1. INVESTX PLC
Portfolio Risk Report
Credit Metrics analysis on bonds’ credit risk and portfolios values
M.Zhang
3/7/2015

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Spreadsheet in Excel Page in Report
Overview P1
Method "Data Analysis", "Portfolio 1",
"Portfolio 2"
P2-4
Results "Comparison" P5-7
Recommendation P8
Limitations P9-10
Overview
This report analyses two bond portfolios’ credit risk with Credit Metrics methodology, which
considers how each bond’s credit rating changes given the correlation between them. So we
can do simulations and see the effect of upgrading/downgrading of bonds on the whole
portfolio’s value in one year and the credit risk is quantified by calculating Value at Risk for
the portfolio.
There are 4 states for the bonds: A rated, B rated, C rated and Default.
Nine bonds are available for investment: 6 B rated bonds (from companies: R, S, T, U, V, W)
and 3 C rated bonds (from companies: X, Y, Z) with 10 years’ term and 5% annual coupon.
Our client wishes to invest in B rated bonds only for his portfolio as he thinks it is safer.
However, considering the correlations between these bonds, there may be potential big losses
occurring. Therefore, this report looks at the following 2 portfolios:
 A portfolio of 6 B rated bonds in same industry (R, S, T, U, V, W)
 A portfolio of 3 B rated bonds in same industry (R, S, T) and 3 C rated bonds in
different industries (X, Y, Z)
Being in the same industry implies higher correlation between one bond to another.
By simulating portfolio values in one year’s time, we analyse the distribution pattern and
99.5% VaR for the 2 portfolios and make a recommendation about which portfolio is safer to
invest in.

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Method
Credit Metrics model assesses credit risk due to changes of credit rating of bonds in a
portfolio, e.g. downgrading and upgrading’s effect on credit quality. By simulating different
scenarios considering the correlation between bonds, we can obtain the VaR for the portfolio
in one year.
Steps taken:
 Obtain Covariance Matrix and Cholesky Matrix for bonds in the portfolio
 Simulate multivariate Normal random variables using Cholesky Matrix
 Return the simulated values back to the according credit rating states using transition
matrix
 Calculate the values for a bond in each credit rating state
 Obtain the simulated portfolio values in one year’s time
 Obtain VaR 99.5% for the simulations for each portfolio
Correlation between the bonds in the portfolio is obtained by analysing the equity returns for
the companies for the past 3 years.
To do this, we first need to find the Covariance Matrix as seen in ‘Data Analysis’ spreadsheet.
Then the Cholesky Matrix( C ) is obtained from the Covariance Matrix ( ∑ ) by performing
Cholesky decomposition1
to the Covariance Matrix.
The result is checked by multiplying Cholesky Matrix with its transpose to fit equation:
CC’=∑
Cholesky Matrix allows us to generate multivariate Normal random variables xi with same
correlation as the equity returns between the companies, which we assume to represent the
bonds’ correlations in the portfolio.
The values in Excel is copied and pasted values for illustration. The first 10 rows contain
formula used, which can be easily reproduced if more simulations are needed. Then we
multiply the standard Normal random variables by transpose of Cholesky Matrix.
These correlated random variables would then be returned back to the credit ratings.
Since each simulation trial contains variables xi from standard Normal distribution with same
correlation as the bonds, it determines the new ratings in one year for each bond. Given the
Transition Matrix, we can return xi to the credit rating it belongs to in one year’s time by
inversing the cumulative transition probability with standard Normal function to set the
boundary for categorising the simulated values.
1
See Paul Sweeting- Enterprise Risk Management Formula 10.97

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The following diagrams show how the simulated results can be categorised into four states:
A rated, B rated, C rated, or Default for a bond starts as B rated and C rated respectively.
We can see a B rated bond has largest probability to remain B rated.
-3 -2 -1 0 1 2 3
B Rated Bond
Bond remains
B rated
ACDefault
1.47-1.126-1.881
If xi falls
below
-1.881
If xi falls
between
-1.881
and
-1.126
If xi falls between
-1.126 and 1.476
If xi is
above 1.476
-3 -2 -1 0 1 2 3
C Rated Bond
Bond remains
C rated
ADefault B
1.641.036-1.036
If xi falls
below
-1.036
If xi falls between
-1.036 and 1.036
If xi falls
between
-1.036 and
1.644
If xi is
above 1.644

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C rated bond is most likely to remain C rated and has larger probability to default.
The next step is to calculate the values of the bonds in different states. Recovery rate of a
defaulted bond is 30% of face value. Current risk free rate is 2.5% and credit spread for A
rated, B rated and C rated bond are 0.5%, 2%, and 6% respectively. With coupons payable
for 10 years in arrears, using discounted cash flow methods, the current bond value and
values at the end of first year for each rating can be obtained by annuity formula.
Below is a summary of implied risk discount rate for each rating and the values of the bonds:
Credit Rating Credit risk
Current value
Implied Discount
Rate
End of first
year
A High quality, low credit risk 117.06 3.0% 120.57
B
Speculative, moderate credit
risk 103.96
4.5% 108.63
C High credit risk 77.04 8.5% 83.58
D 30.00
At the start of year, 100 is invested in each bond. Both portfolios starts with same initial
value of 600. Assuming the bond is divisible and is priced as the present value, the holding at
the start is:
Portfolio 1
R S T U V W portfolio total
initial investment 100 100 100 100 100 100 600
number of bonds 0.962 0.962 0.962 0.962 0.962 0.962
Portfolio 2
R S T X Y Z portfolio total
initial investment 100 100 100 100 100 100 600
number of bonds 0.962 0.962 0.962 1.298 1.298 1.298
3,000 simulations has been performed representing different possible scenarios of portfolio
values in one year. And the losses for each simulation are calculated by taking the difference
between portfolio value at the start (600) and simulated portfolio values (sum of the 6 bond
values in one year). A negative loss indicates a gain.
The VaR at 99.5% representing the loss that we are 99.5% confident that our loss would not
exceed over one year, would be obtained from the simulations and used to compare the 2
portfolios. In this project, it is obtained by finding the highest 99.5 percentile of the losses
incurred as shown below:
VaR 99.5%
Portfolio 1 Portfolio 2
375.306 223.956

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This means, for Portfolio 1, 99.5% of the time the loss wouldn’t exceed 375.306. For
Portfolio 2, 99.5% of the time the loss wouldn’t exceed 223.956. This result indicates that,
Portfolio 1 has a higher potential loss than Portfolio 2.

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Results
Below is a summary statistics of the results:
Portfolio 1 Portfolio 2
Average Loss -4.39 -12.57
Average Portfolio Value 604.39 612.57
Variance 5237.32 4336.05
Max Loss 426.85 327.04
Max Gain 95.90 171.01
VaR 99.5% 375.31 223.96
Even though the average credit rating for Portfolio 1 is higher, the results show that Portfolio
2 generates a higher average return.
Also, Portfolio 2 has a lower variance, this means the portfolio values are less volatile than
Portfolio 1. The 99.5% Value at Risk for Portfolio 2 is much lower, this means at given
confidence level, the loss from Portfolio 2 is capped at a much lower level than Portfolio 1 in
one year’s time.
This is because Portfolio 2 contains 3 C rated bonds that has low correlation to each other and
to the 3 B rated bonds. The low correlation means that the portfolio is more diversified. So
when one specific market is bad, bond values would not drop together and create huge losses.
Below is a histogram of Portfolio 1:
We can see that in most case (57%) we would have a gain of 20, this is because the bonds are
relatively highly correlated, they tend to behave the same way as each other, i.e. if one bond
stays in B rating, the others are likely to stay in B rating, which causes the leptokurtic shape.
0
200
400
600
800
1000
1200
1400
1600
1800
-100
-80
-60
-40
-20
0
20
40
60
80
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120
140
160
180
200
220
240
260
280
300
320
340
360
380
400
420
more
Loss
Portfolio 1 Histogram

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However, if one bond performs badly, the others are likely to behave the same way, which
causes the potential huge loss (greatest loss lies beyond 420 for a probability of 0.43%) in the
tail area. The tail is long and fat, which means there is a chance of catastrophic risk.
Below is a histogram for Portfolio 2:
Values for portfolio 2 is spreat more evenly. With four peaks at -70, -30, 40, and 110, these
may be caused by the 3 correlated B rated bonds (R, S, T) behaving in similar ways. But the
effect is offset by the 3 less correlated bonds (X, Y, Z).
It has a positive skew showing a chance of general trend of gain. The lagest loss is around
330 with a probability of 0.07% which is a much lower change for a lower loss compared to
Portfolio 1.
0
100
200
300
400
500
600
700
800
-180
-160
-140
-120
-100
-80
-60
-40
-20
0
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80
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120
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180
200
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300
320
More
Loss
Portfolio 2 Histogram

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The following diagram shows a comparison between loss distributions from both portfolios.
Both portfolios peak around zero, but Portfolio 1 has a fatter and longer tail than portfolio 2.
This means a higher potential loss if we invest solely in B rated correlated bonds. Also
Portfolio 2 has a higher potential gain with lower potential loss than Portfolio 1.
This diagram indicates that Portfolio 2 is a less risky choice than Portfolio 1.
0
500
1000
1500
2000
2500
-150 -100 -50 0 50 100 150 200 250 300 350 400 450 More
Loss Comparison
Portfolio 1
Portfolio 2

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Recommendation
Depending on how much risk the client would like to take, he should consider investing in
some C rated bonds for diversification in the portfolio. This would reduce the risk of large
losses occurring due to the high correlation of B rated bonds.
According to the results above, Portfolio 2 offers a less volatile return over the year and has a
lower VaR than Portfolio 1. This means if the worst case scenario happens, our client would
lose significantly less if he chooses Portfolio 2.
Since he wants a ‘safer’ portfolio, his statement of ‘invest solely in B rated bonds’ is not an
appropriate action to take as it is more risky due to concentration risk. The safer approach
should be adding some bonds with lower correlation, i.e.the C rated bonds in this case, so that
the tail of loss distribution is not as fat and portfolio values less concentrated. In this way,
there is less chance of huge losses due to all the correlated bonds performing badly together
which is safer.
According to his actual risk appetite and preference, he should choose how many C rated
bonds to add to the portfolio, or bonds other than the 9 bonds given that has low or negative
correlation to the existing invested bonds. This can reduce credit risk by diversification
without extra costs.

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Limitations
The results should, however, be used with caution with consideration of the following
limitations:
 Using equity returns’ correlation for bonds
The bonds’ credit quality’s correlation may not behave the same way as equity returns. Price
for shares can be affected by many demand and supply factors including subjective opinions
of investors while bonds may be affected by other factors. Therefore the Cholesky Matrix
may not represent the bonds’ correlations perfectly and this may lead to the inaccuracy of the
model.
 Assume future performance would be the same
We used empirical data for modelling correlation and assume future behaviour to be the same,
which might not be the case. And there may be new risks or other factors effecting the credit
quality of bonds.
 Difficulty to capture fat tail nature
This needs large amount of data and large number of simulations to capture the whole shape
of portfolio distribution.
 Credit rating may not reflect the actual credit quality
I t is not necessary the highly rated investments have better credit quality as rating agency are
paid by the companies being rated. Therefore as the method relies on credit rating’s accuracy
to reflect credit risk, it may not be reliable.
 Unavailability of rating
This method cannot be performed if a company has no credit rating available. E.g. Some
unlisted firms’ bonds would not be suitable to be used with this method.
 Data quality
The quality of results depends hugely on the quality of the data given. The results would not
be reliable if there are any data errors.
 Only 4 credit rating state
This is overly simplified as in reality there would be more categories for credit rating, which
leads to the only 4 values the bonds can take, giving cluster in certain values.
 Limited amount of data
The data are only for the past 3 years and may not capture the whole picture for the
correlation between each bonds.
 Assuming a multivariate Normal
It may not be the case in reality therefore introducing inaccuracy.

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 Simplified assumptions
Flat yield curve, all bonds starts at the same time, and other simplified assumptions may
mean that the modelling doesn’t follow reality closely and the results may not be very
accurate.