Econ 424/CFRM 462
Statistical Analysis of Eﬃcient Portfolios

Eric Zivot

August 14, 2014

The CER Model and Eﬃcient Portfolios
Let  denote the return on asset  in month  and assume that  follows
CER 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 errorKey result: Sharpe ratios and eﬃcient portfolios are functions of ̂ ̂2 ̂ ;they are random variables and are subject to error

Statistical Properties of Eﬃcient portfolios• Inputs to portfolio theory are estimates from CER model ̂ and Σ̂• Sharpe ratios and eﬃcient portfolios are functions of ̂ and Σ̂• The estimated Sharpe ratio isd =d )• No easy formula for (̂ − ̂

• The estimated global minimum variance portfolio isΣ̂−11m̂ =10Σ̂−11m̂ is estimated with error because we estimate Σ using Σ̂

• No easy analytic formulas for the standard errors of the elements of m̂ =(̂1     ̂)0; i.e. no easy formula for (̂)• In addition, the expected return and standard deviation of ̂ = m̂0R
have additional sources of error due to the error in m̂ That is,̂̂ = m̂0̂̂̂ = (m̂0Σ̂m̂)12
No easy analytic formulas for SE(̂̂) and SE(̂̂)Optimizers are Error Maximizers• From our analysis of the CER model,  is estimated less precisely than That is, there is more estimation error in ̂ than ̂• Large estimation error in ̂ greatly impacts eﬃcient portfolios— Large positive errors (̂ much greater than ) leads to eﬃcient portfolios being concentrated in asset — Large negative errors (̂ much less than ) leads to eﬃcient portfoliosthat avoid asset  or shorts asset • Constraints on portfolio weights can oﬀset the impact of estimation errorin ̂Bootstrapping Eﬃcient Portfolios
The bootstrap can be used to evaluate the sampling uncertainty of Sharpe
ratios and eﬃcient portfolios.
Portfolio statistics to boostrap:• Portfolio weights• Portfolio expected returns and standard deviations

Are Eﬃcient Portfolios Constant Over Time?
Result: We have seen evidence that the parameters of the CER model forvarious 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 ̂ ̂ and ̂ imply time variation in eﬃcient
portfolios

Rolling Eﬃcient Portfolios
Idea: Using rolling estimates of  and Σ 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

Rolling Global Minimum Variance Portfolio
Idea: compute estimates of portfolio weights m over rolling windows of length :min m()0Σ̂()m() s.t. m()01 = 1m() =      Σ̂() = rolling estimate of Σ in month IfΣ̂() ≈ Σ̂+1() ≈ · · · ≈ Σ̂ ()thenm() ≈ m+1() ≈ · · · ≈ m ()Rolling Eﬃcient Portfolios
Idea: compute estimates of portfolio weights x over rolling windows of length   for  =       :min x()0Σ̂()x()x()s.t. x()01 = 1 x()0̂() = target̂() = rolling estimate of  in month Σ̂() = rolling estimate of Σ in month Ifthen̂() ≈ ̂+1() ≈ · · · ≈ ̂ ()Σ̂() ≈ Σ̂+1() ≈ · · · ≈ Σ̂ ()x() ≈ x+1() ≈ · · · ≈ x ()