Skip to main content
Skip table of contents

1.4.2 Equity Risk Premia

JUNE 2024

Key points

  • Standard approaches in public markets consists of estimating risk factor loadings (betas) first, then risk factor prices (lambdas). This is the standard Fama-McBeth approach used to develop numerous factor models. 

  • With private, illiquid assets, factor loadings can be estimated using a bottom-up approach and the firm's financials and other relevant and observable characteristics.

  • Next, using realised secondary market transactions or a well-defined listed proxy, observable expected returns (e.g. deal IRRs) can be decomposed into time series of risk factor premia

  • Finally, once risk factor premia have been estimated for each valuation date, a firm-specific, mark-to-market risk-premia can be computed for any private assets for which factor loadings (e.g. financials) are observable at that time. 

  • InfraMetrics® model uses 5 key risk factors (Size, Profits, Investment, Leverage and Term) and a range of control variables to estimate the price of systematic risk factors in the private infrastructure equity market. 

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 (see (Fama and McBeth,2017)) 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 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).

The fair market value of any unlisted infrastructure equity investment is a function of three components:

1- Future stream of dividends (cash flows),

2- Term structure of risk free rates at the relevant horizon,

3- Risk premium.

As long as sufficient information about the expected cash flows to equity or debt holders can be obtained or estimated, at any time , the price of unlisted infrastructure asset can be obtained by discounting the future cash flows as follows:

for primary or secondary investment  in asset , paying  until time .  is the discount rates that the infrastructure company should be discounted at for time.

When we observe a secondary market transaction for the equity of company , given the access to secondary market transaction prices , the expected dividends stream through the company’s cash flows  and the risk-free , we estimate the deal risk premia through a numerical solver.

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.

The cross section of factor prices

Once the risk premium is obtained, we calibrate a risk factor model to understand the sensitivity of the risk premia to different risk factors. This factor risk premia is common to all infrastructure companies. The risk premia can be regressed against individual asset factor loadings (betas) to estimate individual factor prices Consider the following risk factor model of the risk premia:

Once a cross section of approximate expected returns is known, it can be regressed against individual asset factor loadings to estimate individual factor prices. Thus, we have:

where  is the measurement noise introduced when estimating the risk premia . In other words, using the APT equation, we can write estimated excess returns at time  as a function of factor loading and factor prices plus some measurement error.

Where represents the to risk factor that company is exposed to at time and is the price the market is willing to bear for risk factor . The risk factors include the company size, leverage ratio, profitability, investment, country risk and a range of sector and business model related variables.

As the relationship between the risk factors and the equity risk premia evolve over time as the investor preferences and market condition evolve, we use a dynamic model with time-varying coefficients to capture this relationship through the use of Kalman filter in the estimation.

The Kalman filter model consists of two key components: an observation equation and a state equation. The observation equation is given by:

This illustrates how the equity risk premia are linked to a set of risk factors modulated by the model's coefficients .

The state equation is given by:

where (an identity matrix), models the coefficients as an autoregressive process of order one, AR(1), capturing their time-varying nature.

The estimation of this model utilises the Kalman filter, which involves two stages: prediction and updating. In the prediction stage, the previous state estimate is used to forecast the current state based on previous data. The updating stage involves adjusting the state estimate based on the discrepancy between the predicted and realised equity risk premia, using this error to refine the state estimate and its variance.

The model is dynamically re-estimated for each transaction within a month, and the coefficients are averaged across these transactions to derive a robust estimate for that month. Finally, the updated coefficients are used to compute the marked-to-market discount factor for valuing assets .

Choice of Factors for Unlisted Infrastructure Equity

The following systematic five key risk factors are used by infraMetrics® model 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.

Factor Name

Factor Definition

Factor Interpretation


Total Assets

Larger infrastructure companies are more illiquid and complex than relatively smaller ones. The size factor systematically attracts a positive premia. 


Return on Assets before Tax

More profitable firms are more valuable. Higher profits thus tend to attract a lower premia. The factor premia or  is negative.  As a result, during the greenfield phase, when profits are also negative, or in periods of distress, this factor carries a positive premia.  


Capex over Total Assets

During the development or greenfield phase of infrastructure companies, relatively large capital costs are incurred and sunk. This is a riskier period and a higher ration of capital expenditure to size attracts a higher expected return i.e. a positive risk premia.


Total Liabilities over Total Assets 

Likewise, controlling for other effects, higher leverage signals higher risk and is characterised by a positive risk premia


20-year public bond yield minus 3-month public bond yield

The slope of the yield curve can be a good proxy of country risk, both political and macro-economic. The term spread is computed as the time of each valuation,  using the relevant curve in the country of the investment. A higher term spread is characterised by a positive risk premia and thus a higher aggregate risk premia.

TICCS® Business Risk

Merchant, Regulated or Contracted control variables

Controlling for business risk families as defined in TICCS® shows that merchant companies systematically attract a higher risk premia i.e. expected returns are higher in riskier segments of the infrastructure market.

TICCS® Sector 

Industrial activity superclass or class control variables

Likewise, a few sectors are found to have systematically higher or lower expected returns even after controlling for the effect of the factors described above e.g. renewable energy projects have systematically lower returns (or higher prices) even for similar size, profits, leverage. etc.

Summarising diagram for the estimation of risk factor premia for unlisted infrastructure assets

ERP Figure.png

click to enlarge image.

Video Tutorial

Watch a 2-minute video highlighting our approach to asset pricing using a multi-factor model in illiquid markets.

[1] Fama, E. F., MacBeth, J. D., 1973. Risk, Return, and Equilibrium: Empirical Tests. Journal of Political Economy, 81(3), 607–636.

JavaScript errors detected

Please note, these errors can depend on your browser setup.

If this problem persists, please contact our support.