Portfolio Optimization with Many Risky Assets — Econ 133 (2024)

Portfolio Returns¶

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

• 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 themean and variance of the portfolio return.

• Recognize that the \(N\) asset returns, \(r_i\), are randomvariables.

• Denote the means of \(r_i\) as \(\mu_i\).

Portfolio Moments¶

• The \(N \times N\) covariance matrix of the returns contains thevariances, \(\sigma^2_i\), 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 riskyfrontier), 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 meanreturn.

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 fromthe constraint: \(\omega_N = 1 - \sum_{i=1}^{N-1} \omega_i\).

• This is a challenging problem that is only tractable with linearalgebra (we won’t solve it).

Risky Minimum-Variance Frontier¶

\(\qquad\)

Risky Minimum-Variance Frontier¶

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

• The lower portion of the frontier is inefficient since a higher meanportfolio exists with the same volatility on the upper portion ofthe frontier.

• The efficient MV frontier is generated by allowing investment in arisk-free asset and finding the CAL which is tangent to the riskyefficient MV frontier.

Efficient Minimum-Variance Frontier¶

\(\qquad\)

Optimization: Efficient MV Frontier¶

To determine the tangency portfolio, the investor solves the sameproblem 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 minimizingvariance subject to a particular expected return.

2. To determine the tangency portfolio, by maximizing the SharpeRatio subject to constraints on the mean and standarddeviation.

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

3. Maximize utility, \(U(\mu_p, \sigma_p)\), subject to investingin 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 bothbecause 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 estimationburden.

Portfolio Optimization Recipe¶

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

1. Specify (estimate) the return characteristics of all securities(means, variances and covariances).

2. Establish the optimal risky portfolio.

• Calculate the weights for the tangency portfolio.

• Compute mean and std. deviation of the tangency portfolio.

Portfolio Optimization Recipe¶

3. Allocate funds between the optimal risky portfolio and therisk-free asset.

• Calculate the fraction of the complete portfolio allocated to thetangency portfolio and to the risk-free asset.

• Calculate the share of the complete portfolio invested in eachasset of the tangency portfolio.

Separation Property¶

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

• The optimal risky (tangency portfolio) is the same for allinvestors, regardless of preferences.

• The tangency portfolio is simply determined by estimation and amathematical formula.

• Individual preferences determine the exact proportions of wealtheach investor will allocate to the two assets.

• This is known as The Separation Property (or Two FundSeparation).

Separation Property¶

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

• Determining the optimal risky portfolio (preference independent).

• Deciding what proportion of wealth to invest in the risk-free assetand the tangency portfolio (preference dependent).

Separation Property¶

The separation property will not hold if

• Individuals produce different estimates of asset returncharacteristics (since differing estimates will result in differenttangency portfolios).

• Individuals face different constraints (short-sale, tax, etc.).

The Power of Diversification¶

Let’s formalize the benefits of diversification. The variance of aportfolio 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 correlatedon average (\(\overline{Cov} = 0\)), there is no limit todiversification: \(\sigma^2_p = 0\).

• If there are systemic sources of risk that affect all assets(\(\overline{Cov} > 0\)) there will be a lower bound on abilityto diversify: \(\sigma^2_p > 0\).