MRM FRM-GARP Oct-2001 Zvi Wiener 02-588-3049 mswiener/zvi.html Market Risk Management.

MRM FRM-GARP Oct-2001

Zvi Wiener

02-588-3049http://pluto.mscc.huji.ac.il/~mswiener/zvi.html

Market Risk Management


Introduction to Market Risk Measurement

Following Jorion 2001, Chapter 11

Financial Risk Manager Handbook

Zvi Wiener - MRM slide 3http://www.tfii.org

Old ways to measure risk

• notional amounts

• sensitivity measures (duration, Greeks)

• scenarios

• ALM, DFA

assume linearity

do not describe probability


1938 Bonds duration

1952 Markowitz mean-variance

1963 Sharpe’s CAPM

1966 Multiple risk-factors

1973 Black-Scholes option pricing

1983 RAROC, risk adjusted return

1986 Limits on exposure by duration

1988 Risk-weighted assets for banks;

exposure limits by Greeks

1993 VaR endorsed by G-30

1994 Risk Metrics

1997 CreditMetrics, CreditRisk+


How much can we lose?

Everything

correct, but useless answer.

How much can we lose realistically?


What is the current Risk?

duration, convexity

volatility

delta, gamma, vega

rating

target zone

• Bonds

• Stocks

• Options

• Credit

• Forex• Total ?


Standard Approach


Modern Approach

Financial Institution


Definition

VaR is defined as the predicted worst-case

loss at a specific confidence level (e.g. 99%)

over a certain period of time.


Definition (Jorion)

VaR is the maximum loss over a target

horizon such that there is a low, prespecified

probability that the actual loss will be larger.


-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8

1

Profit/Loss

VaR

1% VaR1%


Meaning of VaR

A portfolio manager has a daily VaR equal $1M at 99% confidence level.

This means that there is only one chance in 100 that a daily loss bigger than $1M occurs,

1%VaR

under normal market conditions.


Returns

year

1% of worst cases


Main Ideas

• A few well known risk factors

• Historical data + economic views

• Diversification effects

• Testability

• Easy to communicate


History of VaR

• 80’s - major US banks - proprietary

• 93 G-30 recommendations

• 94 - RiskMetrics by J.P.Morgan

• 98 - Basel

• SEC, FSA, ISDA, pension funds, dealers

• Widely used and misused!


FRM-99, Question 89

What is the correct interpretation of a $3 overnight VaR figure with 99% confidence level?

A. expect to lose at most $3 in 1 out of next 100 days

B. expect to lose at least $3 in 95 out of next 100 days

C. expect to lose at least $3 in 1 out of next 100 days

D. expect to lose at most $6 in 2 out of next 100 days


FRM-99, Question 89

What is the correct interpretation of a $3 overnight VaR figure with 99% confidence level?

A. expect to lose at most $3 in 1 out of next 100 days

B. expect to lose at least $3 in 95 out of next 100 days

C. expect to lose at least $3 in 1 out of next 100 days

D. expect to lose at most $6 in 2 out of next 100 days


VaR caveats

• VaR does not describe the worst loss

• VaR does not describe losses in the left tail

• VaR is measured with some error


Other Measures of Risk

• The entire distribution

• The expected left tail loss

• The standard deviation

• The semi-standard deviation


-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8

1

Profit/Loss

Risk Measures


Properties of Risk Measure

• Monotonicity (X<Y, R(X)>R(Y))

• Translation invariance R(X+k) = R(X)-k

• Homogeneity R(aX) = a R(X) (liquidity??)

• Subadditivity R(X+Y) R(X) + R(Y)

the last property is violated by VaR!


No subadditivity of VaR

Bond has a face value of $100,000, during the target period there is a probability of 0.75% that there will be a default (loss of $100,000).

Note that VaR99% = 0 in this case.

What is VaR99% of a position consisting of 2

independent bonds?


FRM-98, Question 22

Consider arbitrary portfolios A and B and their combined portfolio C. Which of the following relationships always holds for VaRs of A, B, and C?

A. VaRA+ VaRB = VaRC

B. VaRA+ VaRB VaRC

C. VaRA+ VaRB VaRC

D. None of the above


FRM-98, Question 22

Consider arbitrary portfolios A and B and their combined portfolio C. Which of the following relationships always holds for VaRs of A, B, and C?

A. VaRA+ VaRB = VaRC

B. VaRA+ VaRB VaRC

C. VaRA+ VaRB VaRC

D. None of the above


Confidence levellow confidence leads to an imprecise result.

For example 99.99% and 10 business days will require history of

100*100*10 = 100,000 days in order to have only 1 point.


Time horizonlong time horizon can lead to an imprecise result.

1% - 10 days means that we will see such a loss approximately once in 100*10 = 3 years.

5% and 1 day horizon means once in a month.

Various subportfolios may require various horizons.


Time horizon

When the distribution is stable one can

translate VaR over different time periods.

TdayVaRdaysTVaR )1()(

This formula is valid (in particular) for iid

normally distributed returns.


FRM-97, Question 7

To convert VaR from a one day holding period to a ten day holding period the VaR number is generally multiplied by:

A. 2.33

B. 3.16

C. 7.25

D. 10


FRM-97, Question 7

To convert VaR from a one day holding period to a ten day holding period the VaR number is generally multiplied by:

A. 2.33

B. 3.16

C. 7.25

D. 10


Basel Rules

• horizon of 10 business days

• 99% confidence interval

• an observation period of at least a year of

historical data, updated once a quarter


Basel Rules MRC

Market Risk Charge = MRC

SRC - specific risk charge, k 3.

tti

itt SRCVaRVaRk

MaxMRC

1

60

1

,60

10%)99,1( dVaRVaR tt


FRM-97, Question 16

Which of the following quantitative standards is NOT required by the Amendment to the Capital Accord to Incorporate Market Risk?

A. Minimum holding period of 10 days

B. 99% one-tailed confidence interval

C. Minimum historical observations of two years

D. Update the data sets at least quarterly


VaR systemRisk factors

Historical data

Model

Distribution ofrisk factors

VaRmethod

Portfolio

positions

Mapping

Exposures

VaR


FRM-97, Question 23

The standard VaR calculation for extension to multiple periods also assumes that positions are fixed. If risk management enforces loss limits, the true VaR will be:

A. the same

B. greater than calculated

C. less then calculated

D. unable to determine


FRM-97, Question 23

The standard VaR calculation for extension to multiple periods also assumes that positions are fixed. If risk management enforces loss limits, the true VaR will be:

A. the same

B. greater than calculated

C. less then calculated

D. unable to determine


FRM-97, Question 9

A trading desk has limits only in outright foreign exchange and outright interest rate risk. Which of the following products can not be traded within the current structure?

A. Vanilla IR swaps, bonds and IR futures

B. IR futures, vanilla and callable IR swaps

C. Repos and bonds

D. FX swaps, back-to-back exotic FX options


FRM-97, Question 9

A trading desk has limits only in outright foreign exchange and outright interest rate risk. Which of the following products can not be traded within the current structure?

A. Vanilla IR swaps, bonds and IR futures

B. IR futures, vanilla and callable IR swaps

C. Repos and bonds

D. FX swaps, back-to-back exotic FX options

No limits!


Stress-testing

• scenario analysis

• stressing models, volatilities and

correlations

• developing policy responses


Scenario Analysis

• Moving key variables one at a time

• Using historical scenarios

• Creating prospective scenarios

The goal is to identify areas of potential vulnerability.


FRM-97, Question 4

The use of scenario analysis allows one to:

A. assess the behavior of portfolios under large moves

B. research market shocks which occurred in the past

C. analyze the distribution of historical P&L

D. perform effective back-testing


FRM-98, Question 20

VaR measure should be supplemented by portfolio stress-testing because:A. VaR measures indicate that the minimum is VaR, they do not indicate the actual lossB. stress testing provides a precise maximum loss levelC. VaR measures are correct only 95% of timeD. stress testing scenarios incorporate reasonably probable events.


FRM-00, Question 105

VaR analysis should be complemented by stress-testing because stress-testing:A. Provides a maximum loss in dollars.B. Summarizes the expected loss over a target horizon within a minimum confidence interval.C. Assesses the behavior of portfolio at a 99% confidence level.D. Identifies losses that go beyond the normal losses measured by VaR.


Identification of Risk Factors




Absolute and Relative Risk

• Absolute risk - measured in dollar terms

• Relative risk - measured relative to

benchmark index and is often called tracking

error.


Directional Risk

Directional risk involves exposures to the direction of movements in major market variables.

beta for exposure to stock market

duration for IR exposure

delta for exposure of options to undelying


Non-directional Risk

Non-linear exposures, volatility exposures, etc.

residual risk in equity portfolios

convexity in interest rates

gamma - second order effects in options

vega or volatility risk in options


Market versus Credit Risk

Market risk is related to changes in prices,

rates, etc.

Credit risk is related to defaults.

Many assets have both types - bonds, swaps,

options.


Risk Interaction

You buy 1M GBP at 1.5 $/GBP, settlement in

two days. We will deliver $1.5M in exchange

for 1M GBP.

Market risk

Credit risk

Settlement risk (Herstatt risk)

Operational risk


Exposure and Uncertainty

Losses can occur due to a combination of

A. exposure (choice variable)

B. movement of risk factor (external variable)

yPDP *)(

Dollar duration


Exposure and Uncertainty

Market loss =

Exposure * Adverse movement in risk factor

iMiii RR


Specific Risk

iiMii PRPP

Market exposure Specific risk

Specific risk - risk due to issuer

specific price movements

iiMii PRPP 2222


FRM-97, Question 16

The risk of a stock or bond which is NOT correlated with the market (and thus can be diversified) is known as:A. interest rate risk.B. FX risk.C. model risk.D. specific risk.


• Continuous process - diffusion

• Discontinuities

• Jumps in prices, interest rates

• Price impact and liquidity

• market closure

• discontinuity in payoff:• binary options

• loans


Emerging Markets

Emerging stock market - definition by IFC

(1981) International Finance Corporation.

Stock markets located in developing countries

(countries with GDP per capita less than

$8,625 in 1993).


Liquidity Risk

Difficult to measure.

Very unstable.

Market depth can be used as an approximation.

Liquidity risk consists of both asset liquidity

and funding liquidity!


Funding Liquidity

Risk of not meeting payment obligations.

Cash flow risk!

Liquidity needs can be met by• using cash

• selling assets

• borrowing


Highly liquid assets

• tightness - difference between quoted mid market price and transaction price.

• depth - volume of trade that does not affect prices.

• resiliency - speed at which price fluctuations disappear.


Flight to quality

Shift in demand from low grade towards high grade securities.

Low grade market becomes illiquid with depressed prices.

Spread between government and corporate issues increases.


On-the-run

• recently issued

• most active

• very liquid

• after a new issue appears they become off-the-run

• liquidity premium can be compensated by repos/reverse repos


FRM-98, Question 7

Which of the following products has the least liquidity?

A. US on-the-run Treasuries

B. US off-the-run Treasuries

C. Floating rate notes

D. High grade corporate bonds


FRM-98, Question 7

Which of the following products has the least liquidity?

A. US on-the-run Treasuries

B. US off-the-run Treasuries

C. Floating rate notes

D. High grade corporate bonds


FRM-97, Question 54

“Illiquid” describes an instrument which

A. does not trade in an active market

B. does not trade on any exchange

C. can not be easily hedged

D. is an over-the-counter (OTC) product


FRM-97, Question 54

“Illiquid” describes an instrument which

A. does not trade in an active market

B. does not trade on any exchange

C. can not be easily hedged

D. is an over-the-counter (OTC) product


FRM-98, Question 6

A finance company is interested in managing its balance sheet liquidity risk. The most productive means of accomplishing this is by:

A. purchasing market securities

B. hedging the exposure with Eurodollar futures

C. diversifying its sources of funding

D. setting up a reserve


FRM-98, Question 6

A finance company is interested in managing its balance sheet liquidity risk. The most productive means of accomplishing this is by:

A. purchasing market securities

B. hedging the exposure with Eurodollar futures

C. diversifying its sources of funding

D. setting up a reserve


FRM-00, Question 74In a market crash the following is usually true?

I. Fixed income portfolios hedged with short Treasuries and futures lose less than those hedged with IR swaps given equivalent duration.

II. Bid offer spreads widen due to less liquidity.

III. The spreads between off the run bonds and benchmark issues widen.

A. I, II & III C. I & III

B. II & III D. None of the above


FRM-00, Question 74In a market crash the following is usually true?

I. Fixed income portfolios hedged with short Treasuries and futures lose less than those hedged with IR swaps given equivalent duration.

II. Bid offer spreads widen due to less liquidity.

III. The spreads between off the run bonds and benchmark issues widen.

A. I, II & III C. I & III

B. II & III D. None of the above


Sources of Risk




Currency Risk

• free movements of currency

• devaluation of a fixed or pegged currency

• regime change (Israel, Europe)


Currency VolatilityEnd 99 End 96

Argentina 0.35 0.4Australia 7.6 8.5Canada 5.1 3.6Switzerland 10 10Denmark 9.8 7.8Britain 6.5 9.1Hong Kong 0.3 0.3Indonesia 24 1.6Japan 11 6.6Korea 6.9 4.5


Currency VolatilityEnd 99 End 96

Mexico 7.5 7Malaysia 0.1 1.6Norway 7.6 7.6New Zealand 13.4 7.9Philippines 5.5 0.6Sweden 8.5 6.4Singapore 3.8 1.8Thailand 9.7 1.2Taiwan 1.8 0.9Euro 9.8 8.3S. Africa 4.2 8.4


FRM-97, Question 10

Which currency pair would you expect to have the lowest volatility?

A. USD/DEM

B. USD/CAD

C. USD/JPY

D. USD/ITL


FRM-97, Question 10

Which currency pair would you expect to have the lowest volatility?

A. USD/DEM

B. USD/CAD

C. USD/JPY

D. USD/ITL


FRM-97, Question 14

What is the implied correlation between JPY/DEM and DEM/USD when given the following volatilities for foreign exchange rates?

JPY/USD 8%, JPY/DEM 10%, DEM/USD 6%

A. 60%

B. 30%

C. -30%

D. -60%


Cross Rate volatilityJPY/USD = x JPY/DEM = y DEM/USD = z

z

xy zyx lnlnln

)(ln)(ln2)(ln)(ln)(ln lnln222 zyzyx zy

06.01.0206.01.008.0 222

6.0012.0

0072.0

1.006.02

0064.00036.001.0

yzx


Fixed Income Risk

Arises from potential movements in the level and volatility of bond yields.

Factors affecting yields• inflationary expectations

• term spread

• higher volatility of the low end of TS


Volatilities of IR/bond pricesPrice volatility in % End 99 End 96Euro 30d 0.22 0.05Euro 180d 0.30 0.19Euro 360d 0.52 0.58Swap 2Y 1.57 1.57Swap 5Y 4.23 4.70Swap 10Y 8.47 9.82Zero 2Y 1.55 1.64Zero 5Y 4.07 4.67Zero 10Y 7.76 9.31Zero 30Y 20.75 23.53


Duration approximation

What duration makes bond as volatile as FX?

What duration makes bond as volatile as stocks?

A 10 year bond has yearly price volatility of 8% which is similar to major FX.

30-year bonds have volatility similar to equities (20%).

)(* yDP

P


Models of IR

Normal model (y) is normally distributed.

Lognormal model (y/y) is normally distributed.

Note that:

y

yyy )(


Time adjustment

Square root of time adjustment is more questionable for bond prices than for other assets

• there is a strong evidence of mean reversion

• bond prices converge approaching maturity (bridge effect) - strong for short bonds, weak for long.


Volatilities of yieldsYield volatility in %, 99 (y/y) (y) Euro 30d 45 2.5Euro 180d 10 0.62Euro 360d 9 0.57Swap 2Y 12.5 0.86Swap 5Y 13 0.92Swap 10Y 12.5 0.91Zero 2Y 13.4 0.84Zero 5Y 13.9 0.89Zero 10Y 13.1 0.85Zero 30Y 11.3 0.74


FRM-99, Question 86For computing the market risk of a US T-bond portfolio it is easiest to measure:A. yield volatility, because yields have positive skewness.B. price volatility, because bond prices are positively correlated.C. yield volatility for bonds sold at a discount and price volatility for bonds sold at a premium.D. yield volatility because it remains more constant over time than price volatility, which must approach zero at maturity.


FRM-99, Question 86For computing the market risk of a US T-bond portfolio it is easiest to measure:A. yield volatility, because yields have positive skewness.B. price volatility, because bond prices are positively correlated.C. yield volatility for bonds sold at a discount and price volatility for bonds sold at a premium.D. yield volatility because it remains more constant over time than price volatility, which must approach zero at maturity.


FRM-99, Question 80You have position of $20M in the 6.375% Aug-27 US T-bond. Calculate daily VaR at 95% assume that there are 250 business days in a year.Price 98 8/32 Accrued 1.43%Yield 6.509% Duration 13.133Modified Dur. 12.719 Yield volatility 12%A. $291,400B. $203,080C. $206,036D. $206,698


FRM-99, Question 80

Value of the position 936.19$100

143.1

32

89820$

Daily yield volatility

000494.0250

1)(

y

yyy annual

)(645.1* yPDVaR

055,206$000494.0645.1936.19$719.12 MVaR


Correlations

Eurodeposit block

zero-coupon Treasury block

very high correlations within each block and much lower across blocks.


Principal component analysis

• level risk factor 94% of changes

• slope risk factor (twist) 4% of changes

• curvature (bend or butterfly)

See book by Golub and Tilman.


FRM-00, Question 96

Which statement about historic US Treasuries yield curves is TRUE?


FRM-00, Question 96A. Changes in the long-term yield tend to be larger than in short-term yield.B. Changes in the long-term yield tend to be approximately the same as in short-term yield.C. The same size yield change in both long-term and short-term rates tends to produce a larger price change in short-term instruments when all securities are traded near par.D. The largest part of total return variability of spot rates is due to parallel changes with a smaller portion due to slope changes and the residual due to curvature changes.


FRM-00, Question 96A. Changes in the long-term yield tend to be larger than in short-term yield.B. Changes in the long-term yield tend to be approximately the same as in short-term yield.C. The same size yield change in both long-term and short-term rates tends to produce a larger price change in short-term instruments when all securities are traded near par.D. The largest part of total return variability of spot rates is due to parallel changes with a smaller portion due to slope changes and the residual due to curvature changes.


FRM-97, Question 42What is the relationship between yield on the current inflation-proof bond issued by the US Treasury and a standard Treasury bond with similar terms?A. The yields should be about the same.B. The yield on the inflation protected bond should be approximately the yield on treasury minus the real interest.C. The yield on the inflation protected bond should be approximately the yield on treasury plus the real interest.D. None of the above.


• Credit Spread Risk

• Prepayment Risk (MBS and CMO)• seasoning

• current level of interest rates

• burnout (previous path)

• economic activity

• seasonal patterns

• OAS = option adjusted spread = spread over equivalent Treasury minus the cost of the option component.


FRM-99, Question 71You held mortgage interest only (IO) strips backed by Fannie Mae 7 percent coupon. You want to hedge this by shorting Treasury interest strips off the 10-year on-the-run. The curve steepens as 1 month rate drops, while the 6 months to 10 year rates remain stable. What will be the effect on the value of your portfolio?A. Both IO and the hedge appreciate in value.B. Almost no change in both (may be a small appreciation).C. Not enough information to find changes in both.D. The IO will depreciate, the hedge will appreciate.


FRM-99, Question 71You held mortgage interest only (IO) strips backed by Fannie Mae 7 percent coupon. You want to hedge this by shorting Treasury interest strips off the 10-year on-the-run. The curve steepens as 1 month rate drops, while the 6 months to 10 year rates remain stable. What will be the effect on the value of your portfolio?A. Both IO and the hedge appreciate in value.B. Almost no change in both (may be a small appreciation).C. Not enough information to find changes in both.D. The IO will depreciate, the hedge will appreciate.


FRM-99, Question 73A fund manager attempting to beat his LIBOR based funding costs, holds pools of adjustable rate mortgages and is considering various strategies to lower the risk. Which of the following strategies will NOT lower the risk?A. Enter a total rate of return swap swapping the ARMs for LIBOR plus a spread.B. Short US government bondsC. Sell caps based on the projected rate of mortgage paydown.D. All of the above.


FRM-99, Question 73A fund manager attempting to beat his LIBOR based funding costs, holds pools of adjustable rate mortgages and is considering various strategies to lower the risk. Which of the following strategies will NOT lower the risk?A. Enter a total rate of return swap swapping the ARMs for LIBOR plus a spread.B. Short US government bonds.C. Sell caps based on the projected rate of mortgage paydown.D. All of the above.

He should buy caps, not sell!


Fixed income portfolio risk

• Yield curve component (government)

• Credit spread (of the class of similar rating)

• Specific spread


Equity risk

• Market risk (beta based relative to an index)

• Specific risk


FRM-97, Question 43Which of the following statements about SP500 is true?I. The index is calculated using market prices as weights.II. The implied volatilities of options of the same maturity on the index are different.III. The stocks used in calculating the index remain the same for each year.IV. The SP500 represents only the 500 largest US corporations.A. II only. B. I and II.C. II and III. D. III and IV only.


FRM-97, Question 43Which of the following statements about SP500 is true?I. The index is calculated using market prices as weights.II. The implied volatilities of options of the same maturity on the index are different.III. The stocks used in calculating the index remain the same for each year.IV. The SP500 represents only the 500 largest US corporations.A. II only. B. I and II.C. II and III. D. III and IV only.

values


Forwards and Futures

The forward or futures price on a stock.

e-rt the present value in the base currency.

e-yt the cost of carry (dividend rate).

For a discrete dividend (individual stock) we can write the right hand side as St- D, where D is the PV of the dividend.

ytt

rtt eSeF


FRM-97, Question 44A trader runs a cash and future arbitrage book on the SP500 index. Which of the following are the major risk factors?I. Interest rateII. Foreign exchangeIII. Equity priceIV. Dividend assumption riskA. I and II only.B. I and III only.C. I, III, and IV only.D. I, II, III, and IV.


FRM-97, Question 44A trader runs a cash and future arbitrage book on the SP500 index. Which of the following are the major risk factors?I. Interest rateII. Foreign exchangeIII. Equity priceIV. Dividend assumption riskA. I and II only.B. I and III only.C. I, III, and IV only.D. I, II, III, and IV.


In an equilibrium the following holds (Sharpe)

iMiii RR

)(

)(),(,2

M

iMi

M

Mii R

RRRCov

fMifi RRERRE

CAPM


iKiKiii yyR 11

APTArbitrage Pricing Theory


FRM-98, Question 62In comparing CAPM and APT, which of the following advantages does APT have over CAPM?I. APT makes less restrictive assumptions about investor preferences toward risk and return.II. APT makes no assumption about the distribution of security returns.III. APT does not rely on the identification of the true market portfolio, and so the theory is potentially testable.A. I only. B. II and III only.C. I, and III only. D. I, II, and III.


FRM-98, Question 62In comparing CAPM and APT, which of the following advantages does APT have over CAPM?I. APT makes less restrictive assumptions about investor preferences toward risk and return.II. APT makes no assumption about the distribution of security returns.III. APT does not rely on the identification of the true market portfolio, and so the theory is potentially testable.A. I only. B. II and III only.C. I, and III only. D. I, II, and III.


Commodity RiskBase metal - aluminum, copper, nickel, zinc.

Precious metals - gold, silver, platinum.

Energy products - natural gas, heating oil, unleaded gasoline, crude oil.

Metals have 12-25% yearly volatility.

Energy products have 30-100% yearly volatility (much less storable).

Long forward prices are less volatile then short forward prices.


FRM-97, Question 12

Which of the following products should have the highest expected volatility?

A. Crude oil

B. Gold

C. Japanese Treasury Bills

D. DEM/CHF


FRM-97, Question 12

Which of the following products should have the highest expected volatility?

A. Crude oil

B. Gold

C. Japanese Treasury Bills

D. DEM/CHF


FRM-97, Question 23Identify the major risks of being short $50M of gold two weeks forward and being long $50M of gold one year forward.I. Spot liquidity squeeze.II. Spot risk.III. Gold lease rate risk.IV. USD interest rate risk.A. II only. B. I, II, and III only.C. I, III, and IV only. D. I, II, III, and IV.


FRM-97, Question 23Identify the major risks of being short $50M of gold two weeks forward and being long $50M of gold one year forward.I. Spot liquidity squeeze.II. Spot risk.III. Gold lease rate risk.IV. USD interest rate risk.A. II only. B. I, II, and III only.C. I, III, and IV only. D. I, II, III, and IV.

Spot risk is eliminatedby offsetting positions


Hedging Linear Risk




Hedging

Taking positions that lower the risk profile of

the portfolio.

• Static hedging

• Dynamic hedging


Unit Hedging with CurrenciesA US exporter will receive Y125M in 7 months.

The perfect hedge is to enter a 7-months forward contract.

Such a contract is OTC and illiquid.

Instead one can use traded futures.

CME lists yen contract with face value Y12.5M and 9 months to maturity.

Sell 10 contracts and revert in 7 months.


Market data 0 7m P&L

time to maturity 9 2

US interest rate 6% 6%

Yen interest rate 5% 2%

Spot Y/$ 125.00 150.00

Futures Y/$ 124.07 149.00

667,166$125

1

150

1125

MY

621,168$07.124

1

149

15.1210

MY


Stacked hedge - to use a longer horizon and

to revert the position at maturity.

Strip hedge - rolling over short hedge.


Basis Risk

Basis risk arises when the characteristics of

the futures contract differ from those of the

underlying.

For example quality of agricultural product,

types of oil, Cheapest to Deliver bond, etc.

Basis = Spot - Future


Cross hedging

Hedging with a correlated (but different) asset.

In order to hedge an exposure to Norwegian

Krone one can use Euro futures.

Hedging a portfolio of stocks with index

future.


FRM-00, Question 78What feature of cash and futures prices tend to make hedging possible?A. They always move together in the same direction and by the same amount.B. They move in opposite direction by the same amount.C. They tend to move together generally in the same direction and by the same amount.D. They move in the same direction by different amount.


FRM-00, Question 78What feature of cash and futures prices tend to make hedging possible?A. They always move together in the same direction and by the same amount.B. They move in opposite direction by the same amount.C. They tend to move together generally in the same direction and by the same amount.D. They move in the same direction by different amount.


FRM-00, Question 17Which statement is MOST correct?A. A portfolio of stocks can be fully hedged by purchasing a stock index futures contract.B. Speculators play an important role in the futures market by providing the liquidity that makes hedging possible and assuming the risk that hedgers are trying to eliminate.C. Someone generally using futures contract for hedging does not bear the basis risk. D. Cross hedging involves an additional source of basis risk because the asset being hedged is exactly the same as the asset underlying the futures.


FRM-00, Question 17Which statement is MOST correct?A. A portfolio of stocks can be fully hedged by purchasing a stock index futures contract.B. Speculators play an important role in the futures market by providing the liquidity that makes hedging possible and assuming the risk that hedgers are trying to eliminate.C. Someone generally using futures contract for hedging does not bear the basis risk. D. Cross hedging involves an additional source of basis risk because the asset being hedged is exactly the same as the asset underlying the futures.


FRM-00, Question 79Under which scenario is basis risk likely to exist?

A. A hedge (which was initially matched to the maturity of the underlying) is lifted before expiration.

B. The correlation of the underlying and the hedge vehicle is less than one and their volatilities are unequal.

C. The underlying instrument and the hedge vehicle are dissimilar.

D. All of the above.


FRM-00, Question 79Under which scenario is basis risk likely to exist?

A. A hedge (which was initially matched to the maturity of the underlying) is lifted before expiration.

B. The correlation of the underlying and the hedge vehicle is less than one and their volatilities are unequal.

C. The underlying instrument and the hedge vehicle are dissimilar.

D. All of the above.


The Optimal Hedge Ratio

S - change in $ value of the inventory

F - change in $ value of the one futures

N - number of futures you buy/sell

FNSV

FSFSV NN ,2222 2

FSFV N

N

,2

2

22


The Optimal Hedge Ratio

FSFV N

N

,2

2

22

F

SFS

F

FSoptN

,2

,

Minimum variance hedge ratio


Hedge Ratio as Regression Coefficient

The optimal amount can also be derived as the slope coefficient of a regression s/s on f/f:

f

f

s

ssf

f

ssf

f

sfsf

2


Optimal Hedge

One can measure the quality of the optimal hedge ratio in terms of the amount by which we have decreased the variance of the original portfolio.

22

2*

22 )(

sfs

VsR

2* 1 RsV

If R is low the hedge is not effective!


Optimal Hedge

At the optimum the variance is

2

222

*F

SFSV


FRM-99, Question 66The hedge ratio is the ratio of the size of the position taken in the futures contract to the size of the exposure. Denote the standard deviation of change of spot price by 1, the standard deviation of change of future price by 2, the correlation between the changes in spot and futures prices by . What is the optimal hedge ratio?

A. 1/1/2

B. 1/2/1

C. 1/2

D. 2/1


FRM-99, Question 66The hedge ratio is the ratio of the size of the position taken in the futures contract to the size of the exposure. Denote the standard deviation of change of spot price by 1, the standard deviation of change of future price by 2, the correlation between the changes in spot and futures prices by . What is the optimal hedge ratio?

A. 1/1/2

B. 1/2/1

C. 1/2

D. 2/1


FRM-99, Question 66The hedge ratio is the ratio of derivatives to a spot position (vice versa) that achieves an objective such as minimizing or eliminating risk. Suppose that the standard deviation of quarterly changes in the price of a commodity is 0.57, the standard deviation of quarterly changes in the price of a futures contract on the commodity is 0.85, and the correlation between the two changes is 0.3876. What is the optimal hedge ratio for a three-month contract?

A. 0.1893

B. 0.2135

C. 0.2381

D. 0.2599


FRM-99, Question 66The hedge ratio is the ratio of derivatives to a spot position (vice versa) that achieves an objective such as minimizing or eliminating risk. Suppose that the standard deviation of quarterly changes in the price of a commodity is 0.57, the standard deviation of quarterly changes in the price of a futures contract on the commodity is 0.85, and the correlation between the two changes is 0.3876. What is the optimal hedge ratio for a three-month contract?

A. 0.1893

B. 0.2135

C. 0.2381

D. 0.2599


ExampleAirline company needs to purchase 10,000 tons of jet fuel in 3 months. One can use heating oil futures traded on NYMEX. Notional for each contract is 42,000 gallons. We need to check whether this hedge can be efficient.


ExampleSpot price of jet fuel $277/ton.

Futures price of heating oil $0.6903/gallon.

The standard deviation of jet fuel price rate of changes over 3 months is 21.17%, that of futures 18.59%, and the correlation is 0.8243.


Compute

• The notional and standard deviation f the

unhedged fuel cost in $.

• The optimal number of futures contracts to

buy/sell, rounded to the closest integer.

• The standard deviation of the hedged fuel

cost in dollars.


Solution

The notional is Qs=$2,770,000, the SD in $ is

(s/s)sQs=0.2117$277 10,000 = $586,409

the SD of one futures contract is

(f/f)fQf=0.1859$0.690342,000 = $5,390

with a futures notional

fQf = $0.690342,000 = $28,993.


Solution

The cash position corresponds to a liability

(payment), hence we have to buy futures as a

protection.

sf= 0.8243 0.2117/0.1859 = 0.9387

sf = 0.8243 0.2117 0.1859 = 0.03244

The optimal hedge ratio is

N* = sf Qss/Qff = 89.7, or 90 contracts.


Solution

2unhedged = ($586,409)2 = 343,875,515,281

- 2SF/ 2

F = -(2,605,268,452/5,390)2

hedged = $331,997

The hedge has reduced the SD from $586,409

to $331,997.

R2 = 67.95% (= 0.82432)


FRM-99, Question 67In the early 90s, Metallgesellshaft, a German oil company, suffered a loss of $1.33B in their hedging program. They rolled over short dated futures to hedge long term exposure created through their long-term fixed price contracts to sell heating oil and gasoline to their customers. After a time, they abandoned the hedge because of large negative cashflow. The cashflow pressure was due to the fact that MG had to hedge its exposure by:

A. Short futures and there was a decline in oil price

B. Long futures and there was a decline in oil price

C. Short futures and there was an increase in oil price

D. Long futures and there was an increase in oil price


FRM-99, Question 67In the early 90s, Metallgesellshaft, a German oil company, suffered a loss of $1.33B in their hedging program. They rolled over short dated futures to hedge long term exposure created through their long-term fixed price contracts to sell heating oil and gasoline to their customers. After a time, they abandoned the hedge because of large negative cashflow. The cashflow pressure was due to the fact that MG had to hedge its exposure by:

A. Short futures and there was a decline in oil price

B. Long futures and there was a decline in oil price

C. Short futures and there was an increase in oil price

D. Long futures and there was an increase in oil price


Duration Hedging

dyPDdP *

Dollar duration

yFDFySDS FS **

2**

22*2

22*2

ySFSF

yFF

ySS

SDFD

FD

SD


Duration Hedging

FD

SDN

F

S

F

SF

*

*

2*

If we have a target duration DV* we can get it by using

FD

SDVDN

F

SV

*

**


Example 1A portfolio manager has a bond portfolio worth $10M with a modified duration of 6.8 years, to be hedged for 3 months. The current futures prices is 93-02, with a notional of $100,000. We assume that the duration can be measured by CTD, which is 9.2 years.

Compute:a. The notional of the futures contractb.The number of contracts to by/sell for optimal protection.


Example 1The notional is:

(93+2/32)/100$100,000 =$93,062.5

The optimal number to sell is:

4.795.062,93$2.9

000,000,10$8.6*

*

*

FD

SDN

F

S

Note that DVBP of the futures is 9.2$93,0620.01%=$85


Example 2On February 2, a corporate treasurer wants to hedge a July 17 issue of $5M of CP with a maturity of 180 days, leading to anticipated proceeds of $4.52M. The September Eurodollar futures trades at 92, and has a notional amount of $1M.

Compute

a. The current dollar value of the futures contract.

b. The number of futures to buy/sell for optimal hedge.


Example 2

The current dollar value is given by

$10,000(100-0.25(100-92)) =

$980,000

Note that duration of futures is 3 months,

since this contract refers to 3-month LIBOR.


Example 2

If Rates increase, the cost of borrowing will

be higher. We need to offset this by a gain, or

a short position in the futures. The optimal

number of contracts is:

2.9000,980$90

000,520,4$180*

*

*

FD

SDN

F

S

Note that DVBP of the futures is 0.25$1,000,0000.01%=$25


FRM-00, Question 73What assumptions does a duration-based hedging scheme make about the way in which interest rates move?

A. All interest rates change by the same amount

B. A small parallel shift in the yield curve

C. Any parallel shift in the term structure

D. Interest rates movements are highly correlated


FRM-00, Question 73What assumptions does a duration-based hedging scheme make about the way in which interest rates move?

A. All interest rates change by the same amount

B. A small parallel shift in the yield curve

C. Any parallel shift in the term structure

D. Interest rates movements are highly correlated


FRM-99, Question 61If all spot interest rates are increased by one basis point, a value of a portfolio of swaps will increase by $1,100. How many Eurodollar futures contracts are needed to hedge the portfolio?

A. 44

B. 22

C. 11

D. 1100


FRM-99, Question 61

The DVBP of the portfolio is $1,100.

The DVBP of the futures is $25.

Hence the ratio is 1100/25 = 44


FRM-99, Question 109Roughly how many 3-month LIBOR Eurodollar futures contracts are needed to hedge a position in a $200M, 5 year, receive fixed swap?

A. Short 250

B. Short 3,200

C. Short 40,000

D. Long 250


FRM-99, Question 109

The dollar duration of a 5-year 6% par bond is about 4.3 years. Hence the DVBP of the fixed leg is about

$200M4.30.01%=$86,000.

The floating leg has short duration - small impact decreasing the DVBP of the fixed leg.

DVBP of futures is $25.

Hence the ratio is 86,000/25 = 3,440. Answer A


Beta Hedging

represents the systematic risk, - the intercept (not a source of risk) and - residual.

itmtiiit RR

M

M

S

S

A stock index futures contractM

M

F

F

1


Beta Hedging

M

MNF

M

MSFNSV

The optimal N is F

SN

*

The optimal hedge with a stock index futures is given by beta of the cash position times its value divided by the notional of the futures contract.


Example

A portfolio manager holds a stock portfolio worth $10M, with a beta of 1.5 relative to S&P500. The current S&P index futures price is 1400, with a multiplier of $250.

Compute:

a. The notional of the futures contract

b. The optimal number of contracts for hedge.


Example

The notional of the futures contract is

$2501,400 = $350,000

The optimal number of contracts for hedge is

9.42000,350$1

000,000,10$5.1*

F

SN

The quality of the hedge will depend on the size of the residual risk in the portfolio.


A typical US stock has correlation of 50% with S&P.

Using the regression effectiveness we find that the volatility of the hedged portfolio is still about

(1-0.52)0.5 = 87% of the unhedged volatility for a typical stock.

If we wish to hedge an industry index with S&P futures, the correlation is about 75% and the unhedged volatility is 66% of its original level.

The lower number shows that stock market hedging is more effective for diversified portfolios.


FRM-00, Question 93A fund manages an equity portfolio worth $50M with a beta of 1.8. Assume that there exists an index call option contract with a delta of 0.623 and a value of $0.5M. How many options contracts are needed to hedge the portfolio?

A. 169

B. 289

C. 306

D. 321


FRM-00, Question 93

The optimal hedge ratio is

N = -1.8$50,000,000/(0.623$500,000)=289


VaR methods




Risk Factors

There are many bonds, stocks and currencies.

The idea is to choose a small set of relevant economic

factors and to map everything on these factors.

• Exchange rates

• Interest rates (for each maturity and indexation)

• Spreads

• Stock indices


How to measure VaR

• Historical Simulations

• Variance-Covariance

• Monte Carlo

• Analytical Methods

• Parametric versus non-parametric approaches


Historical Simulations

• Fix current portfolio.

• Pretend that market changes are

similar to those observed in the past.

• Calculate P&L (profit-loss).

• Find the lowest quantile.


Example

4.00

4.20

4.20

4.10

4.15

Assume we have $1 and our main currency is SHEKEL. Today $1=4.30.

Historical data:

4.30*4.20/4.00 = 4.515

4.30*4.20/4.20 = 4.30

4.30*4.10/4.20 = 4.198

4.30*4.15/4.10 = 4.352

P&L

0.215

0

-0.112

0.052


432 )063.01(

20

)062.01(

300

)061.01(

200

06.01

100

432 )13.01(

30

)12.01(

20

)11.01(

100

1.01

120

today

USD NIS

2000 100 -120

2001 200 100

2002 -300 -20

2003 20 30


today

432 )073.01(

20

)072.01(

300

)071.01(

200

07.01

100

432 )12.01(

30

)11.01(

20

)11.01(

100

11.01

120

Changesin IR

USD: +1% +1% +1% +1%NIS: +1% 0% -1% -1%

432 )063.01(

20

)062.01(

300

)061.01(

200

06.01

100

432 )13.01(

30

)12.01(

20

)11.01(

100

1.01

120


Returns

year

1% of worst cases


-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8

1

Profit/Loss

VaR

1% VaR1%


Variance Covariance

• Means and covariances of market factors

• Mean and standard deviation of the portfolio

• Delta or Delta-Gamma approximation

• VaR1%= P – 2.33 P

• Based on the normality assumption!


Variance-Covariance VVVaR 33.2%1

2.33

-2.33

1%


Monte Carlo

-1 -0.5 0.5 1

-1

-0.5

0.5

1


Monte Carlo

• Distribution of market factors

• Simulation of a large number of events

• P&L for each scenario

• Order the results

• VaR = lowest quantile


Monte Carlo Simulation

10 20 30 40

-15

-10

-5

5

10

15


Weights

Since old observations can be less relevant, there is a technique that assigns decreasing weights to older observations. Typically the decrease is exponential.

See RiskMetrics Technical Document for details.


Stock Portfolio

• Single risk factor or multiple factors

• Degree of diversification

• Tracking error

• Rare events


Bond Portfolio

• Duration

• Convexity

• Partial duration

• Key rate duration

• OAS, OAD

• Principal component analysis


Options and other derivatives

• Greeks

• Full valuation

• Credit and legal aspects

• Collateral as a cushion

• Hedging strategies

• Liquidity aspects


Credit Portfolio

• rating, scoring

• credit derivatives

• reinsurance

• probability of default

• recovery ratio


Reporting

Division of VaR by business units, areas of

activity, counterparty, currency.

Performance measurement - RAROC (Risk

Adjusted Return On Capital).


Backtesting

Verification of Risk Management models.

Comparison if the model’s forecast VaR with

the actual outcome - P&L.

Exception occurs when actual loss exceeds

VaR.After exception - explanation and action.


Backtesting

Green zone - up to 4 exceptions

Yellow zone - 5-9 exceptions

Red zone - 10 exceptions or more

OK

increasing k

intervention


Stress

Designed to estimate potential losses in abnormal markets.

Extreme events

Fat tails

Central questions:

How much we can lose in a certain scenario?

What event could cause a big loss?


Local Valuation

Simple approach based on linear approximation.

)()*( dyWorstPDdPWorst

Full Valuation

Requires repricing of assets.

)()( 00 yPdyWorstyPdPWorst


Delta-Gamma Method

The valuation is still local (the bond is priced only at current rates).

22

2

)(2

1dy

dy

Pddy

dy

dPdP

2)(5.0* dyCPPdyDdP


FRM-97, Question 13An institution has a fixed income desk and an exotic options desk. Four risk reports were produced, each with a different methodology. With all four methodologies readily available, which of the following would you use to allocate capital?A. Simulation applied to both desks.B. Delta-Normal applied to both desks.C. Delta-Gamma for the exotic options desk and the delta-normal for the fixed income desk. D. Delta-Gamma applied to both desks.


An institution has a fixed income desk and an exotic options desk. Four risk reports were produced, each with a different methodology. With all four methodologies readily available, which of the following would you use to allocate capital?A. Simulation applied to both desks.B. Delta-Normal applied to both desks.C. Delta-Gamma for the exotic options desk and the delta-normal for the fixed income desk. D. Delta-Gamma applied to both desks.

FRM-97, Question 13

Bad question!


Mapping

Replacing the instruments in the portfolio by

positions in a limited number of risk factors.

Then these positions are aggregated in a

portfolio.


Delta-Normal method

Assumes

• linear exposures

• risk factors are jointly normally distributed

The portfolio variance is

xxreturns T)(2Forecast of the covariance matrix for the horizon


Delta-normal Histor. MC

Valuation linear full full

Distribution normal actual general

Extreme events low prob. recent possible

Ease of comput. Yes intermed. No

Communicability Easy Easy Difficult

VaR precision Bad depends good

Major pitalls nonlinearity unstable model

fat tails risk


FRM-97, Question 12Delta-Normal, Historical-Simulations, and MC are various methods available to compute VaR. If underlying returns are normally distributed, then the:A. DN VaR will be identical to HS VaR.B. DN VaR will be identical to MC VaR.C. MC VaR will approach DN VaR as the number of simulations increases. D. MC VaR will be identical to HS VaR.


FRM-97, Question 12Delta-Normal, Historical-Simulations, and MC are various methods available to compute VaR. If underlying returns are normally distributed, then the:A. DN VaR will be identical to HS VaR.B. DN VaR will be identical to MC VaR.C. MC VaR will approach DN VaR as the number of simulations increases. D. MC VaR will be identical to HS VaR.


FRM-98, Question 6

Which VaR methodology is least effective for measuring options risks?

A. Variance-covariance approach.

B. Delta-Gamma.

C. Historical Simulations.

D. Monte Carlo.


FRM-98, Question 6

Which VaR methodology is least effective for measuring options risks?

A. Variance-covariance approach.

B. Delta-Gamma.

C. Historical Simulations.

D. Monte Carlo.


FRM-99, Questions 15, 90

The VaR of one asset is 300 and the VaR of another one is 500. If the correlation between changes in asset prices is 1/15, what is the combined VaR?

A. 525

B. 775

C. 600

D. 700


FRM-99, Questions 15, 90

BABABA 2222

15

5003002500300 222

BA

22 600BA


ExampleOn Dec 31, 1998 we have a forward contract to buy 10M GBP in exchange for delivering $16.5M in 3 months.

St - current spot price of GBP in USD

Ft - current forward price

K - purchase price set in contract

ft - current value of the contract

rt - USD risk-free rate, rt* - GBP risk-free rate

- time to maturity


**

1

1)1(,

1

1)1($

tt

tt r

GBPPVPr

PVP

ttttt

tt KPPS

r

K

r

Sf

*

* 11

KdPSdPdSP

dPdP

dfdP

dP

dfdS

dS

dfdf ttt

t

**

**


The forward contract is equivalent to

a long position of SP* on the spot rate

a long position of SP* in the foreign bill

a short position of KP in the domestic bill

P

dPKP

P

dPSP

S

dSSPdf

*

***


On the valuation date we have

S = 1.6595, r = 4.9375%, r* = 5.9688%

Vt = $93,581 - the current value of the contract

tt

ttt r

M

r

SMGBPQfV

1

5.16$

1

10*

MRM FRM-GARP Oct-2001 Zvi Wiener 02-588-3049 mswiener/zvi.html Market Risk Management.

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