14123_varnew

8/12/2019 14123_VARNew

1/24

Options, Futures, and Other Derivatives

7thEdition, Copyright John C. Hull2008 1

The Question Being Asked inVaR

What loss level is such that we areX%confident it will not be exceeded inN

business days?

8/12/2019 14123_VARNew

2/24

Value at Risk

The Value at Risk measures the potentialloss in value of a risky asset or portfolioover a defined period for a given confidenceinterval. Thus, if the VaR on an asset is $100 million at a one-week, 95% confidencelevel, there is a only a 5% chance that thevalue of the asset will drop more than $ 100

million over any given week

Options, Futures, and Other

Derivatives, 7th Edition, Copyright John C. Hull 2008 2

8/12/2019 14123_VARNew

3/24



VaR and Regulatory Capital(Business Snapshot 20.1, page 452)

Regulators base the capital they requirebanks to keep on VaR

The market-risk capital is ktimes the 10-day 99% VaR where kis at least 3.0

8/12/2019 14123_VARNew

4/24



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 canthings get?

8/12/2019 14123_VARNew

5/24



Time Horizon

Instead of calculating the 10-day, 99% VaRdirectly analysts usually calculate a 1-day99% VaR and assume

This is exactly true when portfolio changeson successive days come from independent

identically distributed normal distributions

day VaR1-day VaR-10 10

8/12/2019 14123_VARNew

6/24

VAR of portfolio

An investor has invested 70000 Rs inasset A and 30000 Rs in asset B. Daily

VaR on asset A is 20000 Rs. and on

asset B is 8000 Rs. Calculate the daily

VaR for overall investment portfolio.



8/12/2019 14123_VARNew

7/24

Variance-Covariance Method

Steps:

Calculate the value of Mean and Std Devfor the return data series.

Portfolio Mean Return,:



8/12/2019 14123_VARNew

8/24

Portfolio Std. Dev.:



8/12/2019 14123_VARNew

9/24

Percentile value at risk,

VAR=|P+ z| V

Where, P= expected return , V = portfolio value , =volatility



8/12/2019 14123_VARNew

10/24

An investor has invested 400,000 Rs inan portfolio based on Index XYZ.

Calculate the 99 percentile, 10 days VaR

for his portfolio by using Variance-

Covariance method. Z value is 2.33 and

Historical prices for index XYZ are given

given as following



8/12/2019 14123_VARNew

11/24

Date 4-1-12 5-1-12 6-1-12 7-1-12 8-1-12 9-1-12 10-1-12 11-1-12 12-1-12Price 4123 4823 4712 5612 4935 5832 5912 5712 5467



8/12/2019 14123_VARNew

12/24

An investor has invested 20000 Rs inasset A and 80000 Rs in asset B. Daily

VaR on asset A is 50000 Rs. and on

asset B is 2000 Rs. Calculate the daily

VaR for overall investment portfolio.

Portfolio VaR = W1 * VaR(A) + W1 * VaR(B)



8/12/2019 14123_VARNew

13/24



Historical Simulation(See Tables 20.1 and 20.2, page 454-55))

Historical simulations represent the simplestway of estimating the Value at Risk formany portfolios. In this approach, the VaRfor a portfolio is estimated by creating a

hypothetical time series of returns on thatportfolio, obtained by running the portfoliothrough actual historical data andcomputing the changes that would have

occurred in each period.

8/12/2019 14123_VARNew

14/24

To run a historical simulation, we begin withtime series data on each market risk factor,

just as we would for the variance-covariance approach. However, we do notuse the data to estimate variances and co-variances looking forward, since thechanges in the portfolio over time yield all

the information you need to compute theValue at Risk



Hi t i l i l ti V R

8/12/2019 14123_VARNew

15/24

Historical simulation VaR

Method-

Historical simulation method includesfollowing Steps-

Define current portfolio value=1000000

Confidence level- 95% for internal purposeAnd 99% for credit rating and reportingpurpose

Forecast horizon is 10 days so we will

calculate 10 day- VaR

10 Days-VaR= SQRT (10)*Daily VaR



8/12/2019 14123_VARNew

16/24

Historical Observation Period-I am takingS&P CNX Nifty data from 1 Jan 2007 to 30Nov 2009

Calculating daily return data Multiply this historical return data with

current portfolio value to create hypotheticalportfolio

Arrange this in descending order andplotting histogram of this data

Calculate 95thand 99thpercentile values

using hypothetical return data.Options, Futures, and Other


8/12/2019 14123_VARNew

17/24



Historical Simulation continued

Suppose we use mdays of historical data Let v

ibe the value of a variable on day i

There are m1simulation trials

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

1i

i

m

v

v

v

8/12/2019 14123_VARNew

18/24



The Model-Building Approach

The main alternative to historical simulationis to make assumptions about theprobability distributions of return on themarket variables and calculate theprobability distribution of the change in thevalue of the portfolio analytically

This is known as the model building

approach or the variance-covarianceapproach

8/12/2019 14123_VARNew

19/24



Daily Volatilities

In option pricing we measure volatility peryear

In VaR calculations we measure volatility

per day

252

year

day

8/12/2019 14123_VARNew

20/24



Daily Volatility continued

Strictly speaking we should define dayasthe standard deviation of the continuouslycompounded return in one day

In practice we assume that it is the standarddeviation of the percentage change in oneday

8/12/2019 14123_VARNew

21/24



Microsoft Example (page 456)

We have a position worth $10 million inMicrosoft shares

The volatility of Microsoft is 2% per day

(about 32% per year) We useN=10 andX=99

8/12/2019 14123_VARNew

22/24



Microsoft Example continued

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,

8/12/2019 14123_VARNew

23/24



Microsoft Example continued

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

SinceN(2.33)=0.01, the VaR is

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

8/12/2019 14123_VARNew

24/24



AT&T Example (page 457)

Consider a position of $5 million in AT&T The daily volatility of AT&T is 1% (approx

16% per year)

The S.D per 10 days is

The VaR is50 000 10 144, $158,

158 114 2 33 405, . $368,

14123_varnew

Documents