2.6.2 Cross-sectional (model) volatility

Overview

In what follows, we provide an overview of the approach taken to estimate the (cross-sectional) volatility of the index at one point in time, given the risk factors driving returns. This provides an alternative understanding of the risk of the index given that observable transaction prices may not capture the full extent of the variance of asset values because of the illiquidity and segmentation of the market for private infrastructure companies.

The volatility of index returns is one of the most important measurements of risk in the Index Universe. 

Two approaches are available to estimate volatility: historical (time series-based) and cross-sectional (risk factor-based).

The historical volatility is simply the standard deviation of the index return within a certain time period. In a liquid and complete market, with reasonably symmetrical returns, historical volatility can be a good proxy of the risk of the index. However, in highly illiquid and segmented markets like unlisted infrastructure, historical volatility can be biased and also fail to capture the extent of the 'disagreement' on asset value that may prevail amongst buyers and sellers. 

In other words, in order to capture the bid-ask spread of unlisted infrastructure assets and integrate it into the variance of prices (i.e. the volatility of returns) a cross-sectional approach can better reflect the range of possible values that a given asset may take at one point in time.

In what follows, we describe how:

1. changes in total returns can be described in terms of changes in duration, risk premia, interest rate and payouts;

2. changes in the risk premia can be described in terms of changes in the risk factor loadings and the risk factor prices; and

3. these elements can be combined to express the pair-wise co-variance of total returns between different assets in the index universe.

Once each pair-wise return co-variance is estimated, a variance, co-variance matrix of returns can be determined for the entire index.  

Total returns and factor movements

The index model volatility is computed based on the total return covariance matrix on the volatility of the underlying firms, whose total return is the sum of the cash return and price return. The total return of a firm can be written as:

The cash return is given by:

According to the definition of the duration, the price return can be written as the product of the asset duration and change in yield-to-maturity or IRR

where the change in yield 

In what follows, we remove the subscript

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 for clarity. Computations are all made at time

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 unless explicitly mentioned.

Using the relationships described above, the total return for company

Clearly, the correlation between 

Variance of the risk premia

Scientific Infra & Private Assets'

where 

Hence, the change in the risk premium 

The first and second terms represent the change in factor loading (or exposure) and the change in factor price, while the third term represents the change of idiosyncratic observation noise. The above equation is only valid when the movements are relatively small, but can be viewed as a reasonable approximation for the purpose of correlation estimation.

Co-variance between risk premia and interest rate factors

The

where

Finally, we can write the total return change of any company

Pair-wise covariance of returns

Based on the above equations, the covariance of the total return for two companies 

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   : is the covariance matrix between factor price movements estimated from the calibrated factor model;

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   : is the covariance matrix between all factor loadings for the two companies (except for interest-rate related rate factors). It could be approximated by a diagonal matrix because these factors are company specific and thus have very low correlation. In fact, this matrix is ignored for the 
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   -factor model because the factors are either dummies or very slowly varying variables quarter-on-quarter (e.g. company size). This approximation is supported in the computation, where it is found that this covariance matrix actually contributes to the final covariance very little.

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    is estimated from historical interest rate movement, where the latest interest rate movement has the largest weight.

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    is the covariance matrix of observation noises or factor price bid-ask spread. It is assumed diagonal matrix because the observation noise is idiosyncratic and estimated from the calibrated residuals in the factor model.

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    is the covariance matrix of cash payouts (e.g. dividends) and is assumed diagonal because the cash payout behavior is company-specific. It can be estimated from historical dividend payouts or debt service (for debt indices).

Hence, the return covariance between

Total return variance-covariance matrix

Once 

where