Primer on Financial Data

Modeling the Returns

Portfolio Basics

Heuristic Portfolios

Markowitz’s Modern Portfolio Theory (MPT)

• Mean-variance portfolio (MVP)
• Global minimum variance portfolio (GMVP)
• Maximum Sharpe ratio portfolio (MSRP)

Risk-Based Portfolios (GMVP, IVP, RPP, MDP, MDCP)

Comparison of Portfolios

Conclusions

For stocks, returns are used for the modeling since they are “stationary” (as opposed to the previous random walk).

Simple return (a.k.a. linear or net return) is \[R_{t} \triangleq\frac{p_{t}-p_{t-1}}{p_{t-1}}=\frac{p_{t}}{p_{t-1}}-1.\]

Log-return (a.k.a. continuously compounded return) is \[r_{t} \triangleq y_{t}-y_{t-1}=\log\frac{p_{t}}{p_{t-1}}=\log\left(1+R_{t}\right).\]

Observe that the log-return is “stationary”: \[r_{t}=y_{t}-y_{t-1}=\mu+\epsilon_{t}\]

Note that \(r_{t}\approx R_{t}\) when \(R_{t}\) is small (i.e., the changes in \(p_t\) are small) (Ruppert 2010).

In practice, we don’t just deal with one asset but with a whole universe of \(N\) assets.

We denote the log-returns of the \(N\) assets at time t with the vector \(\mathbf{r}_{t}\in\mathbb{R}^{N}\).

The time index \(t\) can denote any arbitrary period such as days, weeks, months, 5-min intervals, etc.

\(\mathcal{F}_{t-1}\) denotes the previous historical data.

Econometrics aims at modeling \(\mathbf{r}_{t}\) conditional on \(\mathcal{F}_{t-1}\).

\(\mathbf{r}_{t}\) is a multivariate stochastic process with conditional mean and covariance matrix denoted as (Feng and Palomar 2016) \[\begin{aligned} \boldsymbol{\mu}_{t} &\triangleq\textsf{E}\left[\mathbf{r}_{t}\mid\mathcal{F}_{t-1}\right]\\ \boldsymbol{\Sigma}_{t} &\triangleq\textsf{Cov}\left[\mathbf{r}_{t}\mid\mathcal{F}_{t-1}\right]=\textsf{E}\left[(\mathbf{r}_{t}-\boldsymbol{\mu}_{t})(\mathbf{r}_{t}-\boldsymbol{\mu}_{t})^{T}\mid\mathcal{F}_{t-1}\right]. \end{aligned}\]

The previous models are i.i.d., but there are hundreds of other models attempting to capture the time correlation or time structure of the returns, as well as the volatility clustering or heteroskedasticity.

To capture the time correlation we have mean models: VAR, VMA, VARMA, VARIMA, VECM, etc.

To capture the volatility clustering we have covariance models: ARCH, GARCH, VEC, DVEC, BEKK, CCC, DCC, etc.

Standard textbook references (Lütkepohl 2007; Tsay 2010, 2013):

📘 H. Lutkepohl. New Introduction to Multiple Time Series Analysis. Springer, 2007.

📘 R. S. Tsay. Multivariate Time Series Analysis: With R and Financial Applications. John Wiley & Sons, 2013.

Simple introductory reference (Feng and Palomar 2016):

📘 Y. Feng and D. P. Palomar. A Signal Processing Perspective on Financial Engineering. Foundations and Trends in Signal Processing, Now Publishers, 2016.

Consider the i.i.d. model: \[\mathbf{r}_{t}=\boldsymbol{\mu}+\mathbf{w}_{t},\] where \(\boldsymbol{\mu}\in\mathbb{R}^{N}\) is the mean and \(\mathbf{w}_{t}\in\mathbb{R}^{N}\) is an i.i.d. process with zero mean and constant covariance matrix \(\boldsymbol{\Sigma}\).

The mean vector \(\boldsymbol{\mu}\) and covariance matrix \(\boldsymbol{\Sigma}\) have to be estimated in practice based on \(T\) observations.

The simplest estimators are the sample estimators:

• sample mean: \(\quad\hat{\boldsymbol{\mu}} =\frac{1}{T}\sum_{t=1}^{T}\mathbf{r}_{t}\)
• sample covariance matrix: \(\quad\hat{\boldsymbol{\Sigma}} =\frac{1}{T-1}\sum_{t=1}^{T}(\mathbf{r}_{t}-\hat{\boldsymbol{\mu}})(\mathbf{r}_{t}-\hat{\boldsymbol{\mu}})^{T}.\)
  Note that the factor \(1/\left(T-1\right)\) is used instead of \(1/T\) to get an unbiased estimator (asymptotically for \(T\rightarrow\infty\) they coincide).

Many more sophisticated estimators exist, namely: shrinkage estimators, Black-Litterman estimators, etc.

Minimize the least-square error in the \(T\) observed i.i.d. samples: \[\underset{\boldsymbol{\mu}}{\textsf{minimize}} \quad\frac{1}{T}\sum_{t=1}^{T}\left\Vert \mathbf{r}_{t}-\boldsymbol{\mu}\right\Vert _{2}^{2}.\]

The optimal solution is the sample mean: \[\hat{\boldsymbol{\mu}}=\frac{1}{T}\sum_{t=1}^{T}\mathbf{r}_{t}.\]

The sample covariance of the residuals \(\hat{\mathbf{w}}_{t}=\mathbf{r}_{t}-\hat{\boldsymbol{\mu}}\) is the sample covariance matrix: \[\hat{\boldsymbol{\Sigma}}=\frac{1}{T-1}\sum_{t=1}^{T}\left(\mathbf{r}_{t}-\hat{\boldsymbol{\mu}}\right)\left(\mathbf{r}_{t}-\hat{\boldsymbol{\mu}}\right)^{T}.\]

• \(\boldsymbol{\mu}\in\mathbb{R}^{N}\) is a mean vector that gives the location
• \(\boldsymbol{\Sigma}\in\mathbb{R}^{N\times N}\) is a positive definite covariance matrix that defines the shape.

Given the \(T\) i.i.d. samples \(\mathbf{r}_{t}, \;t=1,\ldots,T,\) the negative log-likelihood function is \[\begin{aligned} \ell(\boldsymbol{\mu},\boldsymbol{\Sigma}) &= -\log\prod_{t=1}^{T}f(\mathbf{r}_{t})\\ &=\frac{T}{2}\log\left|\boldsymbol{\Sigma}\right| + \sum_{t=1}^{T}\frac{1}{2}(\mathbf{r}_{t}-\boldsymbol{\mu})^{T}\boldsymbol{\Sigma}^{-1}(\mathbf{r}_{t}-\boldsymbol{\mu})+\text{const}. \end{aligned}\]

Setting the derivative of \(\ell(\boldsymbol{\mu},\boldsymbol{\Sigma})\) w.r.t. \(\boldsymbol{\mu}\) and \(\boldsymbol{\Sigma}^{-1}\) to zeros and solving the equations yield: \[\begin{aligned} \hat{\boldsymbol{\mu}} &= \frac{1}{T}\sum_{t=1}^{T}\mathbf{r}_{t}\\ \hat{\boldsymbol{\Sigma}} &= \frac{1}{T}\sum_{t=1}^{T}\left(\mathbf{r}_{t}-\hat{\boldsymbol{\mu}}\right)\left(\mathbf{r}_{t}-\hat{\boldsymbol{\mu}}\right)^{T}. \end{aligned}\]

The parameter estimates \(\hat{\boldsymbol{\mu}}\) and \(\hat{\boldsymbol{\Sigma}}\) are only good for large \(T\), otherwise the estimation error is unacceptable.

For instance, the sample mean is particularly a very inefficient estimator, with very noisy estimates (Meucci 2005).

In practice, \(T\) cannot be large enough due to either:

• unavailability of data or
• lack of stationarity of data.

As a consequence, the estimates contain too much estimation error and a portfolio design (e.g., Markowitz mean-variance) based on those estimates can be severely affected (Chopra and Ziemba 1993).

Indeed, this is why Markowitz’s portfolio and other extensions are rarely used by practitioners.

Suppose the capital budget is \(B\) dollars.

The portfolio \(\mathbf{w}\in\mathbb{R}^{N}\) denotes the normalized dollar weights of the \(N\) assets such that \(\mathbf{1}^{T}\mathbf{w}=1\) (so \(B\mathbf{w}\) denotes dollars invested in the assets).

For each asset \(i\), the initial wealth is \(Bw_{i}\) and the end wealth is \[Bw_{i}\left(p_{i,t}/p_{i,t-1}\right)=Bw_{i}\left(R_{it}+1\right).\]

Then the portfolio return is \[R_{t}^{p}= \frac{\sum_{i=1}^{N}Bw_{i}\left(R_{it}+1\right)-B}{B}=\sum_{i=1}^{N}w_{i}R_{it}\approx\sum_{i=1}^{N}w_{i}r_{it}=\mathbf{w}^{T}\mathbf{r}_{t}\]

The portfolio expected return and variance are \(\mathbf{w}^{T}\boldsymbol{\mu}\) and \(\mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}\), respectively.

Capital budget constraint: \[\mathbf{1}^T\mathbf{w} = 1.\]

Long-only constraint: \[\mathbf{w} \geq 0.\]

Dollar-neutral or self-financing constraint: \[\mathbf{1}^T\mathbf{w} = 0.\]

Holding constraint: \[\mathbf{l}\leq\mathbf{w}\leq \mathbf{u}\] where \(\mathbf{l}\in\mathbb{R}^{N}\) and \(\mathbf{u}\in\mathbb{R}^{N}\) are lower and upper bounds of the asset positions, respectively.

Leverage constraint: \[\left\Vert \mathbf{w}\right\Vert _{1}\leq L.\]

Cardinality constraint: \[\left\Vert \mathbf{w}\right\Vert _{0} \leq K.\]

Turnover constraint: \[\left\Vert \mathbf{w}-\mathbf{w}_{0}\right\Vert _{1} \leq u\] where \(\mathbf{w}_{0}\) is the currently held portfolio.

The Net Asset Value (NAV) is the value of the portfolio (aka wealth, cumulative return, or cumulative PnL).

The portfolio return in terms of NAV is \(R_{t}^{p} = \frac{\textsf{NAV}_t - \textsf{NAV}_{t-1}}{\textsf{NAV}_{t-1}}\) (note that \(1 + R_{t}^{p} = \frac{\textsf{NAV}_t}{\textsf{NAV}_{t-1}}\)).

Also recall that the portfolio return is calculated from the assets’ returns as \(R_{t}^{p} = \mathbf{w}^{T}\mathbf{r}_{t}\).

Thus, if we fully reinvest, it follows that we can compute the wealth of a portfolio as \[ \textsf{NAV}_t = \textsf{NAV}_0 \times (1+R_{1}^{p}) \times (1+R_{2}^{p}) \times \dots \times (1+R_{t}^{p}). \] where \(\textsf{NAV}_0=B_0\) is the initial budget.

If, instead, we keep investing the same initial budget \(B_0\) at each period, then \[\begin{aligned} \textsf{NAV}_t &= \textsf{NAV}_0 + \textsf{NAV}_0 \times R_{1}^{p} + \textsf{NAV}_0 \times R_{2}^{p} + \dots + \textsf{NAV}_0 \times R_{t}^{p}\\ &= \textsf{NAV}_0 \times (1 + R_{1}^{p} + R_{2}^{p} + \dots + R_{t}^{p}) \end{aligned}\]

• Buy & Hold (B&H)
• Buy & Rebalance
• equally weighted portfolio (EWP) or \(1/N\) portfolio
• quintile portfolio
• global maximum return portfolio (GMRP).

The simplest investment strategy consists of selecting just one asset, allocating the whole budget \(B\) to it:

• Buy & Hold (B&H): chooses one asset and sticks to it forever.
• Buy & Rebalance: chooses one asset but it reevaluates that choice regularly.

The belief behind such investment is that the asset will increase gradually in value over the investment period.

There is no diversification in this strategy.

One can use different methods (like fundamental analysis or technical analysis) to make the choice.

Mathematically, it can be expressed as \[\mathbf{w} = \mathbf{e}_i\] where \(\mathbf{e}_i\) denotes the canonical vector with a 1 on the \(i\)th position and 0 elsewhere.