Assoc.Prof.Dr. Sevtap Kestel

Winter 2014

VaR and Regulatory Capital

Regulators base the capital they require banks to keep on VaR

The market-risk capital is k times the 10-day 99% VaRwhere k is at least 3.0

Under Basel II capital for credit risk and operational risk is based on a one-year 99.9% VaR

VaR vs. Expected Shortfall VaR is the loss level that will not be exceeded with a

specified probability

Expected shortfall is the expected loss given that the loss is greater than the VaR level (also called C-VaR and Tail Loss)

Two portfolios with the same VaR can have very different expected shortfalls

Distributions with the Same VaR but Different Expected Shortfalls

VaR

VaR

Normal Distribution Assumption

The simplest assumption is that daily gains/losses are normally distributed and independent

It is then easy to calculate VaR from the standard deviation (1-day VaR=2.33s)

The N-day VaR equals times the one-day VaR

Regulators allow banks to calculate the 10 day VaRas times the one-day VaR

N

10

Impact of Autocorrelation: Ratio of N-day VaR to 1-day VaR (Hull,2006)

N=1 N=2 N=5 N=10 N=50 N=250

r=0 1.0 1.41 2.24 3.16 7.07 15.81

r=0.05 1.0 1.45 2.33 3.31 7.43 16.62

r=0.1 1.0 1.48 2.42 3.46 7.80 17.47

r=0.2 1.0 1.55 2.62 3.79 8.62 19.35

Choice of VaR Parameters Time horizon should depend on how quickly portfolio

can be unwound. Regulators in effect use 1-day for bank market risk and 1-year for credit/operational risk. Fund managers often use one month

Confidence level depends on objectives. Regulators use 99% for market risk and 99.9% for credit/operational risk.

A bank wanting to maintain a AA credit rating will often use 99.97% for internal calculations.

(VaR for high confidence level cannot be observed directly from data and must be inferred in some way.)

VaR Measures for a Portfolio where an amount xi is invested in the ith component of the portfolio

Marginal VaR:

Incremental VaR: Incremental effect of ithcomponent on VaR

Component VaR

ix

VaR

i

ix

x

VaR

Properties of Component VaR The component VaR is approximately the same as the

incremental VaR

The total VaR is the sum of the component VaR (Euler’s theorem)

The component VaR therefore provides a sensible way of allocating VaR to different activities

Coherent Risk Measures

A Risk measure is a functionthat assigns a real number toevery risk

Risk measure is calledCoherent if it is

Translation invariant

Subadditive

Positive Homogen

Monotone

Properties of coherent risk measure

If one portfolio always produces a worse outcome than another its risk measure should be greater

If we add an amount of cash K to a portfolio its risk measure should go down by K

Changing the size of a portfolio by l should result in the risk measure being multiplied by l

The risk measures for two portfolios after they have been merged should be no greater than the sum of their risk measures before they were merged

VaR satisfies the first three conditions but not the fourth oneVaR is not Coherent, as it is not particularly subadditiveCoherent risk measure Expected Shortfall (ES) or Conditional VaR

CVaR: Expected Loss Conditioned on the losses exceedingVaR

It is the probability of weighted average of all losses thatexceed VaR.

Expected shortfall satisfies all four conditions.

[ | ( )]ES CVaR E X X VaR X

Source: Uryasev

CVaR We have, let say, 1000 trials

An estimate of would be the largest of the 50 smallest realisations of the net worth X

A coherent ES would be the negative of the avarage ofthese 50 realisations

Thus, ES is a much better alternative for measuring risk in a practical setting

)(VaR 0.05 X

15Source: Uryasev

16

Source: Uryasev

17

Source: Uryasev

Backtesting

Backtesting a VaR calculation methodology involves looking at how often exceptions (loss>VaR) occur

Alternatives: a) compare VaR with actual change in portfolio value and b) compare VaR with change in portfolio value assuming no change in portfolio composition

Suppose that the theoretical probability of an exception is p. The probability of m or more exceptions in n days is

knkn

mk

ppknk

n

)1()!(!

!

Basel Committee Rules for Market Risk VaR If number of exceptions in previous 250 days is less than 5

the regulatory multiplier, k, is set at 3

If number of exceptions is 5, 6, 7, 8 and 9 supervisors may set k equal to 3.4, 3.5, 3.65, 3.75, and 3.85, respectively

If number of exceptions is 10 or more k is set equal to 4

Bunching Bunching occurs when exceptions are not evenly

spread throughout the backtesting period

Statistical tests for bunching have been developed

Stress Testing Considers how portfolio would perform under extreme

market moves

Scenarios can be taken from historical data (e.g. assume all market variable move by the same percentage as they did on some day in the past)

Alternatively they can be generated by senior management

Reading: Wiszniowski, E., 2010

The Model-Building Approach The main alternative to historical simulation is to

make assumptions about the probability distributions of the returns on the market variables and calculate the probability distribution of the change in the value of the portfolio analytically

This is known as the model building approach or the variance-covariance approach

Daily Volatilities

In option pricing we express volatility as volatility per year

In VaR calculations we express volatility as volatility per day

252

year

day

Daily Volatility continued

Strictly speaking we should define day as the standard deviation of the continuously compounded return in one day

In practice we assume that it is the standard deviation of the percentage change in one day

Microsoft Example

We have a position worth $10 million in Microsoft shares

The volatility of Microsoft is 2% per day (about 32% per year)

We use N=10 and X=99

Microsoft Example continued

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

The standard deviation of the change in 10 days is

200 000 10 456, $632,

Microsoft Example continued

We assume that the expected change in the value of the portfolio is zero (This is OK for short time periods)

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

Since N(–2.33)=0.01, the VaR is

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

AT&T Example (Hull 2006)

Consider a position of $5 million in AT&T

The daily volatility of AT&T is 1% (approx 16% per year)

The SD per 10 days is

The VaR is

50 000 10 144, $158,

158 114 2 33 405, . $368,

Portfolio

Now consider a portfolio consisting of both Microsoft and AT&T

Suppose that the correlation between the returns is 0.3

S.D. of Portfolio

A standard result in statistics states that

In this case sX = 200,000 and sY = 50,000 and r = 0.3. The standard deviation of the change in the portfolio value in one day is therefore 220,227

YXYXYX r 222

VaR for Portfolio

The 10-day 99% VaR for the portfolio is

The benefits of diversification are

(1,473,621+368,405)–1,622,657=$219,369

657,622,1$33.210220,227

The Linear Model

We assume

The daily change in the value of a portfolio is linearly related to the daily returns from market variables

The returns from the market variables are normally distributed

The General Linear Model continued

deviation standard sportfolio' theis and

variableofy volatilit theis where

21

222

1 1

2

1

P

i

n

i

ijjiji

ji

iiP

n

i

n

j

ijjijiP

n

i

ii

i

xP

r

r

Alternatives for Handling Interest Rates Duration approach: Linear relation between DP and Dy

but assumes parallel shifts)

Cash flow mapping: Variables are zero-coupon bond prices with about 10 different maturities

Principal components analysis: 2 or 3 independent shifts with their own volatilities

Handling Interest Rates: Cash Flow Mapping (Hull,2006)

We choose as market variables zero-coupon bond prices with standard maturities (1m, 3m, 6m, 1yr, 2yr, 5yr, 7yr, 10yr, 30yr)

Suppose that the 5yr rate is 6% and the 7yr rate is 7% and we will receive a cash flow of $10,000 in 6.5 years.

The volatilities per day of the 5yr and 7yr bonds are 0.50% and 0.58%, respectively

Example continued

We interpolate between the 5yr rate of 6% and the 7yr rate of 7% to get a 6.5yr rate of 6.75%

The PV of the $10,000 cash flow is

540,60675.1

000,105.6

Example continued

We interpolate between the 0.5% volatility for the 5yr bond price and the 0.58% volatility for the 7yr bond price to get 0.56% as the volatility for the 6.5yr bond

We allocate a of the PV to the 5yr bond and (1- a) of the PV to the 7yr bond

Example continued

Suppose that the correlation between movement in the 5yr and 7yr bond prices is 0.6

To match variances

This gives a=0.074

)1(58.05.06.02)1(58.05.056.0 22222

Example continuedThe value of 6,540 received in 6.5 years

in 5 years and by

in 7 years.

This cash flow mapping preserves value and variance

484$074.0540,6

056,6$926.0540,6

Principal Components Analysis Suppose we calculate

where f1 is the first factor and f2 is the second factor

If the SD of the factor scores are 17.49 and 6.05 the SD of DP is

21 40.408.0 ffP

66.2605.640.449.1708.0 2222

When Linear Model Can be Used

Portfolio of stocks

Portfolio of bonds

Forward contract on foreign currency

Interest-rate swap

The Linear Model and Options

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

andS

P

S

Sx

Linear Model and Options continued

As an approximation

Similarly when there are many underlying market variables

where i is the delta of the portfolio with respect to the ith asset

xSSP

i

iii xSP

Example Consider an investment in options on Microsoft and

AT&T. Suppose the stock prices are 120 and 30 respectively and the deltas of the portfolio with respect to the two stock prices are 1,000 and 20,000., respectively

As an approximation

where x1 and x2 are the percentage changes in the two stock prices

21 000,2030000,1120 xxP

Skewness

The linear model fails to capture skewness in the probability distribution of the portfolio value.

Quadratic Model

For a portfolio dependent on a single stock price it is

approximately true that

so that

Moments are

6364243

4242222

22

875.15.4)(

75.0)(

5.0)(

SSPE

SSPE

SPE

2)(2

1SSP

22 )(2

1xSxSP

Quadratic Model continuedWith many market variables and each instrument dependent on only one

where i and i are the delta and gamma of the portfolio with respect to the ith variable

Formulas for calculating moments exist but are fairly complicated

Referance: Hull, 2006. Risk Management and Financial Institutions.

n

i

n

i

iiiiii xSxSP1 1

22)(

2

1

Main Categories of Risk Market

Loss due to adverse changes in the market

Business/Financial Risk affecting the business in terms of financial

development and growth

Operational Management failure, system software failure, human

error etc.

Compliance Requirements to do in order to comply with the law and

regulations

Managing Linear Risk Hedging

An investment made in order to reduce the risk of adverse price movements in another investment, by taking an offsetting position in a related security, such as an option, a short sale or an “inverse fund”. (http://www.confidentstrategies.com/glossary.htm)

Futures market, Forward and Future Contracts Futures market participants buy and sell standardized future contracts

according to the interaction of the competing expectations of both parts.

Forward contract, the seller of a commodity, financial instrument or equity agrees to deliver to the buyer a specified amount of the products at an agreed price at a specified date in the future.

Forward contracts are privately negotiated between two parties and unless the other party agrees, not obeying a forward contract is very difficult.

Option, in which the parties have a choice of not to exercise the option. An option is a contract that gives the investor the right, but not an obligation, to buy or sell a commodity/product at a specified price within a specified time period.

Traders and brokers trade the contracts. who shout the bids and offers on organized exchanges in a variety of commodities are named as futures contracts. Futures contracts are more useful for risk management and hedging, because they allow parties discover prices before the commodities are traded and also the specifications are clearly defined in a futures contract.

Two types of traders: hedgers and speculators.

Hedgers try to reduce the price risk or establish prices for commodities by using adverse price changes.

Speculators try to make profit by anticipating the price changes of a commodity by buying/selling futures or options contracts attempt to profit through buying and selling, based on price changes, and have no economic interest in the underlying commodity.

Hedging is a technique to mitigate the risk of an adverse price movement in a commodity.

Most of the companies which are using raw materialsin their production are using hedging to reduce thelosses as a result of the price fluctuations in themarket.

Hedge Deal: In hedge deal, by using the forecast results and historical analysis of the price of the commodity a price is defined and as soon as this pre-defined price achieved, the commodity is immediately bought from this price and hedging completed.

Stop Loss Order: Again the price of the commodity is limited; however this time limit is set over the current price level and buying the commodity cancelled whenever the price exceeds this limit. This limit is also called as security level. This technique is safer than Limit Order

Limit Order: First the price of the commodity is limited with a pre-defined price level. This limit is set under the current price level and whenever, the price of commodity reduces under this limit, the commodity is bought. However, it can not be known whether the price will fell under this limit or not. As a result of this uncertainty, this technique is not a safe way of hedging.

0

4.500

4.750

5.000

5.250

5.500

8.000

09.08 10.08 11.08 12.08 01.09 02.09 03.09 04.09 05.09 06.09 07.09 08.09 09.09

Limit Order

Order Cancels Order (OCO): The OCO is a combination of Limit-Order and Stop-Loss Order. According to this technique, both an upper and lower level is set to the price of the commodity and hedging is done whenever the price is between this range. OCO is the most common and safest technique of commodity hedging

Factor Models and Hedging

Hedging with Regression

Hedging with Regression

Hedging with Regression

The „Offices“

The „Greeks“

The „Greeks“

The „Greeks“

The „Greeks“

The „Greeks“

The „Greeks“

The „Greeks“

The „Greeks“

The „Greeks“