March 22

1
March 22

• Current events
• Market changes
• Interest rate risk summary problems
• Convexity
• Market Risk concepts Chapter 10
• Future
• VaR Measurement
• Credit risk
• Asia Crisis
• Some Slides adapted from Risk Metrics Group
  and Goldman Sachs presentations to U of Illinois,
  3/2000. Alan Laubsch, Partner Risk Management
  Group (www.riskmetrics.com) and Sid Browne, Head
  Risk Management, Goldman Sachs

2
Convexity Measurement

• duration is slope (first derivative)
• convexity is change in slope (second derivative)

3
Convexity Measurement Book

• Scaling factor 108

4
Convexity Measurement method 2

• Calculationbecause you already have

5
Convexity vs. Price Changes

• Remembercorrection for convexitywhere CX
  is convexity

6
Simple Example

• 2 year bond, 1,000 face value
• annual coupon 10
• YTM 14
• Duration?, Convexity?
• What is expected change if interest rates
  increase 50 basis points?

7
You try

• 3 year bond
• 5 coupon
• YTM 6
• Duration, convexity
• Effect of rate change from 6 to 8
• Components
• Duration
• Convexity
• True

8
Repricing

• Duration
• Equity (net worth) measure
• When it a cash flow measure?

9
Summary Interest Rate Risk Measurment

• Repricing Gap
• Duration and convexity for an instrument
• Duration measurement for a firms B/S
• Duration GAP
• Chapter 8 questions
• 3, 6, 7, 9, 13, 14
• Chapter 9
• 1-9, 12-23, 28

10
Repricing Gap

• Repricing or Maturity Buckets
• one day
• 1 day to 3 months
• 3 months to 6 months
• more than 6 months to 12 months
• more than 1 year to 5 years
• over 5 years
• Rate sensitive assets (RSA) in each category
• Rate sensitive liabilities (RSL) in each category
• Difference (RSA - RSL) is termed GAP
• GAP x r net interest income (NII)
• Estimate (NII) for different buckets
• less than 3 months
• less than 1 year
• less than 5 years

11
Duration Gap

• Identify assets and liabilities
• Calculate duration for each component
• Weight the durations by total assets and total
  liabilities respectively

12
Market Risk concepts

• Chapter 10

13
What is Market Risk?

• The risk of loss due to changes in position
  value associated with market moves.

14
ImportanceMarket Risk Information and Management

• Establish appropriate policies and procedures
• Set risk limits ordinary and catastrophic times
• Resource allocation (risk/return trade off)
• Monitor risk limits and violation follow up
• Understand sources of the risks across an
  organization
• Evaluation of performance
• Regulation

15
Introduction to VaR

• DEAR Daily earnings at risk
• VaR Value at risk
• History of VaR
• VaR was pioneered as derivative markets
  developed, and banks moved from NIE to MTM
• Evolution of IR risk measures Gap -gt Duration
  -gt VaR
• Advantages of VaR
• Equates risk across products, risk takers and
  regions
• Relates risk to capacity
• VaR facilitates standard communication about risk
• from trading desk to corporate office and board
• to outside audiences regulators, shareholders,
  ratings agencies

16
Definition of VaR

• Definition
• VaR is the predicted worst-case loss at a
  specific confidence level (e.g., 95) over a
  certain period of time (e.g., 1 day).

• Assuming 95 confidence and a 1-day horizon, a
  VaR of 11 million means that, on average, only
  1 day in 20 would you expect to lose more than
  11 million due to market movements.
• This definition of VaR uses a 95 confidence
  level Losses exceeding the VaR amount should
  occur 5 of the time

17
Examples Historical DataActual data 250 days MER
12.5 days out of 250 return (5 of the time) lt
0.0412
18
Examples Means/Std. DevActual data 250 days
MER

• Mean 0.00 Std Dev. 0.033
• What is 5 value if normal distribution is
  assumed?

1.65? -0.0544
19
3 Related VaR measures

• Relative VaR
• measures risk of underperformance relative to a
  pre-defined benchmark, such as the SP 500 Index.
  
• Marginal VaR
• measures how much risk a position adds to a
  portfolio
• difference in portfolio risk with and without
  position
• Incremental VaR
• measures the impact of small changes in position
  weighting
• sum of all incremental VaRs diversified VaR

20
Relative VaR

• Assuming 99 confidence, a 1-month relative VaR
  of 8 means that on average, only 1 month in 100
  would you expect to underperform your benchmark
  by more than 8
• Often used by investment managers and traders
  with specific benchmarks
• Example An investment managers report

• Which portfolio would be of most concern to a
  risk manager?

21
Marginal VaR

• Measures how much risk a position adds to a
  portfolio
• Specifically, marginal VaR measures how much
  portfolio VaR would change if the position were
  removed entirely, (i.e., VaR with position minus
  VaR without position).
• May be computed for both absolute and relative
  VaRExample 3 positions from a risk report

• Which position adds most risk to the portfolio?

22
Incremental VaR

• Measures the impact of small changes in position
  weighting.
• For example, we can estimate incremental VaR by
  (a) increasing a position weight by 1 dollar and
  measuring the change in diversified portfolio
  VaR, and (b) multiplying this change by the
  position weighting.
• Incremental VaR may be used to calculate
  percentage contribution to risk because the sum
  of all incremental VaRs adds up to the total
  diversified portfolio VaR.
• Useful for making hedging and portfolio
  re-balancing (e.g., risk optimization)
• Example Global risk contribution report

• How would you re-balance this portfolio?

23
Portfolio risk factor correlation

• Whereas standard deviation shows how risky
  individual assets are, correlations show how
  asset risks are interrelated.
• Correlations are calculated by observing
  historical co-movement in returns and range
  between -1 and 1.
• A correlation of 1 means that returns move
  together perfectly, whereas a correlation of -1
  implies perfect opposite movement. A 0 (zero)
  correlation implies independence.
• Correlations are dynamic and often change during
  volatile market conditions, which may
  significantly affect portfolio risk and hedging
  decisions
• Example

Correlation SP 500 vs US 30Y Zero
Observe how correlations suddenly reversed in
a flight to safety phenomenon between SP 500
US 30 Year Zero coupon
24
Steps for VaR analysis

• 1. Set VaR parameters
• Confidence level between 90 to 99
• Forecast horizon 1 day(DEAR), 10 days (BIS
  Capital), or longer
• Base currency currency of equity capital and
  reporting
• 2. Calculate VaR of individual positions, given
  market volatilities and price sensitivity of
  instruments
• 3. Calculate portfolio VaR, given correlations
  between all variables

Portfolio positions and market data are fed
into a risk engine to generate risk reports, as
illustrated in this chart
25
Total risk and diversification

• To determine the total price risk of financial
  instruments, we aggregate market risk with
  residual risk

Market Risk Interest FX Equity
Commodity
Residual Risk Spread Basis Specific
Volatility
Total Price Risk

• Total risk is less than the sum of its parts
  because of diversification. There is significant
  diversification between FX, Equity Commodity,
  and between market and residual risk
• Diversification benefit is defined as total risk
  minus the sum of all individual risk components.

26
Dimensions for analyzing risk
Market risk affects credit risk through
counterparty exposures
Market Risk
27
Industry perspective of VaR Reporting

• Virtually all global banks have adapted VaR as a
  cornerstone of day-to-day market risk management.
• Over 2000 companies have implemented VaR
• Trends
• Migration to historical simulation for enterprise
  risk reporting (e.g., Chase, J.P. Morgan)
• 99 confidence level as reporting standard (e.g.,
  Citibank Salomon Brothers merger)
• Marginal risk analysis (e.g., Goldman Sachs Hot
  Spots Reports)
• Picture of Risk analysis

28
Enterprise risk communication

• Three levels of reporting corporate, business
  level, desk level

29
Enterprise VaR report
Daily Global VaR Report
March 11, 1999
Market Risk Commentary Global market volatility
has continued to increase, with widening credit
spreads and decreased liquidity for risky assets
across Europe and the Americas. Trading volume
across U.S. fixed income was unusually low, and
corporate bond traders note declining liquidity
and increasing spreads due to a flight to
quality. The firms large inventory of corporates
could suffer from further widening of spreads.
Brazilian markets continue to bleed, with
uncertainty surrounding impending fiscal reforms.
While mostly FX hedged, the Emerging Markets desk
is close to its 2MM VaR limits and could suffer
large bond losses if Brazil is forced to raise
interest rates to stem capital flight. A Latin
American liquidity crunch could put pressure on
Emerging Asia again, where the firm has long
positions in THB, MYR, and SGD government paper.
The FX desk reported significant trading by macro
hedge funds, mostly short interest in long dated
JPY/USD forwards and options. U.S. equity markets
continue to be volatile, with Internet stocks
racing ahead. However the banks direct equity
exposure is currently low due to short SPX
futures positions in proprietary trading which
offset some systemic risk in market making books.
30
Enterprise Picture of Risk report

• Picture of Risk reports can combine VaR with
  Stress Testing

Corporate Picture of Risk Report
July 11, 1999
Stress Scenarios - PV Changes 1. July 1998
Turmoil 1-day -13,478,941 2. Russian
Devaluation (1998) 1-day -8,467,208 3.
Asian Crisis (1997) 1-day -6,597,188 4.
Gulf War (1990) 1-day -2,839,852 5.
Mexican Peso Fallout (1995) 1-day
-2,702,207 6. US Gov 50bp -284,264 7.
Black Monday (1987) 1-day 3,496,649
99 VaR 6,875,408 95 VaR 5,012,459 90 VaR
4,155,481

• Comments
• Firmwide stress results show greatest exposure to
  July 1998 turmoil, which was marked by
  significant swap spread widening, bearish equity
  markets and emerging markets capital flight. The
  firm is exposed to swap and corporate bond swaps
  widening, due to a large inventory of Euro
  issues. The firms Latin American, Eastern
  European and Southeast Asian bond positions are
  vulnerable to an emerging markets liquidity
  crisis.
• The overall book is relatively interest rate
  neutral, as shown by the low PV impact of a
  parallel increase in US yields.
• Short US equity positions by proprietary trading
  contribute to a net positive impact in the event
  of a US equity market sell off

1 2 3 4 5
6 7
-10,000,000 -6,000,000 -2000,000 0
2,000,000 6,000,000
31
Overview of Risk Methodologies

• ParametricEstimates VaR with equation that
  specifies parameters such as volatility,
  correlation, delta, and gamma as input.
• Monte Carlo simulationEstimates VaR by
  simulating random scenarios and revaluing
  positions in the portfolio.
• Historical simulationEstimates VaR by reliving
  history takes actual historical rates and
  revalues positions for each change in the market.

32
Statistics Review
33
Measure VaR for 1 Asset

• Suppose 95
• 1.65 x ? will occur 5 of time (1 out 20)
• Call 1.65 x ? the price volatility
• Example
• US corporation hold a 140 Million FX position in
  DEM (Deutsche Marks)
• What is VaR for one day given 5 loss
• Foreign exchange is 1.4 DEM/USD

34
Data

• ? 0.565
• 1.65 x ? 0.932
• US currency position 140Million / 1.4
  exchange rate
• 100US
• VaR
• 100 million x 0.932
• 932,000
• 95 of time will lose less than 932,000
  over the next day

35
2 Asset Case
VaR1 Value at risk for asset 1 VaR2 Value at
risk for asset 2 ?1,2 correlation coefficient
between assets 1 and 2
Now suppose you hold a DEM 140 million in 10 year
German Bonds (? 0.605, ?1,2 between bond and
FX) What is VaR for bond? What is VaR for
portfolio?
0.605 x 140 / 1.4 100 x 0.605 x 1.65 999,000
36
Answer

• 1.168 Million

37
N Asset Portfolio

• Where
• V VaR1 ,VaR2 ,VaR3 (VaR vector of
  individual positions)
• C correlation matrixVT transposed vector
  of V
• Review of Matrix multiplication

38
Data

• 7 year-zero coupon bond (face value)
• Yield 6.128
• 1,641,483
• Spot position in currency
• 1.7581 DM/US
• 1,600,000 DM
• Stocks in an index fund (Beta 1.0, all
  non-market risk diversified away))
• Correlations
• Bond,FX 0.008688
• Bond,Stocks 0.4109887
• Stocks,FX -0.39754
• Volatilities
• Bond 0.0309
• FX 0.0566
• Stocks 0.0726
• First step PV US currency of each
• Second step Calculate the VaR of each
• Third step Calculate the VaR for portfolio

Diversification Benefit?