SAID BUSINESS SCHOOL MASTERS IN FINANCIAL ECONOMICS

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SAID BUSINESS SCHOOLMASTERS IN FINANCIAL
ECONOMICS

• Asset Pricing Theory
• Lecture 5
• D.P. Tsomocos
• 1/11/07
• Mean-Variance Analysis
• Mutual Fund Separation
• C.A.P.M.

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Construction of the Efficient Frontier
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Construction of the Efficient Frontier
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Construction of the Efficient Frontier (ctd)
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Construction of the Efficient Frontier (ctd)
?
?
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Construction of the Efficient Frontier (ctd)
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Construction of the Efficient Frontier (ctd)
REMARK This is the Minimum Variance Frontier,
since it also includes m-v dominated
portfolios.
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Construction of the Efficient Frontier (ctd)
CASE 3 perfect negative correlation ( ?12 - 1)
?
?
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Construction of the Efficient Frontier (ctd)
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Construction of the Efficient Frontier (ctd)
?p
?
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Construction of the Efficient Frontier (ctd)
CASE 5 n risky assets (imperfectly correlated)

• Efficient frontier will have a bullet shape.
• Analysis is as in Case 2.
• Adding assets improves diversification and
  typically the portfolio frontier moves to the
  left.

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Construction of the Efficient Frontier (ctd)
CASE 6 n risky assets and a risk-free one
T
F
E
rf

• The efficient frontier is the straight line from
  (0,rf) to T.
• T is the tangent portfolio to the risky asset
  frontier.
• If we allow short sales of the risk-free asset,
  then it extends to the right of T.

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Construction of the Efficient Frontier (ctd)
The efficient frontier of n assets (one of them
may be the risk-free asset) is the part of the
undominated portfolios of the minimum variance
frontier (mvf). The mvf is the solution of the
following maximisation problem

• REMARKS
• Alternatively, using duality theory one can
  maximise mean for given levels of standard
  deviation.
• Other constraints may be imposed to the portfolio
  problem
• non-negativity constraints (wi ?0)
• stock holdings should not fall below a certain
  level (i.e. wj ?Vj/Vp)
• home bias puzzle
• transaction costs

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A Separation Theorem

• The optimal portfolio is naturally defined, as in
  micro- economics, as the portfolio that maximises
  the investors mean-variance utility.
• Recall, indifference curves are convex to the
  origin.
• Thus, for the case of risky assets plus a
  risk-free asset, all optimal portfolios should be
  on the linear efficient frontier.
• Given different preferences, they will maximise
  at different points
• BUT
• All optimal portfolios would consist of
  combinations of the risk-free asset and the
  tangency portfolio T.(TWO FUND SEPARATION
  THEOREM)
• Optimal portfolio of risky assets can be
  identified independent of the knowledge of risk
  preferences.

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A Separation Theorem (ctd)
T
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A Separation Theorem (ctd)
THEOREM (2-FUND SEPARATION) If every agents risk
tolerance is linear with common slope ?, then
date-1 consumption plans at any Pareto optimal
allocation lie in the span of the risk-free
payoff and the aggregate endowment.
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C.A.P.M.

• C.A.P.M. is a theory of financial equilibrium.
  There is no attempt to relate returns with
  production. Thus, there is no interaction between
  the real and nominal sectors of the economy.
• No derivation (endogenous) of supply and demand
  functions of assets. It assumes that supply
  equals to demand of assets and, therefore,
  observed prices are equilibrium prices.
• (economy-wide asset holdings are investors
  optimal asset demands)
• It determines asset returns relationships so that
  equilibrium prices coincide with the observed
  asset prices.
• It provides a specific characterisation of risk.

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C.A.P.M. (ctd)

• ASSUMPTIONS
• (A1) Each investor maximises a mean-variance
  utility
• .

with U1gt 0 , U2 lt 0 and U is concave
(A2) All investors have common time horizon and
homogeneous beliefs about expected returns and
variances of existing assets.
(A3) Each asset is infinitely divisible.
(A4) Short sales of the risk-free asset are
allowed.
(A5) If there is no risk-free asset present, no
short sales restrictions on risky assets.
(A6) The endowments of all agents are traded.
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C.A.P.M. (ctd)

• LESSONS FROM C.A.P.M.
• All existing assets MUST belong to T, which will
  hereafter be called the Market Portfolio.

Proof By contradiction. In equilibrium, if
some assets were not in T, there would be no
demand for them. However, all assets exist in
positive supply. Recall that agents should hold T
in equilibrium or combination of T and the
risk-free (if there is one) asset by the 2-fund
separation result. ?
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C.A.P.M. (ctd)

• LESSONS FROM C.A.P.M. (continued)

Proof By contradiction. You violate the
equilibrium condition... ?
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C.A.P.M. (ctd)

• LESSONS FROM C.A.P.M. (continued)
• The market portfolio is efficient, since it is on
  the efficient frontier (Efficiency).
• All optimal portfolios are located on the Capital
  Market Line (CML).

The slope of the CML is the price of (or reward
for) risk. It tells us that an investor
considering a marginally riskier efficient
portfolio would obtain, in exchange, an increase
in the expected return
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C.A.P.M. (ctd)

• REMARKS
• The CML applies to efficient portfolios.
• Recall, every investor holds the market portfolio
  M.
• So, C.A.P.M. identifies the T portfolio as being
  the market portfolio M.
• Question What is the equation of expected
  returns of any asset (i.e. assets that
  may or may not belong to the efficient frontier)?

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C.A.P.M. (ctd)
SECURITY MARKET LINE THEOREM
Proof
?
CML
rM
M
j
?
?M
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C.A.P.M. (ctd)
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C.A.P.M. (ctd)
?
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C.A.P.M. (ctd)
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C.A.P.M. (ctd)
Security Market Line
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The Mathematics of the Portfolio Frontier
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The Mathematics of the Portfolio Frontier(ctd)
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The Mathematics of the Portfolio Frontier(ctd)
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The Mathematics of the Portfolio Frontier(ctd)

• REMARKS
• The portfolio frontier can be derived as a convex
  combination of any two frontier
  portfolios
• 2.

?
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The Mathematics of the Portfolio Frontier(ctd)

• REMARKS
• Any convex combination of frontier portfolios is
  also a frontier portfolio
• The set of efficient portfolios is a convex set
• Thus, M is efficient

LEMMA For any frontier portfolio p, except MVP,
there exists a unique frontier portfolio with
which p has zero covariance (We denote it by
ZC(p)) Hint Just find the unique portfolio that
has this property Take the covariance and set
it equal to zero
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Zero-Beta C.A.P.M.

• The intercept of the line joining p and the
  minimum variance portfolio is the expected return
  on the zero covariance portfolio
• The next step is to describe the expected return
  on any portfolio in terms of frontier portfolios,
  such as M

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Zero-Beta C.A.P.M.
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Standard C.A.P.M.

• 0-beta C.A.P.M. plus risk-free asset
• Efficient frontier is the solution to the problem

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Standard C.A.P.M. (ctd)
Since T is a frontier portfolio, choose p ? T.
But in equilibrium T M. Hence, for any
portfolio q,
Sharpe Litner - Mossin C.A.P.M.