7- 1 © admn 3116, anton miglo admn 3116: financial management 1 lecture 7: portfolio selection...
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© ADMN 3116, Anton Miglo
ADMN 3116: Financial Management 1
Lecture 7: Portfolio selection
Anton Miglo
Fall 2014
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© ADMN 3116, Anton Miglo
Topics Covered
Efficient Set of Portfolios Sharpe ratio and optimal portfolio Optimal portfolio with risk-free asset available Excel: Solver Additional readings: ch. 10-11 B
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© ADMN 3116, Anton Miglo
Investment mistakes
1. “Put all eggs in one basket”
2. Superfluous or Naive Diversification (Diversification for diversification’s sake)
a. Results in difficulty in managing such a large portfolio
b. Increased costs (Search and transaction)3. Many investors think that diversification is
always associated with lower risk but also with lower return
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© ADMN 3116, Anton Miglo
Portfolio of two positively correlated assets
Asset A
0
15
30
-15
Asset B
0
15
30
-15
Asset C=1/2A+1/2B
0
15
30
-15
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© ADMN 3116, Anton Miglo
Portfolio of two negatively correlated assets
-10
15
15
40
4040
15
0
-10
Asset A
0
Asset B
-10
0
Asset C=1/2A+1/2B
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© ADMN 3116, Anton Miglo
Recall: portfolios
For a portfolio of two assets, A and B, the variance of the return on the portfolio is:
Where: xA = portfolio weight of asset A
xB = portfolio weight of asset B
such that xA + xB = 1.
(Important: Recall Correlation Definition!)
)RCORR(Rσσx2xσxσxσ
B)COV(A,x2xσxσxσ
BABABA2B
2B
2A
2A
2p
BA2B
2B
2A
2A
2p
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© ADMN 3116, Anton Miglo
The Markowitz Efficient Frontier
The Markowitz Efficient frontier is the set of portfolios with the maximum return for a given risk AND the minimum risk given a return.
For the plot, the upper left-hand boundary is the Markowitz efficient frontier.
All the other possible combinations are inefficient. That is, investors would not hold these portfolios because they could get either more return for a given level of risk or less risk for a given level of return.
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© ADMN 3116, Anton Miglo
Efficient Portfolios with Multiple Assets
E[r]
s0
Asset 1
Asset 2Portfolios ofAsset 1 and Asset 2
Portfoliosof otherassets
EfficientFrontier
Minimum-VariancePortfolio
Investorsprefer
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© ADMN 3116, Anton Miglo
Example: Solving for a Sharpe-Optimal Portfolio
From a previous chapter, we know that for a 2-asset portfolio:
)R,CORR(Rσσx2xσxσx
r -)E(Rx)E(Rx
σ
r-)E(RRatio Sharpe
)R,CORR(Rσσx2xσxσxσ : VariancePortfolio
)E(Rx)E(Rx)E(R :Return Portfolio
BSBSBS2B
2B
2S
2S
fBBSS
P
fp
BSBSBS2B
2B
2S
2S
2P
BBssp
So, now our job is to choose the weight in asset S that maximizes the Sharpe Ratio.
We could use calculus to do this, or we could use Excel.
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© ADMN 3116, Anton Miglo
Example: Using Excel to Solve for the Sharpe-Optimal Portfolio
DataInputs:
ER(S): 0.12 X_S: 0.250STD(S): 0.15
ER(B): 0.06 ER(P): 0.075STD(B): 0.10 STD(P): 0.087
CORR(S,B): 0.10R_f: 0.04 Sharpe
Ratio: 0.402
Suppose we enter the data (highlighted in yellow) into a spreadsheet.
We “guess” that Xs = 0.25 is a “good” portfolio.
Using formulas for portfolio return and standard deviation, we compute Expected Return, Standard Deviation, and a Sharpe Ratio:
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© ADMN 3116, Anton Miglo
Example: Using Excel to Solve for the Sharpe-Optimal Portfolio, Cont.
Now, we let Excel solve for the weight in portfolio S that maximizes the Sharpe Ratio.
We use the Solver, found under Tools.
Data ChangerInputs: Cell:
ER(S): 0.12 X_S: 0.700STD(S): 0.15
ER(B): 0.06 ER(P): 0.102STD(B): 0.10 STD(P): 0.112
CORR(A,B): 0.10R_f: 0.04 Sharpe
Ratio: 0.553
Solving for the Optimal Sharpe Ratio
Given the data inputs below, we can use theSOLVER function to find the Maximum Sharpe Ratio:
Target Cell Well, the “guess” of 0.25 was a tad low….