Econ 424/CFRM 462
Constant Exected Return Model

Eric Zivot

July 24, 2014

Constant Expected Return (CER) Model = cc return on asset  in month  = 1 · · ·   assets;  = 1 · · ·   months
Assumptions (normal distribution and covariance stationarity) ∼  ( 2) for all  and  = [] (constant over time)2 = var() (constant over time) = cov( ) (constant over time) = cor( ) (constant over time)

Regression Model Representation (CER Model) =  +  = 1 · · ·   ;  = 1 · · ·  ∼ iid (0 2) or  ∼   (0 2)

cov( ) =    = cor( )cov( ) = 0  6=  for all  Interpretation•  represents random news that arrives in month • News aﬀecting asset  may be correlated with news aﬀecting asset • News is uncorrelated over time=  −
unexpected
Actualexpected
news
returnreturnNo news  = 0 =⇒  = Good news   0 =⇒   Bad news   0 =⇒   CER Model Regression with Standardized News Shocks

 =  +  = 1 · · ·   ;  = 1 · · · =  +  ×  ∼ iid (0 1)

cov( ) = cor( ) = cov( ) = 0  6=  for all  Here,  ∼ iid (0 1) is a standardized news shock and  is the volatility of“news”.

Implied Model for Simple Returns
 = exp() − 1⇒ 1 +  ∼ lognormal( 2)

Recallµ¶1[] = exp  + 2 − 12var() = exp(2 + 2)(exp(2) − 1)

Value-at-Risk in the CER Model
For an initial investment of $ for one month, we have  = $0 × ( − 1)
 =  × 100% quantile of Result: In the CER model with  =  +  ×  where  ∼ (0 1)
 =  +  × 

 =  × 100% quantile of  ∼ (0 1) =  +  × Derivation of  we haveLet  ∼ (0 1) Then, by the definition of ) = Pr( ≤ ) = ⇒ Pr( ×  ≤  × ) = ⇒ Pr( +  ×  ≤  +  × ) = ⇒ Pr( ≤  +  ×  = ⇒  +  × CER Model in Matrix Notation
Define the  × 1 vectors  = (1     )0,  = (1      )0  =(1     )0 and the  ×  symmetric covariance matrix⎛12⎜⎜ 12Σ=⎜⎜ ..
⎝⎞12 · · · 1
⎟2
2 · · · 2 ⎟..
.. ⎟⎟
...
⎠2
1 2 · · · 

Then the CER model matrix notation isr =  +  ∼  (0 Σ)
which implies that  ∼  ( Σ)

Monte Carlo Simulation
Use computer random number generator to create simulated values from assumed model• Reality check on proposed model• Create “what if?” scenarios• Study properties of statistics computed from proposed modelSimulating Random Numbers from a Distribution
Goal: simulate random number  from pdf  () with CDF  ()• Generate  ∼ Uniform [0 1]• Generate  ∼  () using inverse CDF technique:−1
 = ()−1= inverse CDF function (quantile function)−1
( ()) = Example - Simulate monthly returns on Microsoft from CER Model• Specify parameters based on sample statistics (use monthly data from June
1992 - Oct 2000)
 = 003 (monthly expected return) = 010 (monthly SD)
 = 003 +   = 1     100
 ∼ iid (0 (010)2)• Simulation requires generating random numbers from a normal distribution. In R use rnorm().

Monte Carlo Simulation: Multivariate Returns
Example: Simulating observations from CER model for three assets• Specify parameters based on sample statistics (e.g., use monthly data fromJune 1992 - Oct 2000)
 = 03   = 03  500 = 01⎛⎞
018 004 002⎜⎟Σ=⎝011 002 ⎠001
 =  +   = 1     100
 ∼ iid (0 2)

cov( ) = • Simulation requires generating random numbers from a multivariate normaldistribution.

• R package mvtnorm has function mvnorm() for simulating data from amultivariate normal distribution.

CER Model and Multi-period cc Returns
 =  +   ∼  (0  2)
() =  + −1 + · · · + −+1 =

−1X

−

=0
¡

= ( + ) + ( + −1) + · · · +  + −+1=  +

−1X

−

=0= () + ()where() = () =

−1X
=0³

− ∼   0  2´¢Result: In the CER model[()] = () = var (()) = 2() = 122√
SD (()) =  () = 12
and() =

−1X
=0

− = accumulated news shocks

The Random Walk Model
The CER model for cc returns is equivalent to the random walk (RW) model
for log stock pricesÃ!

= ln  − ln −1 = ln−1= ln  − ln −1which impliesln  = ln −1 + 

Recursive substitution starting at  = 1 givesln 1 = ln 0 + 1ln 2 = ln 1 + 2= ln 0 + 1 + 2..
ln  = ln −1 + = ln 0 +X

=1Interpretation: Price at  equals initial price plus accumulation of cc returns

In CER model,  =  +  so thatln  = ln 0 += ln 0 +X

=1X( + )

=1= ln 0 +  ·  +X



=1Interpretation: Log price at  equals initial price ln 0, plus expected growth inPprices [ln ] =  · , plus accumulation of news =1 .

The price level at time  is⎛ = 0 exp ⎝ ·  +X

=1⎞

⎠ = 0 exp ( · ) exp ⎝exp ( · ) = expected growth in price⎛exp ⎝X

=1⎞⎛

⎠ = unexpected growth in priceX

=1⎞⎠CER Model for Simple Returns• CER Model can also be used for simple returns
 =  +  ∼  (0  2)
• Main drawbacks: (1) Normal distribution allows   −1; (2) Multiperiod returns are not normally distributed
() = (1 + )(1 + −1) · · · (1 + −+1) − 1¿ ( 2)• However, it can be shown that[()] = (1 + ) − 1var(()) = (1 +  2 + 2 + 2) − (1 + )2

Estimating Parameters of CER modelParameters of CER Model = []2 = var() = cov( ) = cor( )are not known with certainty
First Econometric Task
• Estimate  2    using observed sample of historical monthly
returns

Estimators and Estimates
Definition: An estimator is a rule or algorithm (mathematical formula) forcomputing an ex ante estimate of a parameter based on a random sample.
Example: Sample mean as estimator of [] = {1      } = covariance stationary time series= collection of random variables

1 X̂ = = sample mean =1= random variableDefinition: An estimate of a parameter is simply the ex post value (numericalvalue) of an estimator based on observed data
Example: Sample mean from an observed sample{1 = 02 2 = 01 3 = −01      = 03} = observed sample1̂ = (02 + 01 − 01 + · · · + 03)
= number = 001 (say)Estimators of CER Model Parameters: Plug-in Principle
Plug-in principle: Estimate model parameters using appropriate sample statistics
1X = [] : ̂ = =1

2 = [( − )2] : ̂2 = =q

2 : ̂ =q̂2
1 X( − ̂)2
 − 1 =1
1 X
 = [( − )( −  )] : ̂ =( − ̂)( − ̂ ) − 1 =1̂: ̂ = =̂ · ̂

Properties of Estimators

 = parameter to be estimated̂ = estimator of  from random sample• ̂ is a random variable — its value depends on realized values of randomsample•  (̂) = pdf of ̂ - depends on pdf of random variables in random sample• Properties of ̂ can be derived analytically (using probability theory) or byusing Monte Carlo simulation

Estimation Error(̂ ) = ̂ − Biashi

h ibias(̂ ) =  (̂ ) =  ̂ − ̂ is unbiased if [̂] =  ⇒ bias(̂ ) = 0
Remark: An unbiased estimator is “on average” correct, where “on average”
means over many hypothetical samples. It most surely will not be exactly
correct for the sample at hand!

Precisionhi(̂ ) =  (̂ )2 = = bias(̂ )2 + var(̂)∙³̂ − ´2¸var(̂) = [(̂ − [̂])2Remark: If bias(̂ ) ≈ 0 then precision is typically measured by the standarderror of ̂ defined bySE(̂) = standard error of ̂=qvar(̂) == ̂q[(̂ − [̂])2]

Bias of CER Model Estimates• ̂ ̂2 and ̂ are unbiased estimators:
h[̂i] =  ⇒ bias(̂ ) = 0
 ̂2 = 2 ⇒ bias(̂2 2) = 0hi ̂ =  ⇒ bias(̂   ) = 0• ̂ and ̂ are biased estimators
[̂] 6=  ⇒ bias(̂ ) 6= 0[̂ ] 6=  ⇒ bias(̂   ) 6= 0
but bias is very small except for very small samples and disappears assample size  gets large.

Remarks• “On average” being correct doesn’t mean the estimate is any good foryour sample!

• The value of SE(̂) will tell you how far from  the estimate ̂ typicallywill be.

• Good estimators ̂ have small bias and small SE(̂)Proof that  [̂] = Recall,1X
̂ = =1

 =  +   ∼ iid (0  2)Now[] =  + [] = since [] = 0Therefore,1X[̂] =[] =11X= =11=   = Standard Error formulas for ̂ ̂ ̂  and ̂
SE(̂) = √ 
√ 222

2= √SE(̂ ) ≈ q 2SE(̂) ≈ √ 2
SE(̂ ) : no easy formula!
(1 − 2 )SE(̂ ) ≈ √Note: "≈" denotes "approximately equal to", where approximation error −→ 0as  −→ ∞ for normally distributed data.

Remarks• Large SE=⇒ imprecise estimate; Small SE=⇒ precise estimate• Precision increases with sample size: SE−→ 0 as  −→ ∞• ̂ is generally a more precise estimate than ̂ or ̂• SE formulas for ̂ and ̂ are approximations based on the Central Limit
Theorem. Monte Carlo simulation and bootstrapping can be used to getbetter approximations• SE formulas depend on unknown values of parameters ⇒ formulas are notpractically useful• Practically useful formulas replace unknown values with estimated values:̂c√ ̂ replaces SE(̂) =̂22 replaces  2c 2) ≈ q̂SE(̂ 2
̂cSE(̂) ≈ √  ̂ replaces 2
(1 − ̂2 )c√ ̂ replaces SE(̂ ) ≈

Deriving SE(̂)⎛var(̂) = var ⎝
1X⎞⎠ =1


1 X= 2var() (since  are independent)
 =1

2

1 X= 22 =  (since var() =  2)
 =1
q

√
SE(̂) = var(̂) =

Consistency
Definition: An estimator ̂ is consistent for  (converges in probability to ) iffor any   0lim Pr(|̂ − |  ) = 0

 →∞Intuitively, as we get enough data then ̂ will eventually equal 
Remark: Consistency is an asymptotic property - it holds when we have an
infinitely large sample (i.e, in asymptopia). In the real world we only have afinite amount of data!

Result: An estimator ̂ is consistent for  if

• bias(̂ ) = 0 as  → ∞³ ´• SE ̂ = 0 as  → ∞Result: In the CER model, the estimators ̂ ̂2 ̂ ̂  and ̂ are consistent.

Distribution of CER Model Estimators

 = parameter to be estimated̂ = estimator of  from random sample
KEY POINTS• ̂ is a random variable — its value depends on realized values of randomsample•  (̂) = pdf of ̂ - depends on pdf of random variables in random sample• Properties of ̂ can be derived analytically (using probability theory) or byusing Monte Carlo simulation

Example: Distribution of ̂ in CER Model

Result:1X  =  +   ∼ iid (0 2)
̂ =
 =1

̂ is 1 times the sum of  normally distributed random variables ⇒ ̂ is also
normally distributed with2[̂] =  var(̂) =That is,̂ ∼ Ã2!

 (̂) = (22 )−12 exp(1− 2 (̂ − )22 )

Distribution of ̂ ̂  and ̂Result: The exact distributions (for finite sample size  ) of ̂ ̂  and ̂are not normal.
However, as the sample size  gets large the exact distributions of ̂ ̂  and̂ get closer and closer to the normal distribution. This is the due to thefamous Central Limit Theorem.

Central Limit Theorem (CLT)Let 1      be a iid random variables with [] =  and var() = 2
Then̄ − ̄ −  √√ = =SE(̄) Ã̄ − !

∼ (0 1) as  → ∞Equivalently,³´Ã̄ ∼   SE(̄)2 ∼  2!

for large enough We say that ̄ is asymptotically normally distributed with mean  and varianceSE(̄)2

Definition: An estimator ̂ is asymptotically normally distributed if
̂ ∼ ( SE(̂)2)for large enough 

Result: An implication of the CLT is that the estimators ̂ ̂2 ̂ ̂  and̂ are asymptotically normally distributed under the CER model.

Confidence Intervals

̂ = estimate of = best guess for unknown value of Idea: A confidence interval for  is an interval estimate of  that covers  witha stated probability
Intuition: think of a confidence interval like a “horse shoe”. For a given sample,
there is stated probability that the confidence interval (horse shoe thrown at) will cover Result: Let ̂ be an asymptotically normal estimator for  Then

• An approximate 95% confidence interval for  is an interval estimate ofthe formh³ ´³ ´ic ̂  ̂ + 2 · SEc ̂̂ − 2 · SE
³ ´c ̂
̂ ± 2 · SE

that covers  with probability approximately equal to 0.95. That isn³ ´³ ´occPr ̂ − 2 · SE ̂ ≤  ≤ ̂ + 2 · SE ̂ ≈ 095

• An approximate 99% confidence interval for  is an interval estimate ofthe formh³ ´

³ ´ic ̂  ̂ + 3 · SEc ̂̂ − 3 · SE
³ ´c ̂
̂ ± 3 · SE

that covers  with probability approximately equal to 0.99.

Remarks• 99% confidence intervals are wider than 95% confidence intervals• For a given confidence level the width of a confidence interval depends onc ̂)
the size of SE(In the CER model, 95% Confidence Intervals for   and  are:̂̂ ± 2 · √ ̂̂ ± 2 · √ 2
(1 − ̂2 )̂ ± 2 · √Using Monte Carlo Simulation to Evaluate Bias, Standard Error andConfidence Interval Coverage• Create many simulated samples from CER model• Compute parameter estimates for each simulated sample• Compute mean and sd of estimates over simulated samples• Compute 95% confidence interval for each sample• Count number of intervals that cover true parameterValue-at-Risk in the CER ModelIn the CER model ∼  ( 2) ⇒  =  +  ×   ∼  (0 1)

 may be expressed asThe  · 100% quantile  =  +  × 

 = standard Normal quantile
Then ) − 1) ·   = (exp(0Example:  ∼ (002 (010)2) and 0 = $10 000 Here,  = 002 and = 010 are known values. Then
 = −1645
05

05 = 002 + (010)(−1645) = −01445

VaR05 = (exp(−01145) − 1) · $10 000 = −$1 345

Estimating Quantiles from CER Model = ̂ + ̂  ̂  = standard Normal quantile

Estimating Value-at-Risk from CER Modeld = (exp(̂  ) − 1) · VaR0
 = ̂ + ̂  ̂ h

0 = initial investment in $i³´dd
Q: What is  VaR and SE VaR ?

Computing Standard Errors for VaR ) using• We can compute SE(̂Then³´var(̂) = var ̂ + ̂
³ ´2
 var(̂ ) + 2  cov(̂  ̂ )= var(̂) +  ³ ´2
 var(̂ ) since cov(̂  ̂ ) = 0= var(̂) +  ) =SE(̂r³´2
var(̂) +  var(̂)d ) is not straightforward since• However, computing SE(VaR³´ ) − 1) ·  )dvar VaR = var ((exp(̂0