Portfolio Theory and Financial Engineering

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Portfolio Theory and Financial Engineering

• FIN 428
• Lecture Six Capital Asset Pricing Model
• Thursday, January 25, 2007

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Last Chapter

• In last chapter, we demonstrated the
  diversification benefit of a portfolio in
  reducing return variation.
• Past returns and variances used as a proxy for
  the future expected return and variances. This
  may not always be valid.
• Working in a mean-variance framework
• People are risk averse, but care only about the
  tradeoff between return and risk
• Risk measured by the variance (or standard
  deviation) of return
• Ignoring other measures of risk due to skewness
  of the return. This is OK if return is
  distributed normally.
• Correlations between assets important in
  realizing the benefit of diversification

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Some More Assumptions

• All investors are Markowitz efficient investors
  who want to target points on the efficient
  frontier.
• The exact location on the efficient frontier and,
  therefore, the specific portfolio selected, will
  depend on the individual investors risk-return
  utility function.
• Investors can borrow or lend any amount of money
  at the risk-free rate of return (RFR).
• Clearly it is always possible to lend money at
  the nominal risk-free rate by buying risk-free
  securities such as government T-bills. It is not
  always possible to borrow at this risk-free rate,
  but we will see that assuming a higher borrowing
  rate does not change the general results.

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Some More Assumptions (ii)

• Investors can borrow or lend any amount of money
  at the risk-free rate of return (RFR).
• Clearly it is always possible to lend money at
  the nominal risk-free rate by buying risk-free
  securities such as government T-bills. It is not
  always possible to borrow at this risk-free rate,
  but we will see that assuming a higher borrowing
  rate does not change the general results.
• All investors have the same one-period time
  horizon such as one-month, six months, or one
  year.
• The model will be developed for a single
  hypothetical period, and its results could be
  affected by a different assumption. A difference
  in the time horizon would require investors to
  derive risk measures and risk-free assets that
  are consistent with their time horizons.

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Some More Assumptions (iii)

• All investments are infinitely divisible, which
  means that it is possible to buy or sell
  fractional shares of any asset or portfolio.
• This assumption allows us to discuss investment
  alternatives as continuous curves. Changing it
  would have little impact on the theory.
• There are no taxes or transaction costs involved
  in buying or selling assets.
• This is a reasonable assumption in many
  instances. Neither pension funds nor religious
  groups have to pay taxes, and the transaction
  costs for most financial institutions are less
  than 1 percent on most financial instruments.
  Again, relaxing this assumption modifies the
  results, but does not change the basic thrust.

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Some More Assumptions (iv)

• There is no inflation or any change in interest
  rates, or inflation is fully anticipated.
• This is a reasonable initial assumption, and it
  can be modified.
• Capital markets are in equilibrium.
• This means that we begin with all investments
  properly priced in line with their risk levels.
• All of investors wealth is in market traded
  assets.

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On the assumptions

• Some of these assumptions are unrealistic
• Relaxing many of these assumptions would have
  only minor influence on the model and would not
  change its main implications or conclusions.
• A theory should be judged on how well it explains
  and helps predict behavior, not on its
  assumptions.

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Optimal Portfolio with a Risk-free Asset

• Introduce a risk-free asset
• An asset with zero variance and zero correlation
  with all other risky assets
• Provides the risk-free rate of return (Rf)
• Will lie on the vertical axis of a
  return-standard deviation graph

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Optimal Portfolio with a Risk-free Asset

• Combining a risk-free asset with a risky
  portfolio
• Expected return
• variance of return
• ERp vs sp

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Portfolio Possibilities Combining the Risk-Free
Asset and Risky Portfolios on the Efficient
Frontier
Figure 8.1
D
M
C
B
A
Rf
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Risk-Return Possibilities with Leverage

• To attain a higher expected return than is
  available at point M (in exchange for accepting
  higher risk)
• Either invest along the efficient frontier beyond
  point M, such as point D
• Or, add leverage to the portfolio by borrowing
  money at the risk-free rate and investing in the
  risky portfolio at point M

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Capital Market Line (CML)
CML
Borrowing
Lending
M
Figure 8.2
Rf
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The CML and the Separation Theorem

• The CML leads all investors to invest in the M
  portfolio. The only difference is the location on
  the CML depending on risk preferences
• Risk averse investors will lend part of the
  portfolio at the risk-free rate and invest the
  remainder in the market portfolio
• Investors preferring more risk might borrow funds
  at Rf and invest everything in the market
  portfolio

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The Market Portfolio

• Because Portfolio M lies at the point of
  tangency, it has the highest portfolio
  possibility line
• Everybody will want to invest in Portfolio M and
  borrow or lend to be somewhere on the CML
• Therefore this portfolio must include ALL RISKY
  ASSETS
• Because the market is in equilibrium, all assets
  are included in this portfolio in proportion to
  their market value
• Therefore, Portfolio M must be the market
  portfolio

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The Market Portfolio

• The tangency portfolio M is the market portfolio
• All assets included in this portfolio are
  weighted in proportion to their market value
• Because portfolio M contains all risky assets, it
  is a completely diversified portfolio. Only
  systematic risk remains in the market portfolio.
• Systematic risk may be measured by the standard
  deviation of returns on the market portfolio.

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Diversification and the Elimination of
Unsystematic Risk

• All portfolios on CML are perfectly positively
  correlated with the completely diversified market
  portfolio M.
• Diversification reduces the standard deviation of
  the total portfolio. This assumes that imperfect
  correlations exist among securities
• As you add securities, you expect the average
  covariance for the portfolio to decline. How many
  securities must you add to obtain a reasonably
  diversified portfolio?

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Number of Stocks in a Portfolio and the Standard
Deviation of Portfolio Return
Standard Deviation of Return
Figure 8.3
Unsystematic (diversifiable) Risk
Total Risk
Standard Deviation of the Market Portfolio
(systematic risk)
Systematic Risk
Number of Stocks in the Portfolio
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The Relevant Risk Measure for A Risky Asset

• Its covariance with the market portfolio M
• Suppose you invest 1 dollar in portfolio M, and a
  small amount m in security i. Then the variance
  of the new portfolio is

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Risk and Expected Return for A Risky Asset

• Expected Return Rf aRisk
• Capital Asset Pricing Model

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Capital Asset Pricing Model (I)

• Recall assumptions
• Investors care only about the mean-variance
  trade-off of their portfolios in the next period
• Investors have homogeneous beliefs and equal
  investment opportunities
• There is a risk-free asset and investors can
  borrow and lend at the same risk-free rate
• Markets are frictionless, i.e., with no taxes and
  transaction costs. No limitation on the size of
  trading and short sales
• All of investors wealth is in market traded
  assets

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Capital Asset Pricing Model (II)

• Relates expected return of an asset to its
  exposure to the systematic risk as represented by
  b
• The expected rate of return of a risky asset is
  determined by the RFR plus a risk premium for the
  asset
• The risk premium, which can
  be negative (why?), is determined by the
  systematic risk exposure of the asset (b) and the
  prevailing market risk premium (RM-RFR)
• Security Market Line ( )

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Security Market Line
Figure 8.6
SML
Negative Beta
RFR
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Determining the Expected Rate of Return for
Risky Assets

• Assume RFR 6
• RM 12
• Implied market risk premium 6

E(RA) 0.06 0.70 (0.12-0.06) 0.102
10.2 E(RB) 0.06 1.00 (0.12-0.06) 0.120
12.0 E(RC) 0.06 1.15 (0.12-0.06) 0.129
12.9 E(RD) 0.06 1.40 (0.12-0.06) 0.144
14.4 E(RE) 0.06 (-0.30) (0.12-0.06) 0.042
4.2
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Determining the Expected Rate of Return for a
Risky Asset

• In equilibrium, all assets and all portfolios of
  assets should plot on the SML
• Any security with an estimated return that plots
  above the SML is underpriced (w.r.t. CAPM) (
  )
• Any security with an estimated return that plots
  below the SML is overpriced (w.r.t. CAPM) (
  )

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b for Portfolios

• Expected return of a portfolio

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Estimating b

• where
• Ri,t rate of return for asset i during period t
• RM,t rate of return for the market portfolio M
  during t
• Adjustments to b
• Bloomberg Adjusted Beta 0.66(unadjusted. b)
  0.34
• Time interval problems
• Different holding periods produce different beta
• More pronounced for small company and illiquid
  stocks
• Weekly and monthly returns better for estimation,
  not daily data.

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Other Estimation Issues

• Risk-free rate
• Most use short-term Treasury bill returns
• Notice that bill returns are variable, not truly
  risk-free. May use matching-period T-Strip rate
• Market risk premium
• Historical data of excess return on market index
• But expected return on market index may change
  over time
• Proxies for market portfolio
• SP (U.S. equity only)
• World indices (ignore other assets, like real
  estates, etc.)

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Relaxing the Assumptions of the CAPM

• CAPM assumption all investors can borrow or lend
  at the risk-free rate - unrealistic
• Differential borrowing and lending rates
• Unlimited lending at risk-free rate
• Borrowing at higher rate
• A range of portfolios on the efficient frontier
  may be held (and the market portfolio is also in
  that range, but may not be held by everyone)

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Investment Alternatives When The Cost of
Borrowing is Higher Than The Cost of Lending

• Figure 8.14

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Relaxing the Assumptions of the CAPM

• There may not exist a truly risk-free asset, or
  no borrowing allowed
• Zero-beta portfolio a portfolio that is
  uncorrelated to the market portfolio (b 0)
• The return of the zero-beta portfolio is not
  risk-free
• Usually situated on the lower half of the EF
• Blacks zero-beta model
• E(Ri) E(Rz) biE(Rm) - E(Rz)

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Security Market Line With Transaction Costs

• Figure 8.16

E(R)
SML
E(Rm)
E(RFR) or
E(Rz)
bi
0.0
1.0
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Empirical Tests of the CAPM

• Beta portfolios
• Cross-sectional tests
• The Roll Critique
• Momentum
• Fama-French

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Before the Next Class

• Read
• Chapter 9
• Topics to be discussed in the next class
• Alternative Pricing Models