VAR

The Question Being Asked in VaR “What loss level is such that we are X% confident it will not be exceeded in N business days?” Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

VaR and Regulatory Capital Regulators base the capital they require banks to keep on VaR The market-risk capital is k times the 10-day 99% VaR where k is at least 3.0 Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

VaR vs. C-VaR VaR is the loss level that will not be exceeded with a specified probability C-VaR (or expected shortfall) is the expected loss given that the loss is greater than the VaR level Although C-VaR is theoretically more appealing, it is not widely used Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

Advantages of VaR It captures an important aspect of risk in a single number It is easy to understand It asks the simple question: “How bad can things get?” Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

Time Horizon Instead of calculating the 10-day, 99% VaR directly analysts usually calculate a 1-day 99% VaR and assume This is exactly true when portfolio changes on successive days come from independent identically distributed normal distributions Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

The Model-Building Approach The main alternative to historical simulation is to make assumptions about the probability distributions of return on the market variables and calculate the probability distribution of the change in the value of the portfolio analytically This is known as the model building approach or the variance-covariance approach Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

Description of Value at Risk: Definition: Value at Risk is an estimate of the worst possible loss an investment could realize over a given time horizon, under normal market conditions (defined by a given level of confidence). To estimate Value at Risk a confidence level must be specified.

Choice of confidence level – 95% Investment returns Normal market conditions – the returns that account for 95% of the distribution of possible outcomes. Abnormal market conditions – the returns that account for the other 5% of the possible outcomes.

If a 95% confidence level is used to estimate Value at Risk for a monthly horizon; losses greater than the Value at Risk estimate are expected to occur one in twenty months (5%).

Illustrate Value at Risk: Step 1: Transform simple monthly stock returns into continuously compounded stock returns. Note: Technically, log stock returns are “more likely” to be normally distributed. Step 2: Choose a level of confidence. 90%, 95%, 99%, etc. Banks are required to report Value at Risk estimated with a 99% level of confidence to determine regulatory capital requirements. Step 3: Compute Value at Risk from sample estimates of  and . For example, the largest likely loss in the household industry over the next month under normal market conditions with a 95% level of confidence is: $18,000. Note: It is possible to realize a loss greater than $18,000.

Other Common Interpretations of Value at Risk: “an attempt to provide a single number for senior management summarizing the total risk in a portfolio of assets” – Hull, OF&OD “an estimate, with a given degree of confidence, of how much one can lose from one’s portfolio over a given time horizon” – Wilmott, PWOQF

Conclusions: Value at Risk can be used as a stand alone risk measure or be applied to a portfolio of assets. Value at Risk is a dollar value risk measure, as opposed to the other measurements of risk in the financial industry such as: beta and standard deviation. “We are X percent certain that we will not lose more than V dollars in the next N days.” – Hull

Measures of Risk Standard Deviation (s) Beta (ß) Value at Risk (VaR)

Non-Diversifiable Risk Diversifiable Risk Measured by VAR ß Stand-Alone Risk Or Total Risk = Systematic Risk + Unsystematic Risk Non-Diversifiable Risk Diversifiable Risk Market Risk Company-Specific Risk

Dispersion of Returns – Variances and Standard Deviations Variance (s2) Formula: Variance and Standard Deviation are measures of total (or stand-alone) risk. The larger the variance (or Std. Dev.), the lower the probability that actual returns will be close to the expected return.

Risk Measure - Beta (ß) Beta (ß) formula: Beta measures the portfolio’s systematic risk, that is, the degree to which its return is correlated with the return on the market as a whole. Stock with high beta (ß>1) is more volatile than the market taken as a whole.

Risk Measures – Value at Risk (VaR) VaR is a measure of risk based on a probability of loss and a specific time horizon. VaR translates portfolio volatility into a dollar value. Measure of Total Risk rather than Systematic (or Non-Diversifiable Risk) measured by Beta.

Advantages of VaR VaR provides an measure of total risk. VaR is an easy number to understand and explain to clients. VaR translates portfolio volatility into a dollar value. VaR is useful for monitoring and controlling risk within the portfolio.

Advantages of VaR (Cont.) VaR can measure the risk of many types of financial securities (i.e., stocks, bonds, commodities, foreign exchange, off-balance-sheet derivatives such as futures, forwards, swaps, and options, and etc.) As a tool, VaR is very useful for comparing a portfolio with the market portfolio (S&P500).

Monte Carlo Simulation Calculate DP Repeat many times to build up a probability distribution for DP VaR is the appropriate fractile of the distribution times square root of N For example, with 1,000 trial the 1 percentile is the 10th worst case. Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

Stress Testing This involves testing how well a portfolio performs under some of the most extreme market moves seen in the last 10 to 20 years Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

Back-Testing Tests how well VaR estimates would have performed in the past We could ask the question: How often was the actual 10-day loss greater than the 99%/10 day VaR? Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

VaR is useful also for: Limits of risk control Measure return adjusted for risk  RAROC Pricing CAPITAL Allocation

How to calculate Var VaR of a position is measured by : Market Value of position (VM) Sensitivity of price to market variations () Adverse price variation as the product of: Volatility of instruments () A factor  corrispondin to the confidence levele desired

example BTP 10y for nominal €1mln Ptel quel = 105 DM = 7 anni rend.gg. = 15 b.p. (0,15%)  = 2,326 (c = 99%)

VaR of a portfolio Can be estimated using portfolio theory : MARKET VALUES and sensitivity of all single instruments Volatility of single instruments Correlatons between the N instruments

The limits of VaR models They disregard exceptional events They disregard clint relationships (short termism) They are based on non realistic hypotheses They amplify instability of financial markets They arrive late

The major limit is maybe that ... They might disregard the magnitude of the loss: VaR with confidence level of 99%  probability to lose more than VaR is only 1%… …but HOW MUCH CAN YOU LOSE IF EVENTS INCLUDED IN THAT 1 % HAPPEN ?