Mean variance optimisation Flashcards
In the two asset cse when is the efficient frontier all portfolios? - With reference to dv/dh for variance
Includes all portfolios when dv/dh is zero at h=0.
Summarise for equatorial, northern and southern case where eta infinity lies in two asset case
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
Summarise for equatorial, northern and southern case how Sigma0-CorrelationSigma1 behaves and Sigma1-CorrelationSigma0 behaves
Equatorial: Sigma0-CorrelationSigma1>0, Sigma1-CorrelationSigma0>0
Northern:Sigma0-CorrelationSigma1<=0, Sigma1-CorrelationSigma0>0
Southern:Sigma0-CorrelationSigma1>0, Sigma1-CorrelationSigma0<=0
Summarise for equatorial, northern and southern case where the efficient frontier is in terms of H
Equatorial: hmin<=h<=1
Northern:0<=h<=1
Southern: h=1
How far can we go into positive correlation before slipping from equatorial case to northern/southern
If sigma0<sigma 1 :
Equatorial: p<sigma0/sigma1
Northern: p>=sigma0/sigma1
If sigma0>sigma 1 :
Equatorial: p<sigma1/sigma0
Southern: p>=sigma1/sigma0
If sigma 0<sigma 1 and were in the equatorial case. Where is there no portfolio with standard deviation x
Where x<sigma min
If sigma 0<sigma 1 and were in the equatorial case. Where is there only one unique portfolio with sigma x
MRP, x=sigma min OR if no short selling when x>sigma 0
If sigma 0<sigma 1 and were in the equatorial case. Where is there two unqiue solutions/portfolios with standard deviation x
When x is between sigma min and sigma 0
Define the sharpe ratio
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.
How to check in the equatorial case if there exists a tangency portfolio
If risk free rate exceeds the extrapolated return at asset 1 then there is no tangency portfolio
In the equatorial case if there is no tangency portfolio what is the maximal sharpe ratio
Asset 1
Describe in the equatorial case of risky assets the possible 3 asset frontier with a risk free asset
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
Describe in the northern case of risky assets the possible 3 asset frontier with a risk free asset
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
Describe in the southern case of risky assets the possible 3 asset frontier with a risk free asset
r<mu1 - Frontier is from risk free to asset 1
r>Mu1 - Risk free only
Why do combinations of a risk free asset and a fixed risky portfolio all share a common sharpe ratio?
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.
When solving for a portfolio on efficient frontier between two risky assets and RF asset what must you ensure to do
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
Correlation between portfolios at latitude Lamda 1 and 2
cos(Lamda1-lamda2)
Why is determination of the correlation matrix of efficient portfolios always zero?
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
What is proportions in a minimum risk portfolio for uncorrelated assets
The MRP has holding inversely proportional to the variance
What does the holding vector represent in the multi asset case?
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
What are the difficulties when developing a portfolio theory based on geometric rather than arithmetic mean returns?
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
