basic investment models and their statistical analysiszhu/ams516/handout02.pdfoutline asset returns...

Outline Asset Returns Markowitz’s Portfolio Theory CAPM Multifactor Pricing Models Resampling Methods

Basic Investment Models and TheirStatistical Analysis

Haipeng Xing

Haipeng Xing SUNY Stony Brook

Basic investment models and their statistical analysis


Outline

1 Asset Returns

2 Markowitz’s Portfolio Theory

3 Capital Asset Pricing Model (CAPM)

4 Multifactor Pricing Models

5 Applications of resampling to portfolio management




Discrete returns

Let Pt denote the asset price at time t. Suppose the asset does not havedividends over the period from time t ! 1 to time t.

The one-period net return on this asset is Rt = (Pt ! Pt!1)/Pt!1,and the one-period gross return is Pt/Pt!1 = 1 + Rt.

The gross return over k periods is defined as

1 + Rt(k) = Pt/Pt!k =k!1!

j=0

(1 + Rt!j),

and the net return over these periods is Rt(k). In practice, weusually use years as the time unit. The annualized gross return forholding an asset over k years is (1 + Rt(k))1/k, and the annualizednet return is (1 + Rt(k))1/k ! 1.




Continuously compounded return (log return)

The logarithmic return or continuously compounded return on anasset is defined as rt = log(Pt/Pt!1).

One property of log returns is that, as the time step !t of a periodapproaches 0, the log return rt is approximately equal to the netreturn:

rt = log(Pt/Pt!1) = log(1 + Rt) " Rt.

A k-period log return is the sum of k simple single-period log returns(the additivity of multiperiod returns):

rt[k] = logPt

Pt!k=

k!1"

j=0

log(1 + Rt!j) =k!1"

j=0

rt!j .




Adjustment for dividends

Many assets pay dividends periodically. In this case, the definition ofasset returns has to be modified to incorporate dividends. Let Dt bethe dividend payment between times t! 1 and t. The net return andthe continuously compounded return are modified as

Rt =Pt + Dt

Pt!1

! 1, rt = log(Pt + Dt) ! log Pt!1.

Multiperiod returns can be similarly modified. In particular, ak-period log return now becomes

rt[k] = log

#k!1!

j=0

Pt!j + Dt!j

Pt!j!1

$

=k!1"

j=0

log

%Pt!j + Dt!j

Pt!j!1

&

.




Excess and portfolio returns

Excess return refers to the di!erence rt ! r"t between the asset’s logreturn rt and the log return r"t on some reference asset, which isusually taken to be a riskless asset such as a short-term U.S.Treasury bill.

Suppose one has a portfolio consisting of p di!erent assets. Let wi

be the weight of the portfolio’s value invested in asset i. SupposeRit and rit are the net return and log return of asset i at time t,respectively. The overall net return Rt and a corresponding formulafor the log return rt of the portfolio are

Rt =p"

i=1

wiRit, rt = log'1 +

p"

i=1

wiRit

("

p"

i=1

wirit.




Statistical models for asset prices and returns

A commonly used model for risky asset prices Pt is geometricBrownian motion (GBM), with volatility ! and instantaneous rate ofreturn ", dPt/Pt = "dt + !dwt, where {wt, t # 0} is Brownianmotion. The price process Pt is called, and has the explicitrepresentation Pt = P0 exp{(" ! !2

2)t + !wt}.

The discrete-time analog of this price process has returnsrt = log(Pt/Pt!1) that are i.i.d. N(µ, !2) with µ = " ! !2/2.

Remark 1

The empirical distributions of asset prices and returns are much morecomplicated and voluminous than those summarized here, they areusually characterized by more advanced probabilistic or statistical models.




Portfolio weights

For a single-period portfolio of p assets with weights wi, the return of theportfolio over the period can be represented by R =

)pi=1 wiRi. The

mean µ and variance !2 of the portfolio return R are given by

µ =p"

i=1

wiE(Ri), !2 ="

1#i,j#p

wiwjCov(Ri, Rj).

With the weights wi satisfying the constraints)p

i=1 wi = 1 (and wi # 0if short selling is not allowed).

Remark 2

Such diversification via a portfolio tends to reduce the “risk,” asmeasured by the return’s standard deviation, of the risky investment.




Geometry of e!cient sets

A

P1

P2

!

µ

# = !1

# = !1

# = 1# = !0.6 # = 0.3

Figure 1: Feasible region for twoassets.

Consider the case of p = 2 riskyassets whose returns have meansµ1, µ2, standard deviations !1, !2,and correlation coe"cient #. Letw1 = $ and w2 = 1! $ ($ $ [0, 1]).Then the mean return of theportfolio is µ($) = $µ1+ (1! $)µ2,and its volatility !($) is given by!2($) = $2!2

1 + 2#$(1 ! $)!1!2

+(1! $)2!22 . Figure 1 plots the

curve {(!($), µ($)) : 0 % $ % 1}for di!erent values of #.





Figure 2: Feasible region for p ! 3

assets.

The set of points in the (!, µ) planethat correspond to the returns ofportfolios of the p assets is called afeasible region. For p # 3, thefeasible region is a connectedtwo-dimensional set. It is alsoconvex to the left in the sense thatgiven any two points in the region,the line segment joining them doesnot cross the left boundary of theregion.





!

µEfficientfrontier

Minimum!variancepoint

Figure 3: E!cient frontier andminimum-variance point.

For a given value µ of the meanreturn, the feasible point with thesmallest ! lies on this left boundary,which is the minimum-varianceportfolio (MVP). For a given value! of volatility, investors prefer theportfolio with the largest meanreturn, which is achieved at anupper left boundary point of thefeasible region. The upper portion ofthe minimum-variance set is calledthe e"cient frontier.




Computation of e!cient portfolios

Let r = (R1, . . . , Rp)T denote the vector of returns of p assets,1 = (1, . . . , 1)T , w = (w1, . . . , wp)T , µ = (µ1, . . . , µp)T = (E(R1),. . . , E(Rp))T , and ! = (Cov(Ri, Rj))1#i#j#p. Consider the case whereshort selling is allowed. Given a target value µ" for the mean return ofthe portfolio, the weight vector w of an e"cient portfolio can becharacterized by

we! = arg minw

wT !w subject to wT µ = µ", wT 1 = 1,

The method of Lagrange multipliers leads to the the explicit solution

we! =*

B!!11! A!!1µ + µ"

+C!!1µ ! A!!11

,-.D

when ! is nonsingular, where A = µT !!11 = 1T!!1µ, B = µT!!1µ,C = 1T!!11, and D = BC ! A2.





The variance of the return on this e"cient portfolio is

!2e! =

+B ! 2µ"A + µ2

"C,/

D.

The µ" that minimizes !2e! is given by

µminvar =A

C,

which corresponds to the global MVP with variance !2minvar = 1/C and

weight vectorwminvar = !!11

/C.

For two MVPs with mean returns µp and µq, their weight vectors aregiven by (5) with µ" = µp, µq, respectively. From this it follows that thecovariance of the returns rp and rq is given by

Cov(rp, rq) =C

D

'µp !

A

C

('µq !

A

C

(+

1

C.





When short selling is not allowed, we need to add the constraint wi # 0for all i (denoted by w # 0). Hence the optimization problem (??) hasto be modified as

we! = argminw

wT !w subject to wT µ = µ", wT 1 = 1, w # 0.

Such problems do not have explicit solutions by transforming them to asystem of equations via Lagrange multipliers. Instead, we can usequadratic programming to minimize the quadratic objective functionwT !w under linear equality constraints and nonnegativity constraints.




Estimation of µ and ! and an example

Table 1 gives the means and covariances of the monthly log returns of sixstocks, estimated from 63 monthly observations during the period August2000 to October 2005. The stocks cover six sectors in the Dow JonesIndustrial Average: American Express (AXP), Citigroup Inc. (CITI),Exxon Mobil Corp. (XOM), General Motors (GM), Intel Corp. (INTEL),and Pfizer Inc. (PFE).

Table 1: Estimated mean (in parentheses, multiplied by 102) and covariance

matrix (multiplied by 104) of monthly log returns.

AXP CITI XOM GM INTEL PFE

AXP (0.033) 9.01

CITI (0.034) 5.69 9.64

XOM (0.317) 2.39 1.89 5.25

GM (!0.338) 5.97 4.41 2.40 20.2

INTEL (!0.701) 10.1 12.1 0.59 12.4 46.0

PFE (!0.414) 1.46 2.34 9.85 1.86 1.06 5.33




Estimation of µ and ! and an example

0.0178 0.018 0.0182 0.0184 0.0186 0.0188 0.019!5

0

5

10

15

20

x 10!4

Month

ly log r

etu

rn

Monthly standard deviation

Figure 4: Estimated e!cient frontier of portfolios thatconsist of six assets.

Figure 4 shows the“plug-in” e"cientfrontier for these sixstocks allowing shortselling. By “plug-in”we mean that themean µ andcovariance ! in (5)are substituted by theestimated values 0µand 0! given in Table1.




The CAPM

The capital asset pricing model, introduced by Sharpe (1964) and Lintner(1965), builds on Markowitz’s portfolio theory to develop economy-wideimplications of the trade-o! between return and risk, assuming that thereis a risk-free asset and that all investors have homogeneous expectationsand hold mean-variance-e"cient portfolios.

Suppose the market has a risk-free asset with return rf (interest rate)besides n risky assets. If both lending and borrowing of the risk-free assetat rate rf are allowed, the feasible region is an infinite triangular region.The e"cient frontier is a straight line that is tangent to the originalfeasible region of the n risky assets at a point M , called the tangentpoint; see Figure 5. This tangent point M can be thought of as an indexfund or market portfolio.




The CAPM

!

µ

M

rf

Efficientfrontier

!

µ

M

rf

Efficientfrontier

Figure 5: Minimum-variance portfolios of risky assets and a risk-free asset.Left panel: short selling is allowed. Right panel: short selling is not allowed.




The CAPM

One Fund Theorem

There is a single fund M of risky assets such that any e"cient portfoliocan be constructed as a linear combination of the fund M and therisk-free asset.

When short selling is allowed, the minimum-variance portfolio (MVP)with the expected return µ" can be computed by solving the optimizationproblem

minw

wT !w subject to wT µ + (1 ! wT 1)rf = µ".

The problem has an explicit solution for w when ! is nonsingular:

we! =(µ" ! rf )

(µ ! rf1)T!!1(µ ! rf1)!!1(µ ! rf1)

& !!1(µ ! rf1)/[1T!!1(µ ! rf1)](The fund)




Sharpe ratio and the capital market line

For a portfolio whose return has mean µ and variance !2, its Sharperatio is (µ ! rf )/!, which is the expected excess return per unit ofrisk.

The straight line joining (0, rf ) and the tangent point M in Figure5, which is the e"cient frontier in the presence of a risk-free asset, iscalled the capital market line and given by

µ = rf +µM ! rf

!M!;

i.e., the Sharpe ratio of any e"cient portfolio is the same as that ofthe market portfolio.




Beta and the security market line

The beta, denoted by %i, of risky asset i that has return ri is definedby %i = Cov(ri, rM )/!2

M . The CAPM relates the expected excessreturn (also called the risk premium) µi ! rf of asset i to its betavia

µi ! rf = %i(µM ! rf ),which is referred to as the security market line.

The above linear relationship can be rewritten as

ri ! rf = %i(rM ! rf ) + &i,

in which E(&i) = 0 and Cov(&i, rM ) = 0, it follows that

!2i = %2

i !2M + Var(&i),

decomposing the variance !2i of the ith asset return as a sum of the

systematic risk %2i !2

M , which that is associated with the market, andthe idiosyncratic risk, which is unique to the asset and uncorrelatedwith market movements.




Investment implications

Using % as a measure of risk, the Treynor index is defined by(µ ! rf )/%.

The Jensen index is the $ in the generalization of CAPM toµ ! rf = $ + %(µM ! rf ). An investment with a positive $ isconsidered to perform better than the market.

Jensen (1968) perform an empirical study using the regression modelµ ! rf = $ + %(µM ! rf ) + &. His findings support the e"cientmarket hypothesis, according to which it is not possible tooutperform the market portfolio in an e"cient market.




Estimation and testing

Let yt be a q ' 1 vector of excess returns on q assets and let xt be theexcess return on the market portfolio (or, more precisely, its proxy) attime t. The CAPM can be associated with the null hypothesisH0 : ! = 0 in the regression model

yt = ! + xt" + #t, 1 % t % n, (1)

where E(#t) = 0, Cov(#t) = V, and E(xt#t) = 0.

Letting x = n!1)n

t=1 xt and y = n!1)n

t=1 yt, the ordinary leastsquares (OLS) estimates of ! and " are given by

0" =

)nt=1(xt ! x)yt)nt=1(xt ! x)2

, 0! = y ! x0". (2)





The maximum likelihood estimate of V is

0V = n!1

n"

t=1

(yt ! 0! ! 0"xt)(yt ! 0! ! 0"xt)T . (3)

The properties of OLS can be used to establish the asymptotic normalityof 0" and 0!, yielding the approximations

0" " N

%

",0V

)nt=1(xt ! x)2

&

, 0! " N

%

!,

%1

n+

x2

)nt=1(xt ! x)2

&0V

&

,

from which approximate confidence regions for " and ! can beconstructed.





When #t are i.i.d. normal and are independent of the market excessreturns xt, 0!, 0", and 0V are the maximum likelihood estimates of !, ",and V. Furthermore, the conditional distribution of (0!, 0", 0V) given(x1, . . . , xn) are expressed as

0! ( N'!,

' 1

n+

x2

)nt=1(xt ! x)2

(V

(, (4)

0" ( N'",

V)n

t=1(xt ! x)2

(, n 0V ( Wq(V, n ! 2),

with 0V independent of (0!, 0"). Moreover, we can show that, under H0,'n ! q ! 1

q

(0!T 0V!1 0!

.*1 +

x2

n!1)n

t=1(xt ! x)2

-( Fq,n!q!1. (5)

Note that (5) still holds approximately without the normality assumptionwhen n ! q ! 1 is moderate or large.




An illustrative example

We illustrate the statistical analysis of CAPM with the monthly returnsdata of the six stocks in the previous section using the Dow JonesIndustrial Average index as the market portfolio M and the 3-month U.S.Treasury bill as the risk-free asset. These data are used to estimatequantities in Table 2.

Table 2: Performance of six stocks from August 2000 to October 2005.

AXP CITI XOM GM INTEL PFE

! " 103 0.87 0.81 2.23 !2.41 !4.31 !5.21p-value! 0.72 0.76 0.40 0.59 0.52 0.06" 1.23 1.20 0.52 1.44 2.28 0.46"2#2

M"104 5.77 5.48 1.04 7.91 19.8 0.80

#2! " 104 3.50 4.18 4.22 12.0 26.7 4.64

Sharpe"102!5.49 !5.36 5.05 !12.1 !13.2 !26.4

Treynor"103!1.35 !1.38 2.22 !3.74 !3.96 !13.4

# refers to the p-value of the t-test of H0: !i = 0.




Empirical literature on the CAPM

Since the development of CAPM in the 1960s, a large body of literaturehas evolved on empirical evidence for or against the model. The earlyevidence was largely positive, but in the late 1970s, less favorableevidence began to appear.

Basu (1977) reported the “price–earnings-ratio e!ect”: Firms withlow price–earnings ratios have higher average returns, and firms withhigh price–earnings ratios have lower average returns than the valuesimplied by CAPM.

Banz (1981) noted the “size e!ect,” that firms with low marketcapitalizations have higher average returns than under CAPM.




Empirical literature on the CAPM

Fama and French (1992, 1993) have found that beta cannot explainthe di!erence in returns between portfolios formed on the basis ofthe ratio of book value to market value of equity.

Jegadesh and Titman (1995) have noted that a portfolio formed bybuying stocks whose values have declined and selling stocks whosevalues have risen has a higher average return than predicted byCAPM.

Remark 3

The empirical study for or against CAPM might involves the issues ofdata snooping, selection bias, and proxy bias.




Arbitrage pricing theory (APT)

Multifactor pricing models generalize CAPM by embedding it in aregression model of the form

ri = $i + "Ti f + &i, i = 1, · · · , p, (6)

in which the ri is the return on the ith asset, $i and "i areunknown regression parameters, f = (f1, . . . , fk)T is a regressionvector of factors, and &i is an unobserved random disturbance thathas mean 0 and is uncorrelated with f .Ross (1976) introduced the APT which allows multiple risk factorsfor asset returns. Unlike the CAPM, APT does not requireidentification of the market portfolio and relates the expected returnµi of the ith asset to the risk-free return, or to a more generalparameter '0 without assuming the existence of a risk-free asset,and to a k ' 1 vector $ of risk premiums:

µi " '0 + "Ti $, i = 1, . . . , p. (7)




Arbitrage pricing theory (APT)

While APT provides an economic theory underlying multifactormodels of asset returns, the theory does not identify the factors.

Approaches to the choice of factors can be broadly classified aseconomic and statistical.

The economic approach specifies (i) macroeconomic and financialmarket variables that are thought to capture the systematic risks ofthe economy or (ii) characteristics of firms that are likely to explaindi!erential sensitivity to the systematic risks, forming factors fromportfolios of stocks based on the characteristics.

The statistical approach uses factor analysis or PCA (principalcomponent analysis) to estimate the parameters of model (6) from aset of observed asset returns.




Factor analysis

Letting r = (r1, . . . , rp)T , ! = ($1, . . . , $p)T , # = (&1, . . . , &p)T , and B

to be the p ' k matrix whose ith row vector is "Ti , we can rewrite the

multifactor pricing model (6) as r = ! + Bf + # with E# = Ef = 0 andE(f#T ) = 0. Note that the regressor f is unobservable.

Let rt, t = 1, . . . , n, be independent observations from the model so thatrt = ! + Bf t + #t and E#t = Eft = 0, E(ft#T

t ) = 0, Cov(ft) = ", andCov(#t) = V. Then

E(rt) = !, Cov(rt) = B"BT + V. (8)

The decomposition of the covariance matrix ! of rt in (8) is the essenceof factor analysis, which separates variability into a systematic part dueto the variability of certain unobserved factors, represented by B"BT ,and an error (idiosyncratic) part, represented by V.




The factor analysis — Identifiability

In standard factor analysis, V is assumed to be diagonal; i.e.,V = diag(v1, . . . , vp). Since B and " are not uniquely determined by! = B"BT + V, the orthogonal factor model assumes that " = I sothat B is unique up to an orthogonal transformation andr = ! + B" + # with Cov(f) = " yields

Cov(r, f) = E{(r ! !)fT } = B" = B, (9)

Var(ri) =k"

j=1

b2ij + Var(&i), 1 % i % p, (10)

Cov(ri, rj) =k"

l=1

bilbjl, 1 % i, j % p. (11)




The factor analysis — MLE

Assuming the observed rt to be independent N(!,!), the likelihoodfunction is

L(!,!) = (2()!pn/2(det!)!n/2 exp

1

!1

2

n"

t=1

(rt!!)T !!1(rt !!)

2

,

with ! constrained to be of the form ! = BBT + diag(v1, . . . , vp), inwhich B is p ' k. The MLE 0! of ! is r := n!1

)nt=1 rt, and we can

maximize ! 12n log det(!) ! 1

2tr(W!!1) over ! of the form above,

where W =)n

t=1(rt ! r)(rt ! r)T . Iterative algorithms can be used to

find the maximizer 0! and therefore also 0B and 0v1, . . . ,0vp.




The factor analysis — factor rotation

In factor analysis, the entries of the matrix 0B are called factor loadings.Since 0B is unique only up to orthogonal transformations, the usualpractice is to multiply 0B by a suitably chosen orthogonal matrix Q,called a factor rotation, so that the factor loadings have a simpleinterpretable structure. Letting 0B" = 0BQ, a popular choice of Q is thatwhich maximizes the varimax criterion

C = p!1

k"

j=1

#p"

i=1

0b"4ij !

%p"

i=1

0b"2ij

&2.p

$

&k"

j=1

Var+squared loadings of the jth factor

,.




The factor analysis — factor scores

Since the values of the factors ft, 1 % t % n, are unobserved, it is oftenof interest to impute these values, called factor scores, for modeldiagnostics. From the model r ! ! = Bf + # with Cov(#) = V, thegeneralized least squares estimate of f when B, V, and ! are known is

0f = (BT V!1B)!1BTV!1(rt ! !).

Bartlett (1937) therefore suggested estimating ft by

0ft = (0BT 0V!1 0B)!1 0BT 0V!1(rt ! r). (12)




The factor analysis — the number of factors

The theory underlying multifactor pricing models and factor analysisassumes that the number k of factors has been specified and doesnot indicate how to specify it.

When the rt are independent N(!,!), we may consider a formalhypothesis testing that the k-factor model indeed holds. The nullhypothesis H0 is that ! = BBT + V with V diagonal and B beingp ' k.

The generalized likelihood ratio statistic that tests the H0 againstunconstrained ! is of the form

" = n*

log det+0B0BT + 0V

,! log det

+0!,-

, (13)

where 0! = n!1)n

t=1(rt ! r)(rt ! r)T is the unconstrained MLE of!.




The factor analysis — the number of factors

Under H0, " is approximately )2 with

1

2p(p + 1) !

*p(k + 1) !

1

2k(k ! 1)

-=

1

2

3(p ! k)2 ! p ! k

4

degrees of freedom.

Bartlett (1954) has shown that the )2 approximation to thedistribution of (13) can be improved by replacing n in (13) byn ! 1 ! (2p + 4k + 5)/6, which is often used in empirical studies.




The PCA approach

The fundamental decomposition ! = '1a1aT1 + · · · + 'papa

Tp in

PCA (see Section 2.2.2) can be used to decompose ! into! = BBT + V. Here '1 # · · · # 'p are the ordered eigenvalues of!, ai is the unit eigenvector associated with 'i, and

B = (5

'1a1, . . . ,5

'kak), V =p"

l=k+1

'lalaTl . (14)

PCA is particularly useful when most eigenvalues of ! are small incomparison with the k largest ones, so that k principal componentsof rt ! r account for a large proportion of the overall variance. Inthis case, we can use PCA to determine k.




The Fama-French three-factor model

Fama and French (1993, 1996) propose a three-factor model whichhas the form

E(ri) ! rf = $i + "Ti (rM ! rf , rS ! rL, rH ! rL)T .

The factor rM ! rf is the only factor in the CAPM. The factorrS ! rL captures the risk factor in returns related to size. Here“small” and “large” refer to the market value of equity. The factorrH ! rL, which captures the risk factor in returns related to thebook-to-market equity.

Fama and French (1992, 1993) argue that their three-factor modelremoves most of the pricing anomalies with CAPM. Because thefactors in the Fama-French model are specified, one can use standardregression analysis to test the model and estimate its parameters.




Bootstrap estimate of the CAPM

We illustrate how bootstrap resampling can be applied to estimatethe CAPM in Table 2 based on the monthly excess log returns of sixstocks from August 2000 to October 2005. The $ and % in thetable are estimated by applying OLS to the regression modelri ! rf = $i + %i(rM ! rf ) + &i, in which the market portfolio M istaken to be the Dow Jones Industrial Average index and rf is theannualized rate of the 3-month U.S. Treasury bill.

Let xt = rM,t ! rf,t and yi,t = ri,t ! rf,t. We draw B = 500bootstrap samples {(x"

t , y"i,t), 1 % t % n = 63} from the observed

sample {(xt, yi,t), 1 % t % n = 63} and compute the OLS estimates

0$"i and 0%"

i for the regression model y"i,t = $"

i + %"i x"

t + &"t . We then

report in Table 3 the average values of 0$"i , 0%"

i , Sharpe index andTreynor index of the B bootstrap samples, and their standarddeviations (given in parentheses).




Bootstrap estimate of the CAPM

Table 3: Bootstrapping CAPM.

$ ' 103 % Sharpe '102 Treynor '103

AXP 1.00 (0.28) 1.23 (0.02) !4.51 (1.61) !1.14 (0.40)

CITI 0.84 (0.32) 1.20 (0.02) !5.16 (1.60) !1.36 (0.41)

XOM 2.24 (0.33) 0.53 (0.02) 4.69 (1.59) 2.27 (0.75)

GM !1.99 (0.58) 1.44 (0.04) !12.2 (1.65) !4.02 (0.57)

Intel !4.33 (0.74) 2.29 (0.05) !13.0 (1.44) !3.95 (0.44)

Pfizer !5.23 (0.33) 0.45 (0.02) !26.2 (1.62) !13.5 (4.05)




Michaud’s resampled e!cient frontier

As pointed out before, the estimated (“plug-in”) e"cient frontierbased on the sample mean 0µ and covariance matrix 0! di!ers fromthe true e"cient frontier. Frankfurter, Phillips, and Seagle (1971)and Jobson and Korkie (1980) have found that portfolios thusconstructed may perform worse than the equally weighted portfolio.Michaud (1989) proposes to use instead of 0w the average ofbootstrap weights

w = B!1

B"

b=1

0w"b ,

where 0w"b is the estimated optimal portfolio weight vector based on

the bth bootstrap sample {r"b1, . . . , r"bn} drawn with replacement

from the observed sample {r1, . . . , rn}. Thus, Michaud’s resampled

e"cient frontier corresponds to the plot5

wT 0!w versus wT r = µ"

for a fine grid of µ" values, as shown in Figure 6 (in which we haveused B = 1000) for the six stocks considered in Figure 4.





0.0178 0.018 0.0182 0.0184 0.0186 0.0188 0.019!5

0

5

10

15

20

x 10!4

Month

ly log r

etu

rn


Estimated efficient frontier

Resampled efficient frontier

Figure 6: The estimated e!cient frontier (solid curve) andthe resampled e!cient frontier (dotted curve) of six U.S.stocks.





0.0178 0.018 0.0182 0.0184 0.0186 0.0188 0.019!5

0

5

10

15

20

x 10!4

Month

ly log r

etu

rn


Estimated efficient frontier

Resampled efficient frontier

Figure 6: The estimated e!cient frontier (solid curve) andthe resampled e!cient frontier (dotted curve) of six U.S.stocks.

AlthoughMichaud claimsthat w providesan improvementover 0w, therehave been noconvincingtheoreticaldevelopments andsimulation studiesto support theclaim.




Bootstrap estimates of performance

Whereas simulation studies of performance require specificdistributional assumptions on rt, it is desirable to be able to assessperformance nonparametrically and the bootstrap method provides apractical way to do so.

The bootstrap samples {r"b1, . . . , r"bn; r"b}, 1 % b % B, can be used

to estimate the means E(wTP r) and variances Var(wT

P r) of variousportfolios P whose weight vectors wP may depend on the observeddata (for which E(wT

P r) can no longer be written as wTP E(r) since

wP is random). Details and illustrative examples are given in Lai,Xing, and Chen (2011).



basic investment models and their statistical analysiszhu/ams516/handout02.pdfoutline asset returns...

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