Capital market theory

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Capital market theory

• Assumptions of capital market theory
• 1. All investors are Markowitz efficient
  investors who want to target points on the
  efficient frontier for investment. The exact
  location on the efficient frontier, and
  therefore, the specific portfolio chosen will
  depend on the individual investors risk-return
  tradeoff.

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Capital market theory

• Assumptions of capital market theory
• 2. There exists a risk-free assets such that
  investors can borrow or lend as much as they want
  at the risk-free rate of return (RFR)
• 3. All investors have homogenous expectations
  regarding the probability of future rates of
  returns. that is, they estimate identical
  probability distributions for future rates of
  return.

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Capital market theory

• Assumptions of capital market theory
• 4. All investors have the same one-period time
  horizon. We do not look at how changes through
  time affect investors. The model that we develop
  is a one-period model.
• 5. All assets are infinitely divisible. Thus,
  it is possible to buy or sell fractional shares
  of any asset or portfolio.

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Capital market theory

• Assumptions of capital market theory
• 6. There are no market frictions such as taxes,
  transaction costs, or regulation.
• 7. There is no inflation or any change in
  interest rates is fully anticipated.

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Capital market theory

• Assumptions of capital market theory
• 8. Capital markets are in equilibrium. This
  means that all investments are properly priced
  relative to their risk levels. Any
  disequilibrium situations will be immediately
  corrected in the market.
• 9. All information is costless and
  simultaneously available to all investors.

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Capital market theory

• Assumptions of capital market theory
• The book makes two important points regarding
  these assumptions
• 1. Many of these assumptions can be relaxed
  without affecting the outcome of our results in
  any substantive way.
• 2. A theory should never be judged on the basis
  of its assumptions alone, but rather on how will
  it explains behavior in the real-world.

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Capital market theory

• Assumptions of capital market theory
• Critical assumptions
• 4. All investors have the same one period time
  horizon
• 8. Capital markets are in equilibrium
• 9. All information is costless and available to
  all market participants

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Capital Market Theory

• The risk-free asset
• The major factor that allowed portfolio theory to
  develop into capital market theory was the
  concept of a risk-free asset.
• A risk-free asset is an asset whose expected
  return is certain. Thus, the standard deviation
  of a risk-asset is zero.
• Since the risk-free asset never changes in value,
  its time horizon is irrelevant. It does not
  matter when we purchase the asset, we will always
  know what it return will be.

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Capital Market Theory

• Combining a risk-free and a risky asset
• Covariance of risk-free asset with a risky
  portfolio

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Capital Market Theory

• Combining a risk-free and a risky asset
• Expected return of a portfolio containing a risky
  portfolio/asset and the risk-free asset

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Capital Market Theory

• Combining a risk-free and a risky asset
• Variance/standard deviation of a portfolio
  containing a risky portfolio/asset and the
  risk-free asset

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Capital Market Theory

• Combining a risk-free and a risky asset
• Thus, the expected return of a portfolio
  containing a risky portfolio/asset and the
  risk-free asset is just a linear combination of
  the expected returns on the risky portfolio/asset
  and the risk-free asset.
• The standard deviation of a portfolio containing
  a risky portfolio/asset and the risk-free asset
  is the linear proportion of the standard
  deviation of the risky asset/portfolio.

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Capital Market Theory

• Combining a risk-free and a risky asset
• The expected return/risk combinations of a
  portfolio containing a risky portfolio/asset and
  the risk-free asset will fall on a straight line
  in risk/return space.
• Therefore, the efficient frontier will be a
  straight line between the risk-free asset and the
  risky-asset.

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Lending at the risk free rate

• Examples
• Given the following information
• E(Rr) 0.162, ?r .1208 E(Rrf) 0.04, ?rf
  0
• Port A Port B Port C Port D Port E
• wr 0.00 0.25 0.50 0.75 1.00
• wrf 1.00 0.75 0.50 0.25 0.00

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Lending at the risk free rate

• Examples
• Return
• E(rp) wrE(Rr) wrfE(Rrf)
• Standard deviation
• ?p Xr2?r2 Xrf2?rf2 2XrXrf?r,rf1/2
• ?p Xr2?r21/2
• ?p Xr?r

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Lending at the risk free rate

• Examples
• Return calculations
• rA (0.00)(0.162) (1.00)(0.04) 0.0400
• rB (0.25)(0.162) (0.75)(0.04) 0.0705
• rC (0.50)(0.162) (0.50)(0.04) 0.1010
• rD (0.75)(0.162) (0.25)(0.04) 0.1315
• rE (1.00)(0.162) (0.00)(0.04) 0.1620

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Lending at the risk free rate

• Examples
• Risk calculations
• ?A (0.00)(0.1208) 0.0000
• ?B (0.25)(0.1208) 0.0302
• ?C (0.50)(0.1208) 0.0604
• ?D (0.75)(0.1208) 0.0906
• ?E (1.00)(0.1208) 0.1208

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Lending at the risk free rate

• INVESTING IN BOTH RISKFREE AND RISKY ASSET
• PORTFOLIOS w1 wrf rp ?p
• A .00 1.0 4 0
• B .25 .75 7.05 3.02
• C .50 .50 10.10 6.04
• D .75 .25 13.15 9.06
• E 1.00 .00 16.20 12.08

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rP
E
D
C
B
A
sP
rRF 4
0
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Risk Averse
rP
A
Portfolio A is optimal
A
sP
0
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Not quite as risk averse
rP
B
Portfolio B is optimal
sP
0
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Capital Market Theory

• Adding leverage
• Lending is equivalent to having a positive weight
  for the risk-free asset. We are investing some
  money in the risk-free asset.
• Borrowing occurs when we have a negative weight
  on the risk-free asset. We take the borrowed
  money and invest it in the risky asset. This
  allows us to move to a point of higher return,
  but also higher risk.

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Capital Market Theory

• Adding Leverage
• Leverage extends our set of possibilities beyond
  100 invested in a risky asset. In terms of the
  efficient frontier with the risk-free asset, this
  pushes our set of possible choices to the right
  of the risky asset.

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IV. Adding leverage

• Examples
• Lets take the example that we worked in the last
  section and allow for borrowing
• Port F Port G Port H Port I
• wr 1.25 1.50 1.75 2.00
• wrf -0.25 -0.50 -0.75 -1.00

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Adding leverage

• Examples
• Return calculations
• rF (1.25)(0.162) (-0.25)(0.04) 0.1925
• rG (1.50)(0.162) (-0.50)(0.04) 0.2230
• rH (1.75)(0.162) (-0.75)(0.04) 0.2535
• rI (2.00)(0.162) (-1.00)(0.04) 0.2840

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Adding leverage

• Examples
• Risk Calculations
• ?F (1.25)(0.1208) 0.1510
• ?G (1.50)(0.1208) 0.1812
• ?H (1.75)(0.1208) 0.2114
• ?I (2.00)(0.1208) 0.2416

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rP
I
E
C
A
sP
rRF 4
0
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Adding Leverage and its effect on investors and
the efficient frontier
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Capital Market Theory

• Adding Leverage
• As can be seen in the graph, lending occurs on
  the left side of the risky asset and borrowing
  occurs on the right side of the risky asset.
• Again, borrowing allows us to extend our set of
  possible choices beyond 100 in the risky asset.
  Also, in this graph we can see more clearly that
  the combinations involving the risk-free asset
  and a risky asset dominate all points below them.

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Risk Averse
rP
A
Portfolio A is still optimal
A
sP
0
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Not quite as risk averse
B
rP
B
Portfolio B is optimal
sP
0
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Capital Market Theory

• The capital market line (CML)
• Consider what happens if our efficient frontier
  represents the efficient combinations involving
  all assets. We again add the risk-free asset,
  and again, the efficient frontier for
  combinations involving the risk-free asset
  becomes a straight line between the risk-free
  asset and risky portfolio.

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Capital Market Theory

• The capital market line (CML)
• Eventually we will reach a point at which the
  efficient frontier created between the risk-free
  asset and the risky portfolio is just tangent to
  the old efficient frontier.
• This tangency portfolio is known as the market
  portfolio. Thus, all investors will want to be
  on this efficient frontier created between the
  risk-free asset and the market portfolio.

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Capital Market Theory

• The capital market line (CML)
• -- The line formed by RFR and M is known as the
  capital market line (CML). All portfolios on the
  CML are combinations of the market portfolio and
  the risk-free asset.

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The Capital Market Line
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Capital Market Theory

• The capital market line (CML)
• The market portfolio The market portfolio (M)
  must contain all risky assets. We require
  markets to be in equilibrium. Thus, if an asset
  were not in the market portfolio, it must have
  not demand and hence, no value.
• Therefore, if an asset has value it must be in
  the market portfolio. Also, for equilibrium to
  occur, all assets in the market portfolio should
  be included according to their market value
  weights.

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Capital Market Theory

• The capital market line (CML)
• Risk in the market portfolio Since the market
  portfolio contains all assets, it represents a
  completely diversified portfolio. All individual
  asset risk is diversified away. We are left with
  only systematic risk.
• Given that the we will only invest in the market
  portfolio and given that the only relevant risk
  is market risk, the important risk measure for an
  individual asset is its covariance with the
  market portfolio. This is a risk measure that
  everyone will agree on.

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Capital Market Theory

• Some results of the capital market line
• 1. Slope of the capital market line

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Capital Market Theory

• Some results of the capital market line
• 1. Slope of the capital market line

This is the marginal rate of substitution between
risk and return that all investors will agree
upon.
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Capital Market Theory

• Some results of the capital market line
• 1. Slope of the capital market line
• Managers can use this market determined price of
  risk to evaluate investment projects regardless
  of the risk preferences of individual
  shareholders. Every shareholder will unanimously
  agree on the price of risk, even though they may
  have different degrees of risk aversion.

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Capital Market Theory

• Some results of the capital market line
• 2. Two-fund separation theorem
• The CML shows us that all investors will have a
  utility maximizing portfolio that is a
  combination of the risk-free asset and the market
  portfolio. Hence, investors will choose to
  invest in only two funds, the risk-free asset and
  the market portfolio.

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Capital Market Theory

• Some results of the capital market line
• 3. Separation theorem
• The CML also shows us that a persons investment
  decision is separate from the financing decision.
  
• A person will always invest in some combination
  of the risk-free asset and the market portfolio.
  Your risk preferences will determine if you lend
  money (positive amount in RFR) or borrow money
  (negative amount in RFR).
• Why is this important?