The Quantitative Risk Management Exercise Book

Exercise 2.6 (VaR and expected shortfall) a) Give mathematically precise definitions of value-at-risk VaR (L) and expected shortfall ES (L) for a random loss L at confidence level (0, 1).b) Explain the relative advantages of each risk measure over the other.Exercise 2.7 (Superadditivity scenarios for VaR) Describe some models for financial losses that can lead to situations where VaR is superadditive.Exercise 2.8 (Additivity for two linearly dependent random variables) Consider an arbitrary random variable X and let Y = aX + b for constants a > 0 and b BasicExercise 2.9 (Risk-neutral valuation for interest-rate derivatives) Consider a two-period model.Denote by r t , t {0, 1}, the simple interest rate from t to t + 1, so that 1 monetary unit invested at t is worth 1 + r t at t + 1. Assume that r 0 is 1.5% and that r 1 takes the values 1% and 2% with probability 1/2.Denote by p(t, T ) the price at t of a zero-coupon bond with maturity T and face value 1.a) Write down p(0, 1) and p(1, 2) for the cases r 1 = 0.01 and r 1 = 0.02.b) Suppose a long zero-coupon bond with maturity T = 2 and face value 1 is traded for 0.969729 at t = 0.In this setup an equivalent martingale measure Q is characterized by the probability q = Q(r 1 = 0.01).Compute q from p(0, 2).c) Apply risk-neutral valuation to price a stylized floor contract which pays an amount of 1 if r 1 < r 0 .Note.In general, a floor contract is an option which provides protection against low interest rates.Exercise 2.10 (Mapping of a stock portfolio affected by exchange rates) Consider a portfolio P consisting of two stocks S t,1 , S t,2 , where S t,1 denotes the value of stock 1 in EUR and S t,2 denotes the value of stock 2 in CHF.Let e CHF t denote the CHF/EUR exchange rate at time t.In other words, 1 CHF is worth e CHF t EUR at t. Furthermore, denote by 1 and 2 the number of shares in stocks 1 and 2 in P, respectively.a) Derive the value V t in EUR of P at time t in terms of the risk factors Z t,j = log S t,j , j {1, 2}, and. What is the corresponding mapping?b) Derive the value V t+1 of P at time t + 1 and the one-period loss L t+1 .c) Derive the linearized one-period loss L t+1 and express it in terms of portfolio weights w 1 , w 2 (the values of each stock investment relative to the value V t of the overall portfolio).

Récupéré en direct depuis OpenAlex et désinversé. Les résumés ne sont pas conservés dans cette base de données : les index inversés représentent 8,6 Go des 9,3 Go de texte de la base, et le serveur dispose de 13 Go libres.