1.4.2 Capital Market Theroy

1. Capital Allocation Line(CAL)

Functions of CAL
The portfolios available to an investor through combining the risk-free asset with one risky asset.

R

p

=

w

i

R

i

+

w

r

f

R

r

f

R_p=w_iR_i+w_{rf}R_{rf}

Rp​=wi​Ri​+wrf​Rrf​

σ

p

=

w

i

σ

i

\sigma_p = w_i\sigma_i

σp​=wi​σi​

E

(

R

p

)

=

R

f

+

E

(

R

i

)

−

R

f

σ

i

∗

σ

p

E(R_p) = R_f+\frac{E(R_i)-R_f}{\sigma_i}*\sigma_p

E(Rp​)=Rf​+σi​E(Ri​)−Rf​​∗σp​

Selection among CALs
The CAL with highest Sharpe ratio should be selected, it provides the highest utility among all CALs.

The optimal CAL is tangent to efficient frontier of risky assets.

Optimal portfolio along CAL

• Investor should choose portfolio “m” to invest as it supplies
  the most stratification(utility).
  

• More risk-averse investor (A=4) will select portfolio “j” (less in risky asset), and less risk-averse investor (A=2) will select portfolio “k” (more in risky asset).
  

• CAL ( C ) should be select because it provides the highest utility among these three CALs.
  The Sharpe ratio of CAL ( C ) is the highest.

• Optimal risky portfolio & Optimal investor portfolio.
  The optimal investor portfolio is combined with the optimal risky portfolio and the risk-free asset.
  

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2. Capital Market Line (CML)

• Assuming all investors have a homogeneous expectation.
  • All investors have identical efficient frontier of risky portfolio and identical optimal risky portfilio, which is the market portfolio.
• Capital market line(CML) is a special CAL that includes all possible combinations of risk-free asset and market portfolio.

• CML is tangent to the efficient frontier at a point representing market portfolio.

• The equation of the CML is:
  

  E

  (

  R

  p

  )

  =

  R

  f

  +

  E

  (

  R

  M

  )

  −

  R

  f

  σ

  M

  ∗

  σ

  p

  E(R_p)=R_f+\frac{E(R_M)-R_f}{\sigma_M}*\sigma_p

  E(Rp​)=Rf​+σM​E(RM​)−Rf​​∗σp​

  • The intercept is risk-free rate.
  • The slope is Sharpe Ratio of market portfolio.
  • $S{R}_{market-portfolio}=\frac{E(R_M)-R_f}{{\sigma }_M}$
  • CML is essentially the efficient frontier for all assets under the assumption that all investors have a homogeneous expectation.

• Market portfolio assumes that the market achieves equilibrium and accordingly includes all of the risky assets in the economy weighted by their relative market values.

• In practice, stock market indices are used to represent the market portfolio.

  • E.g., For the United Stated, S&P index or the wider-based Russell 2000.

  • E.g., For the U.K and European markets, FTSE 100 and the Euro Stoxx 50.