Tutorial Analytic Var

8/10/2019 Tutorial Analytic Var

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Introduction to variance-covariance VaR with

VaRworks

Monte Carlo Simulation, Histor ical Simulation, Variance-

Covariance VaR,VaRdeltaand Extreme Value Theory


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Information in this document is subject to change without notice. Companies, names, and data used inexamples herein are fictitious unless otherwise noted. No part of this document may be reproduced ortransmitted in any form or by any means, electronic or mechanical, for any purpose, without the expresswritten permission of Financial Engineering Associates, Inc.

19951999 Financial Engineering Associates, Inc. All rights reserved.

@ENERGY, @EQUITY, @GLOBAL, @INTEREST, DerivaTool, SPAV, SWING, VaRdelta, VaRworks,and Financial Engineering Associates design (logo) are registered trademarks and DTlib, EquiLib, ErgLib,FEA Datepak, FEA Library Modules, GlobLib, IntrLib, MakeVC, SpavLib, SwingLib, VaRlib, ZCURVE,and Zlib are trademarks of Financial Engineering Associates, Inc. in the United States of America.Other brand or product names are trademarks or registered trademarks of their respective holders.

NOTICE: This manual does not constitute financial advice; users should consult their own financialadvisors regarding any such advice.

Carlos Blanco wrote this document.Printed in the United States of America.

Financial Engineering Associates, Inc.

2484 Shattuck Ave., Suite 225Berkeley, CA 94704-2029USATel: +1-510-548-6200Fax: +1-510-548-0332Email: [email protected]://www.fea.com

Printed: 14-Mar-2000
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TABLE OF CONTENTS

What is a cash flow map? ....................................................................................................................... 6What is a vertex?..................................................................................................................................... 6Why Vertices?......................................................................................................................................... 7

Allocating Cash Flows to a Single Vertex .............................................................................................. 7Allocating Cash Flows to more than one Vertex .................................................................................... 7

Inputs .............................................................................................................................................................. 9Volatility data ......................................................................................................................................... 9

Example of a Volatility File........................................................................................................................ 9Correlation data..................................................................................................................................... 10

Example of a Correlation File................................................................................................................... 10Asset Codes File ....................................................................................................................................... 11

Portfolio Data........................................................................................................................................ 12Horizon (days) ...................................................................................................................................... 13Confidence ............................................................................................................................................ 14


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Value at Risk (VaR)

VaR tries to answer the simple question: How much money can we lose due to normal market movements?

However, the above question is far too imprecise. (Usually, any amount of loss is possible.) We can makethis question more precise rephrasing it in the following way:

How much money (or more) might we lose over time period T with probability X?

Value at Risk is a number that represents an estimate of how much our portfolio may lose due to marketmovements for a particular horizon (time period) and for a given confidence level (probability).

The definition of that potential loss depends on three main parameters:

a. The horizonover which the potential loss is measured. It is not the same to measure the expected lossover a one day period than over a one week period.

b. The degree of confidence (probability of ocurrence). The degree of confidence is a measure of thedegree of certainty of the VaR estimate. The most common degree of confidence is 95%, which meansthat 95% of the times our losses will be lower than the VaR number, while 5% of the times, our

portfolio will experience greater losses.c. Thehome currencyin which we measure the VaR number. VaR is a number, but it is expressed inU.S. dollars, British pounds or Japaneses yens. It is not the same to say that our VaR is $1 million, than

1 million or 1 million.

In the following graph, we can see the value at risk concept graphically. Extreme outcomes (large losses, orlarge gains) are less likely to ocurr, and therefore, the number of expected extreme portfolios outcomes, isclearly lower.

Value at Risk

0

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Portfolio's expected profits and losses in $MM.

NumberofOcurrences

Value at Risk at the 95%

level equals 10 $MM.Prob(x < $-10MM) = 5%

5% of

expected

ocurrences

From the distribution of the

portfolio's expected profits

and losses, w e can

determines the probability that

our losses w ill be greater than

a certain number.

For example, if your VaR is US$10MM and you have set the time period to one day and the probability to95% then over the next 24-hour period there is a 5% chance your loss will exceed US$10MM.


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How do we measure the market risk of a portfolio?

A porttfolio may have different instruments (stocks, bonds, swaps, floors, option...) in different markets(fixed income, commodities, equities...). To analyze the overall risk of the portfolio, it is necessary todetermine the risk of the individual assets, and the interactions (correlations) between those assets. Tocompute the volatility for every trade in our portfolio, and then estimate the correlation with the remainingtrades, could be an impossible task.

The VaR methodology provides a way to deal with the problem of measuring the global risk of a portfolio.It is important to point out, that in the VaR analysis, we assume the portfolio remains constant, and only themarket factors will change.

VaR is highly integrative in nature and provides a global picture of the risk of a portfolio. It attempts tobring together into a single framework:

Multiple types of market risk Multiple types of instruments

In a nutshell, VaR is an attempt to quantify risk, given the portfolios current exposure to a certain set ofmarket factors, by asking how much these factors can move (accounting for volatilities and correlations)

over a period of time.

As a first step, we must identify a set of common risk factors, for which we can obtain volatilities andcorrelations, which will represent the Market Factors. To continue the analysis, we have to find a way toexpress all the trades of our portfolio in terms of those risk factors (cash flow mapping). Once we havethe cash flow map in terms of the risk factors, and the volatilities and correlations of those risk factors, weonly need to perform basic matrix manipulation to calculate the VaR of our portfolio.

Analytic VaR simplifies the analysis of market risk in the context of a portfolio.

Vertex Set (Risk Factors)We have to choose a limited number ofrisk factors for which we can obtain:

Individual volatilities

Cross correlations between them Prices (or interest rates)

Portfolio Data

How do we analyze the riskbetween different instruments and

markets inside the portfolio?

Market Factors

How do we define and measure allthe risks of all the instruments in

our portfolio?

Cash Flow MapWe can define all the trades in ourportfolio in terms of a predefined set ofvertices (risk factors). That will allow us

to compare the risks of the heterogenousinstruments inside the portfolio

Value at RiskMeasure of the overall risk of

our portfolio (global picture).

(ANSWER) (ANSWER)


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Cash Flow Mapping

The purpose of cash flow mapping is to find the best replicate of a financial instrument for the purposeof measuring the instruments risk within a portfolio.

What is a cash flow map?

A cash flow mapis the representation of a financial instrument as a stream of one or more zero-couponinstruments marked to market at current market rates and prices. Prior to calculating analytic VaR, financialinstruments must be decomposed into their component cash flows (zero coupon instruments) and allocatedto a pre-determined set of verticesfor which we can obtain volatilities, correlations, and other statistics on aregular basis.

Cash Flow Map of a Sample Portfolio in VaRworks

What is a vertex?

We can think of a vertex as an element which populates a market. For example, USD.GOV.10y would be avertex for the market for 10-year U.S. government bonds.More formally, a vertex is a pre-determined asset class-maturity bucket for which volatilities and other timeseries statistics have been measured. It consists of an asset(a currency or commodity code), an asset class(such as spot currency, interest-rate instrument, equity index, commodity, brady bond), and a tenor ormaturity(a payment date measured in years from today). In VaRworks vertices are represented using thenotation asset.asset_class_and_maturity. A few examples of vertices are the following:The vertex for the 30-year U.S. Government Bond is USD.Z30.

Asset class and Maturity

XS (FX / spot) R (Money Market / days) S (Swap / years) Z (Government / years) SE (Equity index / spot)

C (Commodity / months)

Assets Currencies (USD,CAD,..)

Commodities (GAS, ...)

Total Cash Flows allocatedto the U.S. equity index

(spot) vertex

Total Cash Flowsallocated to the 3-month WTI vertex


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The vertex for the Japanese equities is JPY.SE (notice it is a spot position)The vertex for the three month Natural Gas is GAS.C03

Why Vertices?

Vertices reduce the market description to a manageable size. They may be the only answer tounmanageably huge information sets characterizing markets. To use them properly, two problems need tobe solved:

The systematic measurement of actual markets for the production of data applicable to the vertex setchosen (for example, using Riskmetrics datasets, or creating your own with MakeVC); and

The reduction of portfolio exposures to a form which can be analyzed using vertex datasets (Cash-flow mapping problem)

In order to be compatible with the available data (i.e. vertices), every instrument in a portfolio needs to bereduced to a collection of cahs-flows.

Definition of cash flow in the VaR methodologyA cashflow is a vertex and a number representing the amount (size) of the cashflow. For example, the cash-flow (+100, 10y, USD, GOV) represents a long position of 100 units of 10 year, U.S. dollars, government-credit.

We can think of a cash flow map as a many-to-one association between a portfolio (or trade) and a set ofcash flows lying at pre-selected vertices

Cash flow mapping takes a trade description and produces a series of home currency present values of cashflows marked by 1) a currency, 2) an asset class (which may be indicative of credit rating), 3) a maturity inyears, and 4) a signed amount.

This process has four steps:

1. DECOMPOSITION: Take a description of the trade and shred it into a series of times andsigned cash flows in the local currencies.

2. MARKING-TO-MARKET: Present value the cash flows to the effective date (usually today),

using local currency interest rates and market information.3. HOME BASING: Convert the cash flows (valued at the effective date) to the home currency(currency VaR is calculated for). Spot exchange rates are used since present values of cashflows are converted.

4. ALLOCATION: Allocate the cash flows to the standard vertices defined in the volatilitydataset.

We have to repeat the process for all the trades in our portfolio.

Following certain rules, we will be able to aggregate the cash flows at each vertex and perform the VaRcalculations. All the cash flows in a cash flow map are expressed in the same currency (homecurrency), and are valued at the same date (effective date)

Al locat ing Cash Flows to a Sing le Vertex

If only a single appropriate vertex exists for a cash flow (as in the case of spot foreign exchange, equityindexes, and spot commodities) then the entire cash flow is allocated to that vertex preserving present valueonly.

Al locat ing Cash Flows to more than one Ver tex

If a cash flow falls between two vertices (for example, 2.56 years), we will have to allocate it to the twocloser vertices. For a detailed explanation see the Users Guide.


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Allocating a Cash Flow to Vertices

C

C1 C2

T

T1 T2

Time

Time

Amount

Amount

Original cash

flow

Allocated

cash flows

There are two possible choices of units for cashflow amounts: local currency units and home currency. Bylocal currency, we mean the currency the instrument is originally denominated in. At the portfolio level, wemay have assets denominated in different local currencies, but after the cash-flow mapping process, all thecash-flows will be denominated in the same currency unit, that is, the home currency chosen.


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Inputs

There are three types of inputs to calculate VaR: Market data, Portfolio Data, and Users choices

A. MARKET DATA: CORRELATIONS, VOLATILITIES, PRICES, AND INTERESTRATES

Volatility data

Volatility files define vertices onto which cash flows are mapped. Each record specifies a vertex seriesname, price volatility, exchange rate or yield (where applicable), and other statistics. You can create yourown volatility files with MakeVC or download publicly available ones. The sample volatility file,

dvmmddyy.rm3 (where mmddyyis some date), is a RiskMetrics file containing daily volatilities.

Example of a Volatility File

Example of Volatility File Contents*Estimate of volatilities for a one-day horizon

*COLUMNS=5,LINES=367,DATE=01/30/98,VERSION 2.0

*MakeVC(TM) Version 1.0

*Copyright information.

*SERIES,PRICE/YIELD,DECAYFCTR,PRICEVOL,YIELDVOL

FRF.XS.VOLD,1.000500,0.940,0.035927,NA

JPY.XS.VOLD,0.083867,0.940,0.786864,NA

......................................

WTI.C03.VOLD,0.774500,0.940,0.567395,NA

FIELD Description ExampleSERIES and VOLATILITYESTIMATE

Identification Code of the series and horizon of thevolatility estimate. VOLD for daily, VOLM formonthly, and VOL for other horizons.

WTI.CO3.VOLD

PRICE/YIELD This price or interest rate obtained on the final day of observation period for the time series. The value ofprice/yield depends on the asset class.

0.774500

DECAY FACTOR The exponential moving average decay factor. It isthe same for all series.

0.94 (daily)0.97 (monthly)

PRICE VOLATILITY The forecast price volatility over the horizon -h(not

annualized), expressed as a percent, and multipliedby the confidence factor -m.

0.567395

YIELD VOLATILITY The forecast yield volatility for interest rates (asset

classes R, S, and Z) over the horizon -h(not

annualized), expressed as a percent, and multiplied

by the confidence factor -m.

Only for interestrate series. For theothers NA (no yieldvolatility)

Volatility File

Headers

Volatility Data (separated bycommas)Each record identifies a series (vertex)

and several time series statistics,

including volatility and price.

Market Data

VolatilitiesCorrelationsPrices (interest rates)

Portfolio Data

Description of all thetrades in the portfolio ina articular format

Users Choices

HorizonConfidence LevelHome Currency


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Correlation data

Correlation files contain correlations for the vertices defined in a corresponding volatility file. Each recordspecifies two vertices and their correlation. You may create your own correlation files with MakeVC ordownload publicly available ones. The sample correlation file, dcmmddyy.rm3 (where mmddyyis somedate), is a RiskMetrics file containing daily correlations.

Example of a Correlation File

Correlation File contents*Estimate of correlations for a one-

day horizon

*COLUMNS=2,LINES=67161,DATE=01 /30/98

,VERSION 2.0

*MakeVC(TM) Version 1.0

*Copyright information.

*SERIES,CORRELATION

FRF.XS.FRF.XS.CORD,1.000000

FRF.XS.DEM.XS.CORD,0.144465

.............................

WTI.C01.ZNC.CO3.CORD,0.013546

WTI.C01. ZNC.CO3. CORD , 0.013546

If n is the number of records in the volatility file, then the correlation file contains n (n + 1) / 2 records

(the correlation data includes the diagonal elements, all ones). Each series in the correlation file must havea corresponding record in the volatility file.

Correlation File Headers

Correlation Data

Each record specifies two series

(vertices) and their correlation.

Identification Code for Series1asset1.asset_class1[maturity1].

Identification Code for Series2asset2.asset_class2[maturity2].

CORD for daily correlationCORM for monthly (25-day)COR for other horizons.

Theforecast correlationcoefficientbetween thetwo assets. The value isa real number between

1 and +1, inclusive.

Field Delimiter (comma)


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ASSET CODES FILE

Example of asset codes file*Asset Codes File

*COLUMNS=3,LINES=11,DATE=01/30/98,

VERSION 2.0

*ASSET,ASSET_CLASS,CCYAUD,XS,USD

CAD,XS,USD

DEM,XS,USD

GBP,XS,USD

JPY,XS,USD

CHF,XS,DEM

ELE,C,USD

GAS,C,CAD

GLD,C,HKD

WTI,C,USD

BRENT,C,GBP

The assetfield specifies a currency or commodity code. Examples: USD,DEM, JPY, WTI, GAS.

The asset_classfield indicates if asset is a currency (XS) or a commodity (C).Example: XS or CThe ccyfield indicates the currency in which the original price series aredenominated in .

File Type

Characteristics of the assetcodes file

Column titles


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Portfolio Data

One of the obvious inputs in the VaR calculation is the specification of your portfolio.Portfolio consists of text specifying a portfolio filemay include directory and drive. Theportfolio filedescribes the financial instruments for which VaR is to be calculated. Each record in a portfolio filespecifies either:

the terms of a single trade, or a path pointing to another file containing trades.

VaRworkss sample portfolio file, port.txt, contains the names of files containing different types of trades.

Example of a portfolio file

Example of an instrument file

Portfolio File

Trade Files

Portfolio

File

Headers of the trade file. Each trade file iscomposed of instruments of the same type(swaps, futures, options). The header thatdefines the instrument type is: TYPE=INSTRUMENT_ NAME(for example, TYPE=SWAP. The other headersprovide additional information to value the

instruments defined in that particular file.

Commodities

OptionsInterest Rates

Example of a portfolio file(port.txt) composedby trade files in which the different instrumentsof the portfolio are described in detail. (bond.txt,

bondfut.txt, swaps.txt,...)

Equities

Bonds

Headers of the portfolio file. The most important oneis TYPE=PORTFOLIO, which identifies the file as aportfolio file


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USERS CHOICES

Horizon (days)

In VaRworks, horizon is a number specifying the number of business days of the VaR horizon. Expressthe horizon in number of days (e.g. 1 month is expressed as 30 days) even if you are using monthly (orlonger) volatility and correlation datasetsVaRs are scaled using the square root of the horizon. Theresulting VaR is the expected change in the portfolios value under adverse circumstances in this timeinterval. For example, enter 1 (one) for a one-day horizon.

Choice of horizonThe horizon can be a function either of the position or the investor. In the former case, the longer horizonfor estimating risk can be the result of the time it takes for the position to be liquidated or neutralized. Inless liquid markets, it can take up to a week or longer to significantly modify the market risk profile of aportfolio. In the latter case, it is the investor who defines the horizon. Risk is measured over the period until

investment objectives are reviewed and reassessed.

When choosing a horizon, consider:

Unwind periodhow long, on average, does it take to reverse a market position or individualtrade?

Attention periodhow often, on average, do you re-examine your portfolio and its mark-to-market or hedging trades?

Accounting periodhow long until the next financial reporting must be done?

Common choices of horizon One dayrecommended by the J. P. Morgan RiskMetrics specification (termed DEaR, for

daily earnings at risk). It is thought to be appropriate for banks, clearing houses, traders, etc.,

because of their rapid turnover, short unwind periods. Ten daysrecommended by the Basle Committee on Banking Supervision. It is thought to be

appropriate to banks for capital charges.

25 daysa RiskMetrics alternative. It is thought to be appropriate for funds and triple-A bankderivatives subsidiaries.

65 daysa calendar quarter. It is thought to be appropriate for corporations.

Example of a trade file for swaps (swap.txt). Thisrecord describes a $100 million interest rate swapthat pays floating and receives fixed with

maturity in 1/1/1999.


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Horizon choice and random walksSince Samuelsons pioneering work in 1965, it is widely believed (and infrequently disputed by evidence)that Brownian motion (simple random walk) governs the rate-of-return behavior of most assets. Whenthis is true, VaR for long periods (VaRT) is related to the one-day VaR (VaR1) by:

VaR VaR T T = 1

This is known as the square-root-of-T rule. If this rule holds then the choice of horizon is completelyunimportant since all horizons would scale accordingly. But autocorrelation, mean reversion, nonlinearpositions (options), and other issues can damage the applicability of the square-root-of-T rule.

RecommendationsA long period is merely the sequence of several short periods of risk. You cannot expect to manage riskover long periods unless you are able to manage it over short periods; there is no catching up later in riskmanagement. We recommend that you make the VaR horizon as short as you can, but no shorter than theactual trading decisions or VaR recalculations can be done.

Confidence

In VaRworks, confidence is a number between 0.5 and 1, exclusive, specifying the size of the one-tailedVaR confidence level.

The probability of incurring losses larger than our VaR will be (1-X) %, being X the confidence level. Forexample, for a 95% confidence level (5% one-tail level), the probability that the portfolios losses willexceed the the VaR number will be 5%.

Confidence level selectionCommon choices for a confidence level are 95% (RiskMetrics) and 99% (Basle).

With a 95% level and a one-day horizon, losses in excess of the VaR will occur about once in

every twenty days. With a 99% level and a one-day horizon, losses in excess of the VaR will occur about once in

every one hundred days.

If the probability of loss is normally distributed, the 5% one-tail level is 1.645 standard deviations from themean; the 1% one-tail level is about 2.33 standard deviations.

RecommendationsConfidence levels should be established for solid statistical reasons, not wishful thinking. No choice ismore conservative than the other: it is nonsense to say, We want to experience a serious lossinfrequently, so we use a confidence level which involves a small probability of loss in computing our

VaR.

Therefore, we recommend you choose a confidence level sufficiently large to get one outlier a month.


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For the same portfolio, the Value at Risk will be a different number for different confidence levels, but theunderlying risk that is attempting to measure is the same. The choice of the confidence interval does notmake your portfolio more or less risky.

VaR provides an estimate of maximum likely loss with a given confidence level. The larger the confidencelevel, the larger the VaR number will be.

VaR for dif ferent conf idence levels

0.00%1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

8.00%

9.00%

10.00%

-3-2

.9-2

.7-2

.6-2

.4-2

.3-2

.1 -2-1

.8-1

.7-1

.5-1

.4

Number of standard deviations from mean

P

robabilitythatlosswill

exceedVaR

number

VaR @ 95%

1.645 S.D.

VaR @ 90%

1.28 S.D

VaR @ 97.5%

1.96 S.D.

VaR @ 99%

2.33 S.D.

When choosing a confidence level, we have to take into account that, from a statistical point of view, theless-extreme order statistics have higher significance. Therefore, choose 95% or 92.5%, not 97.5% or 99%.

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