The overall objective for all pension funds is to optimise the long-run expected rate of return. This involves determining the best investment strategy between the different asset classes.

Our objective was to develop a model that takes advantage of the increasing availability of sophisticated computer software. The model should determine an asset allocation and maximum expected rate of return for a chosen risk level suitable for the pension fund. Furthermore, the model should recognise the volatilities of a range of asset classes and their correlations. The study was carried out in co-operation with Morgan Grenfell Asset Management (MGAM), our largest external fund manager.

Our mutual work led to a novel approach to efficient frontiers, namely ternary maps of asset mixes. This is a straightforward generalisation of the standard two-asset efficient frontier. In addition, the model can be probed with a variety of scenarios, to divine messages that are more general than the results of historical analysis only.

We began by modelling the European portfolio mixture with the asset classes: Denmark, UK and rest of Europe. To fully understand the outcome we emphasise that all calculations of return and risk are shown in Danish krone terms, and that all currency positions have been unhedged.

We started out with historical data limited to roughly the last 10 years. Then we put forward a number of plausible scenarios for the likely evolution of the markets. This process leads to an understanding of the trade-offs that are sensible, bearing in mind likely risks and returns.

The result of our work in respect of diversification within Europe is shown in figure 1, which illustrates the extension of the standard two-asset efficient frontier to a three-asset map. It is possible to display the linear return relationship and the quadratic risk relationship for three assets in two dimensions because of the constraint that all proportions must add to one

Lines of equal return are drawn on the left-hand diagram. These are contours of the return surface. They are a set of parallel lines. Similarly, lines of equal risk can also be drawn as contours of a quadratic risk surface. The surface on the right-hand diagram is not planar, but contour bands.

The return chart illustrates that Europe ex UK historically has yielded the lowest returns in Danžsh krone terms. The return can be increased by altering the portfolio towards holding stocks in the UK and Denmark.

With regard to risk, the right-hand diagram illustrates that there is a significant benefit to international diversification from Denmark. Indeed, the lowest risk portion of the portfolio is at 50% UK and 50% Europe ex UK.

The normal risk-averse investor would wish to obtain maximum return for minimum risk. This can be visualised by scanning the return lines on the left-hand figure through the contours of risk on the right-hand figure to draw a balance between tolerable risk and respectable return.

If our pension fund does not want to take more risk than indicated by the second risk contour band - the orange colour - the modelling of diversification within Europe leads to a European mixture portfolio with approximately 60% Danish equities and 20% each in the UK and Europe ex UK

As I indicated earlier, one objective of the more sophisticated modelling was to undertake more general studies than the results of historical analysis alone as it is in no way certain that the future will be like the past.

The starting point of our sensitivity analyses is that the worldwide trend is for markets to become more integrated than less, and that the life cycle of an individual market is likely to move from emerging to mature. Thus risks and returns are likely to decline gradually over time. Furthermore, correlations between markets are more likely to rise than fall as the world financial system converges.

Figure 2 shows one of the results of the efficient frontier sensitivity studies on the European assets. In this example, the base historical case is modified by correlations raised by 0.1, reflecting market convergence in response to the development of the single European market. Here the risk structure has changed, displaying near risk-neutrality between the UK and Europe ex UK assets.

If the risk profile of the pension fund is unchanged this will lead to a slight downweight of Denmark in a European mixture portfolio from the base case.

Some general conclusions may be drawn from our scenario analyses. First, the European mixture portfolio is driven by international diversification away from the domestic Danish assets. The precise portfolio weighting is then driven, primarily, by the view of returns and, secondarily, by the covariance structure. The likely evolution of the covariance structure will tend to tilt the portfolio in the direction of the expected outperforming asset class.

The second level analysis concerns the interaction of the European mixture portfolio, the US and the Far East. Again, various scenarios are considered, more to prompt discussion than to provide a definite view. The result of the base modelling is depicted in figure 3.

If the pension fund is not willing to take any more risk than indicated by the lowest risk contour band - the dark green colour - the modelling of asset mix for the three regions leads to a portfolio with approximately 55% European equities, 25% in the US and around 15% in Pacific ex Japan.

Meanwhile, a key result revealed in this chart is that some admixture of the Far East is not too detrimental to risk. However, it must be emphasised that the emerging markets are high risk/return markets where a high degree of caution is required.

The general statement which can be drawn from the sensitivity studies undertaken is that, as long as the emerging markets remain distinct with respect to the more mature markets worldwide, return can be won for little additional risk by international diversification.

In the studies undertaken all risk/return calculations have been made in Danish krone terms, unhedged. There is, of course, opportunity to manage a currency overlay in addition to the asset classes. The general conclusions to be drawn from the scenarios analysing the currency effects are, first, that currency hedging reduces volatility, but has no systematic effect on return.

The implications for our portfolio strategy of our modelling in conjunction with MGAM can thus be summarised as follows:

q the modelling confirms the current thrust of strategy of diversifying by increasing overseas investments. We have a major commitment to global investing with a target of holding 20% of our investments in foreign equity.

q diversification into Europe reduces risk. The preferred region for us remains Europe, which constitutes a little more than 50% of our international equity investments.

q diversification into the rest of the world is also beneficial, but requires a long-term view on regional growth and a willingness to accept short-term volatility. As a long-term investor, we have set a benchmark of 15% for equity investment in emerging markets.

Steen Jørgensen is managing director of Finanssektorens Pensionskasse in Copenhagen. This article is based on a speech to the EFRP/NAPF international conference in Spain in October