# 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:

Future stream of dividends (cash flows),

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

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 assets can be obtained by discounting the future cash flows as follows:

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

When we observe a secondary market transaction for the equity of a company, given the access to secondary market transaction prices, the expected dividends stream through the companyâ€™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 risk factor that the company is exposed to at time and is the price the market is willing to bear for risk factor . The risk factors include 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 evolves over time as the investor preferences and market conditions evolve, we use a dynamic model with time-varying coefficients to capture this relationship through the use of the 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 market-to-market discount factor for valuing assets : .