 50,000 units must be created or redeemed at a time ○ $27.80 in exchange fees per $1 million transacted on exchange Performance Measurement, Attribution Performance Measurement; Attribution It’s useful to know how well investments are performing Need to do a fair comparison ○ Small cap stock strategy should be compared to small cap stocks: e.g. Russell 2000 Proper benchmarks need to be constructed and used ○ Important to understand how the benchmarks work, not just select it because it’s similar! Typically use index of similar characteristics as a benchmark Finance is about tradeoffs You could invest money with many different strategies, managers, asset classes, time frames Usually, the biggest tradeoff involves risk vs. reward Our final tradeoff is between similar risk investments ○ Similar-risk investments are typically the same asset class being invested in We will compare funds across that asset class Small cap vs. Small cap, LT IG vs. LT IG, etc. ○ Could just invest in the index*! Measurements! Managers need something to do, so they want to measure how good employees are at their job ○ Some managers rarely contribute anything of value, so this is how they feel like they’re contributing Finance people tend to like numbers Finance managers love numbers Oh boy do we have some numbers for you ○ Can numbers ever be misleading? Look at how much money we made, boss Wow we made 15% this year! ○ Ok what about the benchmark? This is a good start! We always want to do fair comparisons! Adjust for risks ○ We can calculate the beta of our portfolio, among other measurements ○ When might beta be inappropriate? (Jensen’s) Alpha If we know the Beta of the portfolio, we can determine how much return the portfolio should have gotten, given that amount of risk. Any additional return beyond that is alpha. What does it imply if alpha is anything besides 0? ○ Folks, this is just CAPM again Alpha is inappropriate to use when beta is inappropriate to use ○ Beta & Alpha are only useful in a well diversified portfolio Sharpe Ratio Omg William Sharpe Pretty similar in spirit to alpha (adjust for risk) Per unit of risk, what is the excess return? A note to remember: S&P500 Sharpe is usually around 0.3-0.4 Sharpe Ratio Sharp Ratio, along with alpha, the two most popular measures of portfolio manager’s performance Each adjust for the risk taken by manager Sharpe Ratio can be more easily used by all asset classes ○ Private Equity likes to take advantage of this! Can’t have volatility if you don’t measure the portfolio’s value We are defining a unit of risk, and how much return that equals Does this measure work if returns are negative? Is higher better or worse? Tracking Error If you are tracking an index, there aren’t many ways you can ‘perform’ well! You should simply return the same amount as the index each period. Tracking Error = Standard Deviation of the difference of returns between the Index and the portfolio ○ Index managers want this to be 0 Information Ratio We have a few ways of identifying ‘good’ management, or at least good returns By now, we know that it is exceedingly rare for active management to be successful Information ratio is a great way to help identify persistent excess returns How consistent do you outperform the benchmark? Is your strategy the result of luck? ○ 𝐼𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 𝑅𝑎𝑡𝑖𝑜 = (𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 − 𝐼𝑛𝑑𝑒𝑥) / (𝑇𝑟𝑎𝑐𝑘𝑖𝑛𝑔 𝐸𝑟𝑟𝑜𝑟)  What might be problematic about this approach? Other Measures Researchers need something to do, and they also like saying that other people did something wrong ○ They also might want their name on a ratio so they can be famous Everybody loves William Sharpe omg Treynor, Sortino, M2 I do not care about these, nor does anybody else other than academics with nothing else to do. We will not cover these, unless someone pisses me off. Value at Risk (VaR) Banks and portfolio managers often operate under specific risk budgets Investment policies at banks may only allow for a specific amount of ‘value’ to be ‘at risk’ for a trading desk Typically, you calculate a standard deviation measure, then allow for risk up to, say, a 2nd percentile move. ○ Often would express our figure in dollars. Risk Modeling Nerds! This is where people who are good at math and statistics end up (risk desks at banks). Many jobs here. Regulators require it. Work with sophisticated models to help determine bank’s current risk measures. ○ Derivatives modeling, pricing, correlations, economic forecasts. Interest rates too Useful to have some computer programming background. Example Spreadsheet on Canvas; 4 desks Determine each desk’s standard deviation ○ Remember that we like annualized numbers in finance Which desks should get a talkin’ to by you (risk manager)? VaR is based on 2nd percentile, with loss for each: ○ -2%; -3%, -6%; -4% ○ 2.33 standard deviations VaR Issues Are returns normally distributed? Does the desk always have the same strategy? Is the track record long enough? Has the market gone through multiple regimes? Just because you did math, does not mean you are correct or sophisticated. Beware No single measure is perfect – they all have individual issues that need to be recognized. ○ Use multiple when making a decision. Use the measure that is most appropriate for that asset class, that strategy, and the use case. Understand the strategies and the useful ways to measure that strategy. Comparisons Always being compared to your peers! Fund managers will be ranked for their performance against the other funds in the same asset class ○ You can imagine those that finish consistently below the others will not have a job for very long Investors don’t need managers to pick their investments ○ Sometimes there’s a list that exists, and you just buy everything on the list! Financial Planning & Analysis (FP&A) Large part of corporate finance. Attribution in portfolio management is very similar to forecast attribution. The index in this case is the baseline forecast. ○ Why did we perform differently than forecast? Are there variables which you know now and didn’t before? Plug those into your forecast and see differences! Attribute Difference Can you find the reason why we earned $6,307.42 less than forecast? Use the FPandA workbook. Small groups, discuss how to find differences. Attribute to: Volume, Price, Cost. ○ Does it add up to the same amount? Breakdown Price: Change in price * Forecast Volume: Change in units * Current Price – Mix Mix/Interaction: Change in units * (Avg forecast price – forecast price) ○ Mix is the hardest conceptually to get Did we sell more units of high-priced items??? Where did the (over/under) performance come from? Attribution can be used for stocks and bonds, etc Useful for manager selection ○ Managers should have consistent strategies that are apparent from attribution  Is the manager really good at picking investments within an industry, or knowing which industry is going to do well? Brinson Attribution Breaks down performance differences as a result of selection and allocation Selection: Security selection; Did the securities in the industry outperform that industry’s overall performance? Overweight a security. ○ Very important for analysts! Allocation: Weighting of industries lead to outperformance? Buy a bunch of oil stocks when oil went up? Interaction: Sometimes included, not intuitive definition. Often part of selection. Did you overweight an industry where you had good selection/areas which outperformed? Example Allocation: Take Over/Under weight, multiply by index performance. You deviated industry weight! Selection: Take Over/Under performance, multiply by portfolio weight. You deviated security weights! Notice: Allocation (+5.5 bps) and Selection (-2.25 bps) add up to the outperformance (+3.25 bps) ○ Remember interaction is part of selection, and if calculated separately will change selection. All three will still add to 3.25 bps. Things can get a little messy if there are transactions during the calculation period! ○ Time-weighted or dollar-weighted return? Hedging, Portfolio Strategies As a manager, you are being paid for your ability to achieve risk-adjusted returns ○ Unless it’s a passive index Arguably, there is some value in being able to ‘time’ markets, though this is not a consistent way to make money, nor is there much evidence that people have skill in doing this. Some strategies will call for ‘market neutral’, meaning there is no market beta. Alpha Porting, Isolation You want to isolate the alpha in a portfolio, and remove it from the beta Imagine a portfolio with 0 beta, and positive alpha – you may be able to find many such strategies and combine them into a larger portfolio  ○ this seems like a great idea! Many folks are trying this with ‘factor models’ like we discussed earlier. Portfolios may short the index to get a beta of 0, and isolate the alpha. Pods, Factor Investing The hope of this market neutral, alpha positive investing, is that the alpha is weakly correlated with other strategies, alphas. Just like a portfolio of assets, a portfolio of strategies will be worth more, return more, if they are not correlated to each other. ○ And worth even more if it is uncorrelated with the market, too. This is what we were doing with APT! Porting If you know a successful stock picker manager, but you need to invest in bonds, you can still use that manager! Invest in the stock manager, and short the index to achieve a beta of 0. ○ Any issues with this? Invest in the bond index. You now have ported that alpha into a bond fund! 𝑅_(𝐵𝑜𝑛𝑑 𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜)=𝑅_(𝑆𝑡𝑜𝑐𝑘 𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜) −𝑅_(𝑆𝑡𝑜𝑐𝑘 𝐼𝑛𝑑𝑒𝑥)+𝑅_(𝐵𝑜𝑛𝑑 𝐼𝑛𝑑𝑒𝑥) 𝑅_(𝐵𝑜𝑛𝑑 𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜)=𝛼_(𝑆𝑡𝑜𝑐𝑘 𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜)+𝑅_(𝐵𝑜𝑛𝑑 𝐼𝑛𝑑𝑒𝑥) This is finance, folks. Get creative. Portfolio Construction, Execution, Personal Portfolios Portfolio Construction and Execution It is difficult to maintain good theses for investments It is sometimes equally as difficult to execute these ideas If you have a good idea, but it will take 3 years to invest in – you may not be able to take advantage of it! Understanding portfolio construction and other constraints is also very important ○ One bad idea may sink the entire portfolio if done incorrectly. Understanding Orders Some markets are quote-driven, meaning market makers will post bids and asks (offers) simultaneously on securities. Bid/ask spreads will be lower in securities with more demand and less risk. Vice Versa. ○ This phenomenon may feed itself. “Depth” in the market is determined by the size of the bid/ask quantities, persistence, and ‘iceberg’ orders. ○ Orders which are not seen to by the public, but will otherwise execute. Market Makers As market makers orders are ‘hit’ or ‘lifted’, their prices will change. Market makers want to earn the spread.  They do not like being adversely selected – if they sell some stock to you, are you really knowledgeable and the price will go up? ○ They lose money! A portfolio manager making an order will experience t