The Gaussian White Noise Return Model
June 5, 2022
Basic Assumptions
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The simplest model regarding asset returns is the Gaussian White Noise model, also known as the Constant Expected Return (CER) model.
Let Rit denote the simple or cc returns of asset i for time period t. Assumptions:
I {Rit} is a weakly stationary and ergodic process with: I E(Rit) = μi, constant for each asset i for all time t
I Var(Rit) = i2, constant
I Cov(Rit,Rjt)= ij,constant I Cor(Rit,Rjt)=⇢ij,constant
I Normality: Rit ⇠ N(μi , i2)
No serial correlation: Cov(Rit , Ris ) = 0 for all t 6= s
Furthermore, as a result of our assumptions, we also have the following conclusions:
I There is no cross-lag correlation, i.e. Cor(Rit , Rjs ) = 0 when i 6= j and t 6= s
I Asset returns are independent of each other across time
Therefore, we can treat asset returns at any particular time period independently of other time periods
Regression Model Representation
Based on the stated assumptions, we can write the model (called the GWN regression model) as:
Rit =μi +”it, {“it}Tt=1 ⇠G⇢WN(0, i2),
ij t=s 0 t6=s
We can also check easily that this regression model satisfies all the assumptions stated on the previous page
cov(“it , “js ) =
What is the noise term? I
Think of ✏it as random (i.e. unexpected) news arriving for asset i in month t
I Nonews: ✏it =0)Rit =μi
I Good news: ✏it >0)Rit >μi I Badnews: ✏it <0)Rit <μi
Meaning of assumptions:
I E(✏it) = 0 is stating our belief that on average, we expect random news to be neither good nor bad
I Var(✏it ) = ✏2 states that the extent to which the random news event a↵ects asset returns is constant over time
I Cor(✏it , ✏is ) = 0 for t 6= s means we don’t expect random news event from this month to be correlated with random news events from other months
What is the noise term? II
I Cor(✏it , ✏jt ) = ⇢ij means that the random news event a↵ect asset i is correlated with the news event a↵ecting asset j but this correlation is constant
I Cor(✏it,✏js) = 0 for i 6= j and t 6= s means we don’t think random news event a↵ecting Apple’s stock return (say) this month has anything to do with the random news event a↵ecting Microsoft’s stock return (say) next month or in the future
We will sometimes write this model in an alternative form:
Rit =μi +"it =μi + i ·Zit {Zit}Tt=1 ⇠ GWN(0,1),
Time Aggregation Property
Suppose t represents months and Rit represents the cc monthly return on asset i on month t.
The annual cc return, Rit(12) can be written as:
Rit (12) =
This can in turn be written as:
Rit(12)=12·μi +"it +"i,t 1 +···+"i,t 11
What are the mean and variance of the annual return?
Following a similar process, we obtain:
I ij (12) = 12 · ij , which means annual covariances are 12 times the monthly covariances
I ⇢ij (12) = 12 · ⇢ij , which means correlations are not changed regardless of whether we’re looking at monthly or annual returns
Extension to k months
We can use the above results to guess the properties of arbitrary k-month returns. The corresponding GWN model would have the form:
R i t ( k ) = μ i ( k ) + " i t ( k ) , w h e r e " i t ( k ) ⇠ G W N ( 0 , i2 ( k ) )
With the following properties:
I μi(k)=k⇤μi isthek-monthreturn,
I "it(k)="i,t+"i,t 1+···(kterms)isthek-perioderror (random news) term,
I i2(k) = k ⇥ i2 is the k-period variance
I which means the k period volatility follows:
I Similarly, the k-period covariance would be: ij (k) = k ⇥ ij , I and the k-period correlation would just be: ⇢ij (k) = ⇢ij , i.e.
unchanged from the single period correlation
The above results are exact for cc returns but only approximate for simple returns
The Random Walk Model for Prices I
I Recall the definition of cc returns:
Rit =log⇣ Pit ⌘=log(Pit) log(Pit 1)
I To make it easier, we’ll represent pit = log(Pit )
I Then we can write: pit = pit 1 + μi + "it
I This is called the Random Walk model for log-prices.
I In this model, μi represents the expected change in the log-price between months: E [ pit ] = E [Rit ] = μi
I and ✏it represents the unexpected change in the log-price: "it = pit E[ pit]
I How do the log-prices evolve over time? Let pi0 denote the log-price of asset i at time 0. Then:
I The expected value of the log-price at time-1 is: E[pi1] =
The Random Walk Model for Prices II
I The variance of the log-price at time-1 is: Var(pi1) =
I Continuing this way, after T time periods, we’ll have: I piT =pi0+T·μi +"i,1+"i,2+···+"i,T
E [piT ] = I and
Var(pi1) =
I Therefore, the log-price process {pit} is non-stationary. I Since ✏it ⇠ N(0, i2), it follows that once we fix pi0:
I Finally, we have been talking of log-prices this whole time. What about the price itself?
The Random Walk Model for Prices III
I Inverting the log function: Pit =epit =
I Note that pit can be negative but Pit must be positive
I eμi t represents the expected exponential growth rate between
times 0 and t
I while ePts=1 "is represents the unexpected exponential growth
rate in prices
I Important consequence: After fixing Pi0, Pit is log-normally distributed
Simulation Study
Goals for simulation:
I Verify theoretical properties
I Study new properties that would be di cult or unfeasible on paper
Steps for a single asset return following GWN process:
I Fix model parameters μ and (Note: we’re dropping the
subscript i because we’re just going to simulate a single asset)
I Decide on number of values (i.e. time) to simulate, T
I Generate random numbers for "t vector following N(0, 2). Denote these as " ̃1,...," ̃T
I C a l c u l a t e R ̃ = μ + " ̃ tt
Example: Let’s simulate daily returns for T = 180 months.
I For μ, we’ll use the historical mean monthly cc return for a stock
I For 2, we’ll use the historcal variance of monthly cc returns
Simulation Error I
I The simulation we just saw was one possible return and price path
I But it is entirely possible that if we only simulate one price path, we might get a ”weird” or ”strange” sample. For instance, if we try the previous simulation with a seed of 35 instead of 100, check out the return and price plots
I The shape of the price path looks pretty much the same but we see that the stock price will cross $3,000, which is 10 times larger than the actual stock price!
I This is why it is important to be careful when using simulations. But why does this happen?
I For this, we need to ask ourselves where we started with the simulation. The only place where we asked for random numbers to be generated was for the error term. After that, every calculation was deterministic.
Simulation Error II
I So let’s see what happened with the error term series "t we generated. Recall that it is supposed to have a mean of 0 and a standard deviation of 0.067. We don’t expect these exact numbers since the values in "t were randomly generated, but they should be close.
I Instead, we find that the sd is close but the mean is way o↵ at 1.5%
I This means we are adding an additional 1.5% return to the stock each month, on top of the 1.35% we started with! So we are more than doubling the monthly return!
I Over 180 months, this will roughly correspond to
e0.015·180 ⇠ 15, i.e. a 15-fold increase in price compared to what we should be getting.
I Instead of relying on luck to get a good ’draw’, a better approach is to draw multiple simulations.
Simulation Error III
I For instance, let’s generate 10 sequences of returns for 180 months. Since each month’s return is independent of the rest, we can generate all 1800 random numbers independently of each other.
I We can think of each of the 10 price paths as an alternate reality for MSFT.
Statistical Properties of Prices I
I Since our stock price follows a lognormal distribution, we can calculate the probability that the stock price after one year lies between $20 and $30
I To obtain this probability in R, we need to use the mean and standard deviation of the log price (and NOT price) after 12 months.
I Using our earlier expressions:
Statistical Properties of Prices II
I The mean will be: log(21.73) + 12(0.0135) = 3.24 I And the sd will be: p12(0.0663) = 0.23
I Then, the area of the shaded region, will be the di↵erence in the values of the corresponding CDF at 20 and 30.
I We can evaluate other probabilities the same way.
I These are the theoretical values.
I We can also use our simulated data to answer these questions. But 10 sample paths aren’t enough, instead we need at least 1000 paths.
I Once we have all those prices, we can estimate the probability that the price after 12 months is between $20 and $30 using the fraction of the 1000 prices that are between those values.
I The more simulations we have, the more our estimated probability will match the theoretical probability.
Multiple Returns I
I To denote the returns of N assets at the same time, we considertheN⇥1vectorRt =(R1t,...,RNt)0.
I We define the mean using: μ = (μ1,...,μN)0,
I the idiosyncratic term (for the GWN model) using:
"t = ("1t,...,"Nt)0
Note: In the three expressions above, the 0 means transpose, so imagine those vectors written vertically instead of horizontally.
I And the covariance matrix using:
0 12 12 ··· 1N 1
var(")=⌃=B 12 2 ··· 2N C t @ . . ... . A
1 N 2 N · · · N2
Multiple Returns II
I We can then define the return vector as: Rt =μ+"t,
"t ⇠iidN(0,⌃),
which then implies that Rt ⇠ N(μ,⌃)
Simulating Multiple Returns I
I Steps for multiple asset returns following a GWN process is similar to that for single asset return:
I Fix model parameters μ, which is now an N ⇥ 1 vector and ⌃, I an N ⇥ N matrix
I Decide on number of values (i.e. time) to simulate, T
Generate T values for "t vector following the N(0,⌃2)
distribution. Denote these as " ̃1 , . . . , " ̃T I CalculateR ̃ =μ+" ̃ fort=1,...,T
Repeat above for each simulation.
I We will use three assets: S&P 500, MSFT, SBUX.
Simulating Multiple Returns II
I For each time period, will need to simulate return values for each asset according to:
0 Rmsft,t 1 0 μmsft 1 Rt =@ Rsbux,t A, μ=@ μsbux A
Rsp500,t μsp500
0 m2 sft ⌃=@ msft,sbux msft,sp500
msft,sbux 2
msft,sp500 1 sbux,sp500 A
sbux sbux,sp500
I Two important considerations apply here (as before):
I We assumed the return process is weakly stationary so the I mean and covariance values don’t have any time dependence
We don’t know the actual values of the mean and covariance matrix so we are going to use the sample values we calculated above. We can only hope that they are accurate enough.
Preview to Estimators I
I We calculated MSFT monthly returns using 15 years of data. I According to our GWN model assumptions, each monthly
return is a random variable with a normal distribution.
I We also assumed that the return for each time period is uncorrelated with the return for other time periods.
I And we assumed that each monthly return follows the same distribution.
I We might ask: What is the mean of the distribution? We don’t know this, of course, but is there a systematic way to guess what it might be?
I Together, all the 180 monthly returns for MSFT comprises a time series, i.e. a collection of random variables.
I As emphasized before, the numbers above are one realization of the time series.
Preview to Estimators II
I Each of the values represents a possible realization of the corresponding monthly return, all these numbers above are realizations of di↵erent random variables.
I However, since there is only one distribution that all the monthly returns are assumed to follow, we can think of the values above as multiple realizations of the same distribution.
I Since we have all this data, we try using that data to give us an estimate of, say, the mean of the underlying distribution.
I Our next topic of study involves studying these estimates and figuring out how reliable they are.
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