12.2 Mean Variance Portfolio: Important concepts

Expected Return

  • Expected return for a group of n assets is calculated as
\[\begin{equation} E(R)=\sum_{i=1}^{n}p_{i}r_{i} \tag{12.1} \end{equation}\]
  • Portfolio construction should use discrete returns and not logarithmic returns

Risk

  • Risk is generally defined as the “dispersion of outcomes around the expected value”
  • Variance and Standard Deviation measure dispersion
  • The variance of a random variable X is
\[\begin{equation} \sigma_{X}^{2}=E(X-E(X)]^{2}=\sum_{i=1}^{n}p_{i}[x_{i}-E(X)]^{2} \tag{12.2} \end{equation}\]

where

  • E(X) is the expected value of a random variable X. This will be financial returns of a security.
  • \(p_{i}\) is the probability of the occurrence of \(x_{i}\)
  • expression applies to the variance of a single asset where \(x_{i}=r_{i}\)

Covariance of Returns

  • One of the measures of association between two (or more) random variables.
  • The covariance is positive if the variables tend to move in the same direction, while it is negative if they tend to move in opposite directions.
  • Denoted by \(\sigma_{i,j}\) or \(Cov(r_{i},r_{j})\)
\[\begin{equation} Cov(r_{i},r_{j})=E\{[r_{i}-E(r_{i})][r_{j}-E(r_{j})]\}=\sum_{s=1}^{n}p_{s}\{[r_{is}-E(r_{i})][r_{js}-E(r_{j})]\} \tag{12.3} \end{equation}\]

Correlation

  • Standardised by standard deviation lies between -1 and 1
\[\begin{equation} \rho_{XY}=\frac{Cov(r_{i},r_{j})}{\sigma_{i}\sigma_{j}} \tag{12.4} \end{equation}\]
  • Within the context of portfolio analysis, diversification can be defined as combining securities with less than perfectly positively correlated returns.
  • In order for the portfolio analyst to construct a diversified portfolio, the analyst must know the correlation coefficients between all securities under consideration.

Portfolio Return

Its a weighted average

\[\begin{equation} R_{p}=\sum_{i=1}^{n}w_{i}r_{i} \tag{12.5} \end{equation}\]

Portfolio Risk

Not a simple weighted average, correlation has to be accounted for.

\[\begin{equation} \sigma_{p}^{2}=E[r_{p}-E(r_{p})]^{2} \tag{12.6} \end{equation}\]

where \(r_{p}=(w_{1}r_{1}+w_{2}r_{2})\) for a two asset portfolio. After substitution.

\[\begin{equation} \sigma_{p}^{2}=E\{w_{1}r_{1}+w_{2}r_{2}-[w_{1}E(r_{1})+w_{2}E(r_{2})]\}^{2} \tag{12.7} \end{equation}\]

After rearranging and expansion

\[\begin{equation} \sigma_{p}^{2}=E\{w_{1}^{2}[r_{1}-E(r_{1})]^{2}+w_{2}^{2}[r_{2}-E(r_{2})]^{2}+2w_{1}w_{2}[r_{1}-E(r_{1})][r_{2}-E(r_{2})]\}\tag{12.8} \end{equation}\] \[\begin{equation} \sigma_{p}^{2}=w_{1}^{2}\sigma_{1}^{2}+w_{2}^{2}\sigma_{2}^{2}+2w_{1}w_{2}\sigma_{12} \tag{12.9} \end{equation}\]

where \(\sigma_{12}=Cov(r_{1},r_{2})\)

Portfolio Risk for N Assets

Here we consider 3 assets but the method is generalisable to N assets.

  • Matrix of mean returns
\[\begin{equation} \mu=\left(\begin{array}{c} \mu_{a}\\ \mu_{b}\\ \mu_{c} \end{array}\right)\tag{12.10} \end{equation}\]
  • Matrix of weights
\[\begin{equation} w=\left(\begin{array}{c} w_{a}\\ w_{b}\\ w_{c} \end{array}\right) \tag{12.11} \end{equation}\]

\(\sum w=1\)

  • Variance-covariance matrix
\[\begin{equation} \Sigma=\left(\begin{array}{ccc} \sigma_{a}^{2} & \sigma_{ab} & \sigma_{ac}\\ \sigma_{ab} & \sigma_{b}^{2} & \sigma_{bc}\\ \sigma_{ac} & \sigma_{bc} & \sigma_{c}^{2} \end{array}\right) \tag{12.12} \end{equation}\]
  • Portfolio expected return
\[\begin{equation} \mu_{p}=w^{\prime}\mu \tag{12.12} \end{equation}\]
  • Portfolio variance
\[\begin{equation} \sigma_{p}^{2}=w^{\prime}\Sigma w \tag{12.13} \end{equation}\]

See chapter-2 and 3 of Francis and Kim (2013) for further details

References

Francis, Jack Clark, and Dongcheol Kim. 2013. Modern Portfolio Theory: Foundations, Analysis, and New Developments. Vol. 795. John Wiley & Sons.