Arbitrage Pricing Theory

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Chapter 11

• Arbitrage Pricing Theory

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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.

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Factor Model Expected Un

• The return on any security consists of two parts.
  
• First, the expected returns
• Second, the unexpected or risky returns
• A way to write the return on a stock in the
  coming month is

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• Any announcement can be broken down into two
  parts, the anticipated (or expected) part and the
  surprise (or innovation)
• Announcement Expected part Surprise.

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Risk Systematic Un

• 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
  single company are examples of unsystematic risk.

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Risk Sys Un
We can break down the total risk of holding a
stock into two components systematic risk and
unsystematic risk

?2
Total risk
Nonsystematic Risk ?
Systematic Risk m
n
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Sys Risk Beta

• The beta coefficient, b, tells us the response of
  the stocks return to a systematic risk.
• In the CAPM, b measures the responsiveness of a
  securitys return to a specific risk factor, the
  return on the market portfolio.

• We shall now consider other types of systematic
  risk.

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Sys Risk and Betars

• For example, suppose we have identified three
  systematic risks inflation, GNP growth, and the
  dollar-euro spot exchange rate, S(,).
• Our model is

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More Sys and Betars..

• Suppose we have made the following estimates
• bI -2.30
• bGNP 1.50
• bS 0.50
• Finally, the firm was able to attract a
  superstar CEO, and this unanticipated
  development contributes 1 to the return.

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SRB

We must decide what surprises took place in the
systematic factors. If it were the case that the
inflation rate was expected to be 3, but in fact
was 8 during the time period, then FI
Surprise in the inflation rate actual
expected 8 3 5
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SRB

• If it were the case that the rate of GNP growth
  was expected to be 4, but in fact was 1, then
• FGNP Surprise in the rate of GNP growth
• actual expected 1 4 3

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SRB

If it were the case that the 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
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• Finally, if it were the case that the expected
  return on the stock was 8, then

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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 has return

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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
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RBRCFER

Excess return
If we assume that there is no unsystematic risk,
then ei 0.
The return on the factor F
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RBRCFER

Excess return
Different securities will have different betas.
The return on the factor F
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Portfolio and Diversification

• We know that the portfolio return is the weighted
  average of the returns on the individual assets
  in the portfolio

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P D

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

• The weighted average of expected returns.

• The weighted average of the betas times the
  factor.

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

• So the return on a diversified portfolio is
  determined by two sets of parameters
• The weighted 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 portfolios sensitivity to the
factor.
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Betas E(R)

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

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Relationship Between b Expected Return

SML
Expected return
D
A
B
C
b
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CAPM APT

• 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.

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APT CAPM

• Both the CAPM and APT are risk-based models.
• Empirical methods are based less on theory and
  more on looking for some regularities in the
  historical record.
• Be aware that correlation does not imply
  causality.
• Related to empirical methods is the practice of
  classifying portfolios by style, e.g.,
• Value portfolio
• Growth portfolio

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Quick Quizzer

• Differentiate systematic risk from unsystematic
  risk. Which type is essentially eliminated with
  well diversified portfolios?
• Define arbitrage.
• Explain how the CAPM be considered a special case
  of Arbitrage Pricing Theory?
• Problems 1, 3, 5, READ Fama-French after
  problems.

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Exam Stuff

• So that you may deserve your well earned A
• Chapters 1- 11
• 20 multiple choice problems (show your work for
  partial credit.) (uh, also write neat)
• Know how to use your calculator (and anything
  else I forgot.)

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Areas of Probable Inquiry

• BS IS construction/deconstruction, ratios (types,
  problems)
• Corp vs. Sole vs, Ltd.
• Agency Problem
• Financial Markets (Primary, Secondary)
• EFN, g
• DCF, NPV
• Bond Calcs, teeter totter.
• Divy Growth Model

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Mo inquiries

• IRR pros/cons
• Real Options
• Return statistics (Var, Cov, Corr)
• CAPM, Efficient Frontier, Beta
• Risk, Sys and Unsys
• Factor Model, APT

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Echos in the Hollow Recesses of Your Mind

• Be able to Construct Basic Financial Statements
• Be Familiar with the Formulas.
• Bring your formula sheet.
• See what you got, where you need to go.
• Plug-n-play what you can first.
• Convert percentages to real s to calc.
• Unless all s are percentages.

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Good Luck!!!!!!!!!

• Bonne chance
• Viel Glueck
• Buona fortuna
• ??????? ???????
• ?? ?????
• Goed Geluk
• Boa sorte