Identification of Risk Factors. Market Risk and Credit risk Market risk is defined as the risk of fluctuations in portfolio values due to volatility in.

Identification of Risk Factors

Market Risk and Credit risk

Market risk is defined as the risk of fluctuations in portfolio values due to volatility in market price.

Absolute and Relative risk

Absolute risk is measured in dollars terms – used for bank trading portfolios.

Relative risk is measured relative to benchmark index – used for pension funds that are given the task of beating a benchmark group.

Source of Market Loss – Bond and Equity

Bond

A bond value is defined as the present value of the bond’s payments discounted by the yield to maturity.

yield to maturity is the annual yield promised to investor who

will hold the bond to maturity.

DRPyy

y1

cP

govcor

n

1tt

t

Bond Valuation

Numerical Example

A bond with $1,000 notional value, 8% coupon and 5 years to

maturity is traded with 10% yield to maturity.

Bond’s Payments

year12345

payment808080801080

2.924$

1.1

080,1)4%,10(PVFA80

1.1

080,1

1.1

80

1.1

80

1.1

80

1.1

80

y1

cP

5

5432

n

1tt

t

Yield to Maturity and Bond Price

y

P

5 years bond

10 years bond

Yield to Maturity and Bond Price

Every things else are equal, the sensitivity of the bond price movement to movements in yields increases as:

The yields are lower

The time to maturity is longer

The coupon rate is lower

Duration

Duration is a measure of the sensitivity of the bond price movement to movement in yields.

Duration is measured as the weighted maturity of each payment, where the weights are proportional to the present value of the cash flow.

n

1t

tt

n

1tt P

y1

c

twt

y1y

PP

D

Thus, PDPy1

D

y

P *

where D* is modified duration and D*P is also known as the dollar duration.

This sensitivity is sometimes expressed in dollar value of a basis point:

DVBP)bp(PΔ

Numerical ExampleA bond with $1,000 notional value and 8% coupon and 3 years to

maturity is traded with 10% yield to maturity.Bond’s Payments

year123

payment80801,080

26.950

1.1

080,1

1.1

80

1.1

80

y1

cP 32

n

1tt

t

777.2

26.950

)1.1(080,13

)1.1(802

1.1801

Py1

c

tD32n

1t

tt

The modified Duration is:

525.21.1

777.2

y1

DD*

24$26.950%1525.2yPDP *

This implies that increasing of 1% in the yield will cause to:

24.0$26.950%01.0525.2DVBP

Spot and Forward Rates

The yield curve is the relationship between the yield to and

the time to maturity.

The yield described by the spot rates, ST , which are derived

from zero-coupon bond prices with different maturity.

Prices of zero-coupon bonds with different maturity

Time to MaturityPrice)$(

1934.58

2857.34

3772.18

%7158.934

000,1S

S1

000,158.934 1

1

%81

34.857

000,1S

S1

000,134.857 22

2

%91

18.772

000,1S

S1

000,118.772 3

333

Yield Curve

5

6

7

8

9

10

1 2 3

Time to Maturity

Sp

ot r

ate)

%(

Forward Rates

Forward rates, Ft,T are the rate on investment that start at a

future date t to time T.

Example

An investor who wishes to invest for 2 years has two

alternatives:

1. Buying two years bond with a spot rate S2

2. Buying one year bond with a spot rate S1, and roll

over the investment by entering to forward contract

to buy in the next year a one year bond with a

forward rate F1,2.

1 2 30

S1

S3

S3

F1,2

F2,3

F1,3

Forward Rates

Since the two portfolios must have the same payoff, we can infer F1,2 form:

and in general:

)F1)(S1()S1( 2,112

2

1)S1(

)S1(F

1

22

2,1

1)S1(

)S1(F tT

tt

TT

T,t

%9107.1

)08.1(F

2

2,1

%111

08.1

)09.1(F 2

3

3,2

%10107.1

)09.1(F

3

3,1

Default Risk Premium – Spreads Over Treasuries

Corporate bonds have an additional risk factor over government bonds - the risk of default.

Default Risk – firm’s failure to pay the coupon payment and/or the par value at maturity.

It causes the yields for corporate bonds to exceed those for Treasury bonds – the difference known as the spread over Treasury

The higher the risk of default, the lower the firm’s bond rating, the lower the bond’s market price, and the higher its yield.

Bond Rating

Moody’sS&PDescription

AaaAAAVery high quality

AaAAHigh quality: Very strong financial position

AAHigh capacity to pay interest and principle, but more sensitive to the economic conditions

BaaBBBMedium quality: the capacity to pay may change with economic conditions

Junk Bonds

BaBBLow quality – provide high yields but are very speculative

BBThe price fluctuations are relatively large

CaaCCCVery speculative bonds

CCVery poor quality: No interest is being paid

DDDebt that is in default

Default Risk Premium – Spreads Over Treasuries

For a given maturity, the lower the bonds rating, the higher

its yield to maturity

0

1

2

3

4

5

6

7

8

9

Gov AAA AA A BBB BB B CCC

DRP

Fixed-Income Risk

Fixed income risk arises from potential movement in the

level of bond yields.

The Fixed income risk can be measured either as return

volatility or yield volatility:

t*

t

tt yD

P

PR

)y(DP

P)R( *

Fixed-Income Portfolio Risk

The major problem with individual bonds is that there may not be sufficient history to measure their risk.

Therefore, we model the movement in each corporate bond yield by:

A movement in Treasury zero-coupon rates with a

closest maturity - zj

A movement in the DRP of the credit rating class to

which it belong - sk.

The remaining component, ei, which assumed to be

independent across the bonds.

iikijiiii eΔDVBPsΔDVBPzΔDVBPyΔDVBPPΔ

The movement in the bond price is:

DVBP is the total dollar value of a basis point for the associated risk factor.

3M 5Y 10Y 20Y

Treasury

BBB

Specific bond

z

z+s

z+s+e

Summing across the portfolio and collecting terms across

the common risk factors:

N

1iii

K

1kkik

j

1jjij

N

1iii eDVBPsDVBPzDVBPyDVBPV

Thus, a portfolio may consist of N=100 corporate bonds, but

we can summarize the yield risk only with j=5 government

bonds.

The total variance:

)e(DVBPRisk_General)V(N

1ii

222

i

Numerical Example

BondYears to Maturity

RatingDVBP

$(M)

19.8A-0.2

21.2A-0.5

34.7BBB-0.1

1510

z-10-1512

ABBB

s815

Change in Basis Points at Time tPortfolio Composition

M1.4$)2.0(12)1.0(15)5.0(10zDVBP3

1jjj

M1.7$)1.0(15))5.0()2.0((8sDVBP k

2

1kk

M3$)M1.7$(M1.4$VΔ

Equity Portfolio Risk

The different market risk can be measured by the volatility

of the major indexes.

0

5

10

15

20

25

0 5 10 15 20 25

Volatily )2006(

Vol

atil

ity

)200

5( S&P500

HSIDAX

Nikkei 225

Equity Portfolio Risk

The diagonal model is a statistical decomposition of the return of the stock i into a market-wide return and a residual which called the specific risk.

02468

101214161820

0 5 10 15 20 25 30

Market Return )%(

Stoc

k R

etur

n )%

(

tMiii eRRtt

M

iM,i2

M

Mii

)R,R(Cov

Equity Portfolio Risk

The diagonal model assumes that all specific risks are

uncorrelated.

Thus, any correlation between two stocks must come from

the joint effect market.

Therefore, with a large portfolio the specific risk should

cancel each other, and the only remaining risk is the general

market risk.

N

1iiiMPp

iMi

N

1iiP

ewRR

eRwR

Equity Portfolio Risk

The portfolio variance is:

2ei

N

1i

22M

2p

2p

iw

Suppose, is equally weighted and the residual variance are the same for all stocks:

NN

1Nw

2e2

e

22ei

N

1i

2i

Factor Model

The one factor model may miss common industry effects.

Adding factors, such as industry factors to the model

improves the precision of the individual stock return, and

decreases the error term.

tktkt11ii eXXRt

.....

The factors X are assumed to be independent

Currency Risk

Currency risk arise from potential movement in the value of foreign currencies.

Currency risk includes currency specific volatility and correlations across currencies, and devaluation risk. It arises in the following environments:

A pure currency float

A currency devaluation

A change in the exchange rate regime

Currency risk is also related to the interest rate risk – Often, interest rate are raised in effort to stem the depreciation of the local currency.

Exchange Rates Volatility Against the USD

Country20052006

Argentina0.350.42

Canada5.073.6

Britain6.59.1

Hong Kong0.270.26

Japan11.16.6

Euro9.88.3

South Afr.4.28.4