Unformatted text preview: P 0 ) ( 2" ai! i , j = # µ j $ r0
i =1 2! 2 = " ( µ P # r0 )
P ( 2! j , P = " µ j # r0 ) 2! 2 = " ( µ P # r0 )
P ( 2! j , P = " µ j # r0 ) Separation or Two Fund
Theorem Formal Optimization Problem
n ( r0 + " a j µ j ! r0 = µ P 2! = " ( µ P # r0 )
2
P ) Any efficient portfolio can be obtained as a combination
of two investments: P and the riskless investment.
Mean
return P r Mean
return
standard
deviation Sharpe Ratio Sharpe Ratio Let j be an asset (or a portfolio of assets). Then the Sharpe
ratio sj of asset j is defined as sj = µ j ! r0
"j Let j be an asset (or a portfolio of assets). Then the Sharpe
ratio sj of asset j is defined as
Find all assets j
µ j ! r0
with the same
sj =
"j
Sharpe ratio as A.
µ
Asset A r0
! Sharpe Ratio Sharpe Ratio Let j be an asset (or a portfolio of assets). Then the Sharpe
ratio sj of asset j is defined as
Find all assets j
µ j ! r0
with the same
sj =
"j
Sharpe ratio as A.
µ
Asset A r0
! Let j be an asset (or a portfolio of assets). Then the Sharpe
ratio sj of asset j is defined as sj = µ j ! r0
"j Sharpe Ratio Sharpe Ratio Let j be an asset (or a portfolio of assets). Then the Sharpe
ratio sj of asset j is defined as sj = µ j ! r0
"j µ j = r0 + s j! j Let j be an asset (or a portfolio of assets). Then the Sharpe
ratio sj of asset j is defined as
Find all assets j
µ j ! r0
with the same
sj =
"j
Sharpe ratio as A.
µ
Asset A µ j = r0 + s j! j
r0
! Sharpe Ratio
sj = µ j ! r0
"j
µ Sharpe Ratio µ j = r0 + s j! j sj =
Along which line
do assets have
a higher Sharpe
ratio? r0 µ j ! r0
"j µ j = r0 + s j! j µ
er
igh
h io
rat
e
arp
Sh Along which line
do assets have
a higher Sharpe
ratio? r0
! ! Sharpe Ratio
sj = µ j ! r0
"j Sharpe Ratio µ j = r0 + s j! j µ
er
igh
h sj = io
rat
e
arp
Sh What is better:
A higher or a
lower Sharpe
ratio? r0 µ j = r0 + s j! j µ
er
igh
h io
rat
e
arp
Sh Higher Sharpe
Ratio: Offers
Higher return
with same stdv. r0
! Sharpe Ratio ! Estimating Sharpe Ratios Efficient Portfolios have the highest Sharpe
ratio among all feasible Portfolios!
Mean
return µ j ! r0
"j P r standard
deviation We now estimate Sharpe Ratios for MSFT
and AAPL, using weekly data from Feburary
25, 2011 to February 25, 2013. Estimating Sharpe Ratios Estimating Sharpe Ratios We now estimate Sharpe Ratios for MSFT
and AAPL, using weekly data from Feburary
25, 2011 to February 25, 2013. We now estimate Sharpe Ratios for MSFT
and AAPL, using weekly data from Feburary
25, 2011 to February 25, 2013. Assume r0=2%, i.e., the weekly rate is
1.021/52"1. Assume r0=0.2%, i.e., the weekly rate is
1.0021/52"1.
Sharpe Ratios: MSFT 0.081
AAPL 0.046
Looks like Apple gets higher riskadjusted
return. Sharpe Ratio
µ j ! r0
sj =
"j Estimating Sharpe Ratios µ j = r0 + s j! j
AAPL has higher
Sharpe ratio µ Problem: There is uncertainty about the
estimates. Need confidence bands... In other
words, it could be pure “luck of the draw”
that resulted in a higher Sharpe Ratio for
Apple. PL
AA r0 Sharpe Ratios: MSFT 0.046
AAPL 0.081
Looks like Apple is better managed than
Microsoft  Apple gets higher riskadjusted
return. T
MSF !...
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Full Document
 Spring '12
 GOTTHEIL
 Modern portfolio theory, sharpe ratio, sj, R0, lower Sharpe, Formal Optimization Problem

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