Cross-sectional Volatility

Given a time series of prices for any relevant unlisted infrastructure asset (debt or equity), returns are computed as

$$//<![CDATA[ \begin{array}{l}\displaystyle \hat{r}_{i,t,t+1} = \log \frac{\hat{P}_{i,t+1} + CF_{i,t+1}}{\hat{P}_{i,t}}\end{array} //]]>$$

where  $//<![CDATA[ \begin{array}{l}\hat{P}_{i}\end{array} //]]>$ denotes model-implied prices and  $//<![CDATA[ \begin{array}{l}CF_{i}\end{array} //]]>$ is the realised cashflow.

Model-implied prices for the next period, $//<![CDATA[ \begin{array}{l}\hat{P}_{i,t+1}\end{array} //]]>$, are computed using predicted variables, such as prices of risk factors and risk-free rates for the next period. Hence, these prices are uncertain and fall in a range depending on the range of predicted values for the underlying variables. As a result, the standard deviation of price can be written as:

$$//<![CDATA[ \begin{array}{l}\displaystyle \sigma(P_{i,t}) = \frac{\partial P_{i,t}}{\partial R_{f}} \sigma(R_{f}) + \sum_{k} \frac{\partial P_{i,t}}{\partial \lambda_{k}} \sigma(\lambda_{k}) + \sum_{k} \frac{\partial P_{i,t}}{\partial \beta_{i,k}} \sigma(\beta_{k})\end{array} //]]>$$

where the first term, $//<![CDATA[ \begin{array}{l}\frac{\partial P_{i,t}}{\partial R_{f}} \sigma(R_{f})\end{array} //]]>$, reflects the uncertainty in the predicted price arising from uncertainty in risk-free rates, the second term, $//<![CDATA[ \begin{array}{l}\sum_{k} \frac{\partial P_{i,t}}{\partial \lambda_{k}} \sigma(\lambda_{k})\end{array} //]]>$, reflects the uncertainty in the predicted price arising from uncertainty in estimated prices of k risk factors, and the third term, $//<![CDATA[ \begin{array}{l}\sum_{k} \frac{\partial P_{i,t}}{\partial \beta_{i,k}} \sigma(\beta_{k})\end{array} //]]>$, reflects the uncertainty in the predicted price arising from the uncertainty in the project's loading to k risk factors.

Thus, the estimated volatility of returns, $//<![CDATA[ \begin{array}{l}\sigma_{i,t}\end{array} //]]>$, reflects:

• the volatility of the risk-free-rate term structure, $//<![CDATA[ \begin{array}{l}\sigma(R_{f})\end{array} //]]>$;
  volatility of forecasted cash flows, $//<![CDATA[ \begin{array}{l}\sigma(CF_{i})\end{array} //]]>$;
• the uncertainty with which the prices of risk factors are estimated, $//<![CDATA[ \begin{array}{l}\sigma(\lambda)\end{array} //]]>$; and
• the uncertainty with which the factor loadings are estimated, $//<![CDATA[ \begin{array}{l}\sigma(\beta)\end{array} //]]>$.

These sources of uncertainty – cash flows, risk-free rates, expected risk premia, and factor loadings – capture the full range of uncertainty for the future price of an asset, and the volatility estimate summarises the combined effect of all these sources of uncertainty.

This volatility measure is thus constructed in a 'bottom-up' fashion, by estimating and aggregating individual sources of uncertainty. Some assets may be risky because their cash flows are riskier (higher $//<![CDATA[ \begin{array}{l}\sigma(CF)\end{array} //]]>$), some assets may be risky because they originate in countries where risk-free rates are volatile ( $//<![CDATA[ \begin{array}{l}\sigma(R_{f})\end{array} //]]>$), some assets may be risky because their factor loadings are uncertain ( $//<![CDATA[ \begin{array}{l}\sigma(\beta_{k})\end{array} //]]>$), and some may be risky because they have a higher exposure to more volatile risk-factor prices ( $//<![CDATA[ \begin{array}{l}\sigma(\lambda)\end{array} //]]>$).

Depending on the investment objective, different sources of uncertainty may matter more or less for different investors, and they may justify different allocations to assets with different underlying sources of risk, even if they exhibit similar levels of aggregate volatility.

A more detailed discussion of the cross-sectional volatility of the market index is available here.