# Risk Budgeting: Concept, Interpretation and Applications

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1 Risk Budgeting: Concept, Interpretation and Applications Northfield Research Conference 005 Eddie Qian, PhD, CFA Senior Portfolio Manager 60 Franklin Street Boston, MA 00 (67) //005

2 The Concept Risk Contribution Risk contribution attribution of total risk to individual underlying components of a portfolio in percentage terms Examples Fixed income portfolio: sector risk, yield curve risk, Equity portfolio: systematic risk (risk indices/industries), specific risk, Asset allocation: TAA risk (stock/bond, cap rotation, ), sleeve active risk, Manager selection, strategy allocation Asset allocation portfolio: beta risk from stocks, bonds, commodities, Parity portfolios //005

3 The Concept An Example A simple illustration Two active strategies strategy A at % active risk, strategy B at % active risk, two sources uncorrelated Total active risk Equals % + % =.4% What is the risk contribution from strategy A and B? Answer A contributes 0% and B at 80% Contribution to variance Strategy A + 4, Strategy B Contribution based upon variances and covariances //005

4 The Concept Mathematical Definition Marginal contribution - Risk contribution - VaR contribution - w i w i w i VaR w i w i Well defined in financial engineering terms But objected by some financial economists for a lack of economic interpretation //005 3

5 The Concept Objections Risk is not additive in terms of standard deviation or VaR You can only add risk when returns are uncorrelated A mathematical decomposition of risk is not itself a risk decomposition Risk budgeting only make sense if considered from meanvariance optimization perspective Marginal contribution to risk is sensible, but not risk contribution //005 4

6 Interpretation Loss Contribution Question: For a given loss of a portfolio, what are the likely contribution from the underlying components? Answer: The contribution to loss is close to risk contribution Risk contribution can be interpreted as loss contribution Risk budgeting ~ loss budgeting Risk budgets do add up //005 5

7 Interpretation An Asset Allocation Example Balanced benchmark 60% stocks, 40% bonds Is it really balanced? Risk contribution Stock volatilities at 5%, bond volatilities at 5% Stocks are actually 9 times riskier than bonds Stocks contribution is roughly 95% and bonds at 5% = 95% The balanced fund a misnomer //005 6

8 Interpretation Loss Contribution of a 60/40 Portfolio Loss > Stocks Bonds % 95.6% 4.4% 3% 00.% -0.% 4% 0.9% -.9% Stocks contribution is almost 00% It is greater than the theoretical value of 95%, because stocks have greater tail risk //005 7

9 //005 Interpretation Question: For a given loss of a portfolio, what are the likely contribution from the underlying components? Answer: Conditional expectation Mathematical Proof Total risk: Risk contribution: Loss contribution? ρ w w w w + + = ρ ρ w w w w w p w w w w w p + = = + = = ( ) / i i i c E wr wr wr L L = + =

10 Interpretation Mathematical Proof The loss contribution is approximately risk contribution Cases where they are identical Zero expected returns Large losses Mean-variance optimal weights pw µ pw µ D c = p+ p+ L L pwµ pwµ D c = p + p + L L //005 9

11 Interpretation Mean-variance Optimal Risk contribution is equivalent to loss contribution Risk contribution is also equivalent to expected return contribution wµ wµ = =λ p p The interpretation is valid and very close even if the portfolio is not MV optimal //005 0

12 Interpretation Loss Contribution Figure The value of D over standard deviation for asset allocation portfolios 3.0%.0%.0% 0.0% -.0% -.0% -3.0% -4.0% 0% 0% 40% 60% 80% 00% Stock Weight The difference between risk contribution and loss contribution is quite small //005

13 Application Loss Contribution of a 60/40 Portfolio Loss Predicted c Realized c N Predicted Std Realized Std -4% to -3% 93.5% 89.8% % 6.% -5% to -4% 9.8% 9.7% 3.0% 0.7% -6% to -5% 9.3% 88.% 6.8% 6.% -7% to -6% 9.0% 99.5% 9 4.0% 8.7% -8% to -7% 9.8% 90.% 8.0% 8.6% -9% to -8% 9.3% 0.4% 0.5%.3% Realized loss contribution from stocks increases as losses grow It could be due to higher tail risks of stocks //005

14 Application Fat Tails S&P 500 US LT Gvt 60/40 Portfolio Avg Return 0.98% 0.46% 0.78% Stdev 5.6%.7% 3.6% Skewness Kurtosis Corr w/ S&P Ibbotson Risk contribution in terms of standard deviation assumes normal distribution We need to extend the risk contribution to non-normal return portfolio Hedge funds //005 3

15 Application Hedge Fund Returns Equity Fixed Convertible Dedicated Emerging E.D. Multi- Risk Global Long/Short Managed Multi- Mkt Distressed Income Arbitrage Short Markets Strategy Arbitrage Macro Equity Futures Strategy Neutral Arb Average Stdev IR Skewness Kurtosis CSFB Tremont 0/994 03/005 Some strategies have high IR, but also high negative skewness and high positive kurtosis //005 4

16 VaR Contribution Same Interpretation Value at Risk Maximum loss at a given probability 95% VaR, 99% VaR Contribution to VaR Interpretation: the expected contribution to a portfolio loss equaling the size of VaR It changes with different level of VaR It is not easy to calculate because analytic formula is rarely available for VaR //005 5

17 Calculating VaR Contribution Cornish-Fisher Approximation Cornish-Fisher approximation for VaR Example VaR = µ + ~ z α 99% VaR for the 60/40 portfolio ( z ) ( ) ( ) s + z 3 3z k z 3 5z ~ z z s a a + α α a α a Normal assumption VaR at 99% is -7.6%, Approximate VaR is -3%, z % = α 3.8 z α = //005 6

18 Calculating VaR Contribution Cornish-Fisher Approximation Approximation for VaR contribution We have an analytic expression of VaR Algebraic function, linear, quadratic, cubic and quartic polynomials We can then calculate w i VaR w i //005 7

19 Calculating VaR Contribution Application to 60/40 Portfolio Loss Predicted VaR % Predicted c Realized c -3.50% 84.90% 93.5% 89.8% -4.50% 90.50% 9.8% 9.7% -5.50% 94.0% 9.3% 88.% -6.50% 97.0% 9.0% 99.5% -7.50% 99.0% 9.8% 90.% -8.50% 00.90% 9.3% 0.4% Stocks VaR contribution increases as VaR increases Capture the high tail risks Contribution to standard deviation shows declining instead //005 8

20 Calculating VaR Contribution Other Applications Need to calculate skewness and kurtosis from historical returns Risk management and risk budgeting Asset allocation with traditional assets and hedge funds High moments strategies: event driven, distressed securities, fixed income arbitrage, etc Strategy allocation among managers with non-normal alphas //005 9

21 Summary Risk Budgets Do Add Up Financial interpretation of risk contribution Loss contribution It applies to both standard deviation and VaR Cornish-Fisher approximation can be used to calculate VaR contribution It should clear up some doubts about the concept Applications Risk budgeting for active risks Risk budgeting for beta portfolios parity portfolios Efficient method for risk budgeting of hedge funds //005 0

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