Large deviation theory and applications Application II The Theorem of Bahadur and Rao and Large Portfolio Losses

The Theorem of Bahadur an Rao is a refinement of the classical Cramér’s Theorem. Instead of just calculating the exponential rate of decay of large deviation events, it gives a precise estimate of the large deviation probabilities. We use this theorem as well as Cramér’s Theorem to calculate the tail distributions of total financial losses of a large portfolio and compare the quality of both approximations. 1 The Theorem of Bahadur and Rao Lemma 1 Let X be a real valued non-degenerate random variable, Λ(λ) := logE[exp(λX)] and Λ∗(x) := sup{λ ∈ R : λx − Λ(λ)}. Define DΛ := {λ ∈ R : Λ(λ) <∞}. Then Λ is real analytic in Int{DΛ} with Λ′(λ) = E [ X exp ( λX − Λ(λ) )] (1) Λ′′(λ) = E [ X exp ( λX − Λ(λ) )] − E [ X exp ( λX − Λ(λ) )]2 > 0 (2) Λ′(λ) = x⇒ Λ∗(x) := λx− Λ(λ) (3) If moreover X is positive and bounded, then DΛ = R, Λ∗(x) = sup{λ ≥ 0 : λx− Λ(λ)}, Λ′(λ) is strictly nondecreasing for λ ≥ 0 and Λ′(0) = E[X] (4) lim λ→∞ Λ′(λ) = ess supX (5) Theorem 1 (Theorem of Bahadur and Rao) Let X1, . . . , Xn be real valued non-degenerate i.i.d. random variables with non-lattice law μ and Sn := 1 n ∑n i=1Xn. Let ζ ∈ Int{DΛ}, ζ > 0 and z = λ′(ζ). Then lim n→∞ Jn(z)P(Sn ≥ z) = 1 where Jn(z) = ζ √ Λ′′(ζ)2πn exp(nΛ∗(z)) 1if X1 has a lattice law with lattice constant d−1 and P(X1 = z) > 0, then the theorem holds with modified limit ζd 1−exp(−ζd) ; confer [2] page 110 et seqq.