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Return and Risk : CAPM and APT Reference: RWJ Chp. 11. Arbitrage Pricing Theory. Arbitrage - arises if an investor can construct a zero investment portfolio with a sure profit. Since no investment is required, an investor can create large positions to secure large levels of profit.

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arbitrage pricing theory
Arbitrage Pricing Theory

Arbitrage - arises if an investor can construct a zero investment portfolio with a sure profit.

  • Since no investment is required, an investor can create large positions to secure large levels of profit.
  • In efficient markets, profitable arbitrage opportunities will quickly disappear.
factor models announcements surprises and expected returns
Factor Models: Announcements, Surprises, and Expected Returns
  • The return on any security consists of two parts.
    • First the expected returns
    • Second is the unexpected or risky returns.
  • A way to write the return on a stock in the coming month is:
factor models announcements surprises and expected returns4
Factor Models: Announcements, Surprises, and Expected Returns
  • Any announcement can be broken down into two parts, the anticipated or expected part and the surprise or innovation:
  • Announcement = Expected part + Surprise.
  • The expected part of any announcement is part of the information the market uses to form the expectation, R of the return on the stock.

The surprise is the news that influences the unanticipated return on the stock, U.

risk systematic and unsystematic
Risk: Systematic and Unsystematic
  • A systematic risk is any risk that affects a large number of assets, each to a greater or lesser degree.
  • An unsystematic risk is a risk that specifically affects a single asset or small group of assets.
  • Unsystematic risk can be diversified away.
  • Examples of systematic risk include uncertainty about general economic conditions, such as GNP, interest rates or inflation.
  • On the other hand, announcements specific to a company, such as a gold mining company striking gold, are examples of unsystematic risk.
risk systematic and unsystematic6

Risk: Systematic and Unsystematic

We can break down the risk, U, of holding a stock into two components: systematic risk and unsystematic risk:

Total risk; U

Nonsystematic Risk; 

Systematic Risk; m

n

systematic risk and betas
Systematic Risk and Betas
  • The beta coefficient, b, tells us the response of the stock’s return to a systematic risk.
  • In the CAPM, b measured the responsiveness of a security’s return to a specific risk factor, the return on the market portfolio.
  • We shall now consider many types of systematic risk.
systematic risk and betas8
Systematic Risk and Betas
  • For example, suppose we have identified three systematic risks on which we want to focus:
    • Inflation
    • GDP growth
    • The dollar-euro spot exchange rate, S($,€)
  • Our model is:
systematic risk and betas example
Systematic Risk and Betas: Example
  • Suppose we have made the following estimates:
    • bI = -2.30
    • bGDP = 1.50
    • bS = 0.50.
  • Finally, the firm was able to attract a “superstar” CEO and this unanticipated development contributes 1% to the return.
systematic risk and betas example10
Systematic Risk and Betas: Example

We must decide what surprises took place in the systematic factors.

If it was the case that the inflation rate was expected to be by 3%, but in fact was 8% during the time period, then

FI = Surprise in the inflation rate

= actual – expected

= 8% - 3%

= 5%

systematic risk and betas example11
Systematic Risk and Betas: Example

If it was the case that the rate of GDP growth was expected to be 4%, but in fact was 1%, then

FGDP = Surprise in the rate of GDP growth

= actual – expected

= 1% - 4%

= -3%

systematic risk and betas example12
Systematic Risk and Betas: Example

If it was the case that dollar-euro spot exchange rate, S($,€), was expected to increase by 10%, but in fact remained stable during the time period, then

FS = Surprise in the exchange rate

= actual – expected

= 0% - 10%

= -10%

systematic risk and betas example13
Systematic Risk and Betas: Example

Finally, if it was the case that the expected return on the stock was 8%, then

portfolios and factor models
Portfolios and Factor Models
  • Now let us consider what happens to portfolios of stocks when each of the stocks follows a one-factor model.
  • We will create portfolios from a list of N stocks and will capture the systematic risk with a 1-factor model.
  • The ith stock in the list have returns:
relationship between the return on the common factor excess return
Relationship Between the Return on the Common Factor & Excess Return

Excess return

If we assume that there is no unsystematic risk, then ei = 0

The return on the factor F

relationship between the return on the common factor excess return16
Relationship Between the Return on the Common Factor & Excess Return

Excess return

If we assume that there is no unsystematic risk, then ei = 0

The return on the factor F

relationship between the return on the common factor excess return17
Relationship Between the Return on the Common Factor & Excess Return

Excess return

Different securities will have different betas

The return on the factor F

portfolios and diversification
Portfolios and Diversification
  • We know that the portfolio return is the weighted average of the returns on the individual assets in the portfolio:
portfolios and diversification19

The weighed average of expected returns.

  • The weighted average of the betas times the factor.
  • The weighted average of the unsystematic risks.
Portfolios and Diversification

The return on any portfolio is determined by three sets of parameters:

In a large portfolio, the third row of this equation disappears as the unsystematic risk is diversified away.

portfolios and diversification20
Portfolios and Diversification

So the return on a diversified portfolio is determined by two sets of parameters:

  • The weighed average of expected returns.
  • The weighted average of the betas times the factor F.

In a large portfolio, the only source of uncertainty is the portfolio’s sensitivity to the factor.

betas and expected returns
Betas and Expected Returns

The return on a diversified portfolio is the sum of the expected return plus the sensitivity of the portfolio to the factor.

relationship between b expected return
Relationship Between b & Expected Return
  • If shareholders are ignoring unsystematic risk, only the systematic risk of a stock can be related to its expected return.
the capital asset pricing model and the arbitrage pricing theory
The Capital Asset Pricing Model and the Arbitrage Pricing Theory
  • APT applies to well diversified portfolios and not necessarily to individual stocks.
  • With APT it is possible for some individual stocks to be mispriced - not lie on the SML.
  • APT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolio.
  • APT can be extended to multifactor models.