# Index Return Volatility

Standard deviation , also known as volatility, measures how dispersed returns are around the average. A higher standard deviation indicates that returns are spread out over a larger range of values, hence, more volatile.

Asset-level 1.5.5 Cross-sectional Volatility is computed as an output of the valuation model.

At the index level, the portfolio risk is estimated by relying on the multi-factor model used to compute asset values and returns.

In the multi-factor model, the expected excess return of each asset  can be represented as:

where  is the asset exposure to factor , $f_k$ is the factor return to be estimated, and  is the specific return.

Portfolio risk can then be calculated as:

where  is a  weights matrix of the assets,  is a  covariance matrix of assets

Covariance between assets is computed as follows:

Since each asset's return is described by factors, first the covariances () between these factors are computed. It requires estimating correlations between the factors.

Sample correlation estimates are subject to dimensionality issues: limited data history and sparse frequency, non-stationary and rank deficient.

A Bayesian dynamic model is used to produce time-varying estimates of factor correlation (see Asset Pricing).

Then the covariances of each asset are derived at each point in time

where,

is the covariance of assets n and m.
is the exposure of asset n to factor k1.
is the covariance of factor k1 with factor k2.
the specific covariance of assets and m, which is zero by design unless n=m, in which case it is the specific variance of the asset.

This can be represented in matrix notation as follows:

where,

is an  covariance matrix of assets.

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is a  covariance matrix of factors.
is an  matrix of asset exposures to the k factors.
is an  matrix of specific covariances of assets.

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