ASX Sector Risk Return Portfolio Optimisation

Sector performance since 2013 is summarised int he following chart. There have been significant differences between sectors:

• top performing Health Care performance very strong;
• technology sector is the next stronger performer, especially in the period since the start of COVID;
• the energy sector has been particularly weak; and
• similarly there have been no significant gains in the communication sector which reflects tough environment for the major media companies.

Use the filter to compare more recent relative performance.

Normalized returns over the period from to :

Measures of Asset Returns

Returns on financial assets are measured in terms of log differences over a time span or horizon h. For daily returns h is equal to 1, for weekly h equals 7. Due to patterns in financial asset volatility, for example mean reversion, patterns of financial market performance can vary based on the choice of horizon.

Average Returns

Average return for a sector (S) is the simple average of the log difference returns over the time period (T):

One benefit of log differences over percentage changes is that the simple sum of the returns accumulates to the total return over the observation period T.

Return Variance

The sector return variance is squred sum of the deviations from the mean return :

Sharpe Ratio

The Sharpe Ratio is a simple ratio of the average return to the variance of the returns. In general higher returns should be accompanied by higher volatility.

Covariance

The covariance measures the relative co-movement in the variance of the sector to the overall variance of the market (M).

Beta

A measure of the co-movement and hence diversification benefits of individual sectors.

• : most diversifying, the stock is often increasing while the rest of the market is falling;
• : diversifying, the stock is less volatile than the market overall and may fall less than market in a crisis;
• : more volatile than the market overall and is likely to increase more than the market overall.

The sector Beta is measured as the ratio of the sector covariance to the market variance :

Beta estimates, as with other measures of share performance are not necesary constant through time. Time variation in beta estimates illustrates both broader macroeconomic trends as well as structural differences between sectors in response to shocks. Time varying estimates for ASX sector betas, constructed through Kalman Filtering are provided here.

Sector Premium

The extent to which the realised returns in a sector exceed the market returns adjusting for covariance (beta), and estimate of the sector specific premium, often also called alpha.

Note these returns exclude dividends and hence are biased in favour of stocks with strong capital growth, which explains why sectors such as finance and utilities which have strong dividends have relatively lower sector average returns.

Autocorrelation

The autocorrelation of returns is a measure of the correlation between the returns of a stock and the returns of the same stock lagged by a certain number of periods. For the Australian sectors the latest autocorrelation estimates are close to zero, the first lag of the partial autocorrelation function is provided below. In efficient markets the autocorrelation of returns should be zero, as share prices should have no memory of past returns.

Risk Return Portfolio Optimisation

The selection of an optimal portfolio requires the selection of a combination of assets that generate the maximum return at a given risk appetite, or the minimum risk at a given desired return, subject to a budget contraint which represents the available investment capital. The problem is summarised as:

Where:

• is the portfolio risk represented which is to be minimised as the weighted sum of the return covariance ;
• is the portfolio return is the weight sum of the expected returns which yields the desired return ;
• and includes the normalised budget constraint that the weights should sum to 1

The solution is available through a constrained minimization problem, using a Lagrangian as where is the slack on the first condition and is the slack on the second.

The differentiating the Lagrangian with respect to the unknowns and setting them equal to zero, the first order conditions for the minimum are the following linear equations. There is one equation for each asset in the weighting vector :

This is a fairly simple system of N + 2 linear equations and N + 2 unknowns. Key to the solution is the inverse of the return covariance matrix. For a complete outline of the problem and the solution with worked examples Introduction to Computational Finance and Financial Econometrics with R.

Sector weights on the efficient frontier at the selected expected return are compared with the composition of the ASX 200 as at March 2024.

Negative weights imply short (sold) exposures, positive weights are long (purchased) positions.