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## Investment Analysis and Portfolio Management

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The Capital Asset Pricing Model (CAPM)

- The CAPM is a model of equilibrium in the market for securities.
- Previous lectures have addressed the question of how investors should choose assets given the observed structure of returns.
- Now the question is changed to:
- If investors follow these strategies, how will returns be determined in equilibrium?

The Capital Asset Pricing Model (CAPM)

- The simplest and most fundamental model of equilibrium in the security market
- Builds on the Markowitz model of portfolio choice
- Aggregates the choices of individual investors
- Trading ensures an equilibrium where returns adjust so that the demand and supply of assets are equal
- Many modifications/extensions can be made
- But basic insights always extend

Assumptions

- The CAPM is built on a set of assumptions
- Individual investors
- Investors evaluate portfolios by the mean and variance of returns over a one period horizon
- Preferences satisfy non-satiation
- Investors are risk averse
- Trading conditions
- Assets are infinitely divisible
- Borrowing and lending can be undertaken at the risk-free rate of return
- There are no taxes or transactions costs

Assumptions

- The risk-free rate is the same for all
- Information flows perfectly
- The set of investors
- All investors have the same time horizon
- Investors have identical expectations

Assumptions

- The first six assumptions are the Markowitz model
- The seventh and eighth assumptions add a perfect capital market and perfect information
- The final two assumptions make all investors identical except for their degree of risk aversion

Direct Implications

- All investors face the same efficient set of portfolios

Direct Implications

- All investors choose a location on the efficient frontier
- The location depends on the degree of risk aversion
- The chosen portfolio mixes the risk-free asset and portfolio M of risky assets

Separation Theorem

- The optimal combination of risky assets is determined without knowledge of preferences
- All choose portfolio M
- This is the Separation Theorem
- M must be the market portfolio of risky assets
- All investors hold it to a greater or lesser extent
- No other portfolio of risky assets is held
- There is a question about the interpretation of this portfolio

Equilibrium

- The only assets that need to be marketed are:
- The risk-free asset
- A mutual fund representing the market portfolio
- No other assets are required
- In equilibrium there can be no short sales of the risky assets
- All investors buy the same risky assets
- No-one can be short since all would be short
- If all are short the market is not in equilibrium

Equilibrium

- Equilibrium occurs when the demand for assets matches the supply
- This also applies to the risk-free
- Borrowing must equal lending
- This is achieved by the adjustment of asset prices
- As prices change so do the returns on the assets
- This process generates an equilibrium structure of returns

The Capital Market Line

- All efficient portfolios must lie on this line
- Slope =
- Equation of the line

Interpretation

- rf is the reward for "time"
- Patience is rewarded
- Investment delays consumption
- is the reward for accepting "risk"
- The market price of risk
- Judged to be equilibrium reward
- Obtained by matching demand to supply

Security Market Line

- Now consider the implications for individual assets
- Graph covariance against return
- The risk on the market portfolio is
- The covariance of the risk-free asset is zero
- The covariance of the market with the market is

Security Market Line

- Can mix M and the risk-free asset along the line
- If there was a portfolio above the line all investors would buy it
- No investor would hold one below
- The equation of the line is

M

Security Market Line

- Define
- The equation of the line becomes
- This is the security market line (SML)

Security Market Line

- There is a linear trade-off between risk measured by and return
- In equilibrium all assets and portfolios must have risk-return combinations that lie on this line

Market Model and CAPM

- Market model uses
- CAPM uses
- is derived from an assumption about the determination of returns
- it is derived from a statistical model
- the index is chosen not specified by any underlying analysis
- is derived from an equilibrium theory

Market Model and CAPM

- In addition:
- I is usually assumed to be the market index, but in principal could be any index
- M is always the market portfolio
- There is a difference between these
- But they are often used interchangeably
- The market index is taken as an approximation of the market portfolio

Estimation of CAPM

- Use the regression equation
- Take the expected value
- The security market line implies
- It also shows

CAPM and Pricing

- CAPM also implies the equilibrium asset prices
- The security market line is
- But

where pi(0) is the value of the asset at time 0 and pi(1) is the value at time 1

CAPM and Pricing

- So the security market line gives
- This can be rearranged to find
- The price today is related to the expected value at the end of the holding period

CAPM and Project Appraisal

- Consider an investment project
- It requires an investment of p(0) today
- It provides a payment of p(1) in a year
- Should the project be undertaken?
- The answer is yes if the present discounted value (PDV) of the project is positive

CAPM and Project Appraisal

- If both p(0) and p(1) are certain then the risk-free interest rate is used to discount
- The PDV is
- The decision is to accept project if

CAPM and Project Appraisal

- Now assume p(1) is uncertain
- Cannot simply discount at risk-free rate if investors are risk averse
- For example using

will over-value the project

- With risk aversion the project is worth less than its expected return

CAPM and Project Appraisal

- One method to obtain the correct value is to adjust the rate of discount to reflect risk
- But by how much?
- The CAPM pricing rule gives the answer
- The correct PDV of the project is

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