Performance and Adjustments (Week 4)

This post will take a first look into the actual realized performance of the constructed portfolio over the first 4 week period. Afterwards, the impact of the adjustments discussed in the previous post will also be looked into in more detail.

Below is the plotted returns of the client's portfolio compared to the benchmark:

The client's portfolio achieved a 1.31% return in this period compared to the 1.67% return of the benchmark. Looking at the plot of the performance difference, it can be seen that there was variation both above and below that of the benchmark. 

At this stage it would seem premature to read too much into the differences or to over analyse the small sample of data. Instead this first period of returns can act as a yardstick against which further adjustments made to the portfolio can be compared. What this first period does seem to imply is that under a simple 1/N passive investment strategy, the performance of the portfolio is similar to the benchmark.

Recall from the previous post that it has been decided to re-weight the allocation of funds within the portfolio on the basis of the mean-variance optimization results. Below are the adjusted weightings for the new allocation:

Lets also take a brief look at how this has changed the structure of the portfolio. Average maturity is now 9.5 years (up from 8.9 years in week 0), and similarly duration has also increased slightly. This means that over the next period we can expect the portfolio to react a bit more to changes in interest rates.

The characteristics by country are also provided below:

It seems that going forward the performance of the portfolio will be stronger tied to the performance of corporate bonds in Canada and Japan, as they now each contribute 35% weightings.

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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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Portfolio Characteristics (Week 0)

This post will look in more detail at the characteristics of the portfolio of corporate bonds constructed and also see how the portfolio performs in a one-year backdate test.

The table below shows the full details of each of the bonds hand picked for the portfolio. Individual bonds were not selected based on their coupon rates or yield-to-maturity. Instead, bonds were selected according to the diversification criteria discussed in the previous blog post.

Looking at the summary of holdings below gives a quick indication that bonds are fairly evenly spread across denominations in 6 different currencies. Furthermore, each individual bond has been assigned the same weighting; each of the 40 stocks chosen were allocated weightings of 2.5%. This follows the simple 1/N allocation strategy.

The view below in Bloomberg has been customized to analyse the portfolio by currency segments. It can be seen that the Australian segment of bonds has significantly lower average maturity (Mty=5.23 yrs) and thus also lower modified duration (ModDur=4.45). Modified duration gives a first order approximation for how the bond's price will react to a 100 basis point change in interest rates. If interest rates in Australia were to rise by 1%, this figure predicts a 4.45% drop in the price of the bonds. 

The Australian bonds had lower maturities since there were no AUD denominated bonds at longer maturities. Most Australian bonds at longer maturities were denominated in USD. So this is largely an artifact of the selection rules applied to the portfolio. It will be interesting to note whether the AUD segment performs comparatively better if a general decline in interest rates is observed.

A replicate of the portfolio has also been constructed one year prior so that the backdated performance of the portfolio can be analysed between 17/02/14 and 17/02/15. Backdating is certainly not the most reliable measure of future performance, but it is a useful procedure for getting a feel for what expectations we should have about the portfolio. 

Furthermore, backdating is useful to determine a suitable choice of benchmark. Without a benchmark for comparison it is not possible to evaluate the performance of a portfolio. For instance, say the value of the portfolio falls by 2% over the next month. A stand-alone absolute figure like this does not inform us on whether that is a good or bad result. If other comparable portfolios had lost 5% over the same period, even this loss could indicate a well managed fund.

Below is the backdated performance of the portfolio against the Bloomberg Global Investment Grade Corporate Bond Index (BCOR). This portfolio was chosen since it holds the most diversified holdings among the indices considered, with considerable holdings across each of the countries included in the client's portfolio.

Looking at these return graphs performance shows that there is a high degree of correlation between the two funds (in fact that correlation is 0.89, as shown in the statistical summary below). This is a good sign as it seems the benchmark chosen is an appropriate one. While both funds showed solid performance, over the past year the returns of the constructed portfolio fell short of the benchmark by approximately 1%.

The attribution tab in Bloomberg can help break down the performance further:

According to the attribution results there was a large difference in returns attributable to currency exposure. It seems despite issues from a wide selection of countries, the benchmark contains mostly (~60%) bonds denominated in USD. Since the USD performed comparatively well over this period the benchmark achieved 5.15% greater returns from exchange rate exposure (performance is being measured in pounds sterling).

As for the allocation of stocks among sectors and the selection of individual stocks, Bloomberg's analysis is very favorable of the constructed portfolio. In fact, it outperformed the benchmark by a significant 3.16% in terms of selecting individual stocks.

Looking at the statistical summary above, it also seems that the constructed portfolio experienced lower volatility in its returns. Running a simulation analysis, supports this evidence further. The 5% Value at Risk (daily) for the portfolio is simulated at £5,150 compared to £5,706 for the benchmark index.

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