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
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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
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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
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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
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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
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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
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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
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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
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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
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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 +
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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
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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
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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
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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
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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
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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
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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
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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
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∂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
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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
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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
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