Why is the Markowitz (mean variance) model impractical (4)

- Large amoutn of data required (returns, variances, covariances)
- The need to forecast the large number of inputs
- The inability of individual analysts to have the expertise to forecast all the required covariances (given their focus on a specialised industry or stocks)
- Errors in estimation of correlation coefficients can lead to nonsensical results

Optimisation procedure, single index model (10)

- compute initial position in active portfolio
- Scale initial positions to force portfgvolio weights to sum to one by dividing by their sum
- Compute the alpha of the active portfolio
- Compute the beta of the active portfolio
- Compute the initial position in the active portfolio
- Compute the beta of the active portfolio
- Adjust the initial position in the active portfolio
- Note; th e optimal portfolio now has weights
- Calcualte the risk premium of the optimal riskiy portfolio from the risk premium of the index portfolio and the alpha of the active portfolio
- Compute the variance of the optimal risky portfolio from the variance of the index portfolio and the residual variance of the active portfolio

CAPM setting (6)

- Investors are price takers - with a wealth that is small relative to the total wealth of all invetsors
- Investors plan for a single period investment horizon
- The investment universe is limited to publicly traed financial assets and a risk free asset (such that borrowers and lenders do so at the risk free rate)
- Investors do not incur taxes or transaction costs when trading
- Investors are mean-variance optimisers
- Investors hold homogeneous expectations about returns and covariances

Market equilibrium - attributes (2)

- All investors will hold portfolios containing all traded assets in the same weights as the market portfolio (M)
- The market portfolio is the tangency portfolio (M). Its capital allocation line is the best available and is called the Capital Market Line (CML)

Mutual Fund theorem

If the market portfolio is the ultimate risky portfolio, then investors can avoid fundamental security analysis and sim[ply hold a combination of the marklet portfolio and the risk free asset.

Capital Market Line is constructed:

CML is cnostructed as the Capital Allocation Line constructed from money markets and the market portfolio.

Variance of a portfolio =

Weighted sum of the elements of the covariance matrix

Measure the contribution of a single stock to overall portfolio risk

calculate the single stock’s covariance with the market portfolio.

Difference between security market line (SML) and capital market line (CML)

CML: graphs risk premiums of efficient portfolios (ie comp[rised of mkt portfolio and risk free asset) as a function of STANDARD DEVIATION.

SML: charts INDIVIDUAL ASSET risk premiums as a function of asset risk; the relevant risk measure for individual assets is the contribution of the asset to portfolio varianec - ie the BETA of the asset

SML is valid for the entire portfolio and for individual assets

Applications for CAPM (4)

- Provides a benchmark for investment performance. Recognise beta of portfolio , then investment returns are appropriately adjusted for market risk
- Consensus estimates to support stock selection. recognise a stocks exposure to market risk, then determine a fair rate of return
- Provides inputs for calculating the required rate of return on a capital budgeting project or equity valuation.
- Guide to pricing instruments in (and services of) non traded entities

Regression equation of single index model, what is the intercept

Intercept of the single index regression model is (alpha), which is the security’s expected excess turn when the market excess return is zero

Single index regression model, what is the slope coefficient

The slope coefficient is beta, ie the security’s sensitivity to the index. The amount by which the security return tends to increase or decrease for every 1% increase or decease in the return on the index.

Single index model, what is e

E is the zero mean, firm specific surprise in security returning time t, also called the residual.

Total risk =

Total risk = systematic risk + firm specific risk

COVARIANCE =

COVARIANCE = product of betas x market index risk

Correlation =

Correlations = product of correlations with the market index

Limitations of CAPM

Assumes unlimited capacity to borrow and lend, but when there are restrictions there is no unique point tendency betw CML and efficient frontier.

There is no truly risk free asset

Zero beta model and CAPM

- Developed by Black (1972)
- Zero beta model assumes beta of zero, relies on unrestricted shrt selling
- Short sell because there is another portfolio on the inefficient portion of the efficent frontier, their returns are uncorreltated so the companion portfolio’s beta must be zero
- CAPM assumes unrestricted borrowing and lending at the risk free rate, with restrictions there is no unique point of tengency between the efficient frontier and the capital market line

Multifactor SML

Attempt to recognize various sources of risk

Arbitrage pricing theory: assumptions

- Security returns can be described by a factor model
- There a sufficient securities to diversify away idiosyncratic risk
- Well functioning securities markets do not allow for the persistence of arbitrage opportunities

Arbitrage

Riskless profits

Law of one price

If two assets are equivalent in all economically relevant respects, then they should have the same market price

What does R squared measure

R squared measures the fraction of the total variance explained by the market return.

Single factor model of the economy classifies sources of uncertainty as:

Systematic (macro economic) factors or firm specific (microeconomic) factors

How is the index model (single factor model) estimated

Apply regression analysis to excess rates of return. The slope of the regression curve is the beta of the Wasset, while the assets alpha is the intercept. The regression Lne is called the SECURITY CHARACTERISTIC LINE

Estimate index model using total rather than excess rates of turn:

Substitute estimation of alpha with alpha + r(f)(1-beta)

Beta tendency

Betas have a tendency to evolve to 1 over time

Beta

Beta is the sensitivity of the stocks return to the market return, ie the change in stock return per unit of market return.

Calculate each stock beta by calculating the difference in return across the scenarios divided by the difference in market return

Alpha ( or excess return) =

Alpha = actually expected return minus required return.

Calculate the required return given risk, eg beta

Project hurdle rate, using beta=

Project hurdle rate is determined by the PROJECT not the FIRM’S beta. Use the rate of return that equates to the fair rate of return for that project as the discount rate

When is CAPM valid (6)

When provided with betas, the stock with the highest beta should have the highest return in equilibrium

When provided with standard deviations, if CAPM is valid the expected rate of return compensates for systematic risk (market risk) represented by beta, rather than standard deviation, which includes non systematic risk. As such, a lower rate of return can be paired with Higher standard deviation as long as the beta is lower.

Each security’s beta represents

The security’s exposure to the common market factor

Betas capture the way securities COVARy with each other.

Security market line - GRAPH

- Draw expected returns on the vertical axis, stock betas on the horizontal axis

SML uses BETA

All fairly priced assets will sit on the SML, if not, the price neil adjust until it does.

If a security is correctly priced in CAPM, it’s alpha is zero. Otherwise will have positive or negative alpha

Uses of CAPM (4)

- CAPM provides a benchmark for investment performance. By recognizing the BETA of the portfolio, the investment returns are appropriately adjusted for the portfolios exposure to market risk
- It also provides consensus estimates to support stock selection. Can establish a fair rate of return. An active process would need to forecast a return higher than this to warrant a higher than benchmark/market weighting
- Provides inputs for calculating the required rate of return on a capital budgeting project or equity valuation
- Provides a guide to pricing investments in (and the services of) non traded entities.

Compare SML and CML

- Measure of risk
- Dependent variable
- Purpose

- Measure of risk, CML is standard deviation, SML is beta
- Dependent variable ???????
- Purpose: CML: graphs risk premiums of efficient portfolios, SML: graphs individual assets risk premium as a function of asset risk.
- On the CML, different portfolios can have the same expected return and different portfolio standard deviations, AS LONG AS THEY HAVE THE SAME SYSTEMATIC RISK (ie beta)

Chen Roll & Ross multifactor model - which macroeconomic factors are used (5)

Industrial production Expected inflation unanticipated inflation Corporate to govt long term bond spread Government long term bond to t-bill spread

Fama & French multi factor model - estimate firm style characteristics. Factors include: (2)

Size returns (ie small stocks less large stocks) Value returns (ie low book to market value less high book to market value)

What is a factor model called when it uses an index of security returns to proxy a common macro factor?

This model is called a single index model

Problems with mean variance (3)

- Number of inputs
- Estimating inputs (forecasting error)
- Sector specialists (difficult to be across all assets

Single index solution - advantages (2)

- Single factor (sensitivity to market risks. 1 beta for each asset
- Fewer inputs

If the CML is the best Sharpe ratio - why would some investors overweight particular stocks

- Maximising overall Sharpe raio could be a LONG TERM focus; active management could incorproate short term views.
- If short term opinions dictated overall portfolio weights, this could be representative of a New Sharpe ratio (new CML)

Security Market Line; what happens if a stock lies off the SML? (4)

- The beta of the security can be dupicated using a combination of the risk free asset and the security with a beta of 1
- Therefore if a security lies off the line, then the price will adjust accordingly until the security sits on the line (if below, price will drop)
- Actively managed portfolios with o/w positive alpha stocks and u/w negative alpha stocks will sit above the SML
- In equilibrium all securities sit on the SML

APT depends on___________

while CAPM depends on __________

Implications?

APT depends on arbitrage

CAPM depends on rational equilibrium (based on mean variance dominance)

Implications: it takies only a few players to enforce APT, whereas CAPM requires all participants to make adjustments to their portfolios

Stephen Ross’ Arbitrage Pricing Theory (1976)

3

- security returns can be described by a factor model
- there are sufficient securities to diversify away risk
- well functioning security markets do not allow for the persistence of arbitrage opportunities

Law of One Price

If two assets are equivalent in all economically relevant respects, then they should have the same market price.

Security Characteristic Line - chart observations

- Firm specific risk (how to observe)
- Systematic risk (measured by?)
- R^2 = ?
- Intercept of SCL = ?
- Correlation coefficient

- Firm specific risk: deviations from the SCL
- Systematic risk is measured by BETA, which is the slope of the SCL
- R^2 = squared correlation coefficient; = ratio of explained variance of stock’s return to total variance. Total variance is explained + unexplained variance
- Intercept of SCL on expected return axis = alpha
- Correlation coefficient = square root of R^2

Chen Roll & Ross multifactor model - which macroeconomic factors are used (5)

Industrial production Expected inflation unanticipated inflation Corporate to govt long term bond spread Government long term bond to t-bill spread

Fama & French multi factor model - estimate firm style characteristics. Factors include: (2)

Size returns (ie small stocks less large stocks) Value returns (ie low book to market value less high book to market value)

What is a factor model called when it uses an index of security returns to proxy a common macro factor?

This model is called a single index model

Problems with mean variance (3)

- Number of inputs
- Estimating inputs (forecasting error)
- Sector specialists (difficult to be across all assets

Single index solution - advantages (2)

- Single factor (sensitivity to market risks. 1 beta for each asset
- Fewer inputs

If the CML is the best Sharpe ratio - why would some investors overweight particular stocks

- Maximising overall Sharpe raio could be a LONG TERM focus; active management could incorproate short term views.
- If short term opinions dictated overall portfolio weights, this could be representative of a New Sharpe ratio (new CML)

Security Market Line; what happens if a stock lies off the SML? (4)

- The beta of the security can be dupicated using a combination of the risk free asset and the security with a beta of 1
- Therefore if a security lies off the line, then the price will adjust accordingly until the security sits on the line (if below, price will drop)
- Actively managed portfolios with o/w positive alpha stocks and u/w negative alpha stocks will sit above the SML
- In equilibrium all securities sit on the SML

APT depends on___________

while CAPM depends on __________

Implications?

APT depends on arbitrage

CAPM depends on rational equilibrium (based on mean variance dominance)

Implications: it takies only a few players to enforce APT, whereas CAPM requires all participants to make adjustments to their portfolios

Stephen Ross’ Arbitrage Pricing Theory (1976)

3

- security returns can be described by a factor model
- there are sufficient securities to diversify away risk
- well functioning security markets do not allow for the persistence of arbitrage opportunities

Law of One Price

If two assets are equivalent in all economically relevant respects, then they should have the same market price.

Security Characteristic Line - chart observations

- Firm specific risk (how to observe)
- Systematic risk (measured by?)
- R^2 = ?
- Intercept of SCL = ?
- Correlation coefficient

- Firm specific risk: deviations from the SCL
- Systematic risk is measured by BETA, which is the slope of the SCL
- R^2 = squared correlation coefficient; = ratio of explained variance of stock’s return to total variance. Total variance is explained + unexplained variance
- Intercept of SCL on expected return axis = alpha
- Correlation coefficient = square root of R^2