Financial risks faced by investors and financial institutions

Mean Absolute Deviation
Conditional Value at Risk
(cVaR)
Value at Risk
(VaR)
Measures the potential losses of a specific investment or portfolio.
Algebraically: VaR (x, α) = min {u: F (x, u) ≥ 1−a} = min {u: P {R (x, r̃) ≤ u} ≥ 1 − a}
Risk assessment measure that quantifies the amount of tail risk an investment has.
Algebraically: CVaR = (1 / 1 – c) ∫VaR xp(x) dx
Uses the absolute deviation of the rate of return of a portfolio instead of the variance to measure the risk.
Algebraically: MAD = Σ | xi – x | / n
Stressed VaR and cVaR
Are estimated under the use of a historical data frame window of a stressed and high volatile period in prices of underlying assets.
Should be able to quantify the capital that a financial institution should keep meeting its liabilities in the case things go wrong and potential losses come true.
A risk measure is categorized as coherent if satisfies the next four axioms:
Monotonicity: The worst the output of a portfolio the higher the level of risk measure should be.
Algebraically: ∀ x1, x2 with x1< x2 →f (x1) < f(x2)
Coherent Risk Measures:
Translation Invariance: Portfolio f(xi) = A, C is added, risk measure of the new portfolio Xi′= xi + C should be decreased by C and be f(Xi′) = A − C, C work as a buffer when is needed.
Homogeneity: Portfolio f(xi) = A, changes by a factor k (∀ k ∈ ℝ+), relative amount of each asset remains the same, risk measure of the new portfolio (Xi′=kXi) should be f(kxi)=kA
Subadditivity: If Xi (∀ i ϵ ℕ) portfolio has a f(Xi)=Ai amount of risk measure then the merged portfolio should have risk amount up to f(Σnxi) ≤ Σnf(xi)
With Modern Portfolio Theory (MPT) Markowitz introduced a new kind of risk measurement based on two components, the maximization of return and the minimization of risk of every investment which is totally described by variance.
Var(x) = E(x−E(x))2 = E(x2) − E(x)2
Cov(x,y) = E(x−E(x))(y−E(y)) = E(xy)−E(x)E(y)
Beta: Basic pillar for developing pricing models such as CAMP and arbitrage pricing model which calculate the expected return of an asset (or a portfolio) using beta and expected market returns over a specific period.