INTF 6010 - Lecture 2 Revised

Introduction to Market Risk Measurement

INTF 6010 – Lecture 2

Market Risk

• Market Risk – is the risk of loss

due to movements in market factors

such as interest rates, exchange

rates, equity and commodity prices

• Market risk can affect instruments

and portfolios that contain bonds or

other fixed income instruments,

equities, foreign exchange positions

or instruments, derivatives,

commodity positions, structured

products.

Notes

Fixed Income

• Fixed income instruments or bonds

are a form of debt.

• The price of a bond is the present

value of all its future cash-flows

• The formula below shows that price

(P) is a function of yield (y)

Ct represents cash-flows (Principal or interest)in period t

t represents the number of periods(e.g. half years) to each payment

T is the number of periods to final maturity

y is the discounting factor per period (e.g. ys/2)

T

t

t

y

CP t

1 )1(

The chart below shows the relationship between Price and

Yield for a fixed income instrument and one with an option.

Price Yield Relationship

Market Risk – Fixed Income

• Fixed income risk arises from

potential movements in the level of

volatility of the risk factors,

usually taken as bond yields.

• Movements in yields reflect economic

fundamentals.

• The primary factor in determining

the level of interest rates is

inflationary expectations.

• For corporate and agency bonds or

foreign currency sovereign bonds

Credit Spread risk is another factor

that affects bond yields.

Notes

Bond Sensitivity Measures

• Two common derivatives of the price-

yield function are Duration and

Convexity

• Duration and Convexity are the first

and second derivatives of the price

yield function

• From a mathematical perspective:

– Duration represents a tangent to the

price yield function

– Convexity represents the curvature of

the price yield function

Notes

Duration

• Duration represents an average of

the time to wait for all cash-flows

• Duration is a measure of the

interest rate sensitivity of a bond.

• Duration and modified duration are

calculated using the formulas:

)1(*

/1 )1(

y

DD

PtC

DT

tt

t

y





P is the price of the bond

Convexity

• Convexity can be used together with

duration to give the full impact of

interest rate changes on the price

of a bond

• Convexity is calculated as follows:

PCtt

CT

ttt

y/

)1(

12

)1(





P is the price of the bond

The Duration and Convexity Effect

• The duration effect can be

summarized as follows:

• Duration represents the percentage

change in value of a bond from a 100

basis point (1%) change in yield.

• The combined duration and convexity

effects are as follows:

Convexity actually reduces the duration impact.

yPDPP 00 *

2021

00 * yCPyPDPP

Duration and Convexity

• Duration of a Zero Coupon instrument

is the same as its time to maturity.

• The duration of a floating rate

instrument is the time to the next

coupon reset.

• The duration of a callable bond is

calculated using the call date as

the maturity date.

• Convexity is always positive for

regular coupon paying instruments.

• All else equal duration and

convexity both increase for longer

maturities, lower coupons and lower

yields .

Notes

Market Risk - Equities

• Equity holders make returns from the

appreciation in price and payment of

dividends.

• Both price movement and dividend

payments are dependent on the

performance of the company.

• There are also preferred stocks

(preference shares) where the

dividend payments are fixed i.e.

specific rate and dates

Notes

Market Risk - Equities

• Equity Risk arises from potential

movements in the value of stock

prices

• Equity Risk can be decomposed into:

– Market wide Risk

– Stock specific Risk

• Volatility can be used as a measure

of the risk of a stock or stock

index

• Value at risk (VaR) can also be

calculated for a stock or portfolio

of equities.

Market Risk - Currencies

• Foreign Exchange (FX) or currency

market is the most actively traded

in the world.

• FX trading consists of Spot

Transactions, Forward Contract and

Currency Swaps.

• Currency conversion rates are

typically quoted in European Terms

i.e. units of the currency per US

dollar.

• The exceptions to this convention

are the British Pound and the Euro

which are quoted in American Terms

i.e. units of US dollar per unit of

foreign currency.

Notes


• Currency risk arises from potential

movements in the value of foreign

exchange rates.

• This can occur in the following

situations:

– Pure Currency Float where market

demand and supply determine FX rate

movements

– Fixed Currency System where rates are

subject to one off adjustments

(devaluations or revaluations).

– Change in currency regime where a

fixed currency system is changed to

floating or vice versa.

See example 11.3 on pg 263


• The risk associated with spot

transactions can be measured using

volatilities and value at risk

(VaR).Exercise – analyze the currency swap on pgs 264 – 266.

Market Risk - Commodities

• Commodities typically involve the

trading of contracts on:

– Agricultural Products e.g. wheat,

corn

– Livestock and meats e.g pork bellies

– Base Metals e.g. copper, aluminum

– Precious Metals e.g. gold, silver,

platinum

– Energy Products e.g. crude oil,

natural gas

• Commodity contracts include Spot,

Futures and Options on Futures.

Notes


Important concepts:

• Convenience Yield – benefit of

holding an inventory of a commodity

that is used in production.

• Lease Rate – When a commodity can be

lent out for profit.

• Contango – When the futures price of

a commodity is higher than the spot

price.

• Backwardation – when the spot price

is higher than the futures price.

If the lease rate on a commodity is ‘y’ and the risk free market interest rate is ‘r’ then:Contango occurs when y<rBackwardation occurs when y>r


• Commodity Risk arises from potential

movements in the value of commodity

prices.

• Volatility, correlations and value

at risk can be calculated for

commodity contracts.

• Energy commodities are more volatile

than other commodities since they

are less storable than metals and as

a result are more affected by

variations in demand and supply.

See Example 11.8 on pg 273

Portfolio Sensitivity Measures

• All of the concepts covered above

can be applied a portfolio level

– Portfolio Duration and Convexity can

be found by calculating the weighted

average.

Wi represents the weight of each security in the portfolio

See examples 6.15 and 6.16 on pg 150

N

iiip

N

tiiP

wCC

wDD

1

1

**

Portfolio Sensitivity Measures

• Portfolio volatility can be

calculated using the following:

• Correlations among the assets in the

portfolio is also very important. A

portfolio of assets that is highly

correlated will tend to move up and

down together. Correlation matrices

can be used to determine the

correlation among securities.

w represents the matrix of weights

Sigma represents the covariance matrix.

wwp '2

Derivatives

• A Derivative instrument is a private

contract that derives its value from

some underlying asset price, rate or

index such as a stock, bond,

currency or commodity.

• Derivatives can be traded in private

over the counter (OTC) markets or on

organized exchanges

• The most common derivatives include

Forward Contracts, Futures

Contracts, Swaps and Options

Notes

Forward Contracts

• Forward contracts are contracts to

buy or sell an asset, currency or

commodity at a specified time in the

future.

• Example: A corn farmer can enter

into a forward contract today to

sell corn at a specified price for

delivery in the future.

• Forward contracts are contracts

between parties and typically do not

involve a clearinghouse. This gives

rise to counterparty risk.

Forward Contracts

• The valuation of a forward contracts

can be calculated using the

following:

contractin units

ofnumber or quantity n

rate freerisk

foreignor yield dividend *r

rate freerisk current r

priceasset g UnderlyinK

Contract of ValueCurrent V

Price ForwardCurrent F

PriceSpot Current S

t-T

Delivery of Time T

TimeCurrent t

t

t

t

rt

rrt

rrtt

rt

rt

tr

t

tr

t

eKFKeeFKeeSV

eSeF

DPVSeF

SeF

)(

)(

*

*

Futures Contracts

• Futures are very similar to Forward

contracts in that they both allow

the ability to buy or sell something

using a price determined today but

executed at a date in the future.

• The key differences with Futures

include:

– Trades on organized exchanges

– Standardization i.e. fixed contact

sizes and limited expiration dates

– A Clearinghouse assumes the

counterparty risk

– Marked to Market Daily

– Margins are required

Valuation of Futures are done with the same formulas as Forward Contracts.

Swaps

• Swaps are OTC agreements to exchange

a series of cash-flows according to

pre-specified terms.

• Swaps are typically longer in tenor

than Forwards or Futures.

• Common Swaps include:

– Interest Rate

– Currency

– Credit Default Swaps (CDS)

– Commodity

– Total Return etc.

Review:Interest Rate Swap pg 239Currency Swap Pg 264

To be discussed in class

The diagram below shows the a typical interest rate swap

structure where

Interest Rate Swap Structure

The diagram below shows the a typical credit default swap

structure

Credit Default Swap

Options

• Options are instruments that gives a

holder the right (but not the

obligation) to buy or sell an asset

at a specific price (strike price)

usually on a specific date.

• Options to buy are called Call

Options while options to sell are

called Put Options.

• European Options can only be

exercised at maturity while American

Options can be exercised at anytime

before or at maturity.

Common notations:

Option Premium

Strike price or exercise price

Long or short (i.e. buy or sell)

Call and Put Options

• The two most common options are call

and put options

• Call option gives the buyer the

right to buy and asset while a Put

option gives the buyer the ability

to sell an asset.

• Holding a short position (i.e. you

sold options) then you become

obligated to buy or sell the asset

if the buyer exercises the option.

Notes

Payoff on Options

• The payoff profile of a long

position in a call option is:

• The payoff profile of a long

position in a put option is:

• From the payoff profiles when is it

best to use call and put options?

)0,( Ktt SMaxC

Ct = Value of the Call option

Pt = Value of the Put option

St = Current price of the asset

K= strike price of the option )0,( tt SKMaxP

Payoff for a Long Call

Combining Options

• Combinations of call and put options

to create various kinds of payoff

profiles.

• A long position in a call option

plus a short position in a put

option in the same asset with the

same strike price and maturity dates

is equivalent to a long position in

the underlying asset.

• Other payoff structures include

Straddles, Bull Spread, Bear Spread,

Butterfly Spread etc.

See pages 183 to 185 for more option combinations

Combining Options

• The link in the relationship between

the value of a call and put option

is known as Put-Call Parity. The

relationship can be expressed as:

• This works well for European options

but does not hold exactly fro

American Options as there is the

possibility of early exercise.

Put-Call Parity is demonstrated on table 8.1 on pg 181

rrrpc eKFKeSe )(*

General Relationships

• The value of an option consists of

two components:

– The Intrinsic Value which is the

value if exercised today.

– The Time Value which is the portion

of the option premium that is

attributable to the amount of time

remaining until the expiration of the

option contract.

Notes

General Relationships

Some other general option terminology:

• At-the-money – when the current spot

price is close to the strike price

• In-the-money – when the intrinsic

value is large

• Out of the money – when the spot

price is much lower than the strike

price

Notes

Upper and Lower Bounds

• The following are some general

bounds for European options:

• If these do not hold then there will

be arbitrage opportunities.

A American Call Option on a non-dividend paying stock should never be exercised early

An American Put Option on a non dividend paying stock may be exercised early

tr

t

tt

rtt

SCt

SKep

KPp

KeSc

c tt

Caps and Floors

• A cap is a call option on interest

rates with value:

• A floor is a put option on interest

rates with value:

• A Collar is a combination of buying

a cap and selling a floor.

Notes

]0,[ KiMaxC TT

]0,[ TT iKMaxP

Introduction to VaR

• Value at Risk or VaR was developed

to measure how much an investor

could lose on an investment over a

specified horizon with a specific

probability.

• It addresses the shortcomings in

other sensitivity measure such as

Duration.

• While sensitivity measures such as

Duration are useful they do not give

a probability of occurrence and

cannot be combined for different

types of assets.

Value at Risk

• VaR provides one number that that

aggregates the risks across the

whole portfolio, taking into account

leverage, diversification and

providing a risk measure with an

associated probability.

• VaR is defined as the worst expected

loss over a target horizon under

normal market conditions at a given

confidence level.

VaR is usually expressed as a dollar loss.

Visual Representation of VaR

Steps in Calculating VaR

1. Mark to market of the security or

portfolio

2. Measure the variability of the risk

factors (e.g. standard deviation of

returns)

3. Set the time horizon or holding

period.

4. Set the confidence level.

5. Calculate and report the worst case

loss using the information above

(e.g. $7 million VaR)

Notes

Steps in Calculating VaR

• The general VaR formula is as

follows:

• For bonds the VaR formula is as

follows:

• Basel 2 recommends a 10 day VaR at

99% confidence level as the Market

Risk Charge.

There are three dominant VaR Methods:

Parametric

Historical

Monte Carlo

teMarketValuVaR

IncreaseWorstYieldMDurationeMarketValuVaR

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INTF 6010 - Lecture 2 Revised