value at risk final ppt

7/28/2019 Value at Risk Final Ppt

http://slidepdf.com/reader/full/value-at-risk-final-ppt 1/27

Anu Bumra (14054)

Pratibha Virdi

Ramandeep



Value at Risk is an estimate of the worstpossible loss an investment could realizeover a given time horizon, under normal

market conditions (defined by a givenlevel of confidence).

To estimate Value at Risk a confidencelevel must be specified.



• “We are X percent certain that we will notlose more than V dollars in time T .”

• Function of confidence level X and time T



Normal market conditions – the returns that accountfor 95% of the distribution of possible outcomes.

Abnormal market conditions – the returns that accountfor the other 5% of the possible outcomes.



If a 95% confidence level is used toestimate Value at Risk for a monthlyhorizon;

losses greater than the Value at Riskestimate are expected to occur one intwenty months (5%).



Step 1: Transform simple monthly stock returns intocontinuously 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 regulatorycapital 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 marketconditions with a 95% level of confidence is: $18,000.

Note: It is possible to realize a loss greater than $18,000.



It captures an important aspect of risk in a single number.

It is easy to understand.

Firm wide summary and handles all the futures , optionsand other complications

Relatively model free. Deviations from normal distributions.

VaR translates portfolio volatility into a dollar value.

VaR is useful for monitoring and controlling risk within the

portfolio. As a tool, VaR is very useful for comparing a portfolio with

the market portfolio (S&P500).

It asks the simple question : “How bad can things get?”



VaR does not measure "event" (e.g.,market crash) risk. That is why portfoliostress tests are recommended tosupplement VaR.

VaR does not readily capture liquiditydifferences among instruments.

VaR doesn't readily capture model risks,

which is why model reserves are alsonecessary.



Instead of calculating the 10-day, 99% VaRdirectly ,analysts usually calculate a 1-day 99% VaR and assume

This is exactly true when portfolio

changes on successive days come fromindependent identically distributednormal distributions.

day VaR1-day VaR-10 10



Historical simulation

Model building approach /Variance- Covariance approach

Linear approachQuadratic model

Monte Carlo simulation



Create a database of the daily movementsin all market variables.

The first simulation trial assumes that

the percentage changes in all marketvariables are as on the first day

The second simulation trial assumes thatthe percentage changes in all market

variables are as on the second day and soon



Suppose we use m days of historical data

Let vi be the value of a variable on day i

There are m-1 simulation trials

The ith trial assumes that the value of themarket variable tomorrow (i.e., on daym+1) is

1i

i

m

v

vv



The main alternative to historicalsimulation is to make assumptions aboutthe probability distributions of return on

the market variables and calculate theprobability distribution of the change inthe value of the portfolio analytically

This is known as the model building

approach or the variance-covarianceapproach



We have a position worth $10 million inMicrosoft shares

The volatility of Microsoft is 2% per day

(about 32% per year)We use N =10 and X =99



The standard deviation of the change in theportfolio in 1 day is $200,000

The standard deviation of the change in 10

days is

200 000 10 456, $632,



We assume that the expected change in thevalue of the portfolio is zero (This is OK forshort time periods)

We assume that the change in the value ofthe portfolio is normally distributed

Since N (–2.33)=0.01, the VaR is

2 33 632 456 473 621. , $1, ,



We assume

The daily change in the value of a portfolio islinearly related to the daily returns from

market variables The returns from the market variables are

normally distributed



Portfolio of stocks

Portfolio of bonds

Forward contract on foreign currency

Interest-rate swap



Consider a portfolio of options dependent ona single stock price, S. Define

and S P

S

S x



As an approximation

Similarly when there are many underlyingmarket variables

where i is the delta of the portfolio withrespect to the ith asset

xS S P

i

iiixS P



Consider an investment in options onMicrosoft and AT&T. Suppose the stockprices are 120 and 30 respectively and thedeltas of the portfolio with respect to thetwo stock prices are 1,000 and 20,000respectively

As an approximation

where x1 and x2 are the percentagechanges in the two stock prices

21 000,2030000,1120 x x P



For a portfolio dependent on a single stockprice it is approximately true that

this becomes2)(

21 S S P

22 )(21 xS xS P



With many market variables we get anexpression of the form

where

This is not as easy to work with as thelinear model

n

i

n

i jiij jiiii x xS S xS P 1 1 2

1

ji

ij

i

iS S

P

S

P

2



To calculate VaR using M.C. simulation we

Value portfolio today

Sample once from the multivariate

distributions of the xi Use the x

ito determine market variables at

end of one day

Revalue the portfolio at the end of day



Calculate P

Repeat many times to build up a probabilitydistribution for P

VaR is the appropriate fractile of thedistribution times square root of N

For example, with 1,000 trial the 1percentile is the 10th worst case.



Model building approach assumes normaldistributions for market variables. It tends togive poor results for low delta portfolios

Historical simulation lets historical datadetermine distributions, but iscomputationally slower

value at risk final ppt

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