In this section, we provide a number of formulas that enable the decomposition of systematic risk across seventeen risk contributions, each associated with an intuitive and meaningful risk dimension.

Risk

As we have seen in the model specification, the return

of a portfolio from time

to

is given by

where

is the vector of weights of each instrument within the portfolio. Consequently, the portfolio's volatility over a given period is approximated by

where

with

and

the matrix containing the characteristics of the

instruments and

characteristics. The matrices

and

are defined within the Risk Model methodology. The dynamical nature of the exposures

​ (which change over time) adds an extra dimension to the definition of the covariance matrix compared to more traditional—and static—risk models.

The above formula can be understood as a sum of point-in-time covariance matrices that must be summed to generate the covariance matrix over the full period. We exploit this idea in the SP platform to show how any sub-period contributes to the risk of the full period.

Relative risk and rewards

In many cases, the absolute performance of an instrument is not as critical as its relative performance where it is compared to a reference portfolio or benchmark. The relative returns of a portfolio are not treated differently than the absolute returns: the exposures still must be evaluated and inputted into the risk model in order to obtain the specific returns.

Because all risk factors are orthogonal to the market risk factor, the characteristics associated with relative returns simply corresponds to the difference between the characteristics of the portfolio and those of the reference, that is

Once the relative characteristics have been evaluated, any risk-based calculation that uses absolute returns is also applicable to relative returns.

Decomposing risk

Risk can be decomposed into contributions that are attributed to an instrument's exposures. In other words, decompositions provide an insight into what actually drives the risk. The decomposition of risk is obtained by use of the application of a mathematical formula called the Euler decomposition.

Before delving into the application of the Euler formula used to decompose risk, it is important to note that the function is sensitive to leverage. Leverage sensitivity means that when portfolio weights are leveraged (multiplied) by a factor

, the risk also scales by a factor proportional to

. This is expected as increasing leverage amplifies returns but also increases risk.

The fact that the function measuring risk is sensitive to leverage is not only desirable from a financial point of view but also results in making the function "decomposable". This surprising fact is a consequence of the well-known Euler theorem that states whenever a function scales with its parameter (here denoted by

), that is,

it can be decomposed into contributions attributable to the parameters. Here, the variable

can be any number and impacts the proportionality between the value of the scaling parameter

and response from the function. Any function with the above mathematical property is called homogeneous.

In our case, we can use both the portfolio's weights or the characteristics as parameters. For ease of notation, we group the instrument characteristics and specific risks into one variable

. For portfolios, the characteristics are transformed into weighted characteristics. Although they are two different parameters, we do not need to separate characteristics from weights. This is because scaling the characteristics has the same effect as scaling the weights: we can scale the characteristics while maintaining

fixed, and vice-versa to obtain the same effect. Because we are interested in understanding how exposures to fundamental factors affect risk, we focus on characteristics and hence use

as a parameter. Note that for the risk function,

, which means that the leverage actually scales linearly.

The Euler decomposition is based on the following observation. First recall the definition of a function's total derivative:

​The Euler formula is obtained by introducing a variable

so that

and

. Using the homogeneity of the function

,

If we cancel the effect of the leverage by setting

, then

and we are left with

​The interpretation of the Euler formula is very simple: if a function scales with its parameters, then it is decomposable into an addition of contributions. Each contribution is composed of the function's first-order sensitivity (partial derivative) to a given parameter times the parameter itself. Because this decomposition is reminiscent of a linear combination where the sensitivities are the coefficients and the parameters the variables, we can interpret these as the contribution of the

parameter to the function's total value. This is why the Euler formula is used extensively in the field of finance to understand the drivers of risk.

In the next section, we show how to apply the Euler theorem to decompose risk.

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