Constant Expected Return ModelEcon 424Eric ZivotFall 2014Fall 2014

Updated: October 23, 2014

© Eric Zivot 2006

Monthly Closing Prices of Microsoft Stock

8012

0

Monthly Prices on MSFT

040

8

pric

e

1993 1994 1995 1996 1997 1998 1999 2000

Index

54.

5

ce

1.5

2.5

3.

log

pric

© Eric Zivot 2006

1993 1994 1995 1996 1997 1998 1999 2000

Index

Conjecture: Monthly CC Returns follow CER Model

r = + iid N(0 2) t=July 1992 Oct 2000rt = + t, t ~ iid N(0,2), t=July 1992,…,Oct 20000.3

Monthly cc returns on Microsoft

0.1

0.2

0.1

0.0

0

cre

turn

3-0

.2-0

.

cc

-0.4

-0.3

© Eric Zivot 2006

1993 1994 1995 1996 1997 1998 1999 2000

Index

Distribution Summary for Monthly CC Returns on MSFTMonthly cc returns on MSFTy

ency

030

40

0.0

0.2

frequ

e

010

20

-0.4

-0.2

return

-0.4 -0.2 0.0 0.2

smoothed densiy Normal Q-Q Plot

34

smoothed densiy

ty 0.0

0.2

Normal Q Q Plot

uant

iles

01

2

dens

i

-0.4

-0.2

0

Sam

ple

Qu

© Eric Zivot 2006

-0.4 -0.2 0.0 0.2 0.4

return

-2 -1 0 1 2

Theoretical Quantiles

Simulating Random Data for X with CDF FX

1.0

0.8 Simulate U ~ Uniform(0,1)

CDF FX(x)

u

40.

60.

20.

4

1( )Xx F u

0.0

© Eric Zivot 2006

x

-4 -2 0 2 4

R Code for MC Simulation of CER Model

# set model parameters> mu = 0.03> sd.e = 0.10> nobs = 100obs 00

# generate random numbers for errors> set seed(111)> set.seed(111)> sim.e = rnorm(nobs, mean=0, sd=sd.e)

# simulate cc returns> sim.ret = mu + sim.e

© Eric Zivot 2006

Simulated Returns from CER Model with Same Mean and SD as Microsoft

0.3

0.2

0

retu

rn

0.0

0.1

re

-0.1

0-0

.3-0

.2

© Eric Zivot 2006months

0 20 40 60 80 100

Distribution Summary for Simulated Data from CER Model

Simulated returns from CER modelcy

3040

0.1

0.3

frequ

enc

010

20

-0.3

-0.1

return

-0.3 -0.1 0.1 0.2 0.3 0.4

smoothed densiy Normal Q-Q Plot

23

4

dens

ity

0.1

0.3

ple

Qua

ntile

s

-0.4 -0.2 0.0 0.2 0.4

01

d

-2 -1 0 1 2

-0.3

-0.1

Sam

p

© Eric Zivot 2006

return Theoretical Quantiles

Monthly Returns on MSFT, SBUX and S&P 500

0.0

0.2

sbux

-0.4

-0.2

s

0.2

-0.2

0.0

msf

t

-0.4

0.00

0.05

0.10

00

-0.1

5-0

.05

sp50

© Eric Zivot 2006

1993 1994 1995 1996 1997 1998 1999 2000

Index

Monthly Returns on SBUX, MSFT and SP500

0.2

0.0

0.2

retu

rns.

z

4-0

1993 1994 1995 1996 1997 1998 1999 2000

-0.4

sbuxmsftsp500

sbuxmsftsp500

© Eric Zivot 2006

1993 1994 1995 1996 1997 1998 1999 2000

Index

Monthly Returns on SBUX, MSFT and SP500-0.4 -0.2 0.0 0.2

0.0

0.2

sbux

-0.4

-0.2

20.

00.

2

msft

-0.4

-0.

0.05

0.10

5-0

.05

0.00

0

sp500

© Eric Zivot 2006

-0.4 -0.2 0.0 0.2 -0.15 -0.05 0.00 0.05 0.10

-0.1

5

R Code for MC Simulation for Multiple Assets# multivariate simulation> library("mvtnorm")> mu = c(0.03,0.03,0.01)> sig2.msft = 0.018> sig2.msft 0.018> sig2.sbux = 0.011> sig2.sp500 = 0.001> sig.msft.sbux = 0.004

00 0 002> sig.msft.sp500 = 0.002> sig.sbux.sp500 = 0.002> Sigma = matrix(c(sig2.sbux, sig.msft.sbux, sig.sbux.sp500,+ sig.msft.sbux, sig2.msft, sig.msft.sp500,g , g , g p ,+ sig.sbux.sp500, sig.msft.sp500, sig2.sp500), + nrow=3, ncol=3, byrow=TRUE)> nobs = 100> t d(123)> set.seed(123)> returns.sim = rmvnorm(nobs, mean=mu, sigma=Sigma)

© Eric Zivot 2006

Simulated Return Data

© Eric Zivot 2006

Simulated Returns

© Eric Zivot 2006

Simulated Data from Random Walk Model for Log Prices

Simulated log prices

34

p(t)E[p(t)]p(t)-E[p(t)]

2

Expected growth in log prices

1

Deviation from expected growth in log prices

0

© Eric Zivot 2006

0 20 40 60 80 100

Simulated Prices from Random Walk Model

Simulated prices

100

80

stP P e e

60

0tP P e e

2040

02

© Eric Zivot 2006

0 20 40 60 80 100

CER Model Estimates> muhat.vals = apply(returns.mat,2,mean)> muhat.vals

b ft 500 1ˆT

sbux msft sp500 0.02777 0.02756 0.01253

> sigma2hat vals = apply(returns mat 2 var)

1

1ˆ tt

rT

> sigma2hat.vals = apply(returns.mat,2,var)> sigma2hat.vals

sbux msft sp500 2 21ˆ ˆ( )T

tr 0.01846 0.01141 0.001432

> sigmahat.vals = apply(returns.mat,2,sd)

1( )

1 tt

rT

> sigmahat.valssbux msft sp500

0.1359 0.1068 0.037852ˆ ˆ

© Eric Zivot 2006

CER Model Estimates1 T

> cov.mat = var(returns.ts)> cor.mat = cor(returns.ts)

1

1ˆ ˆ ˆ( )( )1ij it i jt j

tr r

T

ˆ

ˆ i j

> covhat.vals = cov.mat[lower.tri(cov.mat)]> rhohat.vals = cor.mat[lower.tri(cor.mat)]

ˆ ˆi ji j

[ ( )]> names(covhat.vals) = names(rhohat.vals) =+ c("sbux,msft","sbux,sp500","msft,sp500")

> covhat.valssbux,msft sbux,sp500 msft,sp500 0 00403 0 00215 0 002240.00403 0.00215 0.00224

> rhohat.valsb ft b 500 ft 500

© Eric Zivot 2006

sbux,msft sbux,sp500 msft,sp500 0.2777 0.4197 0.5551

Estimated Standard Errors> se.muhat = sigmahat.vals/sqrt(nobs)> rbind(muhat.vals,se.muhat)

sbux msft sp500 pmuhat.vals 0.0277 0.0275 0.01253

se.muhat 0.0135 0.0106 0.00378

i 2h i 2h l / ( b /2)> se.sigma2hat = sigma2hat.vals/sqrt(nobs/2)> rbind(sigma2hat.vals,se.sigma2hat)

sbux msft sp500 sigma2hat.vals 0.01845 0.01141 0.00143

se.sigma2hat 0.00261 0.00161 0.00020> se.sigmahat = sigmahat.vals/sqrt(2*nobs)g g / q ( )> rbind(sigmahat.vals,se.sigmahat)

sbux msft sp500 sigmahat.vals 0.1358 0.1068 0.0378

© Eric Zivot 2006

gse.sigmahat 0.0096 0.0075 0.0026

Estimated Standard Errors

h h (1 h h l ^2)/ ( b )> se.rhohat = (1-rhohat.vals^2)/sqrt(nobs)> rbind(rhohat.vals,se.rhohat)

sbux,msft sbux,sp500 msft,sp500 p prhohat.vals 0.2777 0.4197 0.5551

se.rhohat 0.0922 0.0823 0.0691

© Eric Zivot 2006

Sample means computed with increasing sample sizes

Consistency of Sample Mean from CER Model

sizes0.

10

1ˆT

r

ple

Mea

n

60.

08

1t

tr

T

Sam

p

040.

06

Sample mean is converging to true mean as T -> ∞0.

0

= 0.03

mean as T > ∞

© Eric Zivot 2006

Sample Size, T

0 200 400 600 800 1000

95% Confidence Intervals For

> mu.lower = muhat.vals - 2*se.muhat> h t l + 2* h t> mu.upper = muhat.vals + 2*se.muhat> mu.width = mu.upper - mu.lower

> cbind(mu.lower,mu.upper,mu.width)mu.lower mu.upper mu.width

sbux 0.0006 0.0549 0.0543msft 0.0061 0.0489 0.0427

sp500 0.0049 0.0201 0.0151

Wide 95% confidence intervals for the mean => imprecise estimate. Note: width of CI is large relative to

© Eric Zivot 2006

Note: width of CI is large relative to size of estimate for sbux and msft

95% Confidence Intervals for

> sigma.lower = sigmahat.vals - 2*se.sigmahat> i i h t l + 2* i h t> sigma.upper = sigmahat.vals + 2*se.sigmahat> sigma.width = sigma.upper - sigma.lower

> cbind(sigma lower sigma upper sigma width)> cbind(sigma.lower,sigma.upper,sigma.width)sigma.lower sigma.upper sigma.width

sbux 0.1166 0.1550 0.0384ft 0 0917 0 1219 0 0302

N 95% fid i t l

msft 0.0917 0.1219 0.0302sp500 0.0324 0.0431 0.0107

Narrow 95% confidence intervals for the sd => precise estimate. Note: width of CI is small relative

© Eric Zivot 2006

to value of estimate

95% Confidence Intervals for

> rho.lower = rhohat.vals - 2*se.rhohat> h h h t l + 2* h h t> rho.upper = rhohat.vals + 2*se.rhohat> rho.width = rho.upper - rho.lower

> cbind(rho.lower,rho.upper,rho.width)rho.lower rho.upper rho.width

sbux,msft 0.0931 0.4622 0.3691sbux,sp500 0.2549 0.5845 0.3295msft,sp500 0.4167 0.6934 0.2767

95% confidence interval for rho is moderately large => somewhat imprecise estimate for rho

© Eric Zivot 2006

imprecise estimate for rho.

St li d F t f th E ti ti f CERStylized Facts for the Estimation of CER Model Parameters

• The expected return is not estimated very preciselyprecisely– Large standard errors relative to size of mean

estimates• Standard deviations and correlations are

estimated more precisely than the expected p y preturn

© Eric Zivot 2006

Monte Carlo Simulation Loop

> mu = 0.05> sd = 0.10> sd 0.10> n.obs = 100> n.sim = 1000> set seed(111)> set.seed(111)> sim.means = rep(0,n.sim)> sim.vars = rep(0,n.sim)> sim.sds = rep(0,n.sim)> for (sim in 1:n.sim) {

sim.ret = rnorm(n.obs,mean=mu,sd=sd)sim.ret rnorm(n.obs,mean mu,sd sd)sim.means[sim] = mean(sim.ret)sim.vars[sim] = var(sim.ret)sim sds[sim] = sqrt(sim vars[sim])

© Eric Zivot 2006

sim.sds[sim] = sqrt(sim.vars[sim])}

10 simulated samples from CER model

0.0

0.2

Serie

s 1

10.

10.

3

Serie

s 6

-0.2

3-0

.10.

10.

3

Serie

s 2

-0.1

-0.1

0.1

Serie

s 7

-0.3

-0.1

0.1

0.3

Serie

s 3

-0.1

0.0

0.1

0.2

Serie

s 8

0.3

-0.2

0.0

0.2

Serie

s 4

0.2

-0.1

0.1

0

Serie

s 9

0.3

0 20 40 60 80 100

-0.2

0.0

0

Serie

s 5

Index0 20 40 60 80 100

-0.1

0.1

Serie

s 10

Index

© Eric Zivot 2006

Index Index

Histograms of 1000 Monte Carlo Estimates

mu hat sigma2 haten

cy

100

150

g

ency

5025

0

Freq

ue

050

1

Freq

ue

050

15

True values:

mu hat values

0.02 0.04 0.06 0.08

sigma2 hat values

0.006 0.008 0.010 0.012 0.014

sigma hat True values:

E[R] = = 0.05

SD(R) = = 0.10

g

uenc

y 200

300

Var(R) = 2 = 0.01Freq

ue

050

100

© Eric Zivot 2006

sigma hat values

0.08 0.09 0.10 0.11 0.12

Monte Carlo Evaluation of Bias

> mean(sim.means) # true mean = 0.05[1] 0.04969> mean(sim.means) – mu # estimate of bias[1] -0.0003105

> mean(sim.vars) # true variance = 0.01[1] 0.00999> mean(sim.vars) - sd^2 # estimate of bias[1] -9.865e-06

> mean(sim.sds) # true SD = 0.10[1] 0 09972[1] 0.09972> mean(sim.sds) – sd # estimate of bias[1] -0.0002782

© Eric Zivot 2006

Monte Carlo Evaluation of Estimated Standard Error

> sd(sim.means) # SD of mu estimates across 1000 [1] 0.01041 # Monte Carlo experiments> sd/sqrt(nobs)> sd/sqrt(nobs)[1] 0.01 # true SE estimate from formula

d( i ) # S f i ^2 ti t 1000> sd(sim.vars) # SD of sigma^2 estimates across 1000[1] 0.001352 # Monte Carlo experiments> sd^2/sqrt(nobs/2)[1] 0 001414 # approx SE estimate from formula[1] 0.001414 # approx SE estimate from formula

> sd(sim.sds) # SD of sigma estimates across 1000[1] 0.006764 # Monte Carlo experiments> sd/sqrt(2*nobs)[1] 0.007071 # approx SE estimate from formula

© Eric Zivot 2006

Monte Carlo Evaluation of 95% Confidence Interval CoverageInterval Coverage

> u = 0.05> sd = 0.10> n.sim = 1000> set.seed(111)> mu.lower = rep(0,n.sim) # initialize vectors> mu.upper = rep(0,n.sim)> for (sim in 1:n.sim) {

sim.ret = rnorm(n.obs,mean=mu,sd=sd)mu.hat = mean(sim.ret)

/se.muhat = sd(sim.ret)/sqrt(n.obs)mu.lower[sim] = mu.hat - 2*se.muhatmu.upper[sim] = mu.hat + 2*se.muhat

}}> in.interval = (mu >= mu.lower) & (mu <= mu.upper)> sum(in.interval)/n.sim

© Eric Zivot 2006

> 0.934 # coverage probability of 95% CI for mu

Monte Carlo Simulation Loop to Evaluate Distribution of correlation estimate

# generate 1000 samples from CER and compute correlations# use estimated parameters as true parameters for MC# use estimated parameters as true parameters for MC> n.obs = 100> n.sim = 1000> set.seed(111)set.seed( )> sim.corrs = matrix(0,n.sim,3) # initialize vectors> colnames(sim.corrs) = c("sbux,msft", "sbux,sp500",

"msft,sp500")

> for (sim in 1:n.sim) {sim.ret = rmvnorm(n.obs, mean=muhat.vals,

cov=cov.mat)cor.mat = cor(sim.ret)sim.corrs[sim,] = cor.mat[lower.tri(cor.mat)]

}

© Eric Zivot 2006

}

Histograms of 1000 Monte Carlo Estimates of Correlationsbux,msft sbux,sp500,

ency

015

020

0

, p

ency 150

200

250

Freq

ue

050

100

Freq

ue

050

100

rhohat(sbux,msft)

-0.1 0.1 0.2 0.3 0.4 0.5 0.6

rhohat(sbux,sp500)

0.1 0.2 0.3 0.4 0.5 0.6

msft,sp500

True values:

Corr(SBUX,MSFT) = 0.28

Corr(SBUX SP500) = 0 42

, p

ency 20

0

Corr(SBUX,SP500) = 0.42

Corr(MSFT,SP500) = 0.56

Freq

ue

050

100

© Eric Zivot 2006

rhohat(msft,sp500)

0.3 0.4 0.5 0.6 0.7

Monte Carlo Evaluation

# true correlation valuessbux,msft sbux,sp500 msft,sp500sbux,msft sbux,sp500 msft,sp500

0.2777 0.4198 0.5551# Averages across 1000 Monte Carlos> appl (sim corrs 2 mean)> apply(sim.corrs,2,mean)sbux,msft sbux,sp500 msft,sp500

0.277 0.4176 0.5505# Monte Carlo Standard Deviations> apply(sim.corrs,2,sd)sbux,msft sbux,sp500 msft,sp500sbux,msft sbux,sp500 msft,sp500 0.09606 0.08148 0.07244

# Analytic SE values for rhohatb ft b 500 ft 500

© Eric Zivot 2006

sbux,msft sbux,sp500 msft,sp500 0.09229 0.08238 0.06919

Estimating VaR in CER Model

# estimate quantiles from CER model> qhat.05 = muhat.vals + sigmahat.vals*qnorm(0.05)> qhat.05

sbux msft sp500 p-0.19571 -0.14815 -0.049717

# estimate 5% VaR> W0 = 100000> VaRhat.05 = (exp(qhat.05)-1)*W0> VaRhat.05

sbux msft sp500 -17775 -13769 -4850.1

© Eric Zivot 2006