Quantitative Analysis Summary of Quantitative Analysis 1

Use More Than One Basket for Your Eggs Don’t put all your eggs in one basket. Failure to diversify may violate the terms of fiduciary trust. Risk aversion seems to be an instinctive trait in human beings. Question – what is the best way to diversify? 4

Preliminary Steps in Forming a Portfolio First Step: Identify a collection of eligible investments known as the security universe. My portfolio’s security universe: Let Value Line do the heavy lifting on securities analysis Documented success in this area, so why reinvent the wheel? Included stocks in Model Portfolio I: Stocks with Above-Average Year-Ahead Price Potential My value added comes from best assembling these stocks into an “efficient” portfolio 5

Preliminary Steps in Forming a Portfolio Second Step: Compute statistics for the chosen securities. e.g. mean of return variance / standard deviation of return matrix of correlation coefficients Key Question – where / how to obtain these statistics? e.g. historical averages? (what I used for exp. return) CAPM / Single-Index Model? (what I used for risk) APT / Multi-Factor Model? Inductive ad hoc Factor Model? Models for risk vs. models for expected return 5

Preliminary Steps in Forming a Portfolio Interpret the statistics. 1. Do the values seem reasonable? 2. Is any unusual price behavior expected to recur? 3. Are any of the results unsustainable? 4. Low correlations: Fact or fantasy? 6

The Role of Uncorrelated Securities The expected return of a portfolio is a weighted average of the component expected returns. where xi = the proportion invested in security i

The Role of Uncorrelated Securities The total risk of a portfolio comes from the variance of the components and from the relationships among the components. two-security portfolio risk = riskA + riskB + interactive risk 7

The Role of Uncorrelated Securities The point of diversification is to achieve a given level of expected return while bearing the least possible risk. expected return risk better performance A portfolio dominates all others if no other equally risky portfolio has a higher expected return, or if no portfolio with the same expected return has less risk.

The Efficient Frontier : Optimum Diversification of Risky Assets The efficient frontier contains portfolios that are not dominated. expected return risk (standard deviation of returns) impossible portfolios dominated efficient frontier

The Efficient Frontier : The Minimum Variance Portfolio The right extreme of the efficient frontier is a single security; the left extreme is the minimum variance portfolio. expected return risk (standard deviation of returns) single security with the highest minimum variance portfolio

The Efficient Frontier : The Effect of a Risk-Free Rate When a risk-free investment complements the set of risky securities, the shape of the efficient frontier changes markedly. expected return risk (standard deviation of returns) dominated portfolios impossible M Rf C efficient frontier: Rf to M to C

The Efficient Frontier : The Effect of a Risk-Free Rate In capital market theory, point M is called the market portfolio. Assumes: Securities universe includes all investable stocks All investors invest in portfolios on the efficient frontier (i.e., use Markowitz optimization) The straight portion of the line is tangent to the risky securities efficient frontier at point M and is called the capital market line. Since buying a Treasury bill amounts to lending money to the U.S. Treasury, a portfolio partially invested in the risk-free rate is often called a lending portfolio.

The Efficient Frontier with Borrowing Buying on margin involves financial leverage, thereby magnifying the risk and expected return characteristics of the portfolio. Such a portfolio is called a borrowing portfolio. expected return risk (standard deviation of returns) dominated portfolios impossible M Rf C efficient frontier: the ray from Rf through M lending borrowing

The Efficient Frontier : Different Borrowing and Lending Rates Most of us cannot borrow and lend at the same interest rate. expected return dominated portfolios impossible M RL N efficient frontier : RL to M, the curve to N, then the ray from N risk (standard deviation of returns) RB

The Efficient Frontier : Naive Diversification Naive diversification is the random selection of portfolio components without conducting any serious security analysis. As portfolio size increases, total portfolio risk, on average, declines. After a certain point, however, the marginal reduction in risk from the addition of another security is modest. total risk nondiversifiable risk number of securities

The Efficient Frontier : Naive Diversification The remaining risk, when no further diversification occurs, is pure market risk. Once you own all the stocks on the market, you cannot diversify any more! Market risk is also called systematic risk and is measured by beta. A security with average market risk has a beta equal to 1.0. Riskier securities have a beta greater than one, and vice versa. i = 1 = Neutral Stock Market Stock i > 1 = Aggressive Stock i < 1 = Defensive Stock

The Efficient Frontier : The Single Index Model Beta is the statistic relating an individual security’s returns to those of the market index. 17

The Efficient Frontier : The Single Index Model The relationship between beta and expected return is the essence of the capital asset pricing model (CAPM), which states that a security’s expected return is a linear function of its beta. 18

The Efficient Frontier : The Single Index Model beta E(Ri) - Rf security market line + -

The Efficient Frontier : The Single Index Model A single index model relates security returns to their betas, thereby measuring how each security varies with the overall market. Ri = Constant + Common-Factor + Firm-Specific News News Ri = i + iI + ei This is the model that is used for estimating betas for CAPM. CAPM is a specific example of a SIM. Now, getting back to calculating the efficient frontier: A pair-wise comparison of the thousands of stocks in existence would be an unwieldy task. To get around this problem, the single index model compares all securities to the benchmark measure of risk.

Why Does the SIM Reduce Computations? It Decreases the Number of Calculations of Covariances From: Cov(Ri, Rj) = Cov(1+ iI + ei,j + jI + ej) Implies 124,750 separate covariances for 500 stocks Typically have 500 x 60 = 30,000 data points Statistical error!! To: Cov (Ri, Rj) = (i j2I) Only need 500 ’s and 1 2I to estimate all 124,750 covariances Assumes: Cov(ei, ej) = 0 and Cov(I, ei) = 0

The Arbitrage Pricing Theory (APT) and Multi-Factor Models But, is the stock market as a whole the only source of co-movements between stocks? Is beta the only driver of expected returns? Relaxing the assumptions of CAPM leads to the Arbitrage Pricing Theory and Multi-Factor Models

APT and CAPM Both models yield similar types of results Both models can be used for estimating either or both expected return and risk Advantage of APT over CAPM: Different Assumptions APT is less restrictive Disadvantage of APT: Fails to identify the common factors As with CAPM, factors chosen are typically based on finance theory.

Multi-Factor APT Some Typically Suggested Factors: Growth rate of GDP Level of interest rates Default premium Term structure Expected inflation Oil prices Ri = a0 + a11 + a22 + … + akk + i E(Ri) = a0 + a11 + a22 + … + akk If markets are not “efficient,” then still other factors may come into play. These other factors must be found by trial and error. This leads to the development of Inductive ad hoc Factor Models.