6.1 GWN Model Assumptions

Let \(R_{it}\) denote the simple or continuously compounded (cc) return on asset \(i\) over the investment horizon between times \(t-1\) and \(t\) (e.g., monthly returns). We make the following assumptions regarding the probability distribution of \(R_{it}\) for \(i=1,\ldots,N\) assets for all times \(t\).

Assumption GWN

  1. Covariance stationarity and ergodicity: \(\{R_{i1},\ldots,R_{iT}\}=\{R_{it}\}_{t=1}^{T}\) is a covariance stationary and ergodic stochastic process with \(E[R_{it}]=\mu_{i},\) \(\mathrm{var}(R_{it})=\sigma_{i}^{2}\), \(\mathrm{cov}(R_{it},R_{jt})=\sigma_{ij}\) and \(\mathrm{cor}(R_{it},R_{jt})=\rho_{ij}\).
  2. Normality: \(R_{it}\sim N(\mu_{i},\sigma_{i}^{2})\) for all \(i\) and \(t\), and all joint distributions are normal.
  3. No serial correlation: \(\mathrm{cov}(R_{it},R_{js})=\mathrm{cor}(R_{it},R_{is})=0\) for \(t\neq s\) and \(i,j=1,\ldots,N\).

Assumptions GWN states that in every time period asset returns are jointly (multivariate) normally distributed, that the means and the variances of all asset returns, and all of the pairwise contemporaneous covariances and correlations between assets are constant over time. In addition, all of the asset returns are serially uncorrelated: \[ \mathrm{cor}(R_{it},R_{is})=\mathrm{cov}(R_{it},R_{is})=0\textrm{ for all }i\textrm{ and }t\neq s, \] and the returns on all possible pairs of assets \(i\) and \(j\) are cross lead-lag uncorrelated: \[ \mathrm{cor}(R_{it},R_{js})=\mathrm{cov}(R_{it},R_{js})=0\textrm{ for all }i\neq j\textrm{ and }t\neq s. \] In addition, under the normal distribution assumption lack of serial and cross lead-lag correlation implies time independence of returns. Clearly, these are very strong assumptions. However, they allow us to develop a straightforward probabilistic model for asset returns as well as statistical tools for estimating the parameters of the model, testing hypotheses about the parameter values and assumptions.

6.1.1 Regression model representation

A convenient mathematical representation or model of asset returns can be given based on Assumption GWN. This is the GWN regression model. For assets \(i=1,\ldots,N\) and time periods \(t=1,\ldots,T\), the GWN regression model is: \[\begin{align} R_{it} & =\mu_{i}+\varepsilon_{it},\tag{6.1}\\ \{\varepsilon_{it}\}_{t=1}^{T} & \sim\mathrm{GWN}(0,\sigma_{i}^{2}),\nonumber \\ \mathrm{cov}(\varepsilon_{it},\varepsilon_{js}) & =\left\{ \begin{array}{c} \sigma_{ij}\\ 0 \end{array}\right.\begin{array}{c} t=s\\ t\neq s \end{array}.\nonumber \end{align}\] The notation \(\varepsilon_{it}\sim\mathrm{GWN}(0,\sigma_{i}^{2})\) stipulates that the stochastic process \(\{\varepsilon_{it}\}_{t=1}^{T}\) is a GWN process with \(E[\varepsilon_{it}]=0\) and \(\mathrm{var}(\varepsilon_{it})=\sigma_{i}^{2}\). In addition, the random error term \(\varepsilon_{it}\) is independent of \(\varepsilon_{js}\) for all assets \(i\neq j\) and all time periods \(t\neq s\).

Using the basic properties of expectation, variance and covariance, we can derive the following properties of returns in the GWN model: \[\begin{align*} E[R_{it}] & =E[\mu_{i}+\varepsilon_{it}]=\mu_{i}+E[\varepsilon_{it}]=\mu_{i},\\ \mathrm{var}(R_{it}) & =\mathrm{var}(\mu_{i}+\varepsilon_{it})=\mathrm{var}(\varepsilon_{it})=\sigma_{i}^{2},\\ \mathrm{cov}(R_{it},R_{jt}) & =\mathrm{cov}(\mu_{i}+\varepsilon_{it},\mu_{j}+\varepsilon_{jt})=\mathrm{cov}(\varepsilon_{it},\varepsilon_{jt})=\sigma_{ij},\\ \mathrm{cov}(R_{it},R_{js}) & =\mathrm{cov}(\mu_{i}+\varepsilon_{it},\mu_{j}+\varepsilon_{js})=\mathrm{cov}(\varepsilon_{it},\varepsilon_{js})=0,~t\neq s. \end{align*}\] Given that covariances and variances of returns are constant over time implies that the correlations between returns over time are also constant: \[\begin{align*} \mathrm{cor}(R_{it},R_{jt}) & =\frac{\mathrm{cov}(R_{it},R_{jt})}{\sqrt{\mathrm{var}(R_{it})\mathrm{var}(R_{jt})}}=\frac{\sigma_{ij}}{\sigma_{i}\sigma_{j}}=\rho_{ij},\\ \mathrm{cor}(R_{it},R_{js}) & =\frac{\mathrm{cov}(R_{it},R_{js})}{\sqrt{\mathrm{var}(R_{it})\mathrm{var}(R_{js})}}=\frac{0}{\sigma_{i}\sigma_{j}}=0,~i\neq j,t\neq s. \end{align*}\] Finally, since \(\{\varepsilon_{it}\}_{t=1}^{T}\sim\mathrm{GWN}(0,\sigma_{i}^{2})\) it follows that \(\{R_{it}\}_{t=1}^{T}\sim iid~N(\mu_{i},\sigma_{i}^{2})\). Hence, the GWN regression model (6.1) for \(R_{it}\) is equivalent to the model implied by Assumption GWN. Interpretation of the GWN regression model

The GWN model has a very simple form and is identical to the measurement error model in the statistics literature.29 In words, the model states that each asset return is equal to a constant \(\mu_{i}\) (the expected return) plus an i.i.d. normally distributed random variable \(\varepsilon_{it}\) with mean zero and constant variance. The random variable \(\varepsilon_{it}\) can be interpreted as representing the unexpected news concerning the value of the asset that arrives between time \(t-1\) and time \(t.\) To see this, (6.1) implies that: \[ \varepsilon_{it}=R_{it}-\mu_{i}=R_{it}-E[R_{it}], \] so that \(\varepsilon_{it}\) is defined as the deviation of the random return from its expected value. If the news between times \(t-1\) and \(t\) is good, then the realized value of \(\varepsilon_{it}\) is positive and the observed return is above its expected value \(\mu_{i}.\) If the news is bad, then \(\varepsilon_{it}\) is negative and the observed return is less than expected. The assumption \(E[\varepsilon_{it}]=0\) means that news, on average, is neutral; neither good nor bad. The assumption that \(\mathrm{sd}(\varepsilon_{it})=\sigma_{i}\) can be interpreted as saying that the volatility, or typical magnitude, of news arrival is constant over time. The random news variable affecting asset \(i\), \(\varepsilon_{it}\), is allowed to be contemporaneously correlated with the random news variable affecting asset \(j,\) \(\varepsilon_{jt}\), to capture the idea that news about one asset may spill over and affect another asset. For example, if asset \(i\) is Microsoft stock and asset \(j\) is Apple Computer stock, then one interpretation of news in this context is general news about the computer industry and technology. Good news should lead to positive values of both \(\varepsilon_{it}\) and \(\varepsilon_{jt}.\) Hence these variables will be positively correlated due to a positive reaction to a common news component. Finally, the news on asset \(j\) at time \(s\) is unrelated to the news on asset \(i\) at time \(t\) for all times \(t\neq s\). For example, this means that the news for Apple in January is not related to the news for Microsoft in February.

6.1.2 Location-Scale model representation

Sometimes it is convenient to re-express the regression form of the GWN model (6.1) in location-scale form: \[\begin{align} R_{it} & =\mu_{i}+\varepsilon_{it}=\mu_{i}+\sigma_{i}\cdot Z_{it}\tag{6.2}\\ \{Z_{it}\}_{t=1}^{T} & \sim\mathrm{GWN}(0,1),\nonumber \end{align}\] where we use the decomposition \(\varepsilon_{it}=\sigma_{i}\cdot Z_{it}\). In this form, the random news shock is the \(iid\) standard normal random variable \(Z_{it}\) scaled by the “news” volatility \(\sigma_{i}\).

This form is particularly convenient for Value-at-Risk calculations because the \(\alpha\times100\%\) quantile of the return distribution has the simple form: \[ q_{\alpha}^{R_{i}}=\mu_{i}+\sigma_{i}\times q_{\alpha}^{Z}, \] where \(q_{\alpha}^{Z}\) is the \(\alpha\times100\%\) quantile of the standard normal random distribution. Let \(W_{0}\) be the initial amout of wealth to be invested from time \(t-1\) to \(t\). If \(R_{it}\) is the simple return then, \[ \mathrm{VaR}_{\alpha}=W_{0}\times q_{\alpha}^{R_{i}} = W_{0}\times \left( \mu_{i}+\sigma_{i}\times q_{\alpha}^{Z} \right), \] whereas if \(R_{it}\) is the continuously compounded return then, \[ \mathrm{VaR}_{\alpha}=W_{0}\times\left(e^{q_{\alpha}^{R_{i}}}-1\right) = W_{0}\times\left(e^{\mu_{i}+\sigma_{i}\times q_{\alpha}^{Z}}-1\right). \]

6.1.3 The GWN model in matrix notation

Define the \(N\times1\) vectors \(\mathbf{R}_{t}=(R_{1t},\ldots,R_{Nt})^{\prime}\), \(\mu=(\mu_{1},\ldots,\mu_{N})^{\prime}\), \(\varepsilon_{t}=(\varepsilon_{1t},\ldots,\varepsilon_{Nt})^{\prime}\) and the \(N\times N\) symmetric covariance matrix: \[ \mathrm{var}(\varepsilon_{t})=\Sigma=\left(\begin{array}{cccc} \sigma_{1}^{2} & \sigma_{12} & \cdots & \sigma_{1N}\\ \sigma_{12} & \sigma_{2}^{2} & \cdots & \sigma_{2N}\\ \vdots & \vdots & \ddots & \vdots\\ \sigma_{1N} & \sigma_{2N} & \cdots & \sigma_{N}^{2} \end{array}\right). \] Then the regression form of the GWN model in matrix notation is: \[\begin{align} \mathbf{R}_{t}=\mu+\varepsilon_{t},\tag{6.3}\\ \varepsilon_{t}\sim\mathrm{iid}~N(\mathbf{0},\Sigma),\nonumber \end{align}\] which implies that \(\mathbf{R}_{t}\sim iid~N(\mu,\Sigma)\).

The location-scale form of the GWN model in matrix notation makes use of the matrix square root factorization \(\Sigma=\Sigma^{1/2}\Sigma^{1/2\prime}\) where \(\Sigma^{1/2}\) is the lower-triangular matrix square root (usually the Cholesky factorization). Then (6.3) can be rewritten as: \[\begin{align} \mathbf{R}_{t}=\mu+\Sigma^{1/2}\mathbf{Z}_{t},\tag{6.4}\\ \mathbf{Z}_{t}\sim \mathrm{GWN}(\mathbf{0},\mathbf{I}_{N}),\nonumber \end{align}\] where \(\mathbf{I}_{N}\) denotes the \(N\)-dimensional identity matrix.

6.1.4 The GWN model for continuously compounded returns

The GWN model is often used to describe continuously compounded (cc) returns defined as \(R_{it}=\ln(P_{it}/P_{it-1})\) where \(P_{it}\) is the price of asset \(i\) at time \(t\). An advantage of the GWN model for cc returns is that the model aggregates to any time horizon because multi-period cc returns are additive. This is particularly convenient for investment risk analysis. The GWN model for cc returns also gives rise to the random walk model for the logarithm of asset prices. The normal distribution assumption of the GWN model for cc returns implies that single-period simple returns are log-normally distributed.

A disadvantage of the GWN model for cc returns is that the model has some limitations for the analysis of portfolios because the cc return on a portfolio of assets is not a weighted average of the cc returns on the individual assets. As a result, for portfolio analysis the GWN model is typically applied to simple returns. However, if the investment horizon is short (e.g. daily, weekly or monthly) then simple returns are typically close to zero on average and there is little difference between simple and cc returns. Hence, the time aggregation results of the GWN model for cc returns can be applied to simple returns and the portfolio results of the GWN model for simple returns can be applied to cc returns. Time Aggregation and the GWN Model

The GWN model for cc returns has the following nice aggregation property with respect to the interpretation of \(\varepsilon_{it}\) as news. For illustration purposes, suppose that \(t\) represents months so that \(R_{it}\) is the cc monthly return on asset \(i\). Now, instead of the monthly return, suppose we are interested in the annual cc return \(R_{it}=R_{it}(12)\). Since multi-period cc returns are additive, \(R_{it}(12)\) is the sum of \(12\) monthly cc returns: \[ R_{it}=R_{it}(12)=\sum_{k=0}^{11}R_{it-k}=R_{it}+R_{it-1}+\cdots+R_{it-11}. \] Using the GWN regression model (6.1) for the monthly return \(R_{it}\), we may express the annual return \(R_{it}(12)\) as: \[ R_{it}(12)=\sum_{t=0}^{11}(\mu_{i}+\varepsilon_{it})=12\cdot\mu_{i}+\sum_{t=0}^{11}\varepsilon_{it}=\mu_{i}(12)+\varepsilon_{it}(12), \] where \(\mu_{i}(12)=12\cdot\mu_{i}\) is the annual expected return on asset \(i\), and \(\varepsilon_{it}(12)=\sum_{k=0}^{11}\varepsilon_{it-k}\) is the annual random news component. The annual expected return, \(\mu_{i}(12)\), is simply \(12\) times the monthly expected return, \(\mu_{i}\). The annual random news component, \(\varepsilon_{it}(12)\), is the accumulation of news over the year. As a result, the variance of the annual news component, \(\sigma_{i}(12)^{2}\), is \(12\) times the variance of the monthly news component: \[\begin{align*} \mathrm{var}(\varepsilon_{it}(12)) & =\mathrm{var}\left(\sum_{k=0}^{11}\varepsilon_{it-k}\right)\\ & =\sum_{k=0}^{11}\mathrm{var}(\varepsilon_{it-k})\textrm{ since }\varepsilon_{it}\textrm{ is uncorrelated over time}\\ & =\sum_{k=0}^{11}\sigma_{i}^{2}\textrm{ since }\mathrm{var}(\varepsilon_{it})\textrm{ is constant over time}\\ & =12\cdot\sigma_{i}^{2}=\sigma_{i}^{2}(12). \end{align*}\] It follows that the standard deviation of the annual news is equal to \(\sqrt{12}\) times the standard deviation of monthly news: \[ \mathrm{sd}(\varepsilon_{it}(12))=\sqrt{12}\times\mathrm{sd}(\varepsilon_{it})=\sqrt{12}\times\sigma_{i}. \] Similarly, due to the additivity of covariances, the covariance between \(\varepsilon_{it}(12)\) and \(\varepsilon_{jt}(12)\) is 12 times the monthly covariance: \[\begin{align*} \mathrm{cov}(\varepsilon_{it}(12),\varepsilon_{jt}(12)) & =\mathrm{cov}\left(\sum_{k=0}^{11}\varepsilon_{it-k},\sum_{k=0}^{11}\varepsilon_{jt-k}\right)\\ & =\sum_{k=0}^{11}\mathrm{cov}(\varepsilon_{it-k},\varepsilon_{jt-k})\textrm{ since }\varepsilon_{it}\textrm{ and }\varepsilon_{jt}\textrm{ are uncorrelated over time}\\ & =\sum_{k=0}^{11}\sigma_{ij}\textrm{ since }\mathrm{cov}(\varepsilon_{it},\varepsilon_{jt})\textrm{ is constant over time}\\ & =12\cdot\sigma_{ij}=\sigma_{ij}(12). \end{align*}\] The above results imply that the correlation between the annual errors \(\varepsilon_{it}(12)\) and \(\varepsilon_{jt}(12)\) is the same as the correlation between the monthly errors \(\varepsilon_{it}\) and \(\varepsilon_{jt}\): \[\begin{align*} \mathrm{cor}(\varepsilon_{it}(12),\varepsilon_{jt}(12)) & =\frac{\mathrm{cov}(\varepsilon_{it}(12),\varepsilon_{jt}(12))}{\sqrt{\mathrm{var}(\varepsilon_{it}(12))\cdot\mathrm{var}(\varepsilon_{jt}(12))}}\\ & =\frac{12\cdot\sigma_{ij}}{\sqrt{12\sigma_{i}^{2}\cdot12\sigma_{j}^{2}}}\\ & =\frac{\sigma_{ij}}{\sigma_{i}\sigma_{j}}=\rho_{ij}=\mathrm{cor}(\varepsilon_{it},\varepsilon_{jt}). \end{align*}\] The above results generalize to aggregating returns to arbitrary time horizons. Let \(R_{it}\) denote the cc return on asset \(i\) between times \(t-1\) and \(t\), where \(t\) represents the general investment horizon, and let \(R_{it}(k)=\sum_{j=0}^{k-1}R_{it-j}\) denote the \(k\)-period cc return. Then the GWN model for \(R_{it}(k)\) has the form:

\[\begin{align} R_{it}(k)=\mu_{i}(k)+\varepsilon_{it}(k), \tag{6.5} \\ \varepsilon_{it}(k)\sim \mathrm{GWN}(0,\sigma_{i}^{2}(k)) \nonumber, \end{align}\]

where \(\mu_{i}(k)=k\times\mu_{i}\) is the \(k\)-period expected return, \(\varepsilon_{it}(k)=\sum_{j=0}^{k-1}\varepsilon_{it-j}\) is the \(k\)-period error term, and \(\sigma_{i}^{2}(k)=k\times\sigma_{i}^{2}\) is the \(k\)-period variance. The \(k\)-period volatility follows the square-root-of-time rule: \(\sigma_{i}(k)=\sqrt{k}\times\sigma_{i}.\) The \(k\)-period covariance between asset \(i\) and asset \(j\) is \(\sigma_{ij}(k) = k\times \sigma_{ij}\), and the \(k\)-period correlation is \(\rho_{ij}(k) = \rho_{ij}\). This aggregation result is exact for cc returns but it is often used as an approximation for simple returns. The random walk model of asset prices

The GWN model for cc returns (6.1) gives rise to the so-called random walk (RW) model for the logarithm of asset prices. To see this, recall that the continuously compounded return, \(R_{it},\) is defined from asset prices via \(R_{it}=\ln\left(\frac{P_{it}}{P_{it-1}}\right)=\ln(P_{it})-\ln(P_{it-1})\). Letting \(p_{it}=\ln(P_{it})\) and using the representation of \(R_{it}\) in the GWN model (6.1), we can express the log-price as: \[\begin{equation} p_{it}=p_{it-1}+\mu_{i}+\varepsilon_{it}.\tag{6.6} \end{equation}\] The representation in (6.6) is known as the RW model for log-prices.30. It is a representation of the GWN model in terms of log-prices.

In the RW model (6.6), \(\mu_{i}\) represents the expected change in the log-price (cc return) between months \(t-1\) and \(t\), and \(\varepsilon_{it}\) represents the unexpected change in the log-price. That is, \[\begin{align*} E[\Delta p_{it}] & =E[R_{it}]=\mu_{i},\\ \varepsilon_{it} & =\Delta p_{it}-E[\Delta p_{it}]. \end{align*}\] where \(\Delta p_{it}=p_{it}-p_{it-1}\). Further, in the RW model, the unexpected changes in the log-price, \(\{\varepsilon_{it}\}\), are uncorrelated over time (\(\mathrm{cov}(\varepsilon_{it},\varepsilon_{is})=0\) for \(t\neq s)\) so that future changes in the log-price cannot be predicted from past changes in the log-price.31

The RW model gives the following interpretation for the evolution of log prices. Let \(p_{i0}\) denote the initial log price of asset \(i\). The RW model says that the log-price at time \(t=1\) is: \[ p_{i1}=p_{i0}+\mu_{i}+\varepsilon_{i1}, \] where \(\varepsilon_{i1}\) is the value of random news that arrives between times \(0\) and \(1.\) The expected log-price at time \(t=1\) is: \[ E[p_{i1}]=p_{i0}+\mu_{i}+E[\varepsilon_{i1}]=p_{i0}+\mu_{i}, \] which is the initial price plus the expected return between times 0 and 1. Similarly, by recursive substitution the log-price at time \(t=2\) is: \[\begin{align*} p_{i2} & =p_{i1}+\mu_{i}+\varepsilon_{i2}\\ & =p_{i0}+\mu_{i}+\mu_{i}+\varepsilon_{i1}+\varepsilon_{i2}\\ & =p_{i0}+2\cdot\mu_{i}+\sum_{t=1}^{2}\varepsilon_{it} \\ & =p_{i0}+2\cdot\mu_{i}+\varepsilon_{it}(2), \end{align*}\] which is equal to the initial log-price, \(p_{i0},\) plus the two period expected return, \(2\cdot\mu_{i}\), plus the accumulated random news over the two periods, \(\sum_{t=1}^{2}\varepsilon_{it} = \varepsilon_{it}(2)\). By repeated recursive substitution, the log price at time \(t=T\) is, \[ p_{iT}=p_{i0}+T\cdot\mu_{i}+\sum_{t=1}^{T}\varepsilon_{it} = p_{i0}+T\cdot\mu_{i}+\varepsilon_{it}(T). \] At time \(t=0,\) the expected log-price at time \(t=T\) is, \[ E[p_{iT}]=p_{i0}+T\cdot\mu_{i}, \] which is the initial price plus the expected growth in prices over \(T\) periods. The actual price, \(p_{iT},\) deviates from the expected price by the accumulated random news: \[ p_{iT}-E[p_{iT}]=\sum_{t=1}^{T}\varepsilon_{it} = \varepsilon_{it}(T). \] At time \(t=0,\) the variance of the log-price at time \(T\) is, \[ \mathrm{var}(p_{iT})=\mathrm{var}\left(\sum_{t=1}^{T}\varepsilon_{it}\right)=T\cdot\sigma_{i}^{2} \] Hence, the RW model implies that the stochastic process of log-prices \(\{p_{it}\}\) is non-stationary because the variance of \(p_{it}\) increases with \(t.\) Finally, because \(\varepsilon_{it}\sim \mathrm{}(0,\sigma_{i}^{2})\) it follows that (conditional on \(p_{i0})\) \(p_{iT}\sim N(p_{i0}+T\mu_{i},T\sigma_{i}^{2})\).

The term random walk was originally used to describe the unpredictable movements of a drunken sailor staggering down the street. The sailor starts at an initial position, \(p_{0}\), outside the bar. The sailor generally moves in the direction described by \(\mu\) but randomly deviates from this direction after each step \(t\) by an amount equal to \(\varepsilon_{t}\). After \(T\) steps the sailor ends up at position \(p_{T}=p_{0}+\mu\cdot T+\sum_{t=1}^{T}\varepsilon_{t}\). The sailor is expected to be at location \(\mu T\), but where he actually ends up depends on the accumulation of the random changes in direction \(\sum_{t=1}^{T}\varepsilon_{t}\). Because \(\mathrm{var}(p_{T})=\sigma^{2}T\), the uncertainty about where the sailor will be increases with each step.

The RW model for log-prices implies the following model for prices: \[ P_{it}=e^{p_{it}}=P_{i0}e^{\mu_{i}\cdot t+\sum_{s=1}^{t}\varepsilon_{is}}=P_{i0}e^{\mu_{i}t}e^{\sum_{s=1}^{t}\varepsilon_{is}}, \] where \(p_{it}=p_{i0}+\mu_{i}t+\) \(\sum_{s=1}^{t}\varepsilon_{s}.\) Since \(P_{it}=e^{p_{it}}\), \(P_{it} > 0\). The term \(e^{\mu_{i}t}\) represents the expected exponential growth rate in prices between times \(0\) and time \(t,\) and the term \(e^{\sum_{s=1}^{t}\varepsilon_{is}}\) represents the unexpected exponential growth in prices. Here, conditional on \(P_{i0},\) \(P_{it}\) is log-normally distributed because \(p_{it}=\ln P_{it}\) is normally distributed.

6.1.5 GWN model for simple returns

For simple returns, defined as \(R_{it}=(P_{it}-P_{it-1})/P_{it-1}\), the GWN model is often used for the analysis of portfolios as discussed in chapters 11 - 15. The reason is that the simple return on a portfolio of \(N\) assets is weighted average of the simple returns on the individual assets. Hence, the GWN model for simple returns extends naturally to portfolios of assets. GWN Model and Portfolios

Consider the GWN model in matrix form (6.3) for the \(N\times1\) vector of simple returns \(\mathbf{R}_{t}=(R_{1t},\ldots,R_{Nt})^{\prime}\). For a vector of portfolio weights \(\mathbf{x}=(x_{1},\ldots,x_{N})\) such that \(\mathbf{x}^{\prime}\mathbf{1}=\sum_{i=1}^{N}x_{i}=1,\) the simple return on the portfolio is: \[ R_{pt}=\mathbf{x}^{\prime}\mathbf{R}_{t}=\sum_{i=1}^{N}x_{i}R_{it}. \] Substituting in (6.1) gives the GWN model for the portfolio returns: \[\begin{equation} R_{pt}=\mathbf{x}^{\prime}\left(\mu+\varepsilon_{t}\right)=\mathbf{x}^{\prime}\mu + \mathbf{x}^{\prime}\varepsilon_{t}=\mu_{p}+\varepsilon_{pt}\tag{6.7} \end{equation}\] where \(\mu_{p}=\mathbf{x}^{\prime}\mu=\sum_{i=1}^{N}x_{i}\mu_{i}\) is the portfolio expected return, and \(\varepsilon_{pt}=\mathbf{x}^{\prime}\varepsilon_{t}=\sum_{i=1}^{N}x_{i}\varepsilon_{it}\) is the portfolio error. The variance of \(R_{pt}\) is given by: \[ \mathrm{var}(R_{pt})=\mathrm{var}(\mathbf{x}^{\prime}\mathbf{R}_{t})=\mathbf{x}^{\prime}\Sigma\mathbf{ x}=\sigma_{p}^{2}. \] Therefore, the distribution of portfolio returns is normal, \[ R_{pt}\sim N(\mu_{p},\sigma_{p}^{2}). \] This result is exact for simple returns but is often used as an approximation for cc returns. GWN Model for Multi-Period Simple Returns

The GWN model for single period simple returns does not extend exactly to multi-period simple returns because multi-period simple returns are not additive. Recall, the \(k\)-period simple return has a multiplicative relationship to single period returns: \[\begin{align*} R_{t}(k) & =(1+R_{t})(1+R_{t-1})\times\cdots\times(1+R_{t-k+1})-1\\ & =R_{t}+R_{t-1}+\cdots+R_{t-k+1}\\ & +R_{t}R_{t-1}+R_{t}R_{t-2}+\cdots+R_{t-k+2}R_{t-k+1}. \end{align*}\] Even though single period returns are normally distributed in the GWN model, multi-period returns are not normally distributed because the product of two normally distributed random variables (e.g., \(R_{t}R_{t-1}\)) is not normally distributed. Hence, the GWN model does not exactly generalize to multi-period simple returns. However, if single period returns are small then all of the cross products of returns are approximately zero (\(R_{t}R_{t-1}\approx\cdots\approx R_{t-k+2}R_{t-k+1}\approx0\)) and, \[\begin{align*} R_{t}(k) & \approx R_{t}+R_{t-1}+\cdots+R_{t-k+1}\\ & \approx\mu(k)+\varepsilon_{t}(k). \end{align*}\] where \(\mu(k)=k\mu\) and \(\varepsilon_{t}(k)=\sum_{j=0}^{k-1}\varepsilon_{t-j}\). Hence, the GWN model is approximately true for multi-period simple returns when single period simple returns are not too big.

Some exact returns can be derived for the mean and variance of multi-period simple returns. For simplicity, let \(k=2\) so that, \[ R_{t}(2)=(1+R_{t})(1+R_{t-1})-1=R_{t}+R_{t-1}+R_{t}R_{t-1}. \] Substituting in (6.1) then gives, \[\begin{align*} R_{t}(2) & =\left(\mu+\varepsilon_{t}\right)+(\mu+\varepsilon_{t-1})+\left(\mu+\varepsilon_{t}\right)(\mu+\varepsilon_{t-1})\\ & =2\mu+\varepsilon_{t}+\varepsilon_{t-1}+\mu^{2}+\mu\varepsilon_{t}+\mu\varepsilon_{t-1}+\varepsilon_{t}\varepsilon_{t-1}\\ & =2\mu+\mu^{2}+\varepsilon_{t}(1+\mu)+\varepsilon_{t-1}(1+\mu)+\varepsilon_{t}\varepsilon_{t-1}. \end{align*}\] The result for the expected return is easy, \[\begin{align*} E[R_{t}(2)] & =2\mu+\mu^{2}+(1+\mu)E[\varepsilon_{t}]+(1+\mu)E[\varepsilon_{t-1}]+E[\varepsilon_{t}\varepsilon_{t-1}]\\ & =2\mu+\mu^{2}=(1+\mu)^{2}-1, \end{align*}\] The result uses the independence of \(\varepsilon_{t}\) and \(\varepsilon_{t-1}\) to get \(E[\varepsilon_{t}\varepsilon_{t-1}]=E[\varepsilon_{t}]E[\varepsilon_{t-1}]=0.\) The result for the variance, however, is more work: \[\begin{align*} \mathrm{var}(R_{t}(2)) & =\mathrm{var}(\varepsilon_{t}(1+\mu)+\varepsilon_{t-1}(1+\mu)+\varepsilon_{t}\varepsilon_{t-1})\\ & =(1+\mu)^{2}\mathrm{var}(\varepsilon_{t})+(1+\mu)^{2}\mathrm{var}(\varepsilon_{t-1})+\mathrm{var}(\varepsilon_{t}\varepsilon_{t-1})\\ & +2(1+\mu)^{2}\mathrm{cov}(\varepsilon_{t},\varepsilon_{t-1})+2(1+\mu)\mathrm{cov}(\varepsilon_{t},\varepsilon_{t}\varepsilon_{t-1})\\ & +2(1+\mu)\mathrm{cov}(\varepsilon_{t-1},\varepsilon_{t}\varepsilon_{t-1}) \end{align*}\] Now, \(\mathrm{var}(\varepsilon_{t})=\mathrm{var}(\varepsilon_{t-1})=\sigma^{2}\) and \(\mathrm{cov}(\varepsilon_{t},\varepsilon_{t-1})=0.\) Next, note that, \[ \mathrm{var}(\varepsilon_{t}\varepsilon_{t-1})=E[\varepsilon_{t}^{2}\varepsilon_{t-1}^{2}]-\left(E[\varepsilon_{t}\varepsilon_{t-1}]\right)^{2}=E[\varepsilon_{t}^{2}]E[\varepsilon_{t-1}^{2}]-\left(E[\varepsilon_{t}]E[\varepsilon_{t-1}]\right)^{2}=2\sigma^{2}. \] Finally, \[\begin{align*} \mathrm{cov}(\varepsilon_{t},\varepsilon_{t}\varepsilon_{t-1}) & =E[\varepsilon_{t}(\varepsilon_{t}\varepsilon_{t-1})]-E[\varepsilon_{t}]E[\varepsilon_{t}\varepsilon_{t-1}]\\ & =E[\varepsilon_{t}^{2}]E[\varepsilon_{t-1}]-E[\varepsilon_{t}]E[\varepsilon_{t}]E[\varepsilon_{t-1}]\\ & =0. \end{align*}\] Then, \[\begin{align*} \mathrm{var}(R_{t}(2)) & =(1+\mu)^{2}\sigma^{2}+(1+\mu)^{2}\sigma^{2}+2\sigma^{2}\\ & =2\sigma^{2}[(1+\mu)^{2}+1]. \end{align*}\] If \(\mu\) is close to zero then \(E[R_{t}(2)]\approx2\mu\) and \(\mathrm{var}(R_{t}(2))\approx2\sigma^{2}\) and so the square-root-of-time rule holds approximately. Implied model for prices

Since \(R_{it}=(P_{it}-P_{it-1})/P_{it-1}\), we can write \[ P_{it} = P_{it-1}(1 + R_{it}) \] Starting at \(t=1\) and assuming \(P_{i0} > 0\) is fixed, by recursive substitution we get \[ P_{it} = P_{i0}(1+R_{i1})(1+R_{i2})\ldots (1+R_{it-1}) \] Prices will be positive provided all returns are greater than \(-1\). This model for prices is not a RW model but behaves similarly to the RW model if all simple returns are close to zero.

  1. In the measurement error model, \(r_{it}\) represents the \(t^{th}\) measurement of some physical quantity \(\mu_{i}\) and \(\varepsilon_{it}\) represents the random measurement error associated with the measurement device. The value \(\sigma_{i}\) represents the typical size of a measurement error.↩︎

  2. The model (6.6) is technically a random walk with drift \(\mu_{i}\) A pure random walk has zero drift (\(\mu_{i}=0\))↩︎

  3. The notion that future changes in asset prices cannot be predicted from past changes in asset prices is often referred to as the weak form of the efficient markets hypothesis.↩︎