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INSTITUTE OF ECONOMIC STUDIES
Faculty of Social Sciences of Charles University
Value at Risk
Lecturer’s Notes No. 3
Course: Portfolio Theory and Investment Management
Teacher: Oldřich Dědek
VIII. VALUE AT RISK MEASURE
value at risk (VaR) is a method that attempts to provide a single number summarising the
total market risk in a portfolio of financial assets (contrary to a huge number of partial
risk measures such as “Greek letters analysis”)
VaR is intended to express the expected maximum loss (worst loss) over a target horizon
within a given confidence level
8.1 Building blocks of VaR
- probability distribution
rates of return on a portfolio R  W /W0 can be summarised by a random
~
variable R whose future outcomes are subject to a given probability distribution
~
density function: f ( x)  P( x    R  x   )
~
distribution function: F ( X )  P( R  X ) 
X
 f ( x)dx

mean (expected value):  

 xf ( x)dx (first moment – location)

variance (volatility):  
2

 (x  )
2
f ( x)dx (second moment –

dispersion)
Probability distribution
f (x )
R
C
(1-C) % of all probability

0
C % of all probability
- target horizon (holding period)
target horizon is a period of time over which changes in portfolio’s value are measured
it is a matter of subjective judgment; ideally it should correspond to the longest period
needed for orderly portfolio liquidation
2
relationship between VaRs for different holding periods
T-day VaR = 1-day VaR  T
formula is exactly true if changes in portfolio value on successive days have
independent identical normal distribution with mean zero
- confidence level
confidence level is a percentage number that determines how far the portfolio should be
protected against unexpected losses (i.e. Basel II uses 99 % confidence level)
a confidence level C determines a cut-off point  C (quantil) such that the area to its
right captures C % of all probability and consequently the area to its left captures a
residual (1-C) % of all probability
~
~
P( R   C )  C % or equivalently P( R   C )  (1  C) %
- definition of VaR
VaR for a given confidence level is defined as the monetary loss relative to the mean; it
is a difference between the expected change in the portfolio’s value and the change
that corresponds to the cut-off rate of return determined by selected confidence level
W0 … initial portfolio’s value (monetary amount)
 … expected rate of return (in percent)
W0 …expected change in portfolio’s value (monetary amount)
 C … cut-off rate of return for given confidence level (in percent)
 CW0 … loss corresponding to the cut-off rate of return (monetary amount)
value at risk = W0   CW0  (   C )W0
VaR is sometimes defined as the absolute monetary loss
VaR  W0  (1   C )W0   CW0
♦
A portfolio has a value 100 mil USD and its average monthly return is 3.5 %. There is only a
5 % chance that the portfolio will fall by more than 1.7 % over a month. Therefore the value
at risk for 95 % confidence level is
3
VaR(95 %) = [0.035  (0.017)]  100 mil  0.06 mil USD
The amount 60 000 USD represents a monthly underperformance relative to the average
outcome that happens with 5 % probability.
♦
VaR serves a number of purposes
- information reporting (senior managers are informed about the risks run by trading and
investment operations)
- resource allocation (setting position limits for trades)
- performance evaluation (benchmark for proper evaluation of risk exposures)
- prudential regulation (maintaining minimum levels of capital reserves against financial
risks)
8.2 Historical approach to the VaR measurement
[see also the excel file dpe02 Value at Risk]
historical approach assumes that the observed frequency distribution approximates quite
well the true probability distribution
for a given confidence level C one can find a cut-off point  C such that the number of
occurrences exceeding this value is C % (or the number of occurrences below this value
is (1-C) %)
VaR is computed according to definition
Frequency distribution and VaR
140
 95  0.012
120
100
80
5 % of occurrence
60
40
20
0
-0,035
-0,025
-0,015
-0,005
0,005
4
0,015
0,025
0,035
♦
An investment company is earning an average daily income 5.1 mil USD. The track record of
260 previous daily observations indicates that the number of cases showing a daily loss 9.6
mil USD and more accounts for 10 % of all observations. Under assumption that the observed
frequency distribution realistically describes the future distribution of earning possibilities, the
value at risk at 90 % confidence level is following
VaR(90 %) = 5.1 – (-9.6) = 14.7 mil USD
♦
disadvantages of historical approach:
- possible lack of a representative number of observations
- past behaviour need not be a good predictor of future behaviour
8.3 Parametric approach to the VaR measurement
[see also the excel file dpe02 Value at Risk]
VaR computation can be simplified considerably if the probability distribution can be
assumed to be normal
- normal distribution of rates of return
normal distribution (bell-shaped distribution) is fully characterised by its first two
moments: mean  and volatility 
~
P( N   ) 

 (x  )2 
exp

 dx
2 2 
 2  
1
66 % of distribution is between   and  
95 % of distribution is between  2 and  2
 2



5
2
- standard normal distribution
SND is a normal distribution with a mean of zero and variance of unity
~
every normal distribution N with parameters  and  can be changed into a standard
~
normal distribution  using a transformation
~
~ R


- relationship between cut-off points
~
if  C is the cut-off point of a standard normal distribution  then  C     C  is the
~
corresponding cut-off point of the associated normal distribution N (for convenience
 C is defined as positive number)
~
N 

~
~

1  C  P(   C )  P
  C   P( N     C  )
 

~
 P( N   C )
- computation formula for the parametric VaR
VaR  W0   CW0  [(   (    C  )]W0
  C W0
if the time horizon for which the VaR is measured differs from the unitary horizon over
which the volatility is measured then
VaR   CW0 t
Most frequent cut-off points
Confidence level
90 %
Cut-off point
1.282
95 %
1.645
99 %
2.326
disadvantages of normal distribution:
- perfect symmetry of normal distribution does not fit some types of risky profiles
(particularly if fails to reflect asymmetries of credit risk)
- „thin tails“ of normal distribution tend to underestimate probability of occurrence of
extreme outcomes
6
♦
Compute the VaR (for confidence level 99 % and holding period 10 days) for each of the two
shares that display following properties:
Share
A
Value
(mil USD)
10
One day
volatility (%)
2
B
5
1
The value of the cut-off point for chosen confidence level is  99  2.326 . Hence
VaR(A)  2.326  0.02  10  10  1.47 mil USD
VaR(B)  2.326  0.01  5  10  0.37 mil USD
♦
diversified and undiversified VaR
~
if rates of return of individual risky assets Ri are normally distributed than the rate of
~
return of a portfolio R P that is composed of these assets is also normally distributed (a
linear combination of normal distributions is also normal distribution)
N
N
W
~
~
R P    i Ri ,  i  i , W P   Wi
WP
i 1
i 1
formula for the portfolio’s VaR can be rewritten in terms of individual VaRs
VaR P   C  PWP0  WP0

 
N
i 1

i 1
N
2
C
N
 i2 i2  2   C2 i j  i j  ij
i 1 j i 1
 iWi 0   2   C  iWi 0  C  jW j0  ij
N
2
C
N
i 1 j i 1
N
 VaR 
i 1
N

i
2
N
 2
N
 VaR VaR
i 1 j i 1
i
j
 ij
portfolio VaR that takes into account a correlation structure among constituent assets is
called diversified VaR (notation is DVaR)
DVaR of a portfolio can be compared with undiversified VaR of the portfolio
(notation is UVaR) which is the sum of VaRs of constituent assets
7
N
UVaR P   VaR i
i 1
UVaR coincides with DVaR only in case of perfect positive correlations among risky
assets
♦
Compare the DVaR and UVaR for a two-share portfolio from the previous example if the
correlation coefficient that measures correlation between the two shares is 0.3.
UVaR P  1.47  0.37  1.84 mil USD
DVaR P  1.47 2  0.37 2  2  1.47  0.37  0.3  1,62 mil USD
The diversification effect decreases the VaR of the portfolio by 1.84 – 1.62 = 0.22 mil USD.
♦
8.4 Monte Carlo simulation
[see also the excel file dpe02 Value at Risk]
Monte Carlo method repeatedly simulates a random process for the financial variable
the method is extensively used where complex financial instruments cannot be priced by
applying analytical formulas
- choice of a particular stochastic model
a variable follows a stochastic process if its values change over time in an uncertain way
Markov property means that the only present value of a variable is relevant for
predicting the future
Wiener process (Brownian motion) is a particular type of Markov stochastic process
with a mean change of zero and a variance rate of unity
geometric Brownian motion (for a price of share)
S t 1  S t
 t   t t
St
t … expected return per unit of time (expected component or drift)
 t t … stochastic component (  t can be seen as random drawing from
standard normal distribution)
 t … volatility per unit of time
8
Monte Carlo simulation is prone to model risk (if the stochastic process chosen in
unrealistic so will be the estimated VaR
- creating random numbers
a computerised random number generator uses a uniform distribution over the interval
[0,1]
a random drawing form uniform distribution must be transformed into a random
drawing from standard normal distribution (by using the inverse function to the
distribution function of the standard normal distribution)
1
2
0
1
1
0
2
0
0
0
random numbers are generated as many times as necessary, number of iterations should
reflect the trade-off between accuracy and computation cost
- computation of VaR
random number are used for creating a simulated frequency distribution
next step in computing VaR is identical with the historical approach to the VaR
measurement
Cholesky factorisation
Cholesky factorisation is a method for adding an observed correlation structure into the
simulated random process
random numbers that simulate stochastic behaviour of individual components of a portfolio
must preserve a given correlated behaviour of these components
~1 ,~2 ,,~N a set of original random variables that are generated by a computer
program (mutually independent standard normal distributions)
9
~1 , ~2 ,, ~N a set of transformed random numbers that respect correlation structure
among components of a multi-asset portfolio
Cholesky transformation is a triangular system of equations:
~  a ~
1
11 1
~2  a 21~1  a 22~2
....................................
~N  a N 1~1  a N 2~2    a NN ~N
where coefficients a ij are found in an iterative way that uses following properties:
1  var( ~i )  var( ai1~1    aii~i ), i  1,, N
 ij  cov(~i , ~j )  cov( ai1~1    aii~, a j1~1    a jj~ j ), i  j
var(~i )  1, cov(~i ,~ j )  0 i  1,, N , i  j
computation of Cholesky factors for two-asset portfolio:
1  var( ~1 )  var( a11~1 )  a112 var(~1 )  a112  a11  1
2
2
1  var(~2 )  var( a21~1  a22~2 )  a21
 a22
  cov(~1 , ~2 )  cov( a11~1 , a 21~1  a 22~2 )  cov( a11~1 , a 21~1 )  cov( a11~1 , a 22~2 )
 a11a 21 var(~1 )  a11a 21  a 21  
2
a 22  1  a 21
 1  2
random numbers that preserve the correlation structure are the product of the following
transformation
~  ~
1
1
~2  ~1  1   2 ~2
8.5 Linear and nonlinear risk
partial approach to the VaR measurement (so called particle finance) tries to decompose
the total risk exposure to individual risk components
linear products – a linear relationship exists between movements in the underlying risk
factor and the profit and loss profile of the dependent variable
nonlinear products – the relationship between the profit and loss profile of the dependent
variable can be represented by a curve (smooth versus kinked non-linearity)
10
value at risk of linear products
- relationship between the price of forward contract and the price of the underlying
asset (see the handout Futures)
f 
1  (r  d )T
 S
1  rT
Linear risk of a forward contract
f
f1
S
f0
S0
S1
- relationship between the percentage change in the price of zero-coupon bond and the
yield to maturity
P
1
(1  r ) T

dP
d (1  r )
 T 
P
1 r
delta of the product is the sensitivity of the value of the financial instrument V with
respect to the change in the price of the underlying risky factor F

V
F
linear products have constant delta hence
V    F
var(V )  var(   F )  2 var(F )
V    F
if F is normally distributed then V is also normally distributed and VaR of the linear
product is
VaR(V )   C   V    ( C F )    VaR(F )
the formula for the linear VaR is frequently used as an approximation of a true
(nonlinear) VAR
11
♦
A bond portfolio has a value of 10 million USD and modified duration of 3.8 years. Interest
rates are currently 10 % and there is a 5 % chance that interest rates will go above 11.2 % at
the end of the year. What would be the VaR for the portfolio at 95 % confidence level?
VaR  3.8  (0.112 - 0.1)  10000000  456000 USD
♦
value at risk of nonlinear products
examples: - relationship between the price of coupon bond and the yield to maturity
(duration-convexity approximation)
P
P

D
1
r  K r 2
1 r
2
- relationship between the price of stock option and the price of an underlying
asset (delta-gamma approximation)
 C     S  12    S 2
Nonlinear kinked risk of an option contract
C
0
X
S
if risky factor F is normally distributed then the value of the product V need not be
normally distributed and the parametric approach to the VaR assessment is only an
approximation of true risk exposure
the formula of nonlinear VaR can be derived in the following way
V    F  12   F 2
12
var(V )  var(   F  12   F 2 )
 2 var(F )  ( 12 ) 2 var(F 2 )  2( 12 ) cov(F , F 2 )
 2 var(F )  12 ( var(F )) 2
the term cov(F , F 2 ) is zero if variable F is normally distributed (all odd
moments are zero for symmetrical probability distribution)
it can be shown for normal distribution that var(F 2 )  2 var 2 (F )
VaR of smooth nonlinear product
VaR (V )   C V V   CV 2 F2  12 ( F2 ) 2
♦
A British exporter opened a long position in sterling currency put options to hedge future
conversion of dollar revenues into pounds. The option portfolio has the actual value of
1200000 GBP. The delta of one currency option is 0.44 and the gamma is 3.1. The exchange
rate dollar-sterling volatility is 3.3 %. What is the VaR of the option portfolio at 95 %
confidence level?
VaR  1.654  1200000  0.44 2  0.0332  0.5  3.12  0.0334  29047 GBP
♦
computation of VaR in case of kinked (broken) nonlinearity must be approximated by
simulation techniques like Monte Carlo simulation
13
IX. CREDIT RISK MEASUREMENT
credit risk refers to potential losses that are experienced when
i) counterparties are unwilling or unable to fulfil their contractual obligations (i.e.
repayment of a bond or coupons of the bond, repayment of loan or charged interest);
ii) debtors are downgraded by credit agencies
iii) credit spreads are widened due to external events beyond debtor’s control (i.e.
increased political uncertainty, a higher macroeconomic instability)
rating agencies provide ratings that describe the creditworthiness of debtors
Moody’s: Aaa, Aa (Aa1, Aa2, Aa3), A (A1, A2, A3), Baa, Ba, B, Caa, Ca, C, D
S&P: AAA, AA (AA+, AA-), A (A+, A-), BBB, BB, B, CCC, CC, C, D
9.1 Expected default loss on bonds
credit spread is an excess of the yield of the risky bond over the yield of the risk-free bond
zero rates extracted from risk-free (AAA) government bond yields
z1 , z 2 ,, zT
zero rates extracted form bond yields of given credit rating X:
1X ,  2X ,,  TX , where X = AA, A, BBB, and so on
spread curve:
1X  1X  z1 ,, TX   TX  zT
Zero yield curves for different classes of credit risk
BBB
Yield
 BBB
A
AA
 AA
AAA
Maturity
14
empirical patterns: - the spread increases as the rating declines
- the spread tends to increase faster with maturity for lower credit
ratings than for higher credit ratings
present value of cost of default is assumed to be equal to the excess of the price of a bond
of a given credit rating over the price of the risk-free bond that has the same maturity
and pays the same coupon
price of risky zero-coupon T-year bond: P 
M
(1   TX ) T
price of risk-free zero-coupon T-year bond: Q 
M
(1  zT ) T
loss from default:
L
(1  zT ) T
QP
 1
 1  (1  ( zT   TX )) T
X T
Q
(1   T )
 1  (1  T  ( zT   TX ))  T  TX
♦
A five year zero-coupon Treasury bond yields 5 % and five year yield of zero-coupon
corporate bond yields 6 %.
Q
100
100
 78.35, P 
 74.73
5
(1  0.05)
(1  0.06) 5
Present value of loss from default is
L
78.35  74.73
 4.62 %  5  (0.06  0.05)  5 %
78.35
♦
extracting default losses from swap rates
swap quotes can be used for extracting default losses on the assumption that LIBOR curve is
the risk-free curve
fixed rate
coupon
Investor
LIBOR
Swap
dealer
by holding the bond and buying the swap an investor is still exposed to the credit risk
15
that the bond may default but is compensated for that risk by receiving an extra yield
over the risk-free rate ( = coupon – fixed rate)
the present value of that compensation can be interpreted as the loss from default
♦
An investor owns a fixed-rate corporate bond that is currently worth par and pays a coupon of
6 % per year. The LIBOR yield curve is flat at 4 % with semiannual compounding. The swap
fixed rate is 4.5 % paid semiannually. The compensation for the bond’s credit risk is thus 150
basis points for five years paid semiannually.
10 1
100  0.015
1
 2 1
 0.0665  6.65 %
loss from default =
100 t 1 (1  2 0.045) t
♦
Investor has the opportunity to buy five year bond with coupon 7.5 % together with the five
year swap paying fixed rate 6 %. The yield curve for zero rates is flat at the level 4.5 %.
9.2 Risk-neutral probability of default
risk-neutral environment means that expected return required by all investors on all
investment is the risk-free interest rate (consequence of the efficient market hypothesis)
risk-neutral probability is the probability that a risky bond will default during the its life
in risk-neutral environment (as opposed to empirical probability of default)
- expected pay-off of the bond at its maturity
qT  R  M  (1  qT )  M
qT … risky-neutral probability that the bond will default at maturity T
R… recovery rate (proportion of the claimed amount in the event of a
default)
M … face value of the bond
- no arbitrage condition in a risk-neutral world
P
q  R  M  (1  qT )  M
M
 T
 Q  (qT R  1  qT )
T
(1   T )
(1  zT ) T
- probability of default
16
qT 
 (1   TX ) T
 T  TX
1
QP
1
 


 

1
1 R
P
1  R  (1  zT ) T
 1 R
♦
A ten year corporate bond rated BBB has a spread 220 bp over the ten year Treasury bond.
Historical recovery rate for bonds of this risky class is 40 %. The risk-neutral probability that
the bond defaults at maturity is therefore
q10 
10  0.022
 0.37  37 %
1  0.4
♦
loss given default (LGD) is an estimation of the percentage loss in the event of default
LGD = 1 – R
T  TX
expected loss = qT  (1  R)  (1  qT )  0  qT  LGD 
 LGD  T  T
1 R
♦
Maturity
1
Riskless
return (%)
5.00
Risky
return (%)
5.25
Spread
(pcp)
0.25
Expected
loss (%)
0.25
2
5.00
5.50
0.50
1.00
3
5.00
5.70
0.70
2.10
♦
9.3 Historical probabilities of default
historical (empirical) probabilities are estimated from historical data and they show default
experience of bonds through time that started with a certain credit rating
marginal mortality rates (MMR) are probabilities that a bond of a given credit rating will
default in a given year of its life
MMR 1X =
total value of grade X bonds defaulting in year 1 of issue
total value of grade X bonds outstanding in year 1 of issue
MMR X2 =
total value of grade X bonds defaulting in year 1 of issue
total value of grade X bonds outstanding in year 1 of issue
17
MMRs are computed as a weighted average of yearly values over some sample of years
MMR Xt   MMR Xt, j  w j , w j are relative issue sizes in different years
j
Rating
Maturity
1
2
3
…
…
15
AAA
0.00
0.00
0.04
…
…
0.67
AA
0.01
0.04
0.10
…
…
1.39
A
0.04
0.12
0.21
…
…
2.59
BBB
0.24
0.55
0.89
…
…
6.85
…
…
…
…
…
…
…
CCC
25.26
34.79
42.16
…
…
66.12
MMR 3BBB  0.89  probability that the bond that was issued with the rating
BBB will default in the third year of its life is 0.89 %
MMRs of high quality bonds tend to increase with the passage of time while
MMRs of low quality bonds tend to decrease with the passage of time
survival rates (SR) are probabilities that the bond of a given credit rating will not default
in a given year of its life
SR Xt  1  MMR Xt
cumulative mortality rates (CMR) are probabilities that the bond of a given credit rating
will default over a given period T
CMR TX  1  SR 1X  SR X2    SR TX
historical default probabilities are significantly less than those derived from an analysis of
bond prices
explanation: - investors want to be compensated for other types of risks (i.e. liquidity risk,
unforeseen bad scenarios, etc.)
- risk-neutral environment may not correspond to the actual market conditions
(an expected return should be discounted by a higher than a risk-free rate)
18
9.4 Credit at Risk
[see also the excel file dpe03 Credit at Risk]
credit at risk (CaR) denotes the value-at-risk approach applied to the measurement of
credit risk
application of parametric formula CaR   C W 0 , in which  denotes volatility of an
underlying asset duet to the credit risk
qualifications: credit returns are typical with skewed returns (higher probability of
downgrading than upgrading) and a fat downside tail (large losses from extreme credit
events)
 normal distribution need not be the appropriate description of credit events and
the parametric formula is only an approximate measure of the credit risk
skewed distribution with
fat tail
symmetrical distribution
with thin tails
CreditMetrics is a registered trademark of J. P. Morgan
calculation of credit risk exposure of each individual bond
a) credit transition matrix contains historical probabilities of credit migration over a time
horizon (mostly one year)
probabilities that a bond with a given credit quality changes its credit ratings (upgrade,
downgrade, remains the same)
BBB
AAA
AA
A
BBB
BB
B
CCC
D
0.02
0.33
5.95
86.93
5.30
1.17
0.12
0.18
b) calculation of a zero credit curve for each credit rating level
application of a standard bootstrap method across the credit rating spectrum
19
Rating
1
2
3
A
6,25
6,75
7,5
BB
9,5
9,7
10,0
c) determination of bond values for each credit migration
♦
A three year BBB bond pays coupon of 8 %. If the bond is upgraded to the rating A, its price
will be
PA 
8
8
108


 101.49
2
1.0625 1.0675
1.0753
If the bond is downgraded to the rating BB, its price will be
PBB 
8
8
108

 3  95.1
2
1.095 1.097
1.1
In a similar manner the bond value for each credit migration can be determined
BBB
AAA
AA
A
BBB
BB
B
CCC
D
104.0
103.0
101.5
100.0
95.1
87.0
75.0
58.0
♦
d) measuring the expected value and volatility of each bond due to credit risk
E ( P)  p AAA PAAA  p AA PAA    pCCC PCCC  p D PD
 2 ( P)  p AAA ( PAAA  E ( P)) 2    p D ( PD  E ( P)) 2
p X … probability of changing the rating to the grade X
PX … bond price at the rating X
a bond value for the rating D (default) can be obtained from empirically observed
recovery rates
e) calculation of the bond CaR
given a confidence level C the standard formula of VaR (now called CaR) can be
applied
CaR   C  P P 0
20
♦
Compute the credit at risk of a bond portfolio whose nominal value is 20 mil USD and
standard deviation of its price fluctuation is 2.68 %. The confidence level is 95 %.
CaR  1.645  0.0268  20 000 000  881720 USD
♦
calculation of credit risk exposure of a two-bond portfolio
a) calculation of portfolio’s end-values
preparing a table of all possible combined values of the two bond portfolio at year end
combined value is the sum of individual values
number of possible combined values is N×N, where N is the number of credit classes
First bond
Second bond
AAA
AA
..........
CCC
D
P1(AAA)
P1(AA)
..........
P1(CCC)
P1(D)
..........
..........
AAA
P2(AAA)
AA
P2(AA)
…….
……….
CCC
P2(CCC)
..........
D
P2(D)
..........
210.81
209.48
209.50
..........
208.16
..........
..........
..........
..........
b) calculation of joint probabilities of portfolio’s end-values
First bond
Second bond
AAA
p2(AAA)
AA
p2(AA)
…….
……….
AAA
AA
..........
CCC
D
p1(AAA)
p1(AA)
..........
p1(CCC)
p1(D)
..........
..........
0,0000
0,0000
0,0001
..........
0,0005
..........
..........
..........
..........
CCC
p2(CCC)
..........
D
p2(D)
..........
21
0,0012
0,0013
0,0018
0,0019
if probabilities of credit migration of the two bonds are independent on each other then
joint probabilities of portfolio’s end-values are simply the product of individual
probabilities
c) measuring the expected value and volatility of each bond due to credit risk
E ( P  Q)  p AAA, AAA ( PAAA  Q AAA )  p AAA, AA ( PAAA  Q AA )    p D , D ( PD  QD )
 2 ( P  Q)  p AAA, AAA ( PAAA  QAAA  E( P  Q)) 2    pD,D ( PD  QD  E( P  Q)) 2
CaR   C    ( P 0  Q 0 )
d) estimating correlation between bond prices
independent transition probabilities are not likely for issues from the same or related
industries or if issues are exposed to common macroeconomic factors
correlations are estimated by means of multifactor models of stock returns for individual
borrowers (stock returns are observed if issuer’s stock is publicly traded)
♦
An issuer of A-rated bond is a universal bank (sensitive to the business climate in banking and
insurance industries) and an issuer of BB-rated bond is a chemical company (sensitive to the
business climate in the chemical industry). Estimated sensitivities of the two issues to
industries’ returns are following
R A  0.15  RBANK  0.74  RINS
RBB  0.9  RCHEM
Estimations of inter-industries’s correlations are following
 CHEM , BANK  0.08
 CHEM , INS  0.16
Estimated proxy correlation between the bond issues is following
 ( A, BB )   (0.15RBANK  0.74 RINS ,0.9 RCHEM )
 0.15  0.9   ( RBANK , RCHEM )  0.74  0.9   ( RINS , RCHEM )
 0.15  0.9  0.08  0.74  0.9  0.16  0.1174
♦
22
e) computing credit rating thresholds
it is assumed that there is a series of threshold levels for changes in firm’s asset value
that determine the firm’s credit rating
Credit rating thresholds for the BB issue
D
B
CCC
AD
ACCC
BBB
BB
AB
ABBB
Bb AA
A
AA
AAA
AAA
AAAA
AX is a threshold rate of return that triggers the change in rating (downgrade or
upgrade)
W W  AD  the BB bond will default
AD  W W  ACCC  the BB bond will be downgraded to the grade CCC
AB  W W  ABBB  the BB bond will retain its rating
AAA  W W  AAAA  the bond will be upgraded to the grade AA
AAAA  W W  the bond will be upgraded to the highest grade AAA
threshold levels can be easily determined under assumption that the probabilities of
~
underlying changes in the firm’s value W / W follow a normal distribution N (  ,  )
W
~
 N (  ,  ) has a normal distribution (with mean  and volatility  )
W
Z
W / W  

follows a standard normal distribution (with mean zero
and unitary volatility)
( )  P( z   ) is the cumulative standard normal distribution function
finding the threshold level for the default
p D  PW W  AD   P  Z  AD 
A 

 P Z  D
  PZ   D    D 
 

23
 D corresponds to the area under the curve of standard normal distribution whose
cumulative probability is equal to the probability of default
 D can be found from the inverse function to the standard normal distribution
in a similar way the threshold level for the grade CCC can be determined
W


pCCC  P AD 
 ACCC   P AD    Z  ACCC 
W


A 
A 
 P D
 Z  CCC
   ( CCC )   ( D )

 

♦
A bond is currently rated at level A. Threshold levels for credit migration can be determined
in the following way:
Migration
probabilities
Cumulative
probabilities
Threshold
levels
D
CCC
B
BB
BBB
A
AA
AAA
0.06
0.01
0.26
0.74
5.52
91.05
2.27
0.09
0.06
0.07
0.33
1.07
6.59
97.64 99.91
100
-3.24 -3.19 -2.72 -2.30 -1.51
1.98
3.12
♦
e) computation of correlated joint probabilities
joint probabilities can be found by integrating the density function of bivariate
normal distribution
number of computations = (number of rating grades)2
P AX  W1 W1  AX 1 , AY  W2 W2  AY 1    X  x   X 1 ,  Y  y   Y 1 

 X 1  Y 1
  x    y    f ( x, y)dxdy
1
X
2
Y
where
f ( x, y) 
1
21 2

 ( x  1 ) 2
( x  1 )( y   2 ) ( y   2 ) 2 
1


exp 

2


2 
2
 1 2
 22
1  2

 2(1   )   1
24
f) CaR of a large bond portfolio
it can be shown that the risk of a portfolio of N bonds depends on the risk of each
pairwise combination as well as the risk of each bond individually
N
var( P1    PN )   var( Pi )  2 cov( Pi , Pj )
i 1
i j
var( Pi  Pj )  var( Pi )  var( Pj )  2 cov( Pi , Pj )
N 1
var( P1    PN )  
N
N
 var( Pi  Pj )  (n  2) var( Pi )
i 1 j i 1
i 1
References
Anson M. P.: Credit Derivatives, Frank J. Fabozzi Associates, 1999.
Butler C.: Mastering value at risk, Prentice Hall, 1999.
Jorion P.: Value at Risk, McGraw-Hill, 1995.
Saunders A.: Credit Risk Measurement. John Wiley & Sons, Inc., 1999.
25