Corner portfolio

[francis]

For example, if there are 4 Cornor portfolios
1. Expected return : 10% ; Sharpe ratio : 0.4
2. Expected return : 8% ; Sharpe ratio : 0.5
3. Expected return : 6.5% ; Sharpe ratio : 0.3
4. Expected return : 5.8% ; Sharpe ratio : 0.32
risk free asset return is 4%
Q. if return requirement is 6%, what portfolio will you choose?
A. choose portfolio 2 + risk free asset, or choose corner portfolio 3 +4. How about if prohibit short positions or the use of margin affect the result. Pls help!
I choose portfolio 2 + risk free asset, no matter permit short position or not, am I right?

 

[rahuls]

Yes, always use the tangency protfolio (Highest sharpe) & Risk free
if you choose CP3 & CP4, Sharpe ratio would lowered than the from combining CP3 & Risk free
1) Combine with Risk free
Expected return : 8% ; Sharpe ratio : 0.5, Risk free = 4%
6% = W2 X 8% + (1-w) X 4%—–> W2 = 0.5, Wf = 0.5
Std Dev -> SR = (Exp retrn - risk free) / Std Dev ….So STD DEV2 = (8-4)/ 0.5 = 8%
Std. dev (using linear approximation) of Combined = W2 * 8% = 0.5* 8% = 4%
Sharpe Ratio of combined = (6 -4 ) / 4 = 0.5
2) Combination of CP3 & CP 4…after solving W3 = 28.57%, W4 = 71.42%
Using linear approximation Std Dev of this portfolio = 0.2857 * 8.33 + 0.7142 * 5.625 = 6.39
Hence Sharpe Ratio = 6-4 / 6.39 = 0.31 < 0.5
You look to combine CP3 & CP4..when there are restriction on borrowing or short selling

 

[francis]

Let me double confirm. When there are restriction on borrowing or short selling, we can still use tangency protfolio (Highest sharpe) & Risk free to form investment portfolio which have highest sharp ratio if we do not need borrowing (like example, risk free asset is positive weight, and CP3 weight is less than or equal to 100%.
Thank you. Just wonder why study materials won’t ask us to check sharp ratio first. Study material or question alwasy use CP3 and CP4 when there are restriction on borrowing. They should check highest Sharp Ratio CP first. Right?

 

[rahuls]

francis wrote:
When there are restriction on borrowing or short selling, we can still use tangency protfolio (Highest sharpe) & Risk free to form investment portfolio which have highest sharp ratio if we do not need borrowing (like example, risk free asset is positive weight, and CP3 weight is less than or equal to 100%.

As in the mentioned case, Required return was lower than the one offered by tangency portfolio (i.e 6 < 8). So we could combine it with risk free asset to produce higher risk adjusted performance.
Problem would have come when return objective say 10% is higher than Tangency portfolio. Here we would have to short sell Risk free asset to invest more than 100% in tangency portfolio
10% = W 8 + (1-W) * 4 or
150% = W….so risk free asset weight = - 50%….To be able to do this, there shouldn’t be any restriction on borrowing/short selling. If there are restriction we would use two corner portfolio between which lies our return requirement.