Modern portfolio theory strives to optimise an investor’s allocation across asset classes and the investor’s risk tolerance. The parameters of risk (measured as volatility), returns and correlations are the input variables to optimise the return for a given risk.

Figure 1 below illustrates the efficient frontier for a traditional portfolio consisting of stocks and bonds. All portfolios along the curve are ‘efficient’ in the sense that investors cannot create a non-levered portfolio with the same return but a lower risk, or the same risk but a higher return.

Herein we will focus on the minimum variance portfolio (such as the portfolio with the lowest risk) and the maximum Sharpe portfolio (such as the portfolio that maximises the Sharpe ratio)**1**. Calculating these optimal allocations for portfolios containing only public stocks and bonds results in a bond allocation of around 80% (minimum variance portfolio) and 75% (maximum Sharpe portfolio) based on historical data**2**.

While modern portfolio theory is still one of the major tools for portfolio construction, it is hotly debated among both practitioners and researchers. People question to what extent the assumptions of standard portfolio optimisation are justified or violated. In addition, the composition of optimal portfolios might not always seem feasible.

When moving along an efficient frontier (for example, in varying risk tolerances), the portfolio composition can change substantially with just a small move of the target risk and the portfolio composition often neglects entire asset classes. This sensitivity makes portfolio optimisation more of an art than a science and the results should often be considered as general guidance rather than a strict directive.

The recent financial crisis painfully demonstrated that asset allocation should not depend solely on quantitative models and historical time series. Additional parameters such as duration and liquidity must be taken into consideration.

Last but not least, investors need to carefully analyse the availability and characteristics of input data.

**Optimising for private equity**

Private company valuations typically reflect fair asset values rather than actual transaction values. Although similar tools are used to value public and private companies, the frequency and focus of valuations varies significantly. As a consequence of private equity valuation mechanics, return series exhibit autocorrelation, which, if not corrected for, may distort the results of portfolio optimisation.

The fact that quarterly returns are, to some extent, dependent on the returns observed in previous quarters, smoothes time series and reduces the measured volatility and the correlation with other asset classes.

Since volatility and correlation are the input variables for portfolio optimisation, auto-correlation in time series would naturally impact the result of portfolio optimisation. There are techniques that allow for the ‘un-smoothing’ of time series (see Conner, for example**3**). Using these methods, one

can determine an adjusted volatility and an adjusted correlation of the underlying economic process and use these underlying parameters as input variables for portfolio optimisation rather than original private market return series.

Even based on adjusted parameters, modern portfolio theory suggests an 11% allocation to private equity for the minimum variance portfolio and a 27% allocation for the maximum Sharpe portfolio (see Table 1 below) based on adjusted historical data.

Interestingly, public equities do not receive any allocation in these two portfolios and private equity basically takes the place of public equity. For long-term investors with corresponding risk profiles and a high tolerance vis-À-vis illiquidity, such a move may actually make sense.

As initially discussed, investors will need to base their portfolio allocation not only on historical data but also on their outlook of expected future returns.

**It’s not the end of the story**

As mentioned before, modern portfolio theory focuses on volatility, returns and correlations as input parameters. In our opinion, there are various dimensions not covered in this framework that are very important in the actual asset allocation decision.

Portfolio optimisation is typically based on long-term, broadly diversified industry data. This does not take into account a potential positive/negative selection bias. As a matter of fact, private equity exhibits a large dispersion between top and bottom performers.

The data from Thomson Reuters illustrated in Figure 2 below underlines the importance of this investment selection. Comparing North American buyout industry returns with the S&P 500 shows that the broad buyout market outperforms the S&P 500 by around 3%. An investor that is able to identify and avoid bottom-quartile investment opportunities is able to increase the outperformance to nearly 5%. If such outperformance is factored in, the investor would naturally increase the allocation to private equity in a maximum Sharpe portfolio. On the other hand, an investor that is not able to identify and access top quartile opportunities is likely to under-perform a public market portfolio.

If such underperformance is factored in, there would consequently be no private equity allocation in the optimal portfolio. The uncertainty of cash flows of private market investments adds another dimension in our portfolio optimisation effort. Using sophisticated modelling, investors need to estimate their future cash flows based on their prevailing portfolio and their unfunded liabilities; actual cash flows will depend on many exogenous factors.

The input data used for the optimisation implicitly assumes that the investor is always fully invested. Given the uncertainty of future cash flows, investors face the difficulty of achieving and maintaining their target investment level over time.

Opportunity costs from being under or overinvested can be significant due to the illiquidity of the asset class and the substantial discounts possible in the case of forced secondary sales, which may erase the entire return benefits. It is clear that portfolio optimisation of industry data does not take these opportunity costs into account.

Further important factors to be considered for portfolio optimisation include liquidity considerations and regulations. Regulators worldwide are imposing new rules for insurance companies that shift the focus from asset-based capital requirements to risk-based capital requirements. Similarly, banks are facing additional capital requirements from Basel III and the Volcker rule. These regulations will further restrict degrees of freedom in asset allocation.

Conclusion

What is the optimal allocation to private equity? From a standard portfolio optimisation point of view, an unconstrained investor may allocate 10% or even up to 30% of overall assets to private equity. Individual investors’ preferences, different levels of investor sophistication and regulations will, however, continue to yield very different answers to this question in practice.

*André Freiis is the chief risk officer and Dr Michael Studer is head of investment risk management at Partners Group*

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**1** The Sharpe ratio is a measure for risk-adjusted performance and is defined as the expected excess return of an asset over the risk-free rate divided by the asset’s volatility

**2** Mutual Fund Performance, William F. Sharpe, Journal of Business, January 1966, p. 119-138

**3** Asset Allocation Effects of Adjusting Alternative Assets for Stale Pricing, A. Conner, The Journal of Alternative Investments, Winter 2003, p. 42-52