Cross-Sectional (model) Volatility

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

Two approaches are available to estimate volatility, i.e. 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 to 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:

$$//<![CDATA[ \begin{array}{l}\displaystyle TotalReturn_t = R_t = CashReturn_t + PriceReturn_t\end{array} //]]>$$

The cash return is given by:

$$//<![CDATA[ \begin{array}{l}\displaystyle CashReturn_t = \frac{CashPayment_t}{Price_{t-1}} \equiv \Delta_{C_t}\end{array} //]]>$$

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 $//<![CDATA[ \begin{array}{l}\Delta_r\end{array} //]]>$. 

$$//<![CDATA[ \begin{array}{l}\displaystyle PriceReturn_t = \frac{Price_t}{Price_{t-1}} - 1 = -D_t \times \Delta r_t = -D_t \times \large(\Delta_{I_t} + \Delta_{\gamma_t} \large)\end{array} //]]>$$

where the change in yield  $//<![CDATA[ \begin{array}{l}\Delta_r\end{array} //]]>$ is itself the combination of the change in risk-free rate $//<![CDATA[ \begin{array}{l}\Delta_{I_t}\end{array} //]]>$ and risk premium $//<![CDATA[ \begin{array}{l}\Delta_{s_t}\end{array} //]]>$.

Using the relationships describes above, the total return for Company $//<![CDATA[ \begin{array}{l}a\end{array} //]]>$ can be written: 

$$//<![CDATA[ \begin{array}{l}\displaystyle R_a = -D_a \large(\Delta_{\gamma_a} + \Delta_{I_a}\large) + \Delta_{C_a}\end{array} //]]>$$

Clearly, the correlation between  $//<![CDATA[ \begin{array}{l}\Delta_{\gamma_a}\end{array} //]]>$ and  $//<![CDATA[ \begin{array}{l}\Delta_{I_a}\end{array} //]]>$ is relevant to the estimation of the variance of $//<![CDATA[ \begin{array}{l}R_a\end{array} //]]>$  – or its /covariance with another asset returns, say, $//<![CDATA[ \begin{array}{l}R_b\end{array} //]]>$ (We return to computing return covariance below).

Variance of the Risk Premia

EDHECinfra's   $//<![CDATA[ \begin{array}{l}K\end{array} //]]>$-factor model expresses the risk premia  $//<![CDATA[ \begin{array}{l}\gamma_a\end{array} //]]>$ of the firm $//<![CDATA[ \begin{array}{l}a\end{array} //]]>$ as:

$$//<![CDATA[ \begin{array}{l}\displaystyle \gamma_a = \sum_{k=1}^K \lambda_{k}\beta_{k,a} + \omega_a\end{array} //]]>$$

where  $//<![CDATA[ \begin{array}{l}\beta_{k,a}\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}\lambda_{k}\end{array} //]]>$ are the factor $//<![CDATA[ \begin{array}{l}k\end{array} //]]>$'s loading (or exposure) factor $//<![CDATA[ \begin{array}{l}k\end{array} //]]>$ and market factor price for the company $//<![CDATA[ \begin{array}{l}a\end{array} //]]>$ respectively. $//<![CDATA[ \begin{array}{l}\omega_a\end{array} //]]>$ is the idiosyncratic observation noise i.e. the estimated bid-ask spread of individual factor prices, given observable transaction prices. 

Hence, the change of the risk premium  $//<![CDATA[ \begin{array}{l}\Delta_{s_a}\end{array} //]]>$ can be written as: 

$$//<![CDATA[ \begin{array}{l}\displaystyle \Delta_{\gamma_a} \approx \sum_k \lambda_{k} \Delta\beta_{k,a} + \sum_k \Delta \lambda_{k} \beta_{k,a} + \Delta_{\omega_a}\end{array} //]]>$$

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 $//<![CDATA[ \begin{array}{l}K\end{array} //]]>$ risk factor that make up the risk premia  $//<![CDATA[ \begin{array}{l}\gamma\end{array} //]]>$ can be split into two types: factors with loading that are company specific (like size or profits) and factors with loadings that are correlated with other quantities like interest rates.  The impact of any interest rate-related $//<![CDATA[ \begin{array}{l}k\end{array} //]]>$ risk factor (e.g. the Term Spread) can be separated from that of other factors. Hence, we have:

$$//<![CDATA[ \begin{array}{l}\displaystyle \Delta_{\gamma_a} = \sum_k \Delta\lambda_k \beta_{ka} + \sum_{k \neq I_a} \lambda_k \Delta \beta_{ka} + \frac{1}{B'_a}\lambda_{I_a} \Delta I_a + \Delta_{\omega_a}\end{array} //]]>$$

where $//<![CDATA[ \begin{array}{l}I_a\end{array} //]]>$ is the risk-free rate in the  $//<![CDATA[ \begin{array}{l}K\end{array} //]]>$-factor model with the transformation $//<![CDATA[ \begin{array}{l}B(.)\end{array} //]]>$ and its derivative $//<![CDATA[ \begin{array}{l}B'(.) = dB/dx\end{array} //]]>$. Here $//<![CDATA[ \begin{array}{l}B'_a = B'(I_a) = dB/dI_a\end{array} //]]>$.

Finally, we can write the total return change of any company $//<![CDATA[ \begin{array}{l}a\end{array} //]]>$'s total return as:

$$//<![CDATA[ \begin{array}{l}\displaystyle \Delta_{R_a} = -D_a [ \sum_k \Delta\lambda_k \beta_{ka} + \sum_{k \neq I_a} \lambda_k \Delta \beta_{ka} + (1 + \frac{1}{B'_a} \lambda_{I_a}) \Delta I_a + \Delta_{\omega_a}] + \Delta C_a\end{array} //]]>$$

Pair-wise Covariance of Returns

Based on the above equations, the covariance of the total return for two companies  $//<![CDATA[ \begin{array}{l}a\end{array} //]]>$ and  $//<![CDATA[ \begin{array}{l}b\end{array} //]]>$ is determined by the covariance of the each type of change, which are summarised as following: 

1. $//<![CDATA[ \begin{array}{l}Cov(\Delta \lambda_a, \Delta \lambda_b) = Cov_{\lambda}(a,b)\end{array} //]]>$: is the covariance matrix between factor price estimated from the calibrated factor model. This matrix reflects the width of bid-ask spread and movement variation.
2.   $//<![CDATA[ \begin{array}{l}Cov(\Delta \beta_{ia}, \Delta \beta_{jb}) = Cov_{\beta}(a,b)\end{array} //]]>$: 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   $//<![CDATA[ \begin{array}{l}K\end{array} //]]>$-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 very little to the final covariance.
3. $//<![CDATA[ \begin{array}{l}Cov(\Delta I_a, \Delta I_b) = Cov_I(a,b)\end{array} //]]>$ is estimated from historical interest rate movement, where the latest interest rate movement has the largest weights.
4. $//<![CDATA[ \begin{array}{l}Cov(\Delta \omega_a, \Delta \omega_b)=Cov_{\omega}(a,b)\end{array} //]]>$ 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.
5. $//<![CDATA[ \begin{array}{l}Cov(\Delta C_a, \Delta C_b)=Cov_C(a,b)\end{array} //]]>$ is the covariance matrix of cash payouts (e.g. dividends) and is assumed to be diagonal because the cash payout behaviour is company specific. It can be estimated from historical dividend payouts or debt service (for debt indices).

Hence, the return covariance between $//<![CDATA[ \begin{array}{l}R_a\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}R_b\end{array} //]]>$ is given by: 

$$//<![CDATA[ \begin{array}{l}\displaystyle Cov(R_a, R_b) = D_a D_b [ \sum_k\sum_l \beta_{ka}\beta_{lb}Cov_{\lambda}(a,b) + (1 + \frac{1}{B'_a} \lambda_{I_a})(1 + \frac{1}{B'_b} \lambda_{I_b}) Cov_I(a,b) + Cov_{\omega}(a,b)] + Cov_C(a,b)\end{array} //]]>$$

Total Return Variance-Covariance Matrix

Once  $//<![CDATA[ \begin{array}{l}Cov(R_a, R_b)\end{array} //]]>$ is known for all assets  $//<![CDATA[ \begin{array}{l}i= a,b,\dots\end{array} //]]>$ the Total Return variance-covariance matrix $//<![CDATA[ \begin{array}{l}\Omega\end{array} //]]>$ can be determined, and portfolio risk can then be calculated as:

$$//<![CDATA[ \begin{array}{l}\displaystyle \hat{\sigma}_P=\sqrt{w^{T}\Omega_{}w}\end{array} //]]>$$

where  $//<![CDATA[ \begin{array}{l}w\end{array} //]]>$ is a  $//<![CDATA[ \begin{array}{l}n \times 1\end{array} //]]>$ weights matrix of the assets,  $//<![CDATA[ \begin{array}{l}\Omega\end{array} //]]>$ is a  $//<![CDATA[ \begin{array}{l}n\times n\end{array} //]]>$ covariance matrix of assets.