chapter 6 – diversification

B383 Finance Thurs. Oct 24-31. 2013.

Chapter 6 – Diversification Diversification: giving something variety manages risk of a portfolio by including variety of assets

How Diversification Works: Portfolio Standard Deviation False: a low-risk portfolio is constructed by selecting low-risk stocks. To build a low-risk portfolio we should collect stock that are negatively correlated Stdev of a 2-asset portfolio: When there are only 2 assets in a portfolio we don’t need 2 portfolio

weights. If we invest fraction w in one asset, then we must be investing fraction ( 1 – w) in the other

o We’ll do an example of 2 correlation scenarios; when the assets have perfect positive correlation, and when the assets have < perfect correlation

o Part b shows that we can combine 2 risky securities and end up with a portfolio that’s less riskyo The lower the correlation coefficient, the lower the portfolio stdev

Stdev of a portfolio with many assets: as the # stocks gets very large (assuming equal weighting across stocks), the risk foe ach stock matters less, and the variance of the portfolio converges toward the average of the pair-wise covariances risk is driven by the average degree of covo Still just pick stocks with low covariances or correlations if you want low risk portfolio

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Types of risk Difficult to find 2 stocks that are perfectly negatively correlated; but diversification works as long as

the stocks are < perfectly positively correlated Risk is divided into 2 parts: non-diversifiable risk + diversifiable risk Non-diversifiable risk (market/systematic risk): wars, oil price shocks, surprise monetary policy

change, sovereign defaults affects all assets to some extent; can’t be eliminated by diversification Diversifiable risk (firm-specific, unsystematic): strikes, input price shocks, loss of major customer,

technological obsolescence affect one/a few firms

Market Portfolio Naïve diversification: investing equal amounts of money in a portfolio of randomly selected stocks

not the most effective approach to portfolio creation Markowitz: maximize return, minimize risk Efficient set: set of all efficient portfolios across all

levels of stdev. Markowitz derived a formula for the portfolio weights for each portfolio in this set.o No unsystematic (firm-specific) risk

William Sharpe added a risk-free asset: asset with no variation in its return and no risk of default (U.S. govt T-Bill) US govt zero coupon bond is risk-free; return is only known after price o Market portfolio: includes every capital asset help in proportion to its market value relative to

the market value of all assets in total large, well diversified, in Markowitz’ efficient seto Investors in Sharpe’s model care only about systematic risk

Invariance of return drives 2 facts about the risk-free asset:1) The stdev of returns = 02) Correlation of risk-free returns with returns of any other asset = 0

All investors hold 2 assets: only difference between investors are portfolio weights for these assets1) Common risky portfolio from Markowit’z efficient set2) The risk-free asset



We’ve learned: 1) Market portfolio is value-weighted: the weights are the relative values of each asset in the portfolio2) All investors should optimally hold a portfolio with the same assets in same proportions (and will

earn the same return)3) Market portfolio is the aggregation of the individual investor portfolios Stock market indexes: a Proxy for the Market Portfolio

o This market includes all risky assets (not just stocks) private corps, real-estate, art, etc.o Many assets are difficult to trade, so use a proxy for the market portfolioo Stock market index: value weighted; most common proxy; statistical indicator showing the

relative value of a basket of stocks compared to their value I base year. Indexes are created by many entities using different baskets of stocks and different weighting schemes (value vs. price). Indexes measure the ups and downs of the stock market.

o Most popular market proxy is S&P 500 index; includes many large firms o Exchange traded fund (ETF): makes it easy for investors to trade market proxies; they’re

investment companies (mutual funds) that are legally classified as open-end companies or trusts. Most ETFs want to achieve the same return as a particular marketing index, so they hold a basket of stocks that mimics composition of the target index. There are many differences between ETFs and indexed mutual funds. ETC can be traded within the day, shorted, and purchased on margin.

SYSTEMATIC RISK William Sharpe developed capital asset pricing model (CAPM):o investors who hold the market portfolio don’t care about unsystematic risk o yields measure of systematic risk beta & equilibrium relationship between beta & expected

returns equilibrium risk-return relationship for individual assets all investors hold the market portfolio. They don’t care about the total risk of the stock cause they

can diversify that in the portfolio. The relevant risk is marginal risk: how will the risk of the market portfolio change if you add one more share of a particular stock?

Marginal risk = covariance between returns on the asset and the market portfolio standardize this to get beta, which is the key measure of risk for large portfolio holders. o Beta also measures the amount of market/systematic risk of an individual asseto Stocks and portfolios with large beta have large systematic risk

Estimating Beta graph the returns on the asset of interest against the returns on the market portfolio beta is the

slope of the characteristic line (line of best fit) firm-specific (unsystematic) errors cause variations (points that don’t fall on the line of best fit)

Properties of Beta market’s beta = 1: beta is defined wrt the market portfolio, so the line is perfect diagonal risk-free asset’s beta = 0: correlation (covariance) of the returns on the risk-free asset with the

returns on any other asset is zero. Since beta’s numerator is covariance, if cov=0, B=0. Logically, the risk-free return doesn’t change as the return on the market changes.



Portfolio beta: beta of a portfolio = the weighted average of the individual betas

EQUILIBRIUM RISK & RETURN Unsystematic risk can be eliminated at no cost by holding a large portfolio (diversifying). So investors

only price the risk that they cannot diversify systematic risk implication is that a security’s expected return should be related to the portion of its risk that cannot be eliminated through diversification

1) calculate return and beta of portfolios that combine the risk-free asset & a risky asset



o Portfolio-possibility lines: graph of the set of risks and returns produced by those portfolios. It’s a line in expected return-risk space. The line shows all risk and return combination possible by forming 2-asset portfolios with the risk-free asset & a risky asset. So each risky asset has its own portfolio possibility line.

2) Calculate the slope of those lines, which is known as the Treynor Index3) Use the index to derive an equilibrium relationship between return and risk

Portfolios with the Risk-Free Asset

this line is called a postfolio possibility line. The y-intercept is the risk/return combination for the T-Bill. The right end of the line is the risk/return combination for the company (RiskCorp.) The middle of the line is the half-half portfolio from above example.

Borrowing: buying on margin is borrowing to buy more than you can afford this would extend the line let’s assume cost of borrowing is the risk-free rate, so that we can model borrowing as just another portfolio of T-Bills & the risky asseto Buying T-Bills means you own them (long), and

you’re lending money to the government. The government is short T-Bills (borrowing).



Adding the risk & return combinations above to the graph, we get this graph>

The portfolio with borrowing (margin portfolio) offers higher expected return, and more risk

The risky asset doesn’t have to be a single asset; the risk and return combinations of a T-Bill and the market portfolio would lie on a straight line connecting the return of the T-Bill and the risk/return points of the market portfolio. powerful result

The Treynor Index Jack Treynor proposed that the slope of the

portfolio possibility line be used as a performance metric for evaluating risky investments rise = risk free rate to expected return, and run = from beta of the risk-free asset to the RiskCorp’s Beta

This slope = the Treynor Index

Treynor Index=E (k i )−k Fβ i

The Treynor index measures excess of asset’s return (risk premium) over the risk-free return per unit of its systematic(beta) risk.

Market Equilibrium Compare which stock is the better investment using Treynor Indexes



Bigger Treynor Index = bigger risk premium per unit of systematic risk = better stock = steeper slope

Capital Market Equilibrium CAPM: buying pressure creates excess demand that pushes price up, and selling pressure creates excess supply which pushes price down. o Ignore dividends; assume that future is fixed, then:

E (k )=E (P t+1 )Pt

−1

o If Pt (the price today) rises, E(k) = expected return falls and vice versa

o As E(k) falls, Treynor index will fall o Eventually, the two lines (of the two stocks) will converge, and there will be no incentive to buy

one stock or sell the other. EQUILIBRIUM o At equilibrium, the Treynor indexes of all assets are equal! They all lie on the same line.

The Security Market Line (SML): at equilibrium, all securities have the same Treynor Index, so all securities plot the same line. The market portfolio’s Treynor Index is:

Treynor Index formarket=E (kM )−kFβM

=E (kM )−kFo Remember, Beta of market portfolio = 1o Since all securities have the same index, pick any security i, and at equilibrium, that index is the

same index as the market portfolio. Rearranging the above, we getCAPM=E (k i )=kF+β i× (E (kM )−kF )

o So the expected return on a risky asset = the return on a riskless asset + additional return to compensate you for the risk of the investment. This additional return = market risk premium * B

o The only type of risk that contributes to the required return is the kind measured by beta (systematic) because firm-specific risk never enters into the calculation of required return

o The graph of CAPM = the security market line (SML) = equilibrium return for various Bo KF is the y=intercept (even if B = 0, you still require a return), and (E (kM )−k F ) is the slope



CAPM theory assumes that the market is filled with many well-diversified portfolio holders who measure systematic risk with B. The investors are rational and constantly search for mispriced securities (where the Treynor index does not equal market risk premium)

Investors move a lot of funds into the mispriced security until the price adjusts and mispricing disappears result = all securities are constantly on the SML and expected return = required return

Conclusion Problem with CAPM = market portfolio is unobservable; difficult/impossible to estimate returns on

many assets (real estate, jewelry, private companies) Problem = beta does not appear to be the only measure of systematic risk Truth = diversification reduces risk; returns are only related to systematic risk in equilibrium, so

CAPM provides a simple measure of systematic risk Not perfect, but CAPM can estimate required return in equity & evaluate mutual fund performance Mutual fund companies, with all their resources, cannot find lots of mispriced securities and so

boost their returns above what CAPM predicts We can achieve risk and return points along a home-made SML by investing in money market

mutual funds (mostly T-Bills) and stock market index ETFs. Since we can achieve the SML, we can outperform most mutual funds with simple homemade portfolios.


chapter 6 – diversification

Documents

asset portfolio

lowrisk portfolio

portfolio weights

portfolio thats

correlated risk

portfolio stdev stdev

lowrisk stocks

price market portfolio