P2.T8. Risk Management & Investment Management Jorion, Value at Risk: The New Benchmark for Managing Financial Risk, 3rd Edition. Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM and Deepa Raju www.bionicturtle.com
Jorion, Chapter 7: Portfolio Risk: Analytical Methods DEFINE, CALCULATE, AND DISTINGUISH BETWEEN THE FOLLOWING PORTFOLIO VAR MEASURES: INDIVIDUAL VAR, INCREMENTAL VAR, MARGINAL VAR, COMPONENT VAR, UNDIVERSIFIED PORTFOLIO VAR, AND DIVERSIFIED PORTFOLIO VAR. EXPLAIN THE ROLE OF CORRELATION ON PORTFOLIO RISK.... 3 2
Jorion, Chapter 7: Portfolio Risk: Analytical Methods Define, calculate, and distinguish between the following portfolio VaR measures: individual VaR, incremental VaR, marginal VaR, component VaR, undiversified portfolio VaR, and diversified portfolio VaR. Explain the role of correlation on portfolio risk. Describe the challenges associated with VaR measurement as portfolio size increases. Apply the concept of marginal VaR to guide decisions about portfolio VaR. Explain the risk-minimizing position and the risk and return-optimizing position of a portfolio. Explain the difference between risk management and portfolio management, and describe how to use marginal VaR in portfolio management. Define, calculate, and distinguish between the following portfolio VaR measures: individual VaR, incremental VaR, marginal VaR, component VaR, undiversified portfolio VaR, and diversified portfolio VaR. Explain the role of correlation on portfolio risk. Before we elaborate upon the various measures of portfolio VaR, let us try to understand the basic concepts of Portfolio VaR: A portfolio can be characterized by positions on a certain number of constituent assets, expressed in monetary form. In simple terms, portfolio VaR determines the predicted losses with a certain confidence (level) that can be incurred over a certain time horizon. Relationship between Portfolio Returns & Portfolio VaR As the returns of a portfolio can be derived as the linear combination of the returns on underlying assets (assuming that their positions are fixed over the selected horizon) similarly, the VaR of a portfolio can be constructed from a combination of the risks of underlying securities. Therefore, the formula to derive portfolio return is: = + + = Portfolio Return,,, = Weights of each asset in the portfolio containing N number of assets, = Return on each asset. In other words, the expected return of the portfolio, ( ), can be depicted as: ( ) = = 3
The variance of the portfolio returns ( ), is derived as: ( ) = = + 2, = Weight of i th asset in the portfolio or = Standard deviation of the returns of i th or j th asset, = Correlation between the returns on asset i and j. We derive the standard deviation of the portfolio returns, of ( ), i.e., by taking the square root = = + 2, Standard deviation is one of the essential factors in deriving VaR as it directly estimates the riskiness of the portfolio. Higher standard deviation may lead to greater losses (or gains). It is important to note that, the correlation plays an important role in defining the magnitude of the calculated VaR. For deriving correlation between each pair of assets in the portfolio, a covariance matrix is used to derive covariance and eventually the correlation coefficient, making it easy to keep track of all covariance terms. To translate the portfolio variance into VaR, we also need to focus upon the distribution of the portfolio return. Consider all individual security returns are normally distributed, such that a linear combination of these normal random variables is also normally distributed. In such a case, we can translate the confidence level c into a standard normal deviate so that the probability of observing losses even more than is c. In other words, is the z-score associated with the confidence level, c. The VaR for a portfolio created with an initial investment of W can be derived as: = W With the derivation of the portfolio VaR, we introduce Individual VaR. Individual VaR It is defined as the VaR of one asset (of a portfolio) measured in isolation. Assuming, w i is the weight of the asset in the portfolio, then individual VaR,, can be derived as: = W = w w i = Proportional weight of the asset in the portfolio W = Total value of the Portfolio The vital point to note here is the usage of the absolute value of position (asset weight). This signifies that the risk exists for both long and short positions and even if the position s weight is negative, the derived risk measure must always be positive. 4
Before we explain the remaining Portfolio VaR measures, let us first understand the role of correlation in estimating portfolio risk. Role of Correlation Correlation coefficient helps in deciding the magnitude of the derived VaR. When used intelligently, correlation can play a vital role in reducing the portfolio risk. Portfolio VaR depends upon the number of assets (comprising the portfolio) and the correlation between each pair of those assets. Correlation measures the extent to which two variables (asset returns) move linearly together. If two variables are independent, their correlation is equal to zero. A positive correlation means that the two variables tend to move in the same direction; a negative correlation means that they tend to move in opposite directions. The correlation coefficient always lies between 1 and +1. Also, when the correlation coefficient between the variables is equal to unity then both are said to be perfectly correlated. When 0, the variables are uncorrelated. The correlation coefficient, is derived as: ρ = σ σ σ Where, σ = Covariance between variable 1 & 2 having standard deviation of σ and σ respectively. Reducing Portfolio Risk Lower portfolio risk can be achieved through low correlations or a large number of assets. To see the effect of large number of assets, N, assume that all assets have the same risk and that all correlations are the same, that equal weight is put on each asset. The figure below shows how portfolio risk decreases with the number of assets. As shown in the figure below, the risk of one security is assumed to be 20 percent. When correlation (ρ) is equal to zero, the risk of a 10-asset portfolio drops to 6.3 percent; increasing the assets to 100 drops the risk even further to 2.0 percent. Risk tends asymptotically to zero. More generally, portfolio risk for large number of assets (N) is derived as: = 1 + 1 1 It is evident from the formula above, that the portfolio risk,, tends to zero as N increases. As shown in the figure below, when ρ = 0.5, risk decreases rapidly from 20 to 14.8 percent as N goes to 10 and afterward converges more slowly toward its minimum value of 14.1 percent. Low correlations help to diversify portfolio risk. The above formula also helps in deriving the standard deviation of portfolio for large number of assets. 5
Now assess the effect of correlation on portfolio risk in detail. Consider a simple example of a portfolio (P) having two assets, 1 & 2, the diversified portfolio standard deviation is: = + + 2 ρ And the portfolio variance, VaR P can be derived as: VaR = W = W + + 2 ρ W = Portfolio Value = z-score associated with the confidence level, c. When the correlation, is zero, the portfolio VaR reduces to (refer Individual VaR): VaR = W + W = VaR + VaR In this case note that the portfolio risk must be lower than the sum of the individual VaRs, VaR < VaR + VaR This reflects the fact that, with the assets that move independently, a portfolio will be less risky than either asset. Thus, VaR is a coherent risk measure for normal and, more generally, elliptical distributions. When the correlation is exactly unity and and are both positive then the portfolio VaR reduces to: VaR = VaR + VaR + 2 VaR VaR = VaR + VaR The portfolio VaR is equal to the sum of the Individual VaR measures if the two assets are perfectly correlated. In general, this will not be the case because correlations typically are imperfect. The benefit from diversification can be measured by the difference between the Diversified VaR and the Undiversified VaR, which is shown in VaR reporting systems. 6