Financial Risk Budgeting Methods

Deep research into how financial institutions decompose portfolio risk to components, and what AI safety can learn from both the mathematics and the failures.

Euler Allocation and Risk Decomposition

Mathematical Foundation

The mathematical foundation rests on Euler’s theorem for homogeneous functions. For any risk measure R(x) that is homogeneous of degree 1:

R(x) = Σᵢ xᵢ · ∂R(x)/∂xᵢ

This enables “full allocation” - component risk contributions sum exactly to total portfolio risk, with no gaps and no waste.

Component VaR, Marginal VaR, Incremental VaR

Measure	Formula	Use
Component VaR	CoVaR_i = x_i · β_i · VaR_p	Dollar contribution to portfolio VaR. Σ CoVaR_i = Total VaR
Marginal VaR	MVaR_i = ∂VaR(P)/∂x_i	Sensitivity to position size changes
Incremental VaR	IVaR = VaR(P+a) - VaR(P)	Actual VaR change from adding position (not additive)

Expected Shortfall vs VaR

Expected Shortfall (ES/CVaR) = expected loss in the worst α% of cases:

ES_α(X) = E[X | X ≥ VaR_α(X)]

Why ES is preferred:

Coherent - satisfies all four axioms (VaR fails subadditivity)
Subadditive - diversification always reduces risk
Tail sensitive - captures severity beyond VaR threshold
Regulatory adoption - Basel FRTB shifted from VaR to ES at 97.5%

Hierarchical Risk Cascading in Banks

Level 1 - Board: 50-100 high-level quantitative metrics defining risk appetite

Level 2 - Business Units: Risk committees allocate limits to lines of business

Level 3 - Trading Desks: Market Risk Management allocates to divisions and desks

Level 4 - Individual Positions: Traders manage within allocated limits, real-time monitoring

Key distinction: Limits are hard constraints (breach blocks business), thresholds require reporting to higher instance.

FSB Risk Appetite Framework (2013)

Required elements:

Written Risk Appetite Statement linked to strategic/capital plans
Quantitative limits under normal and stressed conditions
Hierarchical framework across business lines and legal entities
Clear roles for board, CEO, CRO, CFO
Independent assessment by internal audit or third party

Risk Budgeting in Practice

Risk Parity

Allocate capital so each asset class contributes equally to total portfolio risk:

σ_p² = Σᵢ RC_i where RC_i = w_i · σ_i · ρ_ip · σ_p

For equal risk contribution: RC_i = σ_p²/N for all i

Since bonds have lower volatility than stocks, risk parity uses leverage to scale up bond positions.

Vulnerability: Relies on low-to-negative stock-bond correlation. Failed in March 2020 when both declined simultaneously.

RAROC (Risk-Adjusted Return on Capital)

RAROC = Risk-Adjusted Return / Economic Capital

Applications:

Capital allocation across business units
Performance comparison on like-for-like basis
Limit setting: unit creates value if RAROC > cost of equity
Risk-based pricing

Trading Desk Limits

Types of limits:

VaR Limits: Maximum portfolio VaR (e.g., $10M daily at 99%)
Stress Test Limits: Maximum loss under predefined scenarios
Concentration Limits: Maximum exposure to single issuers/sectors
Position Limits: Maximum notional by instrument type

Research shows limits are “meaningful and costly for traders to breach” - dealers actively manage positions away from limits.

Key Failures and Lessons

LTCM 1998

What went wrong:

Short historical windows - 2-3 years of data, missing regime changes
Extreme leverage - 25:1 on balance sheet, 100:1+ with derivatives ($1.25T notional)
Model overconfidence - worked in normal conditions, failed in crisis
Correlation breakdown - historically loose markets became tightly coupled
Liquidity assumptions - couldn’t exit when Russia default triggered flight to quality

In August 1998 alone, LTCM lost 44% of its value. Fed facilitated $3.6B bailout.

Key insight: “Separation of quantitative analysis and qualitative analysis” - overconfidence in models, ignored embedded risks.

2008 Financial Crisis

VaR failures:

Normal distribution assumption when reality had fat tails
VaR “significantly underestimated probability of extreme losses”
UBS acknowledged “shortcuts” excluding risks from calculations
Correlations assumed stable became 1.0 during crisis
Liquidity risk largely unmodeled

Research finding: “VaR underestimated the risk of loss, while the conditional EVT model performed more accurately” (2010 study).

March 2020 Risk Parity

Risk parity funds suffered 13-43% drawdowns when COVID triggered simultaneous stock and bond declines.

What happened: “Fixed Income as volatility reducer and hedge for equities broke down” - negative stock-bond correlation that persisted since late 1990s rapidly increased to >+0.60.

Basel Evolution

Basel	Year	Key Changes
Basel I	1988	Risk-based capital requirements
Basel II	2004	Internal VaR models permitted
Basel III	2010	Raised capital minimums, added stressed VaR, leverage ratio
FRTB	2025	Replace VaR with Expected Shortfall at 97.5%

Shapley Values for Risk Attribution

The Shapley Value

For cooperative game with value function v:

φ_i(v) = Σ_{S⊆N{i}} [|S|!(n-|S|-1)!/n!] · [v(S∪{i}) - v(S)]

Averages each player’s marginal contribution across all possible coalitions.

Axiomatic Foundation

Shapley is the only solution satisfying:

Efficiency: Σ_i φ_i(v) = v(N) - all risk allocated
Symmetry: Equal contributors get equal shares
Additivity: φ_i(v+w) = φ_i(v) + φ_i(w)
Null Player: Zero contributors get zero

Euler vs Shapley

Property	Euler	Shapley
Basis	Calculus (derivatives)	Combinatorics (coalitions)
Requirements	Differentiable, homogeneous	Any value function
Complexity	O(n)	O(2ⁿ)
Best for	Continuous weights	Discrete components

Computational Challenges

Shapley complexity is O(2ⁿ) - “usually too time expensive” for >25 players.

Approximations:

Monte Carlo sampling
Ergodic sampling with negatively correlated pairs
Machine learning approximators (MLSVA)

Coherent Risk Measures

The Artzner et al. (1999) Axioms

A risk measure ρ is coherent if for all X, Y:

Monotonicity: X ≤ Y ⟹ ρ(Y) ≤ ρ(X)
Translation Invariance: ρ(X + α) = ρ(X) − α
Positive Homogeneity: ρ(λX) = λρ(X) for λ > 0
Subadditivity: ρ(X + Y) ≤ ρ(X) + ρ(Y)

VaR fails subadditivity - can discourage diversification. Expected Shortfall satisfies all four - coherent risk measure.

Subadditivity and Diversification

Subadditivity captures: “a merger does not create extra risk”

Encourages diversification
Prevents perverse incentives
Reflects economic reality: combined risk ≤ sum of individual risks

Applicability to AI Safety

Required Properties for Decomposition

For Euler allocation:

Homogeneity of degree 1: R(λx) = λR(x)
Differentiability: ∂R/∂x_i must exist
Continuity: smooth response to parameters

For coherence:

Monotonicity: more capable → higher harm measure
Translation invariance
Positive homogeneity: scaling deployment scales harm
Subadditivity: two systems together ≤ sum of individual harms

Proposed AI Harm Measures

Multiplicative (CBRA style): System Risk = Criticality × Autonomy × Permission × Impact

Additive (attack surface style): AI_Capability_Surface = Σ (capability_class × damage_potential × accessibility)

Critical Challenges for AI

Systematic vs Random: Finance assumes random failures with known distributions. AI failures are systematic (bugs, misalignment, emergent capabilities).
Non-stationarity: Historical AI behavior may not predict future behavior due to learning/adaptation.
Emergence: Complex interactions may create superlinear risk composition.
Unknown unknowns: AI uncertainty exceeds financial uncertainty.

Lessons for AI Safety

From LTCM/2008/2020 failures:

Don’t trust short historical windows - regime changes happen
Leverage amplifies model errors - conservative margins essential
Correlations break under stress - independence assumptions fail precisely when needed
Tail risks matter - use ES not VaR, capture severity not just probability
Liquidity/capability crises cascade - model interconnections

Key Citations

Seminal Papers

Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D. (1999). “Coherent Measures of Risk.” Mathematical Finance, 9(3), 203-228.
Acerbi, C., & Tasche, D. (2002). “Expected Shortfall: A Natural Coherent Alternative to Value at Risk.” Economic Notes, 31(2), 379-388.
Tasche, D. (2007). “Capital Allocation to Business Units and Sub-Portfolios: the Euler Principle.” arXiv:0708.2542
McNeil, A.J., Frey, R., & Embrechts, P. (2005). Quantitative Risk Management. Princeton University Press.

Regulatory Documents

Financial Stability Board (2013). “Principles for an Effective Risk Appetite Framework.”
Basel Committee. “Fundamental Review of the Trading Book (FRTB).”

Historical Failures

Federal Reserve History. “Long-Term Capital Management and the Federal Reserve’s Response.”
Wikipedia. “2008 Financial Crisis.”
Advisor Perspectives (2020). “Risk Parity in the Time of COVID.”