ECE 307 – Techniques for Engineering Decisions 16. Value–at–Risk or VaRINTRODUCTION TO FUTURESINTRODUCTION TO FUTURESINTRODUCTION TO FUTURESINTRODUCTION TO FUTURESCOMMODITY PORTFOLIOSIMPORTANCE OF DIVERSIFYING THROUGH PORTFOLIO HOLDINGIMPORTANCE OF DIVERSIFYING THROUGH PORTFOLIO HOLDINGPORTFOLIO ANALYSISMARKET UNCERTAINTIESPERFORMANCE PREDICTIONPERFORMANCE PREDICTIONPORTFOLIO VALUE AND RETURNSPORTFOLIO VALUE AND RETURNSDATA COLLECTIONDATA COLLECTIONDATA COLLECTIONDATA COLLECTION‘BUCKETS’ AND FREQUENCYFREQUENCY VS. RETURNS DISTRIBUTIONNORMALIZATIONNORMALIZED FREQUENCY DISTRIBUTIONCUMULATIVE DISTRIBUTION FUNCTION (c.d.f.) of INTERPRETING THE c.d.f.UNDERSTANDING THE c.d.f. c.d.f. OF VALUE – AT – RISK (VaR)CUMULATIVE DISTRIBUTION FUNCTION (c.d.f.)VALUE – AT – RISK (VaR)VALUE – AT – RISK (VaR)A PRACTICAL ASSIGNMENTASSIGNMENT© 2006 - 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 1ECE 307 – Techniques for Engineering Decisions16. Value–at–Risk or VaRGeorge GrossDepartment of Electrical and Computer EngineeringUniversity of Illinois at Urbana–Champaign© 2006 - 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 2INTRODUCTION TO FUTURES Commodity traders trade important commodities such as foodstuff, livestock, metals, oils, and electricity using financial instruments known as forward contracts Standardized forward contracts are known as futures© 2006 - 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 3INTRODUCTION TO FUTURES Futures have finite lives and are primarily used as financial tools either to hedge commodity price–fluctuation risks or to take advantage of price movements, rather than for the purchase or sale of the actual cash commodity The buyer of the futures contract agrees on a fixed purchase price to buy the underlying commodity© 2006 - 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 4INTRODUCTION TO FUTURESfrom the seller upon the expiration of the contract; the seller of the futures contract agrees to sell the underlying commodity to the buyer at expiration at the fixed delivery price The contract's price changes over time in relation to the specified price at which the trade was initiated This creates profits or losses for each trading side© 2006 - 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 5INTRODUCTION TO FUTURES The word "contract" is used because a futures contract requires delivery of the commodity in a stated month in the future unless the contract is liquidated before it expires However, in most cases, delivery never takes place Instead, both the buyer and the seller, usually liquidate their positions before the contract expires; the buyer sells his futures and the sellerbuys futures and thus all the transactions are financial rather than physical© 2006 - 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 6COMMODITY PORTFOLIOS Traders usually hold portfolios of commodities: each portfolio comprises a collection of different commodities, each bought at a certain price/time, and comes with distinct terms and conditions Such holding is done in order to diversify the portfolio and thereby mitigate the overall risk The value of a portfolio, at any given point in time, is determined by the sum of the individual values of each commodity in the ‘basket’© 2006 - 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 7IMPORTANCE OF DIVERSIFYING THROUGH PORTFOLIO HOLDING A well–balanced and diversified portfolio provides benefits to the holder by lowering the overall risk The key reason is because market or other economic conditions that cause one futures contract to perform very well may often cause a different contract to perform rather poorly© 2006 - 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 8IMPORTANCE OF DIVERSIFYING THROUGH PORTFOLIO HOLDING Speculators and hedgers hold diversified positions so as not to have “all eggs in one basket” While diversification is not a guarantee against loss, it is an effective strategy to help manage the risk faced by the holder© 2006 - 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 9PORTFOLIO ANALYSIS We consider a portfolio of investments, say securities, and we quantify and analyze the risk We first define the notation and the key metric We discuss the steps for the determination of the VaR metric for a general case© 2006 - 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 10MARKET UNCERTAINTIES We consider the purchase of a portfolio at time t = 0 at price p0 The value of the portfolio varies over time and we denote its value at an arbitrary time t by pt This portfolio is exposed to the various sources of uncertainty to which the market for each compo-nentor commodity is subjected and consequently its value fluctuates as a result of the impacts of the various sources of uncertainty© 2006 - 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 11PERFORMANCE PREDICTION On any given trading day t = T, the fixed portfolio may either incur a loss or a gain or remain unchanged with respect to its value at t = T – 1 We wish to determine what the worst performance of the portfolio may be from the day t = T – 1 to the day t = T and how to systematically measure the performance over any two consecutive days© 2006 - 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 12PERFORMANCE PREDICTION On day t = T, we cannot lose more than the overall value pTof the portfolio and this statement holds true with a probability of 1 In other words, with a probability of 1, the loss must be less than or equal to pT© 2006 - 2021 George Gross, University of