Basic Investment Models and Their Statistical Analysis

7/27/2019 Basic Investment Models and Their Statistical Analysis

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Ou tl ine A ss et Ret urns Markowit zs Por tf ol io T he ory CAPM Mul ti fa ct or Pri ci ng Mode ls Res amp li ng Met ho ds

Basic Investment Models and TheirStatistical Analysis

Haipeng Xing

Haipeng Xing SUNY Stony Brook

Basic investment models and their statistical analysis

Outline A ss et Ret urns Markowit zs Por tf ol io T he ory CAPM Mul ti fa ct or Pri ci ng Mode ls Res amp li ng Met ho ds

Outline

1 Asset Returns

2 Markowitzs Portfolio Theory

3 Capital Asset Pricing Model (CAPM)

4 Multifactor Pricing Models

5 Applications of resampling to portfolio management




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Outline Asset Returns Markowit zs Por tf ol io T he ory CAPM Mul ti fa ct or Pri ci ng Mode ls Res amp li ng Met ho ds

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 Pt1)/Pt1,and the one-period gross return is Pt/Pt1 = 1 + Rt.

The gross return over k periods is defined as

1 + Rt(k) = Pt/Ptk =

k1j=0

(1 + Rtj),

and the net return over these periods is Rt(k). In practice, weusually use years as the time unit. The annualized gross return for

holding 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/Pt1).

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/Pt1) = 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

Ptk=

k1j=0

log(1 + Rtj) =k1j=0

rtj .




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Adjustment for dividends

Many assets pay dividends periodically. In this case, the definition of

asset 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

Pt1 1, rt = log(Pt + Dt) log Pt1.

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

rt[k] = log k1

j=0 Ptj + DtjPtj1 =k1

j=0 logPtj + DtjPtj1 .Haipeng Xing SUNY Stony Brook



Excess and portfolio returns

Excess return refers to the difference rt rt between the assets logreturn rt and the log return rt 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 different assets. Let wibe the weight of the portfolios value invested in asset i. Suppose

Rit 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 =

pi=1

wiRit, rt = log

1 +

pi=1

wiRit

pi=1

wirit.




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Outline Asset Returns Markowitzs Portfolio Theory CAPM M ult ifactor Prici ng M odels Res ampli ng Metho ds

Geometry of efficient sets

A

P1

P2

= 1

= 1

= 1 = 0.6 = 0 .3

Figure 1: Feasible region for twoassets.

Consider the case of p = 2 risky

assets whose returns have means1, 2, standard deviations 1,2,and correlation coefficient . Letw1 = and w2 = 1 ( [0, 1]).Then the mean return of theportfolio is () = 1+ (1 )2,and its volatility () is given by2() = 221 + 2(1 )12+(1 )222 . Figure 1 plots thecurve {((), ()) : 0 1}

for different values of.





Figure 2: Feasible region for p 3assets.

The set of points in the (, ) 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.




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Efficientfrontier

Minimum!variancepoint

Figure 3: Efficient frontier andminimum-variance point.

For a given value of the mean

return, 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 called

the efficient frontier.




Computation of efficient portfolios

Let r = (R1, . . . , Rp)T denote the vector of returns of p assets,

1 = (1, . . . , 1)T, w = (w1, . . . , wp)T, = (1, . . . , p)

T = (E(R1),. . . , E (Rp))

T, and = (Cov(Ri, Rj))1ijp. Consider the case whereshort selling is allowed. Given a target value for the mean return ofthe portfolio, the weight vector w of an efficient portfolio can becharacterized by

weff = arg minw

wTw subject to wT = , wT1 = 1,

The method of Lagrange multipliers leads to the the explicit solution

weff =

B11A1+

C1A11

D

when is nonsingular, where A = T11 = 1T1, B = T1,C = 1T11, and D = BCA2.




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The variance of the return on this efficient portfolio is

2eff = B 2A + 2CD.The that minimizes 2eff is given by

minvar =A

C,

which corresponds to the global MVP with variance 2minvar = 1/C andweight vector

wminvar = 11

C.

For two MVPs with mean returns p and q, their weight vectors aregiven by (5) with = p, q, respectively. From this it follows that the

covariance of the returns rp and rq is given by

Cov(rp, rq) =C

D

p

A

C

q

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

weff = arg minw

wTw subject to wT = , wT1 = 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 functionwTw under linear equality constraints and nonnegativity constraints.




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Estimation of 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 covariancematrix (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 20.2INTEL (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 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

Monthlylogreturn

Monthly standard deviation

Figure 4: Estimated efficient frontier of portfolios thatconsist of six assets.

Figure 4 shows theplug-in efficientfrontier for these sixstocks allowing shortselling. By plug-inwe mean that the

mean andcovariance in (5)are substituted by theestimated values and given in Table1.




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Outline Asset Returns Markowitzs Portfolio Theory CAPM Multifactor Pricing Models Resampling Methods

The CAPM

The capital asset pricing model, introduced by Sharpe (1964) and Lintner

(1965), builds on Markowitzs portfolio theory to develop economy-wideimplications of the trade-off between return and risk, assuming that thereis a risk-free asset and that all investors have homogeneous expectationsand hold mean-variance-efficient 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 efficient frontier is a straight line that is tangent to the originalfeasible region of the n risky assets at a point M, called the tangent

point; see Figure 5. This tangent point M can be thought of as an indexfund or market portfolio.




The CAPM

M

rf

Efficientfrontier

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.




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The CAPM

One Fund Theorem

There is a single fund M of risky assets such that any efficient 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 can be computed by solving the optimizationproblem

minw

wTw subject to wT+ (1wT1)rf = .

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

weff =( rf)

( rf1)T1

( rf1)1( rf1)

1( rf1)/[1

T1( rf1)](The fund)




Sharpe ratio and the capital market line

For a portfolio whose return has mean and variance 2, its Sharperatio is ( 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 efficient frontier in the presence of a risk-free asset, iscalled the capital market line and given by

= rf +M rf

M;

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




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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)/

2M. The CAPM relates the expected excess

return (also called the risk premium) i rf of asset i to its betaviai rf = i(M 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 = 2i

2M + Var(i),

decomposing the variance 2

i of the ith asset return as a sum of thesystematic risk 2i 2M, which that is associated with the market, and

the 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( rf)/.

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

Jensen (1968) perform an empirical study using the regression model rf = + (M rf) + . His findings support the efficientmarket hypothesis, according to which it is not possible tooutperform the market portfolio in an efficient market.




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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) at

time 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(xtt) = 0.

Letting x = n1n

t=1 xt and y = n1n

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

= nt=1(xt x)ytnt=1(xt x)

2 , = y x. (2)Haipeng Xing SUNY Stony Brook




The maximum likelihood estimate ofV is

V = n1 nt=1

(yt xt)(yt xt)T. (3)The properties of OLS can be used to establish the asymptotic normalityof and , yielding the approximations N, Vn

t=1(xt x)2

, N, 1

n+

x2nt=1(xt x)

2

V,from which approximate confidence regions for and can beconstructed.




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When t are i.i.d. normal and are independent of the market excessreturns xt,

,

, and

V are the maximum likelihood estimates of, ,

and V. Furthermore, the conditional distribution of (,, V) given(x1, . . . , xn) are expressed as N, 1

n+

x2nt=1(xt x)

2

V

, (4)

N, Vnt=1(xt x)

2

, nV Wq(V, n 2),

with V independent of (,). Moreover, we can show that, under H0,

n q 1q TV11 + x2

n1nt=1(xt x)2 Fq,nq1. (5)Note that (5) still holds approximately without the normality assumptionwhen n q 1 is moderate or large.




An illustrative exampleWe 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

22M104 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

Sharpe102 5.49 5.36 5.05 12.1 13.2 26.4Treynor103 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.




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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 priceearnings-ratio effect: Firms withlow priceearnings ratios have higher average returns, and firms withhigh priceearnings ratios have lower average returns than the valuesimplied by CAPM.

Banz (1981) noted the size effect, 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 difference 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.




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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, . . . , f k)

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 returni of the ith asset to the risk-free return, or to a more general

parameter 0 without assuming the existence of a risk-free asset,and to a k 1 vector of risk premiums:

i 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 of

the economy or (ii) characteristics of firms that are likely to explaindifferential 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.




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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(fT) = 0. Note that the regressor f is unobservable.

Let rt, t = 1, . . . , n, be independent observations from the model so thatrt = + Bft + t and Et = Eft = 0, E(ftTt ) = 0, Cov(ft) = , andCov(t) = V. Then

E(rt) = , Cov(rt) = BBT + V. (8)

The decomposition of the covariance matrix ofrt in (8) is the essence

of factor analysis, which separates variability into a systematic part dueto the variability of certain unobserved factors, represented by BBT,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 = BBT + 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) =

kl=1

bilbjl , 1 i, j p. (11)




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The factor analysis MLE

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

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

1

2

nt=1

(rt)T1(rt)

,

with constrained to be of the form = BBT + diag(v1, . . . , vp), inwhich B is p k. The MLE of is r := n1nt=1 rt, and we canmaximize 1

2n log det() 1

2tr(W1) over of the form above,

where W =

nt=1(rt r)(rt r)

T. Iterative algorithms can be used to

find the maximizer and therefore also B and v1, . . . ,vp.Haipeng Xing SUNY Stony Brook



The factor analysis factor rotation

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

C = p1k

j=1

pi=1

b4ij pi=1

b2ij 2p

kj=1

Var

squared loadings of the jth factor

.




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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

f = (BTV1B)1BTV1(rt ).Bartlett (1937) therefore suggested estimating ft by

ft = (

BT

V1

B)1

BT

V1(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 = BB

T + V with V diagonal and B beingp k.

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

= n

log detBBT + V log det , (13)

where = n1nt=1(rt r)(rt r)T is the unconstrained MLE of.




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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

(p k)2 p k

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 = (1a1, . . . ,kak), V =p

l=k+1lalaTl . (14)PCA is particularly useful when most eigenvalues of are small incomparison with the k largest ones, so that k principal componentsofrt r account for a large proportion of the overall variance. Inthis case, we can use PCA to determine k.




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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. Heresmall 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.



Outli ne Asset Returns Markowi tzs Portfoli o Theory CAPM M ult ifactor Prici ng M odels Resampling Methods

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 {(xt , y

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

sample {(xt, yi,t), 1 t n = 63} and compute the OLS estimatesi and i for the regression model yi,t = i + i xt + t . We thenreport in Table 3 the average values of i , i , Sharpe index andTreynor index of the B bootstrap samples, and their standarddeviations (given in parentheses).




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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)




Michauds resampled efficient frontier

As pointed out before, the estimated (plug-in) efficient frontier

based on the sample mean and covariance matrix differs fromthe true efficient 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

w the average of

bootstrap weights

w = B1

B

b=1wb ,

where wb is the estimated optimal portfolio weight vector based onthe bth bootstrap sample {rb1, . . . , r

bn} drawn with replacement

from the observed sample {r1, . . . , rn}. Thus, Michauds resampled

efficient frontier corresponds to the plotwT w versus wTr =

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




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0.0178 0.018 0.0182 0.0184 0.0186 0.0188 0.019!5

0

5

10

15

20

x 10!4

Monthlylogreturn


Estimated efficient frontier

Resampled efficient frontier

Figure 6: The estimated efficient frontier (solid curve) andthe resampled efficient 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

Monthlylogreturn


Estimated efficient frontier

Resampled efficient frontier

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

AlthoughMichaud claimsthat w providesan improvementover

w, there

have been no

convincingtheoreticaldevelopments andsimulation studiesto support theclaim.




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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 {rb1, . . . , rbn; r

b}, 1 b B, can be used

to estimate the means E(wTPr) and variances Var(wTPr) of various

portfolios P whose weight vectors wP may depend on the observeddata (for which E(wTPr) can no longer be written as w

TPE(r) since

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



Basic Investment Models and Their Statistical Analysis

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