Equity positions are a bit more tricky compared to the FX positions. The main problem arises while dealing with the correlations between the different equity positions. As the number of positions in the equity portfolio increases, the correlation matrix can become substantially large and almost impossible to handle.
In case of equity positions, the number of risk factors are reduced by modeling the individual stock returns using a factor model. The simplest way to do so is to use the CAPM model under which the returns from a security can be represented using the equity market returns.
ri = α + βrm + ε
α represents the firm-specific constant
ε is a zero-mean specific constant
β is the stock’s beta
rm is the market return.
If we assume that the equity portfolio of a bank is sufficiently diversified, then the α of different stocks will offset each other and the returns of the portfolio can be simply written as
rp = βprm + ε
βp is the portfolio beta
rm is the market return
Let’s take a simple example to understand how we can use this to calculate the portfolio VaR.
Let’s say we have a portfolio of $100 million invested in three stocks, namely A, B, and C. Also assume that the related stock market index is the S&P500 index.
The details are provided below:
|Stock A||Stock B||Stock C|
|Position||$30 million||$30 million||$40 million|
|Equivalent S&P500 position||$21 million||$36 million||$52 million|
The equivalent S&P500 position is calculated as Position Size * Beta. This means that $100 million invested in three stocks is equivalent to $109 million invested in S&P500, considering the systematic risk.
Using this information, the VaR of the portfolio can be calculated in terms of the index value. Assume that the daily volatility of S&P500 index is 0.90%. The 99% daily VaR of the three stock portfolio will be:
VaR99% = $109 * 2.326 * 0.9% = $2.28 million
An alternative way to calculate the VaR of the same portfolio would be by directly using the stock’s weighted average beta:
VaR99% = $100 * 2.326 * 0.9% * (0.70*3/10+1.20*3/10 +1.30*4/10)
= $2.28 million
As you can see, Mapping can significantly simplify the VaR calculation.
If our portfolio had 1,000 stocks, and referenced 10 different markets, then our VaR calculation could be simplified to 10 different positions on these 10 markets, rather than dealing with the complexity of 1000 different stocks as individual exposures.