Theory of Financial Markets II

Harry Markowitz

Individual who suggested assets be characterized by their return probabilities (or rather, mean and standard deviation of returns)

Finding Variance of Asset

Variance is average of squared deviations (Ri - Mean(R)): Find deviations, square them, and average them into variance

Continuum Hypothesis

States that there is no set of values with a size between the set of all real number and the set of all fractions; assumed true in finance

Measure Theory

Used to quantify the size of an event (value or set of values) on a number line, assuming Continuum Hypothesis is true; single number has measure zero (can be ignored), while range has positive measure (can’t be ignored)

Alpha (in portfolio construction)

Fraction of portfolio of the first asset, with 1 - a being the portion of the second asset

Mean-Variance Diagram

Shows SD on x-axis and mean return on y-axis to plot assets and boundary set of possible portfolios; shows only portfolios with highest mean return for given SD

Covariance

How returns of one asset relate to another; multiplied deviations from all years added and divided by n years; positive = move together, negative = move separately.

Correlation

Standardizes covariance to quantify strength of relationships between stocks: Formula is Cov(X, Y) / (SD(X)*SD(Y))

Markowitz Assumptions

1. Each stock has normal distribution of future expected returns w/ known means and variances

2. Each stock is risky (SD > 0)

3. Investors prefer greater mean return to lower mean return and lower variance to higher variance (all else equal for each)

4. Investors choose the best portfolio according to these assumptions

Markowitz Curve

Shows the feasible portfolios between two assets; all portfolios on the curve are efficient, with the curve’s shape depending on the correlation between each asset

Efficient portfolio

Portfolio with the highest mean return for a given variance or lowest variance for a given mean return

Markowitz Curve Shapes

Corr = 1: Assets connected by linear curve; no benefits are reaped for diversification

Corr = -1: All portfolios provides return of zero

All other Corellations: Curve is bow-shaped, moving leftwards as Corr reaches zero; diversification benefits can be attained

Why are abs(Corr) < 1 assets better for portfolios than Corr = 1 assets?

Because of diversification benefits, providing lower variance (risk) for given mean returns and vice versa; results in investors searching for assets with little to no correlation in returns

Math of Mean-Variance Theory

Variance of Two Assets: Var(aX, (1-a)Y) = a²Var(X) + (1-a)²Var(Y) + 2a(1-a)Cov(X, Y)

When Corr(X, Y) = 1: SD(aX, (1-a)Y) = a(SD(X)) + (1-a)SD(Y)

Portfolio Mean: a*Mean(X) + (1-a)*Mean(Y)

Efficient Frontier

Set of all portfolios that are efficient (assumes no risk-free asset)

James Tobin

Introduces the risk-free asset (asset with SD = 0) to Mean-Variance Theory, conceiving E portfolio as only combination of risky assets to be owned and stated that all investors should own only Rf asset and E-Portfolio (Two-Fund Conclusion)

Risk Free and Risky Assets on Mean-Variance Diagram

Rf on the Y-axis (zero variance); For any risky portfolio Rp on efficient frontier, all portfolios that can be created between Rf and Rp lie on straight line connecting them (preferable w/ risk averse investors due to higher mean for same variance)

Feasible portfolios with risk free asset

Portfolios that lie on the line (from Rf) tangent to the efficient frontier curve

E-Portfolio

Point of tangency for the line from Rf on the efficient frontier; efficient combination of risky assets according to Tobin; changes based on risk-free rate

Tobin conclusion of the E-portfolio

States that any investor who owns risky assets will combine them in accordance with E-portfolio (same proportions of risky assets)

Two-Fund Conclusion

States that all investors should only own combination of E-portfolio and risk-free asset

Borrowing with Portfolios

Investors may spend more than the wealth they own on a portfolio by borrowing money to do so; represented in Tobin model by going beyond E-portfolio point

When Borrowing Rate > Lending Rate (Tobin Model)

Imagine F-portfolio as tangent point of borrowing rate and efficient frontier at higher point than E; beyond F, investors will borrow to buy more of F; below and at E, investors will own combination of E and Rf; Between E and F, investors will own combination of risky asset portfolios based on risk aversiveness (TFC disappears)

How can Investors take on more risk?

Borrowing Rate = Lending Rate: Own higher proportion of E portfolio (can also borrow to obtain more than can afford)

Borrowing Rate ≠ Lending Rate: Increase holdings of E; higher risk = own combination of E and F; even higher risk = Increase holdings in F (can borrow to increase holdings beyond available wealth)

Rf rate and E

Higher Rf rate = E has higher mean and SD or return; High Rf rate requires high mean return from E to entice investment, low Rf coincides with lower expected returns

Portfolio variance with a risk-free asset

Given Y is a risk-free asset: Var(aX, (1-a)Y) = a²Var(X)

Capital Asset Pricing Model (CAPM)

One of the first models concerned with pricing assets developed by Bill Sharpe, which states that asset expected return depend solely on an asset’s beta (relating excess asset returns with excess market returns) in equilibrium

CAPM assumptions

• Finite number of assets with normally distributed returns, with means, variances, and covariances of all stocks being known

• At least one riskless asset present

• Investors prefer higher mean returns and lower return variances

• Number of investors are finite, and these investors can have any risk/reward preferences that satisfy assumption 3 (not present in Tobin formulation)

Beta

Calculated as Cov(Ri, Rm) / Var(Rm), where Rm is market return, Ri is return of stock, and Var(Rm) is the variance of market return; How stock’s returns track with market returns; not the same as volatility/SD

Market Basket (M)

Comrpises all risky assets with value weighed by worth (MVi = Market Val of Stock i = Pi * shares(i)). “Market up or down” means M is up or down

Math of Market Basket

MV = Total Market Val = Sum of all MVi, with each weight wi = MVi/MV

Questions answered by CAPM

What happens when supply and demand of all stocks are in equilbirium? What determines stock prices? What is the best predictor of stock returns?

E portfolio and M

E must consist of M since it contains all risky assets of value (no one would own if it had no val); any risky asset owned by an investor is in M; M of CAPM = E from Tobin Model; Conclusion is that any risky assets would be owned in risky portfolio M with proportions weighed by capitalization

CAPM and Beta

Stock price depends solely on Beta: E[Ri] - Rf = B(E[Rm - Rf)

Security Market Line

Line generated from CAPM equation relating expected market return to expected return of certain stock; Exp. Returns (Market and Assets) on Y-axis and beta on X-axis; slope is market risk premium E[Rm] - Rf; stock return above line is underpriced and below is overpriced

Capital Market Line

Line on Mean-Variance Diagram defining boundary set of feasible portfolios

Limitations of CAPM

Beta alone doesn’t fully explain differences in average returns observed across stocks; no inclusion of info regarding economic/company factors, market portfolio hard to comprehensively measure

Contributions Towards MVT and CAPM

Markowitz: Described how investors should diversify (portfolio valuation rather than individual stocks, interactions between stocks in portfolio (Covariance), stocks characterized by mean and variance of returns, efficient set is curve connecting two assets)

Tobin: Added risk-free asset to Markowitz’s framework (E-portfolio as the most efficient combination of risky assets, Two-Fund Conclusion, efficient set is tangent line to e-portfolio)

Sharpe: If everyone followed Markowitz-Tobin Framework (best risky portfolio is the entire market (market basket), with asset price being determined by beta)

Volatility

Same as standard deviation of returns over a period of time; irrelevant to asset pricing under CAPM since it is diversified away with ownership of M, but plays role in determining optimal port (lowest volatility given a mean return)

Treynor Ratio

E[Rp] - Rf / Bp: Returns that come at the cost of systematic market risk (beta); same as slope of security market line

Sharpe Ratio

E[Rp] - Rf / SD(p): Excess return at cost of risk (SD); same as slope of capital market line; used to evaluate additions to stock to portfolio (increasing sharpe ratio after adding means add it)

Sharpe Ratio and Problems with Normally Distributed Returns

Sharpe ratio works with assumption of normally distributed returns, but almost never true in practice; problem for whole CAPM since this is a key assumption

Benchmarks

Indexes to measure investment manager performance (like S&P 500)

S&P 500

Index of 500 stocks weighed by market capitalization: P1Q1 + P2Q2 + … PnQn / arbitrary divisor; does not include dividends paid from stocks, must be priced into index return

Information Ratio

E[Ra - Rb] / SD(Ra - Rb): Adjusts expected return by that of benchmark; SD(Ra - Rb) is tracking error, large TE = large tracking risk (failing to adhere to benchmark)

Interpretation of the Information Ratio

Degree of imrproved performance gained from increased tracking error to benchmark (Large tracking risk = disproportionate large performance improvement relative to TE)

Jansen’s Alpha

Quantifies superior portfolio performance; residual return not explained by Beta and Rf; ai = Ri - Rf - Bi(Rm - Rf); consist with CAPM if E[ai] = 0

Beta Return

Return attributed to beta or B(E[Rm] - Rf), achievable by default in index funds

Alpha return

Return attributed to investment manager skill in alpha (unlike beta return)

CAPM with asset allocation

Use Mean-Variance framework to allocate funds across asset classes; choose portfolio efficiently

Arbitrage Pricing Theory

Asset pricing model in which unexpected changes in economic factors drive returns, and expected return is a linear function of the asset’s sensitivities to those factors; no assumption of markets in equilibrium for more robust pricing theory

APT Equation

E[Ri] = Bi1F1 + Bi2F2 + … BinFn, where Fj are explanatory economic factors; note betas are different for each stock; only one F in CAPM (market return) where F = E[Rm] - Rf

Intuition of APT

Stock prices reflect that have been anticipated, but cannot price-in events not predicted in advance; Hence, APT is the expected return + influence of unanticipated events = return + error (random)

Four F Factors (Ross)

Unanticipated Inflation, Changes in degree of industrial production, unanticipated shifts in risk premiums, unanticipated movements in shape of interest rate term structure

Determining E[Ri] in APT

Assume E[Ri] - Rf = Bi1y1 + … Binyn, where yj = E[RFj] - Rf, Bij = Cov(RFj, Ri) / Var(RFj), and RFj is return from beta of 1 to jth F; hence, yj = expected excess return attributable to jth F; RFj not stock specfiic.

APT conclusion

Excess returns (on average) are completely determined by sensitivity (beta) to average excess returns of factors they’re sensitive to.

CAPM in APT

One factor in CAPM (Market Return), so E[Ri] - Rf = Bi1y1, where y1 = E[Rm] - Rf and Bi = Cov(M, Ri) / Var(M)

APT Factor models

Macroeconomic (inflation, GDP, interest rate levels), Fundamental (earnings growth, P/E ratio), Statistical (Variance, Covariance); used for stock-picking

APT addressing CAPM Weaknesses

Not necessary to define market basket (impossible in practice), Avoids “same risky portfolio” assumption (efficiency determined by Fs), Uses no arbitrage assumption rather than equilibrium, Use of economic factors to explain behaviors within stocks is biggest advantage

How investor chooses a portfolio

Finds the point of tangency between boundary of feasible set and investor’s indifference curve

Roll’s Critique of CAPM

CAPM is untestable in practice because the returns of the true market portfolio M can’t be observed, as it contains assets like human capital and commodities that aren’t traded or priced and thus can’t be measured. Using a proxy like the S&P 500 for M is not testing the CAPM for this reason

Cost of Capital

Return investors require to hold a security; inverse relationship with security price (Price = Cash Flow / 1 + Required Return) and debt price (yields are lower)