Mean variance optimisation

Question 1

In the two asset cse when is the efficient frontier all portfolios? - With reference to dv/dh for variance

Answer 1

Includes all portfolios when dv/dh is zero at h=0.

Question 2

Summarise for equatorial, northern and southern case where eta infinity lies in two asset case

Answer 2

Equatorial: Lies between Mu0 and Mu1
Northern: Is equal to or less than Mu 0 and less than Mu1
Southern: Is greater than or equal to Mu1 which si greater than Mu0

Question 3

Summarise for equatorial, northern and southern case how Sigma0-CorrelationSigma1 behaves and Sigma1-CorrelationSigma0 behaves

Answer 3

Equatorial: Sigma0-CorrelationSigma1>0, Sigma1-CorrelationSigma0>0
Northern:Sigma0-CorrelationSigma1<=0, Sigma1-CorrelationSigma0>0
Southern:Sigma0-CorrelationSigma1>0, Sigma1-CorrelationSigma0<=0

Question 4

Summarise for equatorial, northern and southern case where the efficient frontier is in terms of H

Answer 4

Equatorial: hmin<=h<=1
Northern:0<=h<=1
Southern: h=1

Question 5

How far can we go into positive correlation before slipping from equatorial case to northern/southern

Answer 5

If sigma0<sigma 1 :
Equatorial: p<sigma0/sigma1
Northern: p>=sigma0/sigma1
If sigma0>sigma 1 :
Equatorial: p<sigma1/sigma0
Southern: p>=sigma1/sigma0

Question 6

If sigma 0<sigma 1 and were in the equatorial case. Where is there no portfolio with standard deviation x

Answer 6

Where x<sigma min

Question 7

If sigma 0<sigma 1 and were in the equatorial case. Where is there only one unique portfolio with sigma x

Answer 7

MRP, x=sigma min OR if no short selling when x>sigma 0

Question 8

If sigma 0<sigma 1 and were in the equatorial case. Where is there two unqiue solutions/portfolios with standard deviation x

Answer 8

When x is between sigma min and sigma 0

Question 9

Define the sharpe ratio

Answer 9

Risk premium (expected returns minus the risk free rate) divided by the standard deviation fo return. It is not defined for a risk free asset.

Question 10

How to check in the equatorial case if there exists a tangency portfolio

Answer 10

If risk free rate exceeds the extrapolated return at asset 1 then there is no tangency portfolio

Question 11

In the equatorial case if there is no tangency portfolio what is the maximal sharpe ratio

Answer 11

Asset 1

Question 12

Describe in the equatorial case of risky assets the possible 3 asset frontier with a risk free asset

Answer 12

r<eta1 - Frontier is from RF to Asset 1 via T
eta1<r<mu1 - Frontier is from RF to Asset 1
r>=Mu1 - Frontier is risk free only

Question 13

Describe in the northern case of risky assets the possible 3 asset frontier with a risk free asset

Answer 13

r<eta0 - frontier is from risk free to asset 1 via asset 0
eta0<r<eta1 - Frontier is from Risk free to asset 1 via T
eta1<r<mu1 - Frontier is from risk free to Asset 1
r>mu1 - Frontier is risk free only

Question 14

Describe in the southern case of risky assets the possible 3 asset frontier with a risk free asset

Answer 14

r<mu1 - Frontier is from risk free to asset 1
r>Mu1 - Risk free only

Question 15

Why do combinations of a risk free asset and a fixed risky portfolio all share a common sharpe ratio?

Answer 15

Consider mixture 1-h in RF and h in risky holding. The way in which the portfolio variance is calculated and the portfolio risk premium we are multiplying by h on top and bottom of fraction - this h will cancel given a constant sharpe ratio.

Question 16

When solving for a portfolio on efficient frontier between two risky assets and RF asset what must you ensure to do

Answer 16

When mixing the risk free asset and 2 risky assets must find the sharpe ratio point on the risky curve for that which you want to estimate
If you have a certain mean/sd then solve using this point and risk free rate as frontier to solve along.
Sharpe ratio must be equal to the share ratio of the target portfolio

Question 17

Correlation between portfolios at latitude Lamda 1 and 2

Answer 17

cos(Lamda1-lamda2)

Question 18

Why is determination of the correlation matrix of efficient portfolios always zero?

Answer 18

Any portfolio along efficient frontier are linear combinations of each other. So all portfolios are lienarly dependent. Linear combination means that the correlation matrix is singular and implies a determinant of zero. This determinant means the matrix does not have an inverse and is singular. This implies that the system of equations is linearly dependent

Question 19

What is proportions in a minimum risk portfolio for uncorrelated assets

Answer 19

The MRP has holding inversely proportional to the variance

Question 20

What does the holding vector represent in the multi asset case?

Answer 20

Does not represent percentages - number of assets which when multiplied by the inital asset prices will add up to the inital amount available to invest

Question 21

What are the difficulties when developing a portfolio theory based on geometric rather than arithmetic mean returns?

Answer 21

Geometric mean returns are intuitive for a long term work as they reflect compounding returns over time - however they lack analytical tractability of arithmetic means. Portfolio mixtures are always linear combinations of constituents so arithmetic mean return of a portfolio is the weighted arithmetic mean of returns for constituents. Optimizations’ is also messy with geometric means