Assoc.Prof.Dr. Sevtap Kestel Winter 2014 · Backtesting Backtesting a VaR ... Stress Testing ... Y...
Transcript of Assoc.Prof.Dr. Sevtap Kestel Winter 2014 · Backtesting Backtesting a VaR ... Stress Testing ... Y...
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“