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