MA634 Financial Risk Management - Basics of Risk Management

A set of vocabulary flashcards covering key concepts in risk management: horizons, loss definitions, risk factors, coherent risk measures, VaR/ES, estimation methods, and common modeling approaches.

Horizon Δ (Delta)

A fixed time interval (e.g., 1 day or 1 week) over which risk is assessed and loss distributions are modeled.

V_t

Portfolio value at time tΔ.

L_{t+1}

Portfolio loss over the period from tΔ to (t+1)Δ: L{t+1} = −(V{t+1} − V_t). Losses are defined as positive.

Z_t

Risk factors vector such that Vt = f(t, Zt). For example, Z_t could be stock prices or transformations of them.

X_{t+1}

Change in risk factors between times: X{t+1} := Zt − Z_{t−1}.

L{t+1}(X{t+1})

Loss expressed as a function of risk-factor changes: L{t+1}(X{t+1}) = −[ f(t+1, Zt + X{t+1}) − f(t, Z_t) ].

First-order Taylor approximation of loss

L̂{t+1}(X{t+1}) = ft(t, Zt)Δ + ∑{i=1}^d f{zi}(t, Zt) X_{t+1,i}, the linear (first-order) approximation.

Small Δ assumption

A situation where Δ is small (e.g., Δ ≈ 1/365) and market volatility is not excessive, making the Taylor approximation more accurate.

Unconditional loss distribution

The distribution of L{t+1} given the portfolio at time t and assuming the risk-factor change distribution FX is FX.

Conditional loss distribution

The distribution of L{t+1} given the portfolio at time t and conditioned on the information set Ft.

i.i.d. Xt

If the X_t’s are i.i.d., then the conditional and unconditional loss distributions coincide.

Coherent risk measure

A risk measure R is coherent if it satisfies subadditivity, positive homogeneity, monotonicity, and translation invariance.

Subadditivity

R(w1 + w2) ≤ R(w1) + R(w2): diversification should not increase risk.

Homogeneity

R(λw) = λR(w) for λ ≥ 0: scaling the portfolio scales risk by the same factor.

Monotonicity

If w1 ≺ w2 (w2 dominates w1 in all scenarios), then R(w1) ≥ R(w2).

Translation Invariance

R(w + m) = R(w) − m for any cash amount m: adding cash reduces risk by m.

Capital hedging

R(w + R(w)) = 0: adding capital equal to the risk measure hedges the risk.

Loss L(w) notation

L(w) = −Pt(w) Rt+h(w): loss expressed via current value and future return; often Pt(w) = 1 for simplicity.

Volatility of the loss

R(w) = σ(L(w)): the standard deviation of portfolio loss.

SD_c risk measure

R(w) = SD_c(w) = E[L(w)] + c · σ(L(w)); a standard deviation-based risk measure with tilt c.

Value-at-Risk (VaR)

VaRα(w) = the α-quantile of the loss distribution L: VaRα(w) = q_α(L).

α-quantile q_α(F)

q_α(F) = inf{x ∈ R : F(x) ≥ α}; the α-quantile of a CDF F.

VaR at level α (VaRα)

VaRα(L) = q_α(L) where L is the portfolio loss; the α-quantile of the loss distribution.

VaR monotonicity

VaRα is monotone: if X ≤ Y almost surely, then VaRα(X) ≤ VaRα(Y) for any α ∈ (0,1).

Positive homogeneity and translation invariance of VaR

VaRα(μ + λX) = μ + λ VaRα(X) for μ ∈ R and λ > 0.

VaR subadditivity remark

VaR is not generally subadditive; diversification may not always reduce VaR.

Expected Shortfall (ES) / CVaR

ESα(L) = (1/(1−α)) ∫{α}^{1} VaRu(L) du = E[L | L ≥ VaRα(L)]. ES measures expected loss given that loss exceeds VaR.

ES ≥ VaR

By definition, ESα(L) ≥ VaRα(L) for any α ∈ (0,1).

Continuous loss distribution implication (Lemma 0.7)

If FL is continuous, ESα = E[L | L ≥ VaRα] = E[L; L ≥ VaRα] / (1−α).

ES for Normal distribution

If L ~ N(μ, σ^2), then ES_α = μ + σ φ(Φ^{-1}(α)) / (1−α), where φ is the standard normal PDF and Φ its CDF.

ES for t distribution

If L ~ t(ν, μ, σ^2), ES involves the t-distribution's CDF and PDF; explicit form uses t{ν}^{-1}(α) and gν (the PDF).

Three approaches to estimate VaR/ES

Historical (empirical) VaR/ES, Analytical (parametric) VaR/ES, and Monte Carlo (simulated) VaR/ES.

Historical simulation

Use a fixed look-back window (e.g., 250 trading days) to replay past market movements and revalue the current portfolio; compute returns R_k and derive VaR/ES.

Monte Carlo simulation

Generate risk-factor scenarios from a specified joint distribution, revalue the portfolio, compute returns, and derive VaR/ES; flexible but computationally intensive.

Variance-Covariance (MVN) approach

Assume risk factors Xt+1 ∼ MVN(μ, Σ) and a linear loss approximation L̂ = −(ct + bt^⊤ X{t+1}); L̂ ∼ N(−ct − b^⊤ μ, b^⊤ Σ b_t).

Portfolio return Π(w)

Π(w) = Wt^⊤ Rt+h, with Wt the exposure vector and Rt+h the asset returns; if Rt+h ∼ N(μ, Σ), then μ(Π) = Wt^⊤ μ and σ^2(Π) = Wt^⊤ Σ Wt.

VaR under MVN (variance-covariance VaR)

VaRα(w; h) = −Wt^⊤ μ + Φ^{-1}(α) √(Wt^⊤ Σ Wt).

Limitations of Gaussian MVN approach

Financial factors often have fat tails/heavy tails; normal distribution underestimates extreme losses.

Multivariate t distribution remedy

Use a multivariate t distribution to capture heavy tails and tail dependence in risk-factor modeling.

Quadratic approximation remedy

For larger horizons or nonlinear portfolios, replace the linear Lt+1 with a quadratic approximation and use Monte Carlo to estimate VaR/ES.