6
the truth if you have a very large portfolio. Risk on the other hand is very hard to quantify since it very much depends on future events that even Nostradamus would struggle to predict, for example, very few people categorically predicted the financial crash of 2007 which was why it hit so many financial institutions, households, businesses (so basically everyone) hard. However, the ability to assess the volatility of listed assets using past data to perhaps try to model future predictions is an extremely important ability, for example, a hedge fund manager called John Paulson made $3.7 billion during the financial crisis 2 as he saw the weakness in the subprime mortgage market first and short sold every appropriate share/collateralized debt obligation (CDO) he could. He’s now worth $11.5 billion 3 . So, now I’ve established being very good at Maths can make you very rich, it’s back to the calculations for risk or, in S1 lingo, the variance. The variance of any security can be written as; ∑ ൫R ୧ െ EሺR ሻ൯ ଶ PሺR ୧ ሻ ୧ୀ୬ ୧ୀଵ From this, we can sub in the equation W A A + W B B = P and then expand the brackets to give; V P = S P 2 = W A 2 S A 2 + W B 2 S B 2 + ሺ∑ ൫R ୧ െ EሺR ሻ൯൫R ୧ െ EሺR ሻ൯pሺR ୧ ሻ ୧ୀ୬ ୧ୀଵ ሻ As you may have noticed, the final part of this long equation is actually the equation for the covariance of returns for A and B (COV AB ). The covariance is the absolute value that measures how one variable changes with respect to another, where positive values indicate that as one variable increases, so does the other variable, and of course the opposite applies for a negative value. The product moment correlation coefficient, as seen in S1, is simply a version of the covariance (so we call the covariance the determinant of the PMCC) that is scaled by the product of the standard deviations of A and B to give the PMCC a value that is always within the parameter; െ1 PMCC 1 Thus, we can say that; COV ൌ PMCC S S = ∑ ൫R ୧ െ EሺR ሻ൯൫R ୧ െ EሺR ሻ൯pሺR ୧ ሻ ୧ୀ୬ ୧ୀଵ And subbing this into the equation for the variance of your portfolio gives; V P = S P 2 = W A 2 S A 2 + W B 2 S B 2 + 2W A W B S A S B PMCC AB This is known in the financial industry as the ‘Portfolio Risk Equation’. The PMCC is now the main player in determining the risk of investing in these two securities, therefore, by using the data from the table on page 1 we can deduce that; V i = S i 2 = ∑ x ୧ ଶ ሺPሺX ൌ x ୧ ሻ െ ሺEሺRሻሻ ଶ = 2W A W B
2 The Long Short – Michael Lewis 3 http://en.wikipedia.org/wiki/John_Paulson
Made with FlippingBook - Online Brochure Maker