Markowitz Mean Variance Optimization

Markowitz (1952) was the first to demonstrate how portfolio managers could effectively minimize the risk of a portfolio for a given a desired return. For any attainable rate of return the portfolio may be weighted among the available assets for investment such that the overall volatility is minimized. 

Net volatility is not simply the sum of the volatilities of the individual assets, because as some rise in value others may tend to fall - such movements in opposing directions may partially cancel each other out and lead to 'smoother', less volatile returns. These many inter-relations are measured by the covariances between each asset and calculated based on historical return data. Thus for any achievable return, there is one or more optimal allocations of funds which minimizes volatility by choosing those assets least correlated with each other. Each such portfolio is termed an efficient portfolio and finding this optimal allocation for each level of achievable return maps out the efficient frontier of portfolios which lie on a parabola of risk-return as shown below.

To find the optimal weighting of assets within a portfolio requires extensive historical data on the return characteristics of the assets under consideration. Thus far, for the first four weeks of investment our portfolio has used equal weightings among each of the 40 corporate bonds among 6 different regions (Australia, Canadian, Europe, Japan, UK, US). Liu (2012) has collected data on the returns and correlations of corporate bonds from these same six regions. This county level data can be used to optimize the weightings within our portfolio and bring allocations closer to the efficient frontier. Our portfolio so far has held 17.5% weightings in Canadian, Europe, UK and US corporate bonds and 15% weightings in Australian and Japanese corporate bonds. 

The mean-variance optimization problem can be expressed mathematically as the following:

Since we require that our weightings are non negative (i.e. not allowing short sales) a computational approach will be applied to solve the above minimization problem. To ensure our portfolio remains internationally diverse let us further restrict the allocation of the portfolio to any one region to be between 5 and 35 per cent. Using the array functions in excel and then the solver function allows the efficient allocation to be determined for any given risk tolerance. The graphic below shows the results of such analysis, showing how the efficient allocation between regions changes as the risk tolerance increases from 0 to 15.

At the lowest risk tolerance levels of 0 or 1 it can be seen that the efficient allocation of the portfolio is given by 35% Australian bonds, 35% Japanese bonds, 15% European bonds, and 5% for each Canadian, US, and UK bonds. As risk tolerance increases to 5 the portfolio shifts away from Australian bonds and towards Canadian bonds. At an even higher risk tolerance levels of 9 the major holdings are in Canadian and European bonds. As risk tolerance continues to increase to the highest levels holdings of US bonds continue to increase until the portfolio is primarily comprised of Canadian and US holdings. The historical data suggests that holdings of UK bonds should remain at the minimum level of 5% as greater returns for a given level of risk can be found in corporate bonds of other regions.

Through our risk tolerance surveys of the client it was determined that they have a below average risk tolerance. For this reason, a risk tolerance of 5 has been chosen as a suitable estimate to rebalance the portfolio more efficiently among regions. This requires an adjustment to our portfolio by increasing Canadian and Japanese bond holdings to 35%, decreasing European bonds slightly to 15% and decreasing all other holdings (Australia, UK, US) down to 5%.

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