# LO 4.2: Explain how the m apping process captures general and specific risks.

LO 4.2: Explain how the m apping process captures general and specific risks.
So how many general risk factors (or primitive risk factors) are appropriate for a given portfolio? In some cases, one or two risk factors may be sufficient. O f course, the more risk factors chosen, the more time consuming the modeling of a portfolio becomes. However, more risk factors could lead to a better approximation of the portfolios risk exposure.
In our choice of general risk factors for use in VaR models, we should be aware that the types and number of risk factors we choose will have an effect on the size of residual or specific risks. Specific risks arise from unsystematic risk or asset-specific risks of various positions in the portfolio. The more precisely we define risk, the smaller the specific risk.
2018 Kaplan, Inc.
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Topic 4 Cross Reference to GARP Assigned Reading – Jorion, Chapter 11
For example, a portfolio of bonds may include bonds of different ratings, terms, and currencies. If we use duration as our only risk factor, there will be a significant amount of variance among the bonds that we referred to as specific risk. If we add a risk factor for credit risk, we could expect that the amount of specific risk would be smaller. If we add another risk factor for currencies, we would expect that the specific risk would be even smaller. Thus, the definition of specific risk is a function of general market risk. 1)] / 2) to evaluate the correlation between each risk factor. To simplify the number of As an example, suppose an equity portfolio consists of 5,000 stocks. Each stock has a market risk component and a firm-specific component. If each stock has a corresponding risk factor, we would need roughly 12.5 million covariance terms (i.e., [5,000 x (5,000
1)] / 2) to evaluate the correlation between each risk factor. To simplify the number of parameters required, we need to understand that diversification will reduce firm-specific components and leave only market risk (i.e., systematic risk or beta risk). We can then map the market risk component of each stock onto a stock index (i.e., changes in equity prices) to greatly reduce the number of parameters needed.
Suppose you have a portfolio of N stocks and map each stock to the market index, which is defined as your primitive risk factor. The risk exposure, /3-, is computed by regressing the return of stock i on the market index return using the following equation:
Ri = oq +(3jRy + j
We can ignore the first term (i.e., the intercept) as it does not relate to risk, and we will also assume that the last term, which is related to specific risk, is not correlated with other stocks or the market portfolio. If the weight of each position in the portfolio is defined as then the portfolio return is defined as follows:
r p = ^ wiR; = y ^ w i(3iRM + y > ii
N
i=i
N
i=i
N
i=i
Aggregating all risk exposures, /T, based on the market weights of each position determines the risk exposure as follows: =X}WiPi p P =X}WiPi
N
i=l
We can then decompose the variance, V, of the portfolio return into two components, which consist of general market risk exposures and specific risk exposures, as follows:
V(Rp) = Pp x V(Rm) + wf x<j2j
N
i= l
General market risk: (3p x V (R m)
Specific risk:
N
i= l
wf x cr^
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2018 Kaplan, Inc.
Topic 4 Cross Reference to GARP Assigned Reading – Jorion, Chapter 11
M a p p i n g A p p r o a c h e s f o r F i x e d -In c o m e P o r t f o l i o s