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

y1cP

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$

1.1080,1)4%,10(PVFA80

1.1080,1

1.180

1.180

1.180

1.180

y1cP

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

y1c

twt

y1y

PP

D

Thus, PDPy1

DyP *

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

1.1080,1

1.180

1.180

y1cP 32

n

1tt

t

777.226.950

)1.1(080,13

)1.1(802

1.1801

Py1

c

tD32n

1t

tt

The modified Duration is:

525.21.1

777.2y1

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 maturityTime to MaturityPrice)$(

1934.58

2857.34

3772.18

%7158.934

000,1SS1

000,158.934 11

%81

34.857000,1S

S1000,134.857 22

2

%91

18.772000,1S

S1000,118.772 3

333

Yield Curve

5

6

7

8

9

10

1 2 3

Time to Maturity

Spot

rat

e)%

(

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(F2

2,1

%111

08.1)09.1(F 2

3

3,2

%10107.1

)09.1(F3

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 RatingMoody’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

0123456789

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

PPR

)y(DPP

)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

zz+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

222i

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 25Volatily )2006(

Vol

atili

ty )2

005( 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 RiskThe portfolio variance is:

2ei

N

1i

22M

2p

2p

iw

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

NN1Nw

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