Equity Risk Premia

Standard Approach in Public Markets

The premise that a limited number of factors explains the majority of investment risk found in financial securities makes the development of robust and persistent factor models of returns an important part of investment-risk management.

With frequent trading and observable prices and returns, factor models can be used to decompose portfolio risk according to common factor exposures and to assess how much of the portfolio's returns are attributable to each common factor exposure.

The standard approach in both academic and industry factor models is the two-step regression method put forward by Fama & McBeth or FMB[1].

• In a first step, asset returns are regressed against one or more factor time series to determine factor exposures or betas.
• In a second step, the cross-section of portfolio returns is regressed against factor exposures at each time step, to give a time series of risk premia coefficients for each factor. FMB then averages each factor coefficient to get a time series of factor prices (lambdas).

Hence, the FMB approach consists of estimating two sets of coefficients, since both the asset betas or factor loadings and the market prices of each risk factor in the APT pricing equation (see (Modern Approach:3)) are unknown and must be estimated using time series of asset prices/returns. Once individual factor loadings have been estimated over time, factor prices (lambdas) are estimated in the cross section of returns given estimated asset betas.

This is possible because individual security prices are observable over time in sufficiently long time series as well as in the cross section in sufficiently large numbers. FMB uses the two dimensions of the data available to estimates first the betas and then the lambdas of the APT framework.

Application to Private Markets

With illiquid financial assets, too few trades are available to decompose individual asset returns into exposures to common sources of risk over time and estimate asset betas. However, if individual factor exposures can be estimated or assumed directly and a minimum number of transaction prices can be observed in each time period, risk-factor prices (lambdas) can still be estimated in the cross section of expected returns.

An approximate cross section of expected returns

Say that, in any given period, a number of primary and secondary transactions are observable in the market for unlisted infrastructure investments (this is a requirement of the condition to be observing a principal market in the sense of IFRS 13).

As long as sufficient information about the expected cash flows to equity or debt holders can be obtained or estimated, at any time $//<![CDATA[ \begin{array}{l}t\end{array} //]]>$ we have:

 
for primary or secondary investment $//<![CDATA[ \begin{array}{l}P_i\end{array} //]]>$ in asset $//<![CDATA[ \begin{array}{l}i\end{array} //]]>$, paying $//<![CDATA[ \begin{array}{l}CF_t\end{array} //]]>$ until time $//<![CDATA[ \begin{array}{l}T\end{array} //]]>$. $//<![CDATA[ \begin{array}{l}R_f+E(\tilde{R_i})\end{array} //]]>$ is the approximate expected internal rate of return (IRR) at time $//<![CDATA[ \begin{array}{l}t\end{array} //]]>$.

Using standard root-finding techniques (see technical appendix), $//<![CDATA[ \begin{array}{l}E(\tilde{R_i})\end{array} //]]>$, an estimate of the expected rate of (excess) return for the entire life of the investment, can be derived.

Hence, a cross section of expected returns is observable in each period. These estimates are noisy because they are solely derived from initial and secondary investment values and expected cash flows. Cash-flow forecasts are characterised by measurement errors, and we know that cash-flow timings and size can have a dramatic impact on IRRs.

Moreover, only a fraction of the investments representing the broad market at time $//<![CDATA[ \begin{array}{l}t\end{array} //]]>$ are the object of observable transactions at that time. Hence, only a subset of approximate excess returns $//<![CDATA[ \begin{array}{l}E(\tilde{R_i})\end{array} //]]>$ is available in each period. 

The cross section of factor prices

Once a cross section of approximate expected returns is known, it can be regressed against individual asset factor loadings (betas) to estimate individual factor prices ( $//<![CDATA[ \begin{array}{l}\hat{\lambda_k}\end{array} //]]>$) in the cross section, at that point in time. Thus, we have:

where $//<![CDATA[ \begin{array}{l}\omega_i\end{array} //]]>$ is the measurement noise introduced when estimating the risk premia  $//<![CDATA[ \begin{array}{l}\gamma_i\end{array} //]]>$. In other words, using the APT equation, we can write estimated excess returns at time $//<![CDATA[ \begin{array}{l}t\end{array} //]]>$ as a function of factor loading and factor prices plus some measurement error.

Using a range of statistical techniques (e.g. Bayesian regressions) the value of $//<![CDATA[ \begin{array}{l}\lambda_i\end{array} //]]>$ can be estimated for observable transactions at time $//<![CDATA[ \begin{array}{l}t\end{array} //]]>$ (the cross section of transactions).

Estimated $//<![CDATA[ \begin{array}{l}\hat{\lambda_k}\end{array} //]]>$ then gives us new estimates of expected excess returns $//<![CDATA[ \begin{array}{l}\hat{\gamma_j}\end{array} //]]>$ for all assets $//<![CDATA[ \begin{array}{l}j\end{array} //]]>$ in the relevant period in accordance with their individual betas, including those assets for which no transaction prices were available at the time, so that:

Hence, our approach consists of predicting the prices of risk factors that apply to assets with certain characteristics even though they have not been traded in the relevant period.

Choice of Factors for Unlisted Infrastructure Equity

The following systematic five key risk factors are used by EDHECinfra to estimate the a model of the expected returns using observable market prices as inputs, as well as several control (dummy) variables that account for sector and business model specific effects.