Portfolio Risk Decomposition using Triangles
Team Latte
May 18, 2005
According to the Pythagorean theorem, the squared length of the hypotenuse equals the sum of the squared lengths of both the sides in a right angled triangle, or z2 = x2 + y2. In every triangle, the square of a triangle side z is equal to the sum of the squares of the sides x and y, less the double product of the sides and the cosine of the angle θ (see figure 1). We can express the above in algebraic terms as:
z2 = x2 + y2 - 2xycoxθ ............................. (1)
We can compare the above equation with the equation for portfolio variance (squared volatility) as well as the portfolio value at risk (VaR). Take the portfolio variance for the two asset case. Asset 1 has a volatility of σ1 and Asset 2 has a volatility of σ2. The proportion of funds invested in asset 1 is w1 and that in asset 2 is w2. The correlation between the two assets is ρ12 In that case the portfolio variance and VaR can be expressed as:
σp2 = w12σ12 + w22σ22 + 2w1w2σ1σ2ρ12 ............................. (2)
VaRp2 = VaR12 + VaR22 + 2 * VaR1VaR2ρ12 ............................. (3)
There is a symmetry between the equations (2) and (3) and equation (1). From the above we can see that the risk of a portfolio comprising two assets can be represented by one side of a triangle and the risk of the individual assets in the portfolio can be represented by the other two sides of the same triangle.
It can be easily seen from the above equations that the correlation between any two assets that make up a portfolio is given by the angle theta between the two sides of the triangle and the relationship is:
θ = cos-1(-ρ12) ............................. (4)
Or in other words, if the angle between the two sides of the triangle is known we can infer the correlation between the assets as:
ρ12 = -cosθ ............................. (5)
This analysis of geometrical decomposition of risk using triangles have a wide ranging applications in portfolio modeling as well as trading.
A series of implied correlations can be derived using the implied volatilities of assets in a portfolio and the impact of varying correlation between assets on the portfolio can be analysed in great (and geometrical) detail.
Another big application is in analyzing the risk decomposition of the tracking error. All fund managers monitor the tracking error of assets and portfolio of assets, with optimizing the tracking error being one of the most common fund management strategy. The above analysis of risk decomposition can be very useful in estimating how orthogonal the returns can be produced without substantially altering the risk profile.
nother application of the above is in analyzing the real and structural information ratios. A structural position is defined as a permanent overweighting of a riskier asset class relative to a pre-specified benchmark portfolio. Real information ratios arises from active positions that are uncorrelated to the benchmark portfolio so that the total risk is only marginally increased.
Disclaimer
"Risk Latte uses proprietary and non-proprietary mathematical and empirical models to measure the volatility and estimate the direction of the market. There is no guarantee of any particular outcome happening and readers must exercise caution while interpreting the conclusions of this article. Risk Latte Company is not a registered stock broker or an SFC registered entity and readers must take advise from their financial advisors, stock brokers, research analysts and bankers while making any buying or selling decisions. Risk Latte Company is not in the business of making stock or asset forecasts whether explicitly or implicitly and shall not be responsible for and/or liable for any losses arising out of any trading decisions based on the above article."
  
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