Lecture 5

Question 1

If the FOC of the utility function is positve and the SOC negative then…

i) there is positive investment in the risky asset and the expected return of the risky asset is higher than the risk free rate
ii) i) there is negative investment in the risky asset and the expected return of the risky asset is lower than the risk free rate

Answer 1

i)

Question 2

When do we say that portfolio A dominates portfolio B

Answer 2

Portfolio A dominates Portfolio B if the expected return is weakly higher and the volatility is strictly lower OR the expected return is strictly higher and the volatility is weakly lower

Question 3

What is the name of the locus of all undominated portfolios in the mu-sigma space

Answer 3

Efficient frontier

Question 4

How does the graphical representation of all possible portfolios in the mu-sigma space look like when we have a two assets case with perfect correlation

Answer 4

Straight line which connects both assets

Question 5

How does the portfolio variance of a two asset case with imperfect correlation compare with the simple weighted average of the two asset’s volatility

Answer 5

The portfolio variance for the case with the imperfect correlation is lower than weighted average
of s1 and s2 due to gains from diversification

Question 6

What is the difference between the minimum-variance frontier and the efficient frontier

Answer 6

The arc between the two assets represents the minimum variance frontier -
combinations of the two assets that have the minimum variance for all arbitrary
levels of expected return.

However portfolios in the top half of this arc dominate those in the bottom half as
they provide a higher expected return for the same levels of risk.

Hence, the efficient frontier is only the top half of the minimum variance frontier, i.e.
the part above the minimum variance portfolio (MVP).

Question 7

What do you study when you like money and maths

Answer 7

Finance

Question 8

What is the return of the MVP for a 2 asset case with perfect negative correlation

Answer 8

The risk free return

Question 9

How do you mathematically construct the efficient frontier in the two asset case (one of them the risk free asset)

Answer 9

You take the formula of the expected return of two assets and insert the rearranged volatility equation

Question 10

How does the Minimum Variance frontier for n assets differ in shape from the case with 2 asset

Answer 10

It does not

Question 11

What happens to the minimum variance frontier when adding more assets to the portfolio of the investor

Answer 11

Improves diversification and typically moves the frontier to the left

Question 12

What is further needed for construction of the tangecy portfolio when having established the minimum variance frontier

Answer 12

A risk free asset

Question 13

What happen to the construction of the tangency portfolio when there is a spread between the risk-free deposit and lending rates

Answer 13

We get two tangency portfolio

For low risk appetites (i.e. low portfolio variance) linear combinations of the long
positions in the risk-free asset and portfolio T1 (tangency portfolio for the lower deposit rate) dominate other portfolios.
I In the medium risk appetite region, between T1 and T2 on the portfolio frontier, it is
optimal to not hold the risk free but to hold efficient combinations of the other n
assets instead. This is because in this region you cannot short the risk free asset at
rate rf ,1 nor go long on the risk-free asset at rate rf ,2.
In the higher risk appetite region, linear combinations of T2 (tangency portfolio with the higher lending rate) and (short positions in)
the risk free asset dominate.

Question 14

What is the two fund separation theorem

Answer 14

All optimal portfolios would consist of combinations of the risk-free
asset and the tangency portfolio T.

The optimal portfolio of risky assets can be identified independent
of the knowledge of individual risk preferences.

Question 15

What is the slope called of the line which combines the tangency portfolio with the risk free rate

Answer 15

The sharpe ratio

Question 16

What is the sharpe ratio

Answer 16

It gives the price of risk, i.e. how much one should be
compensated in expectation for undertaking an additional unit of
risk

The lower the Sharpe Ratio for an asset the less efficient it is by itself

Question 17

What does the CAPM assumes regarding the prices of the assets

Answer 17

No derivation (endogenous) of supply functions of assets. It
assumes that supply equals the demand for assets and, therefore,
observed prices are equilibrium prices (economy-wide asset
holdings are investors’ optimal asset demands)

It determines asset returns relationships so that equilibrium prices
coincide with the observed asset prices

Question 18

What are the assumptions for the CAPM

Answer 18

1 Each investor maximizes a mean-variance utility , which
increases with an increase in the expected value of consumption
and decreases with an increase in the variance of consumption
2 All investors share a common time horizon and homogeneous
beliefs about expected returns and variances of existing assets
3 There exists a risk-free asset and short sales of the risk-free asset are
allowed
4 The endowments of all agents are traded

Question 19

Why do all existing risky asset must belong to T (the tangecy portfolio)

Answer 19

By contradiction. In equilibrium, following from the two fund
separation theorem, all agents would hold either the tangency
portfolio T or a linear combination of T and the risk-free asset. If
some of the risky assets were not in T, then there would be no
demand for them. However, all risky assets exist in positive supply.
Hence, all risky assets must belong to the tangency portfolio

Question 20

The CML equation applies only to efficient portfolios.

What is the equation of expected returns for any asset (i.e. those
that may not belong to the efficient frontier)?

Answer 20

Zero CAPM method

For any frontier portfolio p, except the minimum variance portfolio,
there exists a unique frontier portfolio with which p has zero
covariance. We call this portfolio the zero covariance portfolio
relative to p and denote its vector of portfolio weights by ZC(p) (nothing else than the portfolio on the other side of the frontier line)

Additionally we find that if portfolio p is efficient, then portfolio ZC(p) is innefficient and vice versa

Question 21

How does the diagrammatic derivation of the zero beta capm look like

Answer 21

That’s a bit tricky to say in words. Lecture 5 slide 44 is probably more informative

Question 22

Is the zero capm pareto efficient

Answer 22

In the zero-beta CAPM we do not have a riskless asset

Thus, agents will optimize on the minimum variance frontier

Since the frontier is concave their MRSs will not be equal

The allocation is not Pareto Optimal

Question 23

Is the CAPM pareto efficient

Answer 23

Given that a risk-free rate exists all agents optimize on the Capital
Market Line

Thus, the marginal rates of substitution (the slope of the
indifference curve) are all equal to the Sharpe Ratio, which is the
slope of the CML

Consequently, the allocation is Pareto Optimal

Given their risk-aversion they will choose a different exposure to
the market portfolio