Portfolio Optimization with Many Risky Assets

Portfolio Returns

Suppose you can now invest in an arbitrary number (\(N\)) of risky assets.

  • Index the assets by \(i = 1, \ldots, N\).
  • Let \(\omega_i\) be the fraction of income invested in asset \(i\).
  • We will always assume that \(\sum_{i=1}^N \omega_i = 1\).
  • We will denote the return to asset \(i\) by \(r_i\).
  • The portfolio return is expressed as
\[r_p = \sum_{i=1}^N \omega_i r_i.\]

Portfolio Moments

From the properties of expectation and variance, we can compute the mean and variance of the portfolio return.

  • Recognize that the \(N\) asset returns, \(r_i\), are random variables.
  • Denote the means of \(r_i\) as \(\mu_i\).

Portfolio Moments

  • The \(N \times N\) covariance matrix of the returns contains the variances, \(\sigma^2_i\), and covariances, \(Cov(r_i, r_j) = \sigma_{ij}\):
\[\begin{split}\Sigma_P & = \left[\begin{array}{cccc} \sigma^2_1 & \sigma_{12} & \cdots & \sigma_{1N} \\ \sigma_{21} & \sigma^2_2 & \cdots & \sigma_{2N} \\ \vdots & \vdots & \ddots & \vdots \\ \sigma_{N1} & \sigma_{N2} & \cdots & \sigma^2_N \end{array}\right]\end{split}\]

Portfolio Moments

Thus resulting moments of the portfolio are

\[\begin{split}\mu_p & = \sum_{i=1}^N \omega_i \mu_i \\\end{split}\]
\[\begin{split}\sigma^2_p & = \sum_{i=1}^N \omega^2_i \sigma^2_i + 2 \sum_{i=1}^{N-1} \sum_{j=i+1}^N \omega_i \omega_j \sigma_{ij}.\end{split}\]

What are other ways to express \(\sigma^2_p\)?

Optimization: Risky MV Frontier

To determine the set of efficient risky portfolios (the risky frontier), the investor solves

\[\min_{\{\omega_i\}_{i=1}^{N-1}} \sigma^2_P = \sum_{i=1}^N \omega^2_i \sigma^2_i + 2 \sum_{i=1}^{N-1} \sum_{j=i+1}^N \omega_i \omega_j \sigma_{ij}\]

subject to

\[\mu_p = \sum_{i=1}^N \omega_i \mu_i\]

where \(\mu_p\) is some prespecified value of the portfolio mean return.

Optimization: Risky MV Frontier

Note that

  • The optimization problem has \(N-1\) choice variables: \(\{\omega_i\}_{i=1}^{N-1}\).
  • \(\omega_N\) is not a choice variable because it is found from the constraint: \(\omega_N = 1 - \sum_{i=1}^{N-1} \omega_i\).
  • This is a challenging problem that is only tractable with linear algebra (we won’t solve it).

Risky Minimum-Variance Frontier

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Risky Minimum-Variance Frontier

The frontier generated by multiple risky assets is known as the risky minimum-variance (MV) frontier.

  • The lower portion of the frontier is inefficient since a higher mean portfolio exists with the same volatility on the upper portion of the frontier.
  • The efficient MV frontier is generated by allowing investment in a risk-free asset and finding the CAL which is tangent to the risky efficient MV frontier.

Efficient Minimum-Variance Frontier

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Optimization: Efficient MV Frontier

To determine the tangency portfolio, the investor solves the same problem as before

\[\max_{\mu_p, \sigma_p} SR_p = \frac{\mu_p - r_f}{\sigma_p}\]

subject to

\[\mu_p = \sum_{i=1}^N \omega_i \mu_i\]
\[\sigma_p = \sqrt{\sum_{i=1}^N \omega^2_i \sigma^2_i + 2 \sum_{i=1}^{N-1} \sum_{j=i+1}^N \omega_i \omega_j \sigma_{ij}}.\]

Optimization: Investor Choice

So far we have specified two optimization problems:

  1. To determine the risky minimum-variance frontier by minimizing variance subject to a particular expected return.
  1. To determine the tangency portfolio, by maximizing the Sharpe Ratio subject to constraints on the mean and standard deviation.

Neither of these made use of preferences. A final optimization problem would be the same as before:

  1. Maximize utility, \(U(\mu_p, \sigma_p)\), subject to investing in the tangency portfolio and a risk-free asset.

Estimation

In practice we must estimate \(\mu_i\), \(\sigma^2_i\) and \(\sigma_{ij}\) for \(i=1,\ldots,N\) and \(j=i+1,\ldots,N\).

  • A total of \(N\) estimates of means.
  • How many variances and covariances must we estimate?
  • A total of \(N\) elements on the diagonal (variances).
  • All of the elements above or below the diagonal (not both because of symmetry).

Estimation

  • The resulting number of variance and covariance estimates is
\[\begin{split}N + (N-1) + (N-2) + \ldots + 2 + 1 & = \sum_{i=1}^N i = \frac{N(N+1)}{2}.\end{split}\]

Estimation

The total number of estimates is

\[\begin{split}N + \frac{N(N+1)}{2} & = \frac{N(N+3)}{2}.\end{split}\]
  • As an example, a portfolio of 50 stocks requires \(\frac{50 \times 53}{2} = 1325\) estimates.
  • The models of subsequent lectures will reduce this estimation burden.

Portfolio Optimization Recipe

For an arbitrary number, \(N\), of risky assets:

  1. Specify (estimate) the return characteristics of all securities (means, variances and covariances).
  1. Establish the optimal risky portfolio.
  • Calculate the weights for the tangency portfolio.
  • Compute mean and std. deviation of the tangency portfolio.

Portfolio Optimization Recipe

  1. Allocate funds between the optimal risky portfolio and the risk-free asset.
  • Calculate the fraction of the complete portfolio allocated to the tangency portfolio and to the risk-free asset.
  • Calculate the share of the complete portfolio invested in each asset of the tangency portfolio.

Separation Property

All investors hold some combination of the same two assets: the risk-free asset and the tangency portfolio.

  • The optimal risky (tangency portfolio) is the same for all investors, regardless of preferences.
  • The tangency portfolio is simply determined by estimation and a mathematical formula.
  • Individual preferences determine the exact proportions of wealth each investor will allocate to the two assets.
  • This is known as The Separation Property (or Two Fund Separation).

Separation Property

The separation property implies that portfolio choice can be separated into two independent steps:

  • Determining the optimal risky portfolio (preference independent).
  • Deciding what proportion of wealth to invest in the risk-free asset and the tangency portfolio (preference dependent).

Separation Property

The separation property will not hold if

  • Individuals produce different estimates of asset return characteristics (since differing estimates will result in different tangency portfolios).
  • Individuals face different constraints (short-sale, tax, etc.).

The Power of Diversification

Let’s formalize the benefits of diversification. The variance of a portfolio of \(N\) risky assets is

\[\begin{split}\sigma^2_p & = \sum_{i=1}^N \sum_{j=1}^N \omega_i \omega_j \sigma_{ij} = \sum_{i=1}^N \omega^2_i \sigma^2_i + 2 \sum_{i=1}^{N-1} \sum_{j=i+1}^N \omega_i \omega_j \sigma_{ij}.\end{split}\]

In the case of an equally weighted portfolio,

\[\begin{split}\sigma^2_p & = \frac{1}{N^2} \sum_{i=1}^N \sigma^2_i + \frac{2}{N^2} \sum_{i=1}^{N-1} \sum_{j=i+1}^N \sigma_{ij} \\ & = \frac{1}{N} \overline{Var} + \frac{N-1}{N} \overline{Cov}.\end{split}\]

The Power of Diversification

Where

\[\begin{split}\overline{Var} & = \frac{1}{N} \sum_{i=1}^N \sigma^2_i\end{split}\]

and

\[\begin{split}\overline{Cov} & = \frac{2}{N(N-1)} \sum_{i=1}^{N-1} \sum_{j=i+1}^N \sigma_{ij}.\end{split}\]

These are the average variance and covariance.

The Power of Diversification

The limit of portfolio variance is

\[\begin{split}\lim_{N \to \infty} \sigma^2_p & = \lim_{N \to \infty} \frac{1}{N} \overline{Var} + \lim_{N \to \infty} \frac{N-1}{N} \overline{Cov} = \overline{Cov}.\end{split}\]
  • If the assets in the portfolio are uncorrelated or not correlated on average (\(\overline{Cov} = 0\)), there is no limit to diversification: \(\sigma^2_p = 0\).
  • If there are systemic sources of risk that affect all assets (\(\overline{Cov} > 0\)) there will be a lower bound on ability to diversify: \(\sigma^2_p > 0\).