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