Post on 26-Jun-2020
transcript
Quantitative Finance
LECTURE 26
PORTFOLIO THEORY
&
CAPITAL ASSET PRICING MODEL (CAPM)
Minimum Variance Portfolio and Line
To find a portfolio with the smallest variance in the attainable set. This is called the minimum variance portfolio
To find the portfolio with the minimum variance among all portfolios in the attainable set whose expected return is equal to a given value . The family of such portfolios parameterized by is called the minimum variance line
Minimum Variance Portfolio
The portfolio with the minimum variance has weights
provided the denominator is non zero
Proof:
Need to minimize subject to
Consider
Equating the first partials to zero
Minimum Variance Line
The portfolio with the smallest variance among attainable portfolios with expected return has weights
𝒘 =
1𝜇𝑣
𝒖𝐶−1𝒎𝑻
𝒎𝐶−1𝒎𝑻 𝒖𝐶−1 + 𝒖𝐶−1𝒖𝑻
𝒎𝐶−1𝒖𝑻1𝜇𝑣
𝒎𝐶−1
𝒖𝐶−1𝒖𝑻
𝒎𝐶−1𝒖𝑻𝒖𝐶−1𝒎𝑻
𝒎𝐶−1𝒎𝑻
Minimum Variance Line
Proof: To minimize
FOC:
Constraints
Hence
𝒘 =
1𝜇𝑣
𝒖𝐶−1𝒎𝑻
𝒎𝐶−1𝒎𝑻 𝒖𝐶−1 + 𝒖𝐶−1𝒖𝑻
𝒎𝐶−1𝒖𝑻1𝜇𝑣
𝒎𝐶−1
𝒖𝐶−1𝒖𝑻
𝒎𝐶−1𝒖𝑻𝒖𝐶−1𝒎𝑻
𝒎𝐶−1𝒎𝑻
Example
Find the weights for the MVP and the equation of the MVL for a portfolio of three securities with
Attainable Portfolios
A security with expected return and standard deviation dominates another with return and deviation if &
A portfolio is called efficient
if there is no other portfolio
except itself that dominates
it.
Investors may select different portfolios on the efficient frontier depending on their preferences
Attainable Portfolios
Consider two portfolios on the minimum variance line with weights and
The minimum variance line consists of portfolios of type c + (1-c)
The weights of any portfolio belonging to the efficient frontier except for the minimum variance portfolio satisfy
Here is the gradient of the tangent line to the efficient frontier at the point representing the portfolio and the intercept
Capital Market Line
Consider a risk free security with return in addition to the risky securities
We saw a portfolio consisting of this and a risky security is represented by a broken line on the plane
The efficient frontier of this new portfolio is the upper half line tangent to the Markowitz bullet on the plane passing through
CAPM states that every rational investor will select his portfolio on this line called the capital market line
Capital Market Line
So each investor will be holding the same relative proportion of risky securities.
This means that the portfolio has to contain all risky securities with weights equal to their relative share in the market
This is called the Market
Portfolio
The CML satisfies
Example
In a market of of three securities
consider the portfolio on the efficient frontier with expected return find and such that the weights in the portfolio satisfy
Beta Factor
Measure of how the return on a given portfolio or a single security will react to trends affecting the whole market
Plot vs. For different market scenarios and find the line of best fit
To this end regard as predictions for the return on the given portfolio
The residual is
Condition for best fit is as a function of
attains its minimum at
Beta Factor
This gives the following
Example
Suppose the returns on a given portfolio and on the market take the following values in different scenarios
Compute and
Scenario Probability Return Return
0.1 -5% 10%
0.3 0% 14%
0.4 2% 12%
0.2 4% 16%
Systematic and Unsystematic Risk
We can write the risk as
The first term is the diversifiable or unsystematic risk
The second term is the undiversifiable or systematic risk, so is a measure of systematic risk
Security Market Line
The expected return on a portfolio or an individual security is a linear function of beta