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Arbitrage Pricing Theory

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Title: Arbitrage Pricing Theory


1
Chapter 11
  • Arbitrage Pricing Theory

2
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.

3
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

4
  • Any announcement can be broken down into two
    parts, the anticipated (or expected) part and the
    surprise (or innovation)
  • Announcement Expected part Surprise.

5
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.

6
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
7
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.

8
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

9
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.

10
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
11
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

12
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
13
  • Finally, if it were the case that the expected
    return on the stock was 8, then

14
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

15
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
16
RBRCFER

Excess return
If we assume that there is no unsystematic risk,
then ei 0.
The return on the factor F
17
RBRCFER

Excess return
Different securities will have different betas.
The return on the factor F
18
Portfolio and Diversification
  • We know that the portfolio return is the weighted
    average of the returns on the individual assets
    in the portfolio

19
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.
20
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.
21
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.
22
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.

23
Relationship Between b Expected Return

SML
Expected return
D
A
B
C
b
24
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.

25
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

26
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.

27
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.)

28
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

29
Mo inquiries
  • IRR pros/cons
  • Real Options
  • Return statistics (Var, Cov, Corr)
  • CAPM, Efficient Frontier, Beta
  • Risk, Sys and Unsys
  • Factor Model, APT

30
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.

31
Good Luck!!!!!!!!!
  • Bonne chance
  • Viel Glueck
  • Buona fortuna
  • ??????? ???????
  • ?? ?????
  • Goed Geluk
  • Boa sorte
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