# Risk Budgeting: Concept, Interpretation and Applications

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Risk Budgeting: Concept, Interpretation and Applications

Risk Budgeting: Concept, Interpretation and Applications Northfield Research Conference 2005 Eddie Qian, PhD, CFA Senior Portfolio Manager 260 Franklin Street Boston, MA 02110 (617) 439-6327 217538 8/11/2005 The Concept Risk Contribution § Risk contribution – attribution of total risk to individual underlying components of a portfolio in percentage terms § Examples Ÿ Fixed income portfolio: sector risk, yield curve risk, … Ÿ Equity portfolio: systematic risk (risk indices/industries), specific risk, … Ÿ Asset allocation: TAA risk (stock/bond, cap rotation, …), sleeve active risk, … Ÿ Manager selection, strategy allocation Ÿ Asset allocation portfolio: beta risk from stocks, bonds, commodities, … Ÿ Parity portfolios 217538 8/11/2005 1 The Concept An Example § A simple illustration Ÿ Two active strategies – strategy A at 1% active risk, strategy B at 2% active risk, two sources uncorrelated § Total active risk Ÿ Equals 1% + 2% = 2.24% Ÿ What is the risk contribution from strategy A and B? Ÿ Answer – A contributes 20% and B at 80% Ÿ Contribution to variance 2 Ÿ Strategy A 2 1 1+ 4 , Strategy B 4 1+ 4 § Contribution based upon variances and covariances 217538 8/11/2005 2 The Concept Mathematical Definition § Marginal contribution - ∂∂wσ ∂σ w § Risk contribution - ∂w § VaR contribution - w∂VaR ∂w i i i i i § Well defined in financial engineering terms § But objected by some financial economists for a lack of economic interpretation 217538 8/11/2005 3 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 217538 8/11/2005 4 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 217538 8/11/2005 5 Interpretation An Asset Allocation Example § Balanced benchmark Ÿ 60% stocks, 40% bonds Ÿ Is it really balanced? § Risk contribution Ÿ Stock volatilities at 15%, bond volatilities at 5% Ÿ Stocks are actually 9 times riskier than bonds Ÿ Stocks’ contribution is roughly 95% and bonds at 5% 6 2 ⋅ 32 = 95% 4 2 ⋅ 12 + 62 ⋅ 32 § The balanced fund – a misnomer 217538 8/11/2005 6 Interpretation Loss Contribution of a 60/40 Portfolio Loss > 2% 3% 4% Stocks 95.6% 100.1% 101.9% Bonds 4.4% -0.1% -1.9% § Stocks’ contribution is almost 100% § It is greater than the theoretical value of 95%, because stocks have greater tail risk 217538 8/11/2005 7 Interpretation Mathematical Proof § Question: For a given loss of a portfolio, what are the likely contribution from the underlying components? § Answer: Conditional expectation 2 2 2 2 Total risk: σ = w1 σ 1 + w2σ 2 + 2 ρw1w2σ 1σ 2 Risk contribution: ∂σ w12σ 12 + ρw1 w2σ 1σ 2 σ = p1 = w1 ∂ w σ2 1 w22σ 22 + ρw1 w2σ 1σ 2 ∂σ σ = p 2 = w2 σ2 ∂w2 Loss contribution? ci = E ( wi ri | w1r1 + w2 r2 = L ) / L 217538 8/11/2005 8 Interpretation Mathematical Proof § The loss contribution is approximately risk contribution § Cases where they are identical ŸZero expected returns ŸLarge losses ŸMean-variance optimal weights p2 w1 µ1 − p1w2µ2 D @ p1 + 1 L L p w µ − p2 w1µ1 D @ p2 + 2 c2 = p2 + 1 2 2 L L c1 = p1 + 217538 8/11/2005 9 Interpretation Mean-variance Optimal § Risk contribution is equivalent to loss contribution § Risk contribution is also equivalent to expected return contribution w1 µ1 w2 µ 2 = =λ p1 p2 § The interpretation is valid and very close even if the portfolio is not MV optimal 217538 8/11/2005 10 Interpretation Loss Contribution Figure 1 The value of D 1 over standard deviation for asset allocation portfolios 3.0% 2.0% 1.0% 0.0% -1.0% -2.0% -3.0% -4.0% 0% 20% 40% 60% 80% 100% Stock Weight § The difference between risk contribution and loss contribution is quite small 217538 8/11/2005 11 Application Loss Contribution of a 60/40 Portfolio Loss -4% to -3% -5% to -4% -6% to -5% -7% to -6% -8% to -7% -19% to -8% Predicted c 1 Realized c 1 93.5% 92.8% 92.3% 92.0% 91.8% 91.3% 89.8% 92.7% 88.1% 99.5% 90.1% 102.4% N 45 23 11 9 8 12 Predicted Std 28.0% 21.0% 16.8% 14.0% 12.0% 10.5% Realized Std 26.1% 20.7% 16.1% 18.7% 18.6% 12.3% § Realized loss contribution from stocks increases as losses grow § It could be due to higher tail risks of stocks 217538 8/11/2005 12 Application Fat Tails Avg Return Stdev Skewness Kurtosis Corr w/ S&P 500 S&P 500 0.98% 5.61% 0.39 9.58 1.00 US LT Gvt 0.46% 2.27% 0.66 5.09 0.14 60/40 Portfolio 0.78% 3.61% 0.40 7.64 0.97 Ÿ Ibbotson 1929 - 2004 § Risk contribution in terms of standard deviation assumes normal distribution § We need to extend the risk contribution to non-normal return portfolio ŸHedge funds 217538 8/11/2005 13 Application Hedge Fund Returns Convertible Dedicated Arbitrage Short Emerging Markets Equity Mkt Neutral 9.80 Distressed 13.00 E.D. Multi- Risk Strategy Arbitrage 10.31 7.82 Fixed Income Arb 6.64 Average 8.94 -0.87 8.83 Stdev IR 4.70 17.66 16.88 2.99 6.64 6.13 4.30 1.90 -0.05 0.52 3.28 1.96 1.68 1.82 Skewness Kurtosis -1.36 3.38 0.86 2.03 -0.62 4.30 0.30 0.33 -2.85 17.82 -2.64 17.57 -1.29 6.41 -3.26 17.33 Global Macro Long/Short Managed Equity Futures 13.70 11.77 6.89 3.79 11.46 10.52 12.20 4.35 1.75 1.20 1.12 0.57 2.10 0.01 2.50 0.24 3.75 0.04 0.43 -1.30 3.76 Ÿ CSFB Tremont 01/1994 – 03/2005 § Some strategies have high IR, but also high negative skewness and high positive kurtosis 217538 8/11/2005 14 MultiStrategy 9.13 VaR Contribution Same Interpretation § Value at Risk Ÿ Maximum loss at a given probability Ÿ 95% VaR, 99% VaR § Contribution to VaR Ÿ Interpretation: the expected contribution to a portfolio loss equaling the size of VaR Ÿ It changes with different level of VaR Ÿ It is not easy to calculate because analytic formula is rarely available for VaR 217538 8/11/2005 15 Calculating VaR Contribution Cornish-Fisher Approximation § Cornish-Fisher approximation for VaR VaR = µ + ~zα σ ( ) ( ) ( ) 1 1 3 1 ~ z a ≈ z a + zα2 − 1 s + zα − 3 z a k − 2 zα3 − 5 z a s 2 6 24 36 § Example Ÿ 99% VaR for the 60/40 portfolio Ÿ Normal assumption VaR at 99% is -7.6%, zα = −2.33 Ÿ Approximate VaR is -13%, z% α = −3.81 217538 8/11/2005 16 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 217538 8/11/2005 ∂VaR wi ∂wi 17 Calculating VaR Contribution Application to 60/40 Portfolio Loss -3.50% -4.50% -5.50% -6.50% -7.50% -8.50% Predicted VaR % Predicted c 1 Realized c 1 84.90% 93.5% 89.8% 90.50% 92.8% 92.7% 94.20% 92.3% 88.1% 97.10% 92.0% 99.5% 99.20% 91.8% 90.1% 100.90% 91.3% 102.4% § Stocks’ VaR contribution increases as VaR increases § Capture the high tail risks § Contribution to standard deviation shows declining instead 217538 8/11/2005 18 Calculating VaR Contribution Other Applications § Need to calculate skewness and kurtosis from historical § returns Risk management and risk budgeting Ÿ Asset allocation with traditional assets and hedge funds Ÿ High moments strategies: event driven, distressed securities, fixed income arbitrage, etc Ÿ Strategy allocation among managers with non-normal alphas 217538 8/11/2005 19 Summary Risk Budgets Do Add Up § Financial interpretation of risk contribution Ÿ Ÿ Ÿ Ÿ 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 Ÿ Risk budgeting for active risks Ÿ Risk budgeting for beta portfolios – parity portfolios Ÿ Efficient method for risk budgeting of hedge funds 217538 8/11/2005 20