Transcript שקופית 1 - Bar-Ilan University

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.
n
ct
P
t
t 1 1  y 
y cor  y gov  DRP
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
year
1
2
3
4
5
payment
80
80
80
80
1080
n
ct
80
80
80
80 1,080






t
2
3
4
5
1.1 1.1 1.1 1.1 1.1
t 1 1  y 
1,080
80  PVFA(10%,4) 
 $924.2
5
1.1
P
Yield to Maturity and Bond Price
P
5 years bond
10 years bond
y
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.
ct
P
t
n
n

1  y
P
D
  t  wt   t 
y
P
t 1
t 1
1 y
Thus,
P
D

P   D* P
y
1 y
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:
ΔP(bp)  DVBP
Numerical Example
A bond with $1,000 notional value and 8% coupon and 3 years to
maturity is traded with 10% yield to maturity.
Bond’s Payments
year
1
2
3
payment
80
80
1,080
n
ct
80
80 1,080
P



 950.26
t
2
3
1.1 1.1 1.1
t 1 1  y 
ct
t
n


1 y
D  t

P
t 1
1 80 2  80 3 1,080
 1.1  (1.1) 2  (1.1)3 

  2.777
950.26
The modified Duration is:
D
2.777
D 

 2.525
1 y
1.1
*
This implies that increasing of 1% in the yield will cause to:
P  D*yP  2.5251%  950.26  $24
DVBP  2.525  0.01%  950 .26  $0.24
Spot and Forward Rates
The yield curve is the relationship between the yield to and
the time to maturity.
The yield described by the spot rates, ST , which are derived
from zero-coupon bond prices with different maturity.
Prices of zero-coupon bonds with different maturity
Time to Maturity
Price ($)
1
934.58
2
857.34
3
772.18
1,000
1,000
934.58 
 S1 
 1  7%
1  S1
934.58
1,000
1,000
857.34 
 S2 
 1  8%
2
857.34
1  S2 
1,000
1,000
3
772.18 
 S3 
 1  9%
3
772.18
1  S3 
Yield Curve
Spot rate(%)
10
9
8
7
6
5
1
2
Time to Maturity
3
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
0
2
3
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:
(1 S2 )2  (1 S1 )(1 F1,2 )
(1  S2 ) 2
F1, 2 
1
(1  S1 )
and in general:
Ft ,T  T  t
(1  ST ) T
1
t
(1  St )
(1.08) 2
F1, 2 
 1  9%
1.07
(1.09)3
F2,3 
 1  11%
2
1.08
(1.09)3
F1,3 
 1  10%
1.07
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’s
S&P
Aaa
AAA
Aa
AA
A
A
Baa
BBB
Description
Very high quality
High quality: Very strong financial position
High capacity to pay interest and principle, but
more sensitive to the economic conditions
Medium quality: the capacity to pay may
change with economic conditions
Junk Bonds
Low quality – provide high yields but are very
speculative
Ba
BB
B
B
Caa
CCC
C
C
Very poor quality: No interest is being paid
D
D
Debt that is in default
The price fluctuations are relatively large
Very speculative bonds
Default Risk Premium – Spreads Over Treasuries
For a given maturity, the lower the bonds rating, the higher
its yield to maturity
9
8
7
DRP
6
5
4
3
2
1
0
Gov
AAA
AA
A
BBB
BB
B
CCC
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:
Pt
*
Rt 
 D  y t
Pt
 P
(R )  
 P


  D *   ( y )


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.
Specific bond
z+s+e
BBB
z+s
z
Treasury
3M
5Y
10Y
20Y
The movement in the bond price is:
ΔPi  DVBPiΔyi  DVBPiΔz j  DVBPi Δsk  DVBPi Δei
DVBP is the total dollar value of a basis point for the
associated risk factor.
Summing across the portfolio and collecting terms across
the common risk factors:
N
j
K
N
i 1
j1
k 1
i 1
V   DVBPi yi   DVBPijz j   DVBPik s k   DVBPi ei
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:
N
2 (V)  General_ Risk   DVBPi22 (ei )
i 1
Numerical Example
Portfolio Composition
Change in Basis Points at Time t
Bond
Years to
Maturity
Rating
DVBP
($M)
1
9.8
A
-0.2
2
1.2
A
-0.5
3
4.7
BBB
-0.1
3
 DVBP z
j
j1
j
2
 DVBP s
k 1
k
k
z
s
1
5
10
-10
-15
12
A
BBB
8
15
 10 (0.5)  15 (0.1)  12 (0.2) $4.1M
 8  ((0.2)  (0.5))  15 (0.1)  $7.1M
ΔV  $4.1M  ($7.1M)  $3M
Equity Portfolio Risk
The different market risk can be measured by the volatility of
the major indexes.
25
Volatility (2005)
HSI
20
DAX
S&P500
15
Nikkei
225
10
5
0
0
5
10
15
Volatily (2006)
20
25
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.
Stock Return (%)
R i t  i  i R Mt  et
20
18
16
14
12
10
8
6
4
2
0
Cov(R i , R M )
i
i 
 i , M
2
M
M
0
5
10
15
20
Market Return (%)
25
30
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
R P   w i   i R M  ei 
i 1
N
R p    P R M   w i ei
i 1
Equity Portfolio Risk
The portfolio variance is:
N
 2p  2p  2M   wi2 ei2
i 1
Suppose, is equally weighted and the residual variance are
the same for all stocks:
2

1


2 2
2
e
w


N






i ei
e
N
N
 
i 1
N
2
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.
Rit  i  1X1t  .....k Xkt  et
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
Country
2005
2006
Argentina
0.35
0.42
Canada
5.07
3.6
Britain
6.5
9.1
Hong Kong
0.27
0.26
Japan
11.1
6.6
Euro
9.8
8.3
South Afr.
4.2
8.4