*The mean-variance framework is a good starting point when considering property allocation in a multi-asset fund but the road from there is long and winding, as Parit Jakhria and Henri Vuong report*

Multi-asset funds have grown in popularity since the recent market downturn as investors seek the diversification benefits of spreading their risks across a wide range of asset types. The idea of mixing between assets such as equities, property, cash, corporate bonds and gilts is enticing, but success can only be achieved if the balance of mix is correct. As discussed in the September/October 2010 issue, there is a compelling case for a property allocation in a multi-asset portfolio, and this is largely borne out in practice. Property investments are rarely held in a vacuum, and are nearly always part of a multi-asset exposure and should be viewed in this context. However, the crucial question for a multi-asset portfolio manager is "how much?"

Determining the optimal balance between the asset types is straightforward in theory, but very challenging in practice. The mean-variance (M-V) framework forms an excellent starting point - it has long been a useful and powerful tool in driving asset allocation and is simple and easy to understand. However, in practice there are many issues relating to the nuances of property that must be carefully considered before the asset class can be robustly incorporated into the M-V framework.

**The classical mean variance framework**

To recap, the classical M-V framework characterises investments in terms of two dimensions, risk, represented by standard deviation (the square root of the variance) and return, represented by the mean. A portfolio's expected return is simply the weighted sum of the assets' returns within it but the calculation for portfolio risk is not as straightforward.

It follows that a correlation of less than 1 between any pair of assets could reduce the overall portfolio risk. This simple illustration demonstrates the benefits of diversification and forms the basis of modern portfolio theory, which provides rational investors with a method in which they may use diversification to optimise the risk-return trade-off within their portfolios. It is assumed that a rational investor would prefer a portfolio that provides a higher return for the same level of risk and vice versa - and such a portfolio would be said to dominate the less preferred option.

A portfolio of specific assets into which an investment can be made generates a point in the risk-return space. The collection of all such points derived from portfolios of different combinations of assets defines the investible region. This is illustrated below with three asset classes.

The efficient frontier is the collection of all the points in the investible region that cannot be dominated by any other point within the region. Combinations of assets along the efficient frontier represent portfolios for which there is lowest risk for a given level of return. Conversely, for a given amount of risk, the portfolios lying on the efficient frontier represent the combination offering the best possible return.

If the assets have non-trivial pair-wise correlations, the efficient frontier will be convex because the portfolio return is a linear combination of the asset class returns, whereas the standard deviation is less than that implied by a linear combination, resulting in ‘diversification'.

Thus we can see that property is an ideal asset class to provide diversification benefits to a multi-asset portfolio if one can assume that it is weakly correlated with other asset types.

Finally, the classical theory extends to take into account the inclusion of a risk-free asset in a portfolio. A risk-free asset is one that guarantees a certain level of return for zero additional risk, which when combined with other assets, exhibits changes in risk and return that are linear as a risk-free asset has zero covariance with other assets.

A risk-free asset, combined with the market portfolio, which is the portfolio on the efficient frontier with the highest Sharpe ratio,1 forms the Capital Market Line (CML). Assuming it is possible to borrow and lend at the risk-free rate, and that investors consider the Sharpe ratio to be a robust measure of risk-adjusted performance, rational investors will invest at a point along the CML that satisfies their risk appetite or return requirements.

Thus investments in portfolios that lie on the CML dominate all other previous portfolios (with the exception of the tangency point to the efficient frontier). As such, optimal allocation is a mixture of risk-free assets with the market portfolio, of which property forms some exposure.

**The devil is in the detail!**

In practice, however, getting the mix right is less than straightforward and there are a number of stumbling blocks. We discuss some of the key challenges, although this list is not intended to be exhaustive.

**• Parameter estimation**

We have shown that the Mean-Variance framework is theoretically simple and easy to understand. However, there are a number of considerations that need to be taken into account when adopting the framework in practice, notably estimation of the key sets of assumptions - means, variances and correlations. Projected risks and returns are at risk from contamination of parameter uncertainty. Parameters are difficult to estimate with the mainstream asset classes, but even more so for property.

Property has often been quoted as the "poor cousin" of the asset classes when it comes to -historical representative data. Property is illiquid, therefore data representing the market is based on valuations rather than transactions. But valuation data inherently suffers from valuation smoothing and valuation lag. Both these characteristics, combined with the relatively shorter time series, make parameter estimation especially difficult. So there is a danger that estimates of mean, variance and correlations based purely on empirical analysis tend to estimate inaccurately the characteristics of property assets, and the consequent optimal allocation.

For example, estimates of property risk based on historical IPD annual returns indicate a property volatility of around 12%. However, the IPD annual index is based on appraisals and therefore suffers from valuation smoothness. In order to estimate the ‘true' property volatility, historical data can be de-smoothed**(2)** and volatility re-estimated to be anything between 15-20%. Clearly, an estimate based on smoothed data would result in an underestimate of property risk, the implication being that investors using the M-V framework may accept a lower level of return than is needed to compensate for the risk they are taking in property investment.

Similarly, the same can be shown for property correlation estimates. For example, figure 3 shows estimated correlation of the IPD annual returns with UK equities**(3)** based on raw smoothed data and de-smoothed data. Figure 3 shows that correlation estimates based on the raw smoothed index indicate a lower correlation than that based on the de-smoothed data. This implies that asset allocation based on the under-estimated correlation will overstate the diversification benefits of investing in property assets, and consequently, an over-allocation to property. Clearly, parameter estimation is tricky, and especially so for property assets.

**• The changing future environment**

We have all heard the mantra about the past not being a guide to the future; this is relevant to any parameter estimation. For the property market especially, the emergence of retail funds has shown the asset class to be more liquid and more influenced by global capital market trends than previously thought. Shortening lease lengths has reduced the fixed-income element of property and, overall, the lessons learnt from the most recent downturn indicate that property is synchronised and interlinked with other asset classes to a greater extent than previously assumed.

Property's changing characteristics are only starting to appear in the data. In figure 3 the data show an increasing trend in the correlation between property and UK equities as indicated by a 15-year rolling correlation. A static correlation is less likely to capture the changing dynamics of the property market and its evolving relationship with other asset classes. The danger would be to remain trapped in history, basing parameter estimation on past make-up at the expense of capturing current and, more important, the future make-up of this asset class.

**• Model extensions and nuances**

The Mean-Variance framework provides only a two-dimensional analysis. It assumes that the expected return and the volatility are the only metrics that matter to the investor. This assumption may not fully capture characteristics that are important to the client. For example, insurance companies may be more interested in value at risk (VaR) measures.

For example the standard deviation risk measure results in positive and negative returns receiving equal weighting, whereas in reality investors may be generally more concerned with downside risk than upside risk. As seen in global markets recently, assumptions in the tail of distributions could be significantly different from those applicable for small fluctuations around current market positions. Like many other asset types, property is known to exhibit such characteristics since observations close to the mean and the tail ends tend to occur more frequently than that implied by a normal distribution. The resulting fatter tails and a more peaked centre describe what is known as a leptokurtic distribution. This means that mean variance analysis may under-estimate the true risk of fat-tailed asset classes.

Another consideration is that the M-V framework assumes that the investor is indifferent to other characteristics of the distribution of returns, such as skew. Real world attitudes to risk may lead to high levels of skewness. Therefore, in mean-variance terms, an investor can improve performance by ‘selling' skew, ie, by accepting negatively skewed returns in return for improvements in mean and/or variance.

In summary, mean and variance may not completely describe the complete risk and return properties of any asset class, and one also needs to take into account the metrics that best represent risk and return.

**• Taking the liabilities into account**

Property portfolios are very rarely considered to be completely unencumbered and unconstrained. For example, it is estimated that around 32%**(4)** of UK property holdings can be accounted for by life insurance and pension funds. These funds need to take careful consideration of the liabilities when choosing the optimal asset allocation, as well as the plethora of regulation and reporting measures affecting insurance companies and pension funds. Unit linked funds are not entirely divorced from liabilities, as they are still subject to cash inflows and outflows.

To summarise, although the mean-variance framework provides an excellent starting point, optimal allocation to property is not straightforward and there are complex considerations that should be borne in mind. The characteristics of property as an asset class exacerbate the allocation issue, and actual allocation to property remains the subject of intense research.

**Footnotes:**

*(1) Sharpe ratio is a measure of excess return for a given level of risk
(2) IPF paper on Index Smoothing and the Volatility of UK Commercial Property
(3) IPD Annual Index Digest
(4) IPD Databank size as % of Capital Value by Property Fund Type as at Dec 2009
Parit Jakhria (left) is director of quanti--tative research, portfolio management group,
Prudential.
Henri Vuong is SPR vice-chair, senior property research analyst - quantitative risk, PRUPIM*

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