上财系列 研究生数量风险管理讲义.pdf

Quantitative Risk Management In this course, we will consider the following questions: • What is risk? • How can we measure risk? • How can we model risk? • What are techniques commonly used to analyze risk? • How can we use risk measures to control risk? Part 1 - Introduction to Risk Management 1.1Concepts of risk 1. What is risk? Oxford English Dictionary defi nes risk as “hazard, a chance of bad consequence, loss or exposure to mischance”. In fi nancial risk management, risks are defi ned as “any event or action that may ad- versely aff ect an organization’s ability to achieve its objectives and execute its strate- gies” or “the quantifi able likelihood of loss or less-than-expected returns”. Financial risks are usually refereed as portfolio losses over some time horizon. While insurance risks are often refereed as claim amounts. All these risks can be described by random variables. In terms of probability and statistics, a risk is a random variable, or a random vector, or a random process, representing a loss or payment, or a risky positions in a fi nancial or insurance portfolio. A negative outcome for the loss variable means a gain has occurred. 1 2. The main risk of types in fi nance: (a) Market risk – The risk of a change in the value of a fi nancial position due to changes in the value of the underlying components on which that position de- pends, such as stock and bond prices, and exchanges rates. (b) Credit risk – The risk of not receiving promised repayments on outstanding in- vestments such as loans and bonds, because of the default (failure to repay a debt) of the borrower. What is CreditRisk+? “Credit Suisse Financial Products (CSFP) introduced its own portfolio model CreditRisk+TM(henceforth CreditRisk+) in October 1997. The model was cre- ated by Tom Wilde of CSFP and is still unique among existing portfolio models with respect to the mathematics involved. CreditRisk+applies techniques from actuarial mathematics in an enhanced way to calculate the probabilities for portfolio loss levels (i.e. the portfolio loss distribution, henceforth loss distribu- tion).” (c) Operational risk – The risk of losses resulting from inadequate or failed internal processes, people and systems, or from external events. (d) Model risk – The risk associated with using a misspecifi ed or inappropriate model for measuring risk. 3. The main risk of types in insurance: (a) Liability risk (or technical risk) – The risk that insurance companies are assuming by selling insurance contracts. (b) Asset risk (or investment risk) – The risk associated with insurers’ asset manage- ment, which are subdivided into credit risks (like the issuer of a bond gets ruined) and market risks (like depreciation risk). 2 (c) Operational risk – This risk is subdivided into business risks (like lower produc- tion than expected) and event risks (like system failure). (d) Underlying risk – The risk inherent in insurance policies sold. i. Changes in mortality tables on the life insurance policies such as on a whole life insurance policy. ii. Changing patterns of natural catastrophes (earthquakes, tornados, hurri- canes) on the casualty insurance policy such as home insurance. 4. What is risk management? That is a ‘discipline for living with the possibility that future events may cause adverse eff ects’. In fi nance and insurance, risk management is often related to methodologies and tech- niques used to reduce losses, to make profi ts, and to maintain a comfortable surplus. 5. What is quantitative risk management (QRM)? That is ‘a quantitative science using the language of mathematics in general, and probability and statistics in particular for risk management’. 3 1.2Risk measures 1. What is a risk measure? A risk measure ρ is a mapping from the set of all risks considered to real numbers, namely, for a risk L, ρ(L) ∈ (−∞,∞). The risk measure ρ(L) can be interpret as the amount of capital that should be added to a position, say an investment portfolio, with loss given by L, so that the position becomes acceptable to an external or internal risk controller. Position with ρ(L) ≤ 0 are acceptable without injection of capital. If ρ(L) 0, ρ(λL) = λρ(L). Axiom 4 (Monotonicity) For all L1,L2∈ L∗satisfying L1≤ L2almost surely, ρ(L1) ≤ ρ(L2). ? Clearly, expectation E(·) is a coherent risk measure while variance V ar(·) and stan- dard derivation pV ar(·) are not coherent. Proposition 1.2.2 For a risk measure satisfying Axioms 2 and 3, Axiom 4 is equiv- alent to the requirement that ρ(L) ≤ 0 for all L ≤ 0.? 5 1.3Important risk measures in risk management 1. Defi nition 1.3.1 (Value-at-Risk) Given a confi dence level α ∈ (0,1). The Value-at- Risk (VaR) of a loss X or its distribution function F(x) = P(X ≤ x) at the confi dence level α, denoted by V aRα(X), or xα, or F−1(α), is the smallest number xαsuch that the probability that the loss X exceeds xαis not larger than 1 − α, namely V aRα(X) = inf{x ∈ R : Pr(X x) ≤ 1 − α} = inf{x ∈ R : F(x) ≥ α}. ? Note th。