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Econ 424/Amath 540

Statistical Analysis of Efficient Portfolios

Eric Zivot

August 7, 2012

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The CER Model and Efficient Portfolios

Let denote the return on asset in month and assume that followsCER model:

( 2 )

= 1 (assets)

= 1 (months)

( ) =We estimate the CER model parameters using sample statistics giving

2

Remember, the estimates 2

are are random variables and are subject

to error

Key result: Sharpe ratios and efficient portfolios are functions of2 ;

they are random variables and are subject to error

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Statistical Properties of Efficient portfolios

Inputs to portfolio theory are estimates from CER model and

Sharpe ratios and efficient portfolios are functions of and

The estimated Sharpe ratio is

d=

No easy formula for ( d)

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The estimated global minimum variance portfolio is

m=

11

1011

m is estimated with error because we estimate using

No easy analytic formulas for the standard errors of the elements ofm=(1 )

0; i.e. no easy formula for ()

In addition, the expected return and standard deviation of = m0R

have additional sources of error due to the error in m That is,

= m0

= (m

0m)12

No easy analytic formulas for SE() and SE()

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Bootstrapping Efficient Portfolios

The bootstrap can be used to evaluate the sampling uncertainty of Sharpe

ratios and effi

cient portfolios.

Portfolio statistics to boostrap:

Portfolio weights

Portfolio expected returns and standard deviations

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The CER Model and Efficient Portfolios

Result: We have seen evidence that the parameters of the CER model for

various assets are not constant over time:

Rolling estimates of and show variation over time

Implication: Since estimates of and are inputs to efficient portfolio

calculations, then time variation in andimply time variation in efficient

portfolios

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Rolling Efficient Portfolios

Idea: Using rolling estimates of and compute rolling efficient portfolios

global minimum variance portfolio

efficient portfolio for target return

tangency portfolio

efficient frontier

Look at time variation in resulting portfolio weights

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Rolling Global Minimum Variance Portfolio

Idea: compute estimates of portfolio weights mover rolling windows of length

:

minm()

m()0()m() s.t. m()

01= 1

= () =rolling estimate of in month

If

()

+1()

()then

m() m+1() m()

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Rolling Efficient Portfolios

Idea: compute estimates of portfolio weights x over rolling windows of length

for= :

minx()

x()0()x()

s.t. x()01= 1 x()

0() =target

() = rolling estimate of in month

() =rolling estimate of in month

If

() +1() ()()

+1()

()

then

x() x+1() x()