Portfolio Returns


Kevin Crotty
BUSI 448: Investments

Where are we?

Last time:

  • Calculating returns
  • Fetching data
  • Summarizing returns

Today:

  • Returns of portfolios
  • Portfolio expected return
  • Portfolio standard deviation

Returns of Portfolios

Portfolios

  • Portfolio are combinations of underlying assets

  • Given return properties of the underlying assets, what are the return properties of their combination?

Expected Return of Portfolio of \(N\) Assets

\[ E[r_p] = \sum_{i=1}^{N} w_i E[r_i] \]

  • \(w_i\) is the portfolio weight of asset \(i\)
  • \(E[r_i]\) is the expected return of asset \(i\)
  • The portfolio is fully invested: \(\sum_i w_i = 1\)
  • Notation: \(E(r_p)=\mu_i\)

Variance of Portfolio of \(N\) Assets

\[ \text{var}[r_p] = \sum_{i=1}^{N} \sum_{j=1}^{N} w_i w_j \text{cov}[r_i,r_j] \]

  • \(w_i\) is the portfolio weight of asset \(i\)
  • \(\text{cov}[r_i,r_j]\) is the covariance between assets \(i\) and \(j\)
  • Recall that \(\text{cov}[r_i,r_i]=\text{var}[r_i]\) and \(\text{sd}[r_i]=\sqrt{\text{var}[r_i]}\)
  • Notation: \(\text{var}[r_p]=\sigma^2_p\); \(\text{cov}[r_i,r_j]=\sigma_{i,j}\); \(\text{sd}[r_p]=\sigma_p\)

Variance of Portfolio of \(N\) Assets: A Matrix View

\[ \text{var}[r_p] = \sum_{i=1}^{N} \sum_{j=1}^{N} w_i w_j \text{cov}[r_i,r_j] \]

\(w_1 w_1 \text{cov}[r_1,r_1]\) \(w_1 w_2 \text{cov}[r_1,r_2]\) \(w_1 w_3 \text{cov}[r_1,r_3]\)
\(w_2 w_1 \text{cov}[r_2,r_1]\) \(w_2 w_2 \text{cov}[r_2,r_2]\) \(w_2 w_3 \text{cov}[r_2,r_3]\)
\(w_3 w_1 \text{cov}[r_3,r_1]\) \(w_3 w_2 \text{cov}[r_3,r_2]\) \(w_3 w_3 \text{cov}[r_3,r_3]\)

Variance of Portfolio of \(N\) Assets: A Matrix View

\[ \text{var}[r_p] = \sum_{i=1}^{N} \sum_{j=1}^{N} w_i w_j \text{cov}[r_i,r_j] \]

\(w_1^2 \text{var}[r_1]\) \(w_1 w_2 \text{cov}[r_1,r_2]\) \(w_1 w_3 \text{cov}[r_1,r_3]\)
\(w_2 w_1 \text{cov}[r_2,r_1]\) \(w_2^2 \text{var}[r_2]\) \(w_2 w_3 \text{cov}[r_2,r_3]\)
\(w_3 w_1 \text{cov}[r_3,r_1]\) \(w_3 w_2 \text{cov}[r_3,r_2]\) \(w_3^2 \text{var}[r_3]\)

Variance of Portfolio of \(N\) Assets: A Matrix View

\[ \text{var}[r_p] = \sum_{i=1}^{N} w_i^2 \text{var}[r_i]+ 2 \sum_{j>i} w_i w_j \text{cov}[r_i,r_j] \]

\(w_1^2 \text{var}[r_1]\) \(w_1 w_2 \text{cov}[r_1,r_2]\) \(w_1 w_3 \text{cov}[r_1,r_3]\)
\(w_2 w_1 \text{cov}[r_2,r_1]\) \(w_2^2 \text{var}[r_2]\) \(w_2 w_3 \text{cov}[r_2,r_3]\)
\(w_3 w_1 \text{cov}[r_3,r_1]\) \(w_3 w_2 \text{cov}[r_3,r_2]\) \(w_3^2 \text{var}[r_3]\)

Example: Equal-weighted portfolio of two assets

  • Expected Return

\[\begin{align} E[r_p] =& w_1 E[r_1] + w_2 E[r_2] \\ =& 0.5 E[r_1] + 0.5 E[r_2] \\ \end{align}\]

  • Portfolio Variance

\[\begin{align} \text{var}[r_p] =& w_1^2 \text{var}[r_1]+ w_2^2 \text{var}[r_2]+ 2 w_1 w_2 \text{cov}[r_1,r_2] \\ =& 0.5^2 \text{var}[r_1]+ 0.5^2 \text{var}[r_2]+ 2\cdot 0.5\cdot 0.5 \text{cov}_{12} \\ =& 0.25 \text{var}[r_1]+ 0.25 \text{var}[r_2]+ 0.5 \text{cov}[r_1,r_2] \\ \end{align}\]

Variance of Portfolio of \(N\) Assets: Matrices

\[ \text{var}[r_p] = \sum_{i=1}^{N} \sum_{j=1}^{N} w_i w_j \text{cov}[r_i,r_j] = w'Vw \]

  • Portfolio weights vector \[w'=[w_1\, w_2\,...\,w_N]\]

  • Covariance matrix of returns: \[\begin{equation*} V = \begin{bmatrix} \text{var}[r_1] & \text{cov}[r_1,r_2] & \dots & \text{cov}[r_1,r_N] \\ \text{cov}[r_2,r_1] & \text{var}[r_2] & \dots & \text{cov}[r_2,r_N] \\ \vdots & \vdots & \ddots & \vdots \\ \text{cov}[r_N,r_1] & \text{cov}[r_N,r_2] & \dots & \text{var}[r_N] \\ \end{bmatrix} \end{equation*}\]

Covariance and Correlation

  • Covariance: absolute degree of co-movement between two assets
  • Correlation: relative degree of co-movement between two assets

\[ \text{corr}[r_i,r_j] = \rho_{ij} = \frac{\text{cov}[r_i,r_j]}{\text{sd}[r_i]\cdot\text{sd}[r_j]} \]

  • What are the possible values for \(\rho\)?

Portfolio Statistics in Python

Portfolio expected return in Python

import numpy as np

# Expected returns
mns = np.array([0.10, 0.05, 0.07])

# Portfolio weights
wgts = np.array([0.25, 0.5, 0.25])

#Portfolio expected return
port_expret = wgts @ mns

Portfolio variance: matrix approach

Given a covariance matrix \(V\): \[\begin{equation*} V = \begin{bmatrix} \text{var}[r_1] & \text{cov}[r_1,r_2] & \dots & \text{cov}[r_1,r_N] \\ \text{cov}[r_2,r_1] & \text{var}[r_2] & \dots & \text{cov}[r_2,r_N] \\ \vdots & \vdots & \ddots & \vdots \\ \text{cov}[r_N,r_1] & \text{cov}[r_N,r_2] & \dots & \text{var}[r_N] \\ \end{bmatrix} \end{equation*}\] and a vector of portfolio weights

\[w'=[w_1\, w_2\,...\,w_N]\,,\]

The portfolio variance is the matrix product:

\[ \text{var}[r_p] = w'Vw \,.\]

Portfolio variance in Python: Inputs

import numpy as np

##### Inputs
# Standard deviations
sds = np.array([0.20, 0.12, 0.15])

# Correlations
corr12 = 0.3
corr13 = 0.3
corr23 = 0.3

# Portfolio weights
wgts = np.array([0.25, 0.5, 0.25])

Covariance matrix: method 1

##### Method 1 to calculate covariance matrix
# Covariances
cov12 = corr12 * sds[0] * sds[1]
cov13 = corr13 * sds[0] * sds[2]
cov23 = corr23 * sds[1] * sds[2]
# Covariance matrix
cov = np.array([[sds[0]**2,  cov12,     cov13], \
                [cov12,      sds[1]**2, cov23], \
                [cov13,      cov23,     sds[2]**2]])

Covariance matrix: method 2

##### Method 2 to calculate covariance matrix
# Correlation matrix
C  = np.identity(3)
C[0, 1] = C[1, 0] = corr12
C[0, 2] = C[2, 0] = corr13
C[1, 2] = C[2, 1] = corr23
# Covariance matrix
cov = np.diag(sds) @ C @ np.diag(sds)

Portfolio risk in Python

##### Portfolio risk measures
# Portfolio variance
port_var = wgts @ cov @ wgts

# Portfolio standard deviation
port_sd  = np.sqrt(port_var)

For next time: Equity Markets





BUSI 448