8Description 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.
9Choice of confidence level – 95% Investment returnsNormal 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.
10If 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%).
11Illustrate 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.
12Other 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
13Conclusions: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
14Measures of RiskStandard Deviation (s)Beta (ß)Value at Risk (VaR)
16Dispersion 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.
17Risk 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.
18Risk 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.
19Advantages 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.
20Advantages 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).
24VaR is useful also for: Limits of risk control Measure return adjusted for risk RAROCPricingCAPITAL Allocation
25How 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
26example BTP 10y for nominal €1mln Ptel quel = 105 DM = 7 anni rend.gg. = 15 b.p. (0,15%) = 2,326 (c = 99%)
27VaR of a portfolio Can be estimated using portfolio theory : MARKET VALUES and sensitivity of all single instrumentsVolatility of single instrumentsCorrelatons between the N instruments
28The limits of VaR models They disregard exceptional eventsThey disregard clint relationships (short termism)They are based on non realistic hypothesesThey amplify instability of financial marketsThey arrive late
29The 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 ?