Live index constituents are required to meet minimum size and maturity conditions (see methods section below).

Returns are time-weighted. Volatility is the standard deviation of returns. The Sharpe ratio is equal to excess returns divided by return volatility. In some years, the risk-free rate used to compute excess returns can be negative.
Maximum drawdown is the worst peak to through drop in value in any given year. The returns are computed on a cash-basis i.e. the "effective" rate of interest of each instrument is taken into account whether it is a fixed rate or variable rate instrument. In other words, flexible rate debt is assumed to benefit from a base rate swap.

The corporate bond market reference index is value-weighted. Bond reference metrics are computed using raw iboxx data and the same formulas/models used for the infrastructure debt index.

*by market value

*by market value

VaR is the 99.5%, one-year Value-at-Risk. Volatility is the standard deviation of returns. Maximum drawdown is the worst peak to through drop in value in any given year. Duration expresses asset price sensitivity to interest rate risk and is the option-adjusted (effective) duration.

In each reporting period, we estimate the sensitivity of individual constituent excess returns to year-on-year changes in interest rate level, term structure slope and convexity, as well as cash flow volatility changes. Unexplained variability is attributed to current 'market conditions', including the evolution of investor preferences and country specific factors e.g. regulatory changes in certain constituent countries in the index.

Changes in these factors can have a positive (negative) effect, meaning that a positive change increases (decreases) excess returns. The net effect is always equal to 1.

This map represents the physical infrastructure corresponding to the firms in each index.

Index returns : download a comma-separated file of the annual return values of the index

Index prices : download a comma-separated file of the annual price values of the index

Index constituents and weights download a comma-separated file of the current constituents and weight values in the index

This paper provides an overview of the data, methods and key findings of the first EDHECinfra private infrastructure debt benchmark. It uses data extracted from a 400 sample of firms representing a 2,700 universe of investable infrastructure in Europe from 2000 to 2016. The paper focuse son broad market indices and dicusses the differences between project finance and 'infrastructure corporate' debt, as well as the role of portfolio diversification in private illiquid debt investing.

*by Frederic Blanc-Brude, Aurelie Chreng, Majid Hasan, Qi Wang and Tim Whittaker*

Download the pdf here

This section provides an overview of the technology used to derive the index results presented above. More details can be obtained from individual EDHEC publications describing the theoretical background and technical development of each component of this methodology. These publications are referenced in the different sections available in this page.

Private infrastructure debt is seldom traded and only a limited amount of market price data is observable. Hence, the risk-adjusted performance of the senior debt of each firm in the index sample is derived by forecasting cash flows to debt holders, taking into account future scenarios of default and restructuring, and discounting them on the basis of the volatility of future payouts, and available price information (including the initial value of the investment and comparable transactions taking place each year) .

Once each senior debt tranche has been valued in each period, the derivation of the relevant risk-adjusted performance metrics at the asset level is straightforward.

Individual assets are then combined to represent the performance of a given portfolio or index.

To implement this approach, a number of building blocks are needed:

The latest ’base case’ senior debt service i.e. future principal and debt repayments is either obtained from data contributors or estimated using information available about each senior debt instrument present in the firm’s capital structure

The mean and variance of the firm’s debt service cover ratio (DSCR) are estimated for each firm in all realised periods and forecasted for the remainder of the firm’s debt maturity

Firms are grouped by risk ’clusters’ or buckets, as a function of their free cash flow volatility and time to maturity in each period

Credit risk is assessed for each company and future cash flows to debt holders are forecasted taking into account the impact of future defaults and restructuring scenarios

Within each risk bucket, a term structure of senior debt repayments discount factors (and its range) is derived, reflecting the value of the investment relative to expected payouts and conditional (future) payout volatility, as well as any relevant and observable market prices (credit spreads) in each year

Finally, after individual performance metrics have been obtained for each firm’s senior debt, a return co-variance matrix is estimated for each reference portfolio or index (and sub-index) and individual assets are aggregated following certain inclusion and rebalancing rules

In this section, we describe the approach taken to identify investable infrastructure firms and their private debt and to select a representative sample of these instruments to be included in a “broad market” private infrastructure debt index for Europe.

When selecting constituents and collecting data, we take a ’bottom-up’ approach to identify individual firms and instruments, and to collect the relevant data from a range of public and private sources. Thus, we avoid creating biases in the data collection by overweighting data made available by any one contributor, a common problem with studies involving privately sourced data. Here, the relevant firms and debt instruments are identified first and, in a second step, the relevant data is collected for a representative sample of the investable universe, for which data can be collected.

The private infrastructure debt universe under consideration is a subset of the investable infrastructure market in 14 European countries.

In each national market, “investable infrastructure companies” are identified. These are either the special-purpose entities typically used in infrastructure project financing, or a limited number of ’pure’ regulated or merchant infrastructure businesses such as ports, airports or water companies. The later are selected only if the majority of their commercial activity is related to providing certain infrastructure services in a narrow sense.^{1} All infrastructure firms included in the analysis are privately-owned and operated, hence they are ’investable’ in the sense that they can be acquired or lent to during the period of interest.

Going back to the early-to-mid-90s depending on the country, we identify a population of 2,687 private infrastructure companies that has, at one point, been investable. Not all these firms are investable as of today. Some projects have reached their maturity and disappeared. Some have been acquired and integrated within a larger firm – in which case they are dropped from the universe – and some have been terminated or gone bankrupt.

Amongst these firms, 2,301 are still alive in 2016, representing 790 billion Euros of book value. They are categorised by country, broad industrial sector groups and ’business models’ following the nomenclature put forward in previous EDHEC publications (see for example Blanc-Brude 2014).

Next, 400 firms are selected to create an **index sample** representing around 50% of each national, industrial and business model segment by size at any point in time. We also require that each firm has been operating for at least four years to be included in the index sample. In 2016, 372 firms are alive in the index sample . The indices later described in section

Figure 1 shows the proportion of the live investable European market covered by the index sample from 2000 to 2017. The number of live constituents in the index is indicated by the blue-grey dots, while the pink line indicates the share of total value in the investable universe tracked by the index.

Amongst the investable infrastructure market identified, not all firms have outstanding senior debt provided by third party creditors (as opposed to the firm’s shareholders). Amongst the 400 firms identified, 330 are found to have senior term debt provided by commercial banks, private loan investors or bond holders.

Figure 1 also highlights the evolution of the investable infrastructure market in Europe over the period. A decade and a half ago, fewer private investment opportunities existed in Europe and they were, on average, larger companies such as regulated water and power utilities. Since then, in the wake of the UK, European governments have embarked on a series of public-private partnership programs that have greatly increased the number of investment opportunities but also considerably reduced their average size. Even more recently, the development of renewable energy projects in the wind and solar sectors has also led to the creation of numerous but relatively small infrastructure firms.

As a result, our index sample includes 155 firms in 2000 and 121 senior debt investments, representing 54.6% of the investable firm universe and close to 80% of the investable debt universe at the time by total book value. It peaks at 394 firms or 52.4% of the universe in 2012, and 309 senior debt constituents or 56.4% of the market.

Today, partly as a result of multiple bankruptcies in the Spanish road sector since 2012, our index includes 372 live companies or 49.8% of the investable universe. 243 such firms still have live outstanding senior debt while 87 have fully repaid, pre-paid or failed to repay third party credit.

Current index sample breakdowns by number of firms are given in figures 2, 3 and 4 below

The coverage is computed on the basis of a population in which individual firms and projects are included if we can observe at least their incorporation and investment start (or financial close) dates, basic country, sector and business model categories, as well as corporate status (SPV or ’corporate’) and sufficient information about total assets (book value) and outstanding senior debt.

Detailed information about the firms included in the 400-strong index sample is then collected according to the template described in Blanc-Brude et al. (2016).

Multi-utilities and infrastructure conglomerates are excluded.↩

Blanc-Brude, Frédéric. 2014. “Benchmarking Long-Term Investment in Infrastructure.” *EDHEC-Risk Institute Position Paper*, June. pdf.

Blanc-Brude, Frédéric, Raffaëlle Delacroce, Majid Hasan, Cledan Mandri-Perrot, Jordan Schwartz, and Tim Whittaker. 2016. “Data Collection for Infrastructure Investment Benchmarking: objectives, Reality Check and Reporting Framework.” *EDHEC Infrastructure Institute-Singapore*. pdf.

Detailed financial information is collected for all 400 firms in the index sample, from their incorporation date to year end 2016 or their date of cessation of operations.

Following the EDHEC*infra* template, we collect data about each firm and each debt instrument identified as part of its capital structure. Firms are also the subject of a number of *events*,^{2} firms and instruments also have individual *attributes*,^{3} and they are also attached to *values* (see Blanc-Brude et al. 2016 for a detailed discussion).^{4}

This data is collected from multiple sources and aggregated, cross-referenced, analysed and validated by a team of human analysts. Each firm’s data is reviewed iteratively at three different levels of validation including computer-generated and human checks.

Two issues impact the collection of debt instrument data:

The source of issuance has to be clearly identified: three major debt issuing structures can issue private infrastructure debt: the infrastructure asset owner, a related finance company and a parent as part of portfolio financing. The latter was mainly observed when a company held several infrastructure projects of a similar type ie. power stations or wind farms. Identification of the debt instruments solely associated with the infrastructure asset being researched was necessary before data collection could proceed. Multiple reference sources were consulted to ensure that the debt data was as fully described as possible.

Subsequent corporate actions also have a profound affect on the outstanding debt balance for each infrastructure assets. Following the takeover of an infrastructure SPV’s holding company, project level debt can be repaid early and replaced by parent company financing. Furthermore, infrastructure portfolio companies can issue more debt to add more assets to their portfolios. These changes in the debt outstanding of the infrastructure firms as well the usual observations of debt re-financing have to be logged every year.

The consistency and integrity of each firm’s financials is ensured as well as the details of their financial structure through time, from the creation of the firm, until today.

Sources used include annual audited accounts filed with the relevant regulators in each country, contributed data from asset managers, asset owners and lenders, freedom of information requests, and commercial and open access databases of infrastructure projects and project finance and merger and acquisitions. The physical and spatial characteristics of each infrastructure are also collected and can be used to map the constituents of different sub-indices.

Market benchmark data used to estimate risk-free rates and market comparators is sourced from Datastream and Markit. In the next section, we describe the methodologies used to model the cash flows of each 400 firms before applying a structural credit risk and asset valuation model to each senior debt tranche in index.

e.g. incorporation, construction start and completion, operational phases start, defaults, refinancing and restructuring, pre-payments, end of investment life, etc.↩

e.g. for firms, business model, type of regulation, contracted or index nature of inputs and outputs, etc; for instruments, seniority, currency, repayment profiles, interest rates, maturity date, etc.↩

e.g. for firms, any items of balance sheet, P&L or cash flow statement, forecast or realised; for instruments, realised and future interest and principal repayments.↩

Blanc-Brude, Frédéric. 2014. “Benchmarking Long-Term Investment in Infrastructure.” *EDHEC-Risk Institute Position Paper*, June. pdf.

Blanc-Brude, Frédéric, Raffaëlle Delacroce, Majid Hasan, Cledan Mandri-Perrot, Jordan Schwartz, and Tim Whittaker. 2016. “Data Collection for Infrastructure Investment Benchmarking: objectives, Reality Check and Reporting Framework.” *EDHEC Infrastructure Institute-Singapore*. pdf.

Blanc-Brude, Hasan, and Ismail (2014) have shown that knowing the current or ’base case’ senior debt service of the firm, as well as the statistical characteristics (mean and variance) of the debt service cover ratio of private firms is sufficient to implement a proper structural credit risk model. Next, we summarise the approach taken to obtain these two inputs for each firm in the index sample.

The future senior debt service currently owed by each firm in the index sample is obtained from one of the following sources:

From private contributor data or bond documentation, or

Computed using individual debt instruments’ attributes (outstanding principal, interest rate, maturity date and amortisation profile) collected from contributors or audited accounts, or

Estimated using Bayesian inference after a 4 or 5 years of observed principal and interest payments, and other available information about the firm in question (e.g. average ’tail’ length

^{1}) and similar firms, as well as upper and lower bounds on credit spreads and yield-to-maturity at the time of origination (estimated from market data), typical amortisation profiles used in similar transactions, etc. (see Hasan and Blanc-Brude 2017a for more details)

Using these simple techniques, the future total senior debt service (principal and interest) owed by each firm in the index sample is known with reasonable certainty at the time of computing the index, and can be re-estimated on a regular basis, as new information about the firm’s financial structure becomes apparent (e.g. refinancings, restructuring post default, etc).

Debt Service Cover Ratios or DSCRs provide an economically significant measure of the ability of a firm to service its debt. At each point in time, the \(DSCR_t\) is defined as: \[DSCR_t=\frac{CFADS_t}{DS_t}\] where \(DS_t\) is the senior debt service owned at time \(t\) and \(CFADS_t\) is the cash flow available for debt service (the free cash flow) at time \(t\).

<<<<<<< HEADFor this index, an ’economic’ or cash-based \(DSCR_t\) is computed for each firm in the index sample, using cash flow statement information, so that: \[DSCR_t=\frac{\text{C}_{\text{bank}}+\text{C}_{\text{op}}+\text{C}_{\text{IA}}+\text{C}_{\text{dd}}-\text{C}_{\text{inv}}}{\text{DS}_{\text{senior}}}\] where \(\text{C}_\text{bank}\), \(\text{C}_\text{op}\), \(\text{C}_\text{IA}\), \(\text{C}_\text{dd}\), and \(\text{C}_\text{inv}\) denote cash at bank, cash from operating activities, cash withdrawal from investment account, cash from debt drawdowns, and cash invested physical investments, respectively (see Blanc-Brude, Hasan, and Whittaker 2016 for a more detailed discussion of how DSCRs can be computed).

=======For this index, an ’economic’ or cash-based \(DSCR_t\) is computed for each firm in the index sample, using cash flow statement information, so that: \[DSCR_t=\frac{\text{DS}_{\text{senior}}}{\text{C}_{\text{bank}}+\text{C}_{\text{op}}+\text{C}_{\text{IA}}+\text{C}_{\text{dd}}-\text{C}_{\text{inv}}}\] where \(\text{C}_\text{bank}\), \(\text{C}_\text{op}\), \(\text{C}_\text{IA}\), \(\text{C}_\text{dd}\), and \(\text{C}_\text{inv}\) denote cash at bank, cash from operating activities, cash withdrawal from investment account, cash from debt drawdowns, and cash invested physical investments, respectively (see Blanc-Brude, Hasan, and Whittaker 2016 for a more detailed discussion of how DSCRs can be computed).

>>>>>>> 5ee9ecd151d0e9d4061381817b2d90e1c5a072a9Next, our approach requires modelling and forecasting the expected value and volatility of a firm’s DSCRs at each point in its life.

In a first step, the mean \(\mu\) and variance \(\sigma^2\) parameters (or state) of the DSCR process have to be inferred from observable data. Since between 4 and 20 years of realised values are available for each firm, it is not possible to derive a robust and unbiased estimation of DSCR dynamics at the firm level using standard or ’frequentists’ statistical techniques.

Instead, Bayesian techniques (Monte Carlo Markov Chain) are used to infer the true value of the mean and variance parameters of the DSCR process in each period, based on an initial guess (or prior) and an auto-regressive model expressing a firm’s ability to pay its debt in any given year as a function of its ability to do so in the previous year, and of the effect of various control variables (e.g. time to maturity, future debt service profile, similar projects, etc.).

This ’state-space’ model can be represented by the following two equations:

\[x_t=f_t . x_{t-1}+\epsilon_t \text{ (state equation)}\] \[y_t=g_t . x_t+\eta_t \text{ (observation equation)}\]

where \(x_t\) is the unobserved state of the system at time \(t\), \(y_t\) is the DSCR observation at time \(t\), \(f_t\) is the “evolution” function, and \(g_t\) is the vector containing relevant control inputs. \(\epsilon_t\) and \(\eta_t\) are two independent white noise sequences with mean zero and variance \(\sigma^2\) and \(\omega^2\) respectively, which are the unknown parameters.

With each DSCR observation, the true value of the mean and variance parameters of each firm’s DSCR and their evolution in time is ’learned’ – just like a self-driving car continuously re-assesses its coordinates in an \((x,y)\) plane, we continuously re-assess the position of the DSCR process in the \((\mu,\sigma^2)\) plane.

Figure 1 and 2 illustrate this process for two example project companies in Italy and Germany: the time \(t\) value of the DSCR mean and variance is predicted at time \(t-1\), and effectively tracks the realised DSCR value at time \(t\).

Once the parameters of the DSCR distribution of each firm have been derived for realised time periods, we use these estimates to derive a *forecast* of the mean and variance of the firm’s DSCR until the maturity date of the current senior debt.

This is achieved by implementing Kalman filtering techniques with recursively computed ’innovations’ of the DSCR process as described in Wang and Blanc-Brude (2017) and illustrated in figure 3 and 4. In view of the Markovian (auto-regressive) nature of the state space model, The recursive formulae of the mean and variance of the firm’s DSCR at a future time \(t+k\), given the observed data up to time \(t\), are also derived using Bayesian methods: the \(\mu_t\) and \(\sigma^2_t\) at time \(t\) act like an initial distribution (prior) of the future evolution of the model, which provides a summary of available data that is sufficient for predictive purpose.

Hence, the corresponding posterior distribution contains all the information about the future provided by the available data. As \(k\) becomes larger, depending on the corporate structure and business model of the firm, uncertainty increases in the system, and the forecasts of the future true values of \(\mu\) and \(\sigma^2\), conditional on today’s information, can become less precise, just like long-term prices are forecasted with less certainty by market forces processing all available data today.

Blanc-Brude, Frédéric, Majid Hasan, and Omneia R H Ismail. 2014. “Unlisted Infrastructure Debt Valuation & Performance.” *EDHEC-Risk Institute Publications, EDHEC and NATIXIS Research Chair on Infrastructure Debt Investment Solutions* July. Singapore: EDHEC-Risk Institute. pdf.

Blanc-Brude, Frédéric, Majid Hasan, and Timothy Whittaker. 2016. “Cash Flow Dynamics of Private Infrastructure Project Debt, Empirical Evidence and Dynamic Modelling.” *EDHEC Infrastructure Institute Publications, EDHEC and NATIXIS Research Chair on Infrastructure Debt Investment Solutions* March (March). Singapore: EDHEC Infrastructure Institute. pdf.

Hasan, Majid, and Frédéric Blanc-Brude. 2017a. “A Baysien Approach to Estimating Future Debt Service.” *EDHEC Infrastructure Institute-Singapore*. Singapore: EDHEC Infrastructure Institute-Singapore.

Wang, Qi, and Frédéric Blanc-Brude. 2017. “Estimating and Forecasting the Cash Flow Process in Private Firms.” *EDHEC Infrastructure Institute-Singapore*. Singapore: EDHEC Infrastructure Institute-Singapore.

Next, once a DSCR mean and conditional volatility are known, each firm is assigned to a risk ’cluster’ or ’bucket’ in each year, as a function of its main risk characteristics. Hence, firms that have reasonably similar credit risk (as captured by the variance of \(DSCR_t\)), duration (as proxied by time to maturity), and lifecycle stage (as proxied by the number of years since the firm’s operations began) are assigned to the same risk bucket.

The rationale for this ’bucketing’ of individual firm’s senior debt is that firms with similar risk characteristics are assumed to represent the same combination of priced risk factors and carry – on average and at one point in time – the same risk premia.^{2}

Hence, this grouping of firms into reasonably homogenous volatility and maturity or age groups is useful for two purposes:

Deriving discount rates that correspond to a persistent combination of priced risk factors;

Computing pair-wise return covariances within clusters using the cluster mean return as the expected return for all assets in the same bucket.

This approach improves on previous ones put forward by Blanc-Brude and Hasan (2015) by which ’families’ of infrastructure firms defined more loosely in terms of business model were considered sufficiently homogeneous to capture well-defined combinations of priced risk factors. In practice, some merchant projects may behave more like contracted ones, and some contracted firms like regulated or merchant ones, etc.

The distinction between business models remains valid for the purpose of building sub-indices, but hierarchical clustering allows the derivation of more robust pricing measures and covariance estimates. Hierarchical clustering aims to group a set of objects in such a way that objects within each cluster are more similar to each other than to those in different clusters. It is a bottom-up approach by which, at each level, selected pairs of clusters are recursively merged into a single cluster, thus producing a new grouping at the next step (with one less cluster). The pair chosen for merging consists of the two groups with the smallest intergroup dissimilarity. The number of final groups depends on the heterogeneity of the original data.

Figures 1 and 2 illustrates this process: firms from all three infrastructure business models (contracted, merchant and regulated) can somewhat overlap in terms of DSCR variance, and for the purpose of asset pricing, much more homogenous risk groupings can be made using hierarchical clustering.Once the current senior debt service is known until the maturity of each instrument and the characteristics of the DSCR stochastic process are estimated and forecasted for the remaining of each firm’s senior debt life, the private debt asset pricing framework described in Blanc-Brude and Hasan (2016) and Hasan and Blanc-Brude (2017b) – which we refer to as the BBH framework – can readily be applied.

BBH show that a fully fledged, cash-flow driven structural credit risk model could be applied to infrastructure project debt since the *distance to default* (DD) metric at the heart of the Merton (1974) model can be written: \[DD_t = \frac{1}{\sigma_{DSCR_t}} \frac{\text{DS}^{\text{BC}}_{t-1} }{\text{DS}^{\text{BC}}_{t}} (1- \frac{1}{DSCR_t})\] and the \(DSCR_t\) metric itself provides an unambiguous definition of the various default thresholds that are relevant to predicting default accurately.

BBH also build on the fact that the free cash flow of the firms can be written as: \[\text{CFADS}_{t} = \text{DSCR}_{t} \times \text{DS}^{\text{BC}}_{t}\] to argue that in the case of infrastructure investment, because the value of the firm is solely driven by the value of future free cash flows,^{3} knowledge of the \(DSCR_t\) process and of the current debt service \(DS_t\) is sufficient to value the entire firm and build a stochastic model of the cash flow waterfall.

The value of senior debt is then the discounted value of expected cash flows to senior debt holders, taking into account the different path that such cash flows might take under different DSCR scenarios. BBH adapt the Black and Cox (1976) extension of the Merton model to express the value of the firm as the combination of all possible paths given a set of estimated \(DSCR_t\) dynamics, as discussed previously.

Scenarios under which the \(DSCR_t\) process would breach either a technical or hard default threshold are incorporated using a game theoretical renegotiation model of the restructuring of senior debt, balancing the relative bargaining power of debt and equity holders (see Hasan and Blanc-Brude 2017b for more details). Depending on the corporate structure of the firm (project finance SPV or corporation) different assumptions can be made about the relevant level of default thresholds and renegotiation/ restructuring costs).

Hence, the BBH framework allows taking into account the “option value of the debt tail” found in infrastructure project finance i.e. the embedded option for lenders to either waive default events, walk away from or work out (restructure) the problem, in order to maximise the expected value (or minimise the expected loss) of their investment.

This framework also allows predicting defaults and computing expected recovery rates at the firm level, avoiding the use of sector or regional averages for credit metrics, which can can be a poor approximations of the credit risk of individual exposures. For instance, in our index sample, certain firms exhibit DSCR levels that are either sufficiently high or of low volatility to be assigned expected default frequencies equal to zero. Importantly, these characteristics change over time and need to be tracked at the firm-level as shown above. Hence, other firms which are, on average, considered to be low risk such as “availability payment” public-private partnerships, can exihibt increasingly volatile or decreasing DSCRs, implying an increasing probability of default.

The original BBH framework uses the standard risk-neutral valuation framework, assuming a required price of risk (Sharpe ratio) for specific lenders/investors (see also Kealhofer 2003).

In the context of estimating asset values for a market index, we implement an approach described in Blanc-Brude and Hasan (2015) by which a term structure of discount factors is derived for each firm’s senior debt cash flows. This approach is consistent with the usual inter-temporal capital asset pricing models, such as Brennan and Xia (2003), Parker and Julliard (2005), Dittmar (2002).

In a first step, we use a no-arbitrage asset pricing model (a generic factor model of asset returns), to write discount rates in terms of risk-free rate and a risk premium. Next, we estimate forward-looking risk-free rates and the price of risks to obtain a term structure of risk-adjusted discount rates.

A general factor model of asset returns can be written as:

\[\begin{aligned} r_{i,t+1} &=r_{f,t+1} \\ &+ \sum_{k} \beta_{F_{k},t+1|t} E_{t}\left( r_{F_{k},t+1} - r_{f,t+1} \right)\\ &+ \epsilon_{i,t+1}\end{aligned}\]

where \(r_{i,t+1}\) is the return on \(i^{th}\) asset, \(r_{f,t+1}\) is the return on a riskfree asset, \(\beta_{F_{k},t+1|t}\) is the asset’s exposure to \(k^{th}\) risk factor, and \(E_{t}\left( r_{F_{k},t+1} - r_{f,t+1} \right)\) is the expected excess return on the \(k^{th}\) risk factor. The above equation can be rearranged to write the factor model of asset returns thus: \[r_{i,t+1}=r_{f,t+1}+ \lambda_{i,t+1|t} \sigma_{i,t+1|t} + \epsilon_{i,t+1} \text{.}\] with the excess return on any asset, \(r_{i,t+1}-r_{f,t+1}\), written as the asset’s forward-looking volatility, \(\sigma_{i,t+1|t}\) times the forward-looking ‘price of risk’, \(\lambda_{t+1|t}\), where the price of risk depends on the Sharpe ratio of the risk factor, \(\frac{\left( r_{F,t+1|t} - r_{f,t+1} \right)}{\sigma_{F,t+1|t}}\), and the asset’s correlation with that risk factor, \(\rho_{t+1|t}\).

Thus, the risk-adjusted discount rate for a \(\tau\)-period ahead cashflow is written: \[r_{i,t+\tau} = r_{f,t+\tau} + \lambda_{i,t+\tau|t} \sigma_{i,t+\tau|t} + \epsilon_{i,t+\tau}\] where \(\sigma_{i,t+\tau|t}\) and \(\lambda_{i,t+\tau|t}\) now denote a \(\tau\)-period ahead forecast of asset’s risk and the price of risk, respectively, as seen by the investor, from time \(t\).

One advantage of writing the factor model in this form is that if volatility can be modelled directly – as is the case here, then the price of risk can be inferred from the prices of observed transactions.

That is, given a time-series of volatility estimates, \(\sigma_{i,t}\), a time-series of \(\lambda_{i,t}\) can be estimated such that the observable transaction prices match the prices implied by the asset pricing model. This approach simplifies the task of having to model the expected returns and volatilities of priced factors, and the correlations of the asset with each priced factor.

Indeed, another important advantage of this approach is that it does not require identifying priced risk factors explicitly. As argued above, senior infrastructure debt may be exposed to combinations of priced risk factors that we called risk clusters or buckets, and the price for all risk factors in any given cluster is summarised by \(\lambda_{i,t}\), which can be estimated from observable prices, forecast cash flows and conditional payout volatility.

Since the only asset specific term in the price of risk is the asset correlation with the factors, \(\rho_{k,t}\), all assets with one risk cluster, with identical exposures to a given combination of priced risk factors, should earn identical mean returns. The “risk buckets” described in section Next, to empirically estimate the prices of risk of different risk exposures, we first estimate a term structure of relevant risk-free rates using standard term structure methodologies, such as Ang, Piazzesi, and Wei (2006). Then, we collect observable spread information for senior loans and bonds i.e. available market price data. These spreads can then be expressed in terms of risk premia as: \[spread_{i,t} = \sum_{k} \lambda_{k,t} \sigma_{k,t} + \epsilon_{i,t}\] where \(spread_{i,t}\) is the observed spread on the \(i^{th}\) loan, \(\lambda_{k,t}\) is the price of \(k^{th}\) risk exposure, and \(\sigma_{k,t}\) is the size of the \(k^{th}\) exposure. The different risk exposures that we consider include cashflow risk, as measured by the ‘cluster’ to which the project belongs based on a cluster approach described in Section The prices of risk are estimated by minimising errors between observed spreads and model implied spreads, so that: \[\underset{\lambda_{k,t}}{\min} \left( spread_{i,t} - \sum_{k} \lambda_{k,t} \sigma_{k,t} \right)\] This allows estimating the extent to which different risk exposures are priced. Performing this procedure year by year using instruments originated in each year, allows inferring how risk premia evolve over time. The time series of estimated risk premia is then used to compute a time series of spreads for each project in the same risk bucket.

In other words, the risk premia estimated using instruments originated in a given year are used to recompute current spreads for all live instruments, combining information about the current risk profile of each instrument (the latest iteration of the DSCR state and forecasting models) and prevailing market conditions.

This is as close as we can get to an actual mark-to-market measure of private infrastructure debt.

Hence, we can value each instrument in each year – including those years where the market price for individual instruments could not observed for lack of secondary market transactions – thus overcoming the main data limitation faced in measuring the performance of highly illiquid, private infrastructure projects over time.

A more detailed presentation of the discount factor term structure model and estimation techniques can be found in Hasan and Blanc-Brude (2017c).

Loan tail in project finance: the number of years beyond loan maturity during which the project is still operational↩

Still, the heterogeneity of investor preferences with regards to this otherwise homogenous group of assets implies that there is a range of required risk premia applicable to each bucket, Blanc-Brude and Hasan (see 2015 for a detailed discussion of the role of investor preferences in a illiquid markets).↩

The capital investment is sunk and relationship-specific i.e. it has not alternative use and is often

*de jure*part of the public domain.↩

Ang, Andrew, Monika Piazzesi, and Min Wei. 2006. “What Does the Yield Curve Tell Us About GDP Growth?” *Journal of Econometrics* 131 (1). Elsevier: 359–403.

Black, Fischer, and John C Cox. 1976. “Valuing Corporate Securities: Some Effects of Bond Indenture Provisions.” *The Journal of Finance* 31 (2): 351–67.

Blanc-Brude, Frédéric, and Majid Hasan. 2015. “The Valuation of Privately-Held Infrastructure Equity Investments.” *EDHEC-Risk Institute Publications, EDHEC, Meridiam and Campbell-Lutyens Research Chair on Infrastructure Equity Investment Management and Benchmarking* January (January). Singapore: EDHEC-Risk Institute. pdf.

———. 2016. “A Structural Credit Risk Model for Illiquid Debt.” *Journal of Fixed Income* 26 (1, 1). link.

Blanc-Brude, Frédéric, Majid Hasan, and Omneia R H Ismail. 2014. “Unlisted Infrastructure Debt Valuation & Performance.” *EDHEC-Risk Institute Publications, EDHEC and NATIXIS Research Chair on Infrastructure Debt Investment Solutions* July. Singapore: EDHEC-Risk Institute. pdf.

Brennan, Michael J, and Yihong Xia. 2003. “Risk and Valuation Under an Intertemporal.” *Journal of Finance*.

Dittmar, Robert F. 2002. “Nonlinear Pricing Kernels, Kurtosis Preference, and Evidence from the Cross Section of Equity Returns.” *Journal of Finance*. JSTOR, 369–403.

———. 2017b. “You Can Work It Out! Valuation and Recovery of Private Debt with a Renegotiable Default Threshold.” *Journal of Fixed Income* 26 (4, Spring).

———. 2017c. “Discount Rate Term Structure Estimation with Highly Illiquid Assets.” *EDHEC Infrastructure Institute-Singapore*. Singapore: EDHEC Infrastructure Institute-Singapore.

Kealhofer, Stephen. 2003. “Quantifying Credit Risk I: default Prediction.” *Financial Analysts Journal*. JSTOR, 30–44.

Merton, Robert C. 1974. “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates.” *The Journal of Finance* 29: 449–70.

Parker, Jonathan A, and Christian Julliard. 2005. “Consumption Risk and the Cross Section of Expected Returns.” *Journal of Political Economy* 113 (1). JSTOR: 185–222.

Thus, a combination of cash flow, clustering and asset pricing models allows estimating the full range of performance metrics required for investment benchmarking at the asset level: single-period rates of return, volatility of returns, Sharpe and Sortino ratios, value-at-risk, duration, etc.

To derive performance measures at the portfolio level, it is necessary to estimate the covariance of returns (a.k.a. the variance-covariance matrix) to take into account the effect of portfolio diversification.

Portfolio returns and risk are written in the usual manner: \[R_{P}=\boldsymbol{w'} \ \boldsymbol{R}\] \[\sigma_{P}^{2}=var(\boldsymbol{w'R})=\boldsymbol{w'\Sigma w}\] With \(R\) a vector of constituent returns, \(w\) a vector of portfolio weights (adding up to unity) and \(\Sigma\), the variance-covariance matrix of the portfolio returns.

When estimating \(\Sigma\) the main challenge is dimensionality i.e. estimating the covariance matrix of a portfolio made of a large number of assets is subject to a lot of noise or the ’curse of dimensionality’ (Amenc et al. 2010) i.e. each pair-wise covariance results in some estimation error and the multiplication of the these errors with each other will soon undermine the estimation of portfolio risk as a whole.

One approach is to shrink the dimensionality of the problem by identifying a certain number of common factors driving project returns, and to estimate the covariance matrix of factor returns instead.

In our case, the ultimate factor exposures of private infrastructure debt are what we set out to discover and cannot be assumed *ex ante*. Hence, our approach to group assets by “risk buckets” (defined as statistical clusters volatility and duration) aims to capture persistent but unknown combinations of priced risk factors.

Once covariance is known within each cluster, the covariance matrix can be written as the combination of inter-cluster and intra-cluster covariances and estimated in any given year for the main index or any sub-index of private infrastructure debt.

Thus, consider assets \(x^m\) and \(y^n\) from risk clusters or buckets \(m\) and \(n\), respectively, the relevant covariance between the two assets is written: \[cov(x^m,y^n)=\left\{ \begin{array}{ll} cov(x,y)\text{ if }m=n\\ cov(m,n)\text{ if }m \neq n \end{array} \right.\] Hence, once the covariance of returns relative to the mean return has been estimated within each clusters, and the covariance between clusters is also known, which has largely reduced the dimensionality problem in our case, the covariance component of any index or sub-index constituent is readily known, and the relevant index covariance matrix can be derived.

Portfolio construction methodology consists of two elements: asset selection and weighting scheme design.

Asset selection is done in the context of our effort to document a representative, ’broad market’ index, as described in section Hence the selection of constituents and their rebalancing is largely driven by considerations of sampling and – to some extent – data availability and data quality.

We use two different weighting schemes: value-weights and equal-weights.

Value-weighting is a standard way to proxy “the market” but its overweights the most indebted issuers, and increases risk and issuer concentration. This could be a particular concern in the case of a broad market infrastructure debt index, since very large corporate issuers (utilities) are found side by side with relatively small project finance SPVs, the impact of which on the index is dwarfed by the largest issuers.

Equal-weighting thus represents a simple yet intuitive way to consider the contribution of all index constituents and by maximising the ’effective number of bets’ and, arguably, providing a more representative view on the performance of infrastructure debt.

In the context of traditional and liquid fixed income and equity indices, index weighting schemes are associated with rebalancing decisions requiring buying/selling. In the case of highlly illiquid private infrastructure investments such rebalancing decisions are not possible. In practice, a direct investor or manager in private infrastructure debt cannot easily or speedily adjust their ownership of the senior debt of any given firm.

Here, on a value weighted-basis, each exposure is considered to represent the whole stock of senior debt of the firm. On an equal weight-basis, the size of the exposure is simply ignored. Hence, the indices we produce are **buy-and-hold portfolios** of private infrastructure debt instruments.

In this sense, rebalancing only happens at the issuer selection stage i.e. when building a representative portfolio of the identified investable universe, and each time this sample has to be re-assessed because the underlying population and/or the index sample have changed. For example, certain instruments reach the end of their life, or a limit set by an index inclusion rule in terms of size and maturity.^{4}

Two simple portfolio inclusion rules are implemented to avoid unnecessary noise/distortion of reported performance:^{5}

A minimum outstanding maturity of two years

A maximum outstanding maturity of 30 years

A minimum outstanding face value of one million Euros

Using various geographic, asset selection, weighing schemes and reporting currency, **162 sub-indices have been created** and are available through this website for private infrastructure debt in Europe.

Index constituents weights are computed in a reference currency (here Euros), irrespective of the choice of the reporting currency of the index.↩

These avoids extreme returns and volatility due to very low book or total senior debt outstanding values as a project approaches its maturity, as well as extreme duration effects on some uncharacteristically long instruments.↩

Amenc, Noel, Felix Goltz, Lionel Martellini, and Patrice Retkowsky. 2010. “Efficient Indexation: An Alternative to Cap-Weighted Indices.”

*EDHEC-Risk Institute Publications*.

**Currency:** refers to the reporting currency of the index. Individual projects returns are converted in a reference currency (here Euros), irrespective of the choice of the reporting currency of the index.

**Number of Constituents:** refers to the number of projects in the index as of the end of 2016.

**Cap Coverage:** represents the portion of the identified investable infrastructure debt market captured by the index considering total outstanding debt.

**Regional Universe:** represents the geographical focus of the index.

**Asset Selection:** represents the set of company filters applied in the index construction methodology. Either all firms in the index sample, or filtered by business model, corporate structures, or sector groups.

**Weighting Scheme:** the methodology applied (i.e. value-weighted or equally weighted) to compute the fraction of the index invested in each borrower. All weights are computed in a reference currency (here Euros), irrespective of the choice of the reporting currency of the index.

**Index breakdown:** the split using either number of constituents or weights.

**Annual Performance:** computed as the weighted average of the return on the individual projects at different horizons.

**Sharpe Ratio:** the difference between the average index return and average risk free rate divided by portfolio volatility over the horizon under consideration.

**IRR:** The index Internal Rate of Return (IRR) is inferred following the usual formula:

\[\begin{aligned} 0 &= \frac{\sum_{t=1}^{T} Index Total Seniot Debt Service_{t}}{(1+IRR)^{t}} - Index Price_{t=0} \end{aligned}\]

**Volatility:** the cross-sectional portfolio volatility estimated using a variance-covariance matrix of individual project returns.

**Value-at-Risk:** delta-normal return at 99.5% confidence interval.

**Duration:** the weighted average of the individual constituents duration.

**Maximum Drawdown:** refers to the index maximum loss from a peak to trough.

**PD1:** represents the probability of observing at least one hard default (or default of payment) in the index, taking into account a default correlation matrix.

**Expected Loss:** weighted average of individual project expected losses, weighted according to the index weighing scheme. Individual project expected loss represents the difference between expected debt service taking into account DSCR trajectory and post-default restructuring scenarios, and the "base case" debt service i.e. the future debt service under the current debt contract(s).

**Effective Number of Constituents:** provides a measure of the index diversification and is computed as the inverse of the sum of squared constituent weights. In an equally-weighted portfolio, the effective number of constituents is equal to the number of elligible constituents.

\[\begin{aligned} N &= \frac{1}{\sum_{i=1}^{n} w_{i}^{2}} \end{aligned}\]