SEEM 3580: Risk Analysis for Financial
Engineering
Tutorial 8: Managing Credit Risk and Liquidity Risk
ZENG Yuhao
April 5, 2024
ZENG Yuhao SEEM 3580: Risk Analysis for Financial Engineering
Credit Correlation
Credit correlation measures the degree of dependence
between the change of the credit quality of two
assets/obligors. It answers the question:
”If Obligor A’s credit quality (credit rating) changes, how well
does the credit quality of Obligor B correlate to A?”
The portfolio loss is highly sensitive to the credit correlation,
as shown in the following example.
When the credit correlation coefficient is at 0, the portfolio
loss will not exceed $18 in 99.9% of the cases. However, as
the correlation coefficient increases to 0.6, for the same 99.9%
of the cases, the potential portfolio loss may reach $97.
ZENG Yuhao SEEM 3580: Risk Analysis for Financial Engineering
Credit Metrics
CreditMetrics was introduced in 1997 by J.P. Morgan and its
co-sponsors (Bank of America, UBS, et al.).
On a single bond level, the risk is estimated based on credit
migration analysis, i.e. the probability of moving from one
credit rating class to another within a given time horizon.
On a bond portfolio level, the risk is estimated by
incorporating joint credit quality movements into the single
bond model.
We focus on a risk horizon of one year.
The portfolio loss is calculated by comparing the portfolio
value after one year if the current credit quality (rating) is
maintained against the portfolio value under various credit
quality movement scenarios.
Credit VaR of a portfolio is derived as the percentile of the
portfolio loss distribution corresponding to the desired
confidence level.
ZENG Yuhao SEEM 3580: Risk Analysis for Financial Engineering
CreditMetrics: Single Bond
Let us consider the single bond case.
Procedures:
1 Credit rating migration;
2 Valuation;
3 Credit risk estimation.
We illustrate the above procedures with the following case:
Example
A BBB-rated 5-year senior unsecured bond with face value $100
and pay an annual coupon at the rate of 6%.
ZENG Yuhao SEEM 3580: Risk Analysis for Financial Engineering
CreditMetrics: Single Bond
Step 1. Credit Rating Migration
In this step, we specify rating categories, combined with the
probabilities of migrating from one credit rating class to
another over the credit risk horizon (1 year); see the table on
the next page.
Initial Rating AAA AA A BBB BB B CCC Default
AAA 90.81 8.33 0.68 0.06 0.12 0.00 0.00 0.00
AA 0.70 90.65 7.79 0.64 0.06 0.14 0.02 0.00
A 0.09 2.27 91.05 5.52 0.74 0.26 0.01 0.06
BBB 0.02 0.33 5.95 86.93 5.30 1.17 1.12 0.18
BB 0.03 0.14 0.67 7.73 80.53 8.84 1.00 1.06
B 0.00 0.11 0.24 0.43 6.48 83.46 4.07 5.20
CCC 0.22 0.00 0.22 1.30 2.38 11.24 64.86 19.79
Table: One-year Transition Matrix (%)
ZENG Yuhao SEEM 3580: Risk Analysis for Financial Engineering
CreditMetrics: Single Bond
Step 2. Valuation
In this step, we recalculate the value of the bond at the end of
the risk horizon (1 year) for all possible credit states (from
AAA to Default).
It is assumed all credit rating movements occur at the end of
the risk horizon.
At the state of Default:
Specify the recovery rate (% of the face value can recover
when the bond defaults) for different seniority level.
Value of the bond at default = Face value × Mean recovery
rate.
In our case, the mean recovery is 51.13%, so the value of the
bond when default occurs at the end of one year is 51.13.
ZENG Yuhao SEEM 3580: Risk Analysis for Financial Engineering
CreditMetrics: Single Bond
At the state of Default:
Specify the recovery rate (% of the face value can recover
when the bond defaults) for different seniority level.
Value of the bond at default = Face value × Mean recovery
rate.
In our case, the mean recovery is 51.13%, so the value of the
bond when default occurs at the end of one year is 51.13.
Seniority Class Mean (%) Standard Deviation (%)
Senior Secured 53.80 26.86
Senior Unsecured 51.13 25.45
Senior Subordinated 38.52 23.81
Subordinated 32.74 20.18
Junior Subordinated 17.09 10.90
Table: Recovery Rate by Seniority Class (% of Face Value (Par))
ZENG Yuhao SEEM 3580: Risk Analysis for Financial Engineering
CreditMetrics: Single Bond
At all other credit rating states (from AAA to CCC): Obtain
the one-year forward zero curves (the expected discount rate
at the end of one year over different terms) for each credit
rating class.
Category Year 1 Year 2 Year 3 Year 4
AAA 3.60 4.17 4.73 5.12
AA 3.65 4.22 4.78 5.17
A 3.72 4.32 4.93 5.32
BBB 4.10 4.67 5.25 5.63
BB 5.55 6.02 6.78 7.27
B 6.05 7.02 8.03 8.52
CCC 15.05 15.02 14.03 13.52
Table: Example One-Year Forward Zero Curves by Credit Rating Category
(%)
ZENG Yuhao SEEM 3580: Risk Analysis for Financial Engineering
CreditMetrics: Single Bond
Next, we calculate the price Vj(1) of the bond at the end of
one year in the scenario that the credit rating class becomes j ,
based on the current information.
To see how to do this, consider an N-year bond (as of now)
with face value F and annual coupon rate c .
Denote the one-year forward zero curve for Rating j and Year
i as R ji (which is given in the table on the previous page),
where j ∈ {AAA,AA,A,BBB,BB,B,CCC}, we have:
Vj(1) = cF +
cF
1 + R j1
+
cF
(1 + R j2)
2
+ · · ·+ cF + F
(1 + R jN−1)N−1
.
ZENG Yuhao SEEM 3580: Risk Analysis for Financial Engineering
CreditMetrics: Single Bond
For example, in the scenario that the BBB rated bond will be
upgraded to A, its value at the end of one year is given by:
VA(1) = 6+
6
1.0372
+
6
1.04322
+
6
1.04933
+
6
1.05324
+
106
1.05324
= 108.66.
It should be noted that, at the end of one year, the maturity of
the bond will become four years with one immediate payment.
ZENG Yuhao SEEM 3580: Risk Analysis for Financial Engineering
CreditMetrics: Single Bond
Step 3. Credit risk estimation
The loss of the BBB bond over one year L1 after the transition
to rating j is given by the difference between the bond value
after one year if the BBB is maintained (i.e., VBBB(1)) and
that if the rating is migrated to j (i.e., Vj(1)). In other words,
L1 = VBBB(1)− Vj(1)
ZENG Yuhao SEEM 3580: Risk Analysis for Financial Engineering
CreditMetrics: Single Bond
The table on the previous page provides the discrete
distribution of L1, based on which one can calculate the VaR.
Recall from Lecture Notes 4, page 9 that for a general
distribution of L1, the 1-year α% VaR (α-th percentile) is
given by:
1-year α% VaR = min{ℓ : P(L1 ≤ ℓ) ≥ α%}.
Following this definition, the 1-year 99% VaR is $9.45.
ZENG Yuhao SEEM 3580: Risk Analysis for Financial Engineering
Recall: Definition of Liquidity Risk
Liquidity Risk of a Financial Institution (FI) refers to the risk
of running out of cash and/or unable to raise additional funds
to meet the financial claims from its liability holders (liability
side of the balance sheet) or
to honor the asset purchase agreement (asset side of the
balance sheet).
The DI can usually predict the Net Deposit Drain (NDD),
the difference between deposit withdrawals and additions, over
a short period.
Figure: Assets and Liabilities of Depository Institutions (DIs)
ZENG Yuhao SEEM 3580: Risk Analysis for Financial Engineering
Two Methods of Managing High NDD:
Purchased Liquidity Management funds the positive NDD
through the adjustment on the liability side of the balance
sheet.
This can be done by borrowing funds through issuing
additional fixed-maturity wholesale certificates of deposit
(CDs) or selling notes (bank notes) and bonds (corporate
bonds).
Stored Liquidity Management funds the positive NDD
through the adjustment on the asset side of the balance sheet.
This can be done by liquidating some of its assets such as
using the excess cash reserves or selling its assets.
ZENG Yuhao SEEM 3580: Risk Analysis for Financial Engineering
Two Methods of Managing High NDD:
Before and after a $5 million loan commitment exercise via
purchased or store liquidity management.
ZENG Yuhao SEEM 3580: Risk Analysis for Financial Engineering
Measurement of Liquidity Risk: Liquidity Index
The Liquidity Index measures the Depository Institution’s (DI)
potential losses from sudden or fire-sale disposal of assets,
compared to the fair market value established under normal
market sale conditions.
The liquidity index, I , is defined as:
I =
N∑
i=1
wi
Pi
P∗i
where wi is the percent of asset i in the portfolio;
Pi and P
∗
i are the fire-sale asset price and fair market price of
asset i , respectively.
It is straightforward that 0 ≤ I ≤ 1.
The closer I is to 0, the higher is the liquidity risk of the FI.
ZENG Yuhao SEEM 3580: Risk Analysis for Financial Engineering
Measurement of Liquidity Risk: Liquidity Index
Example 6.5 (Liquidity Index)
Consider a portfolio which consists of:
$30 million in Cash
$20 million in Loans.
$50 million in Securities.
Suppose the DI can only receive 80% and 90% of the fair market
values of the loans and the securities, respectively, under the
situation of a fire sale. What is the liquidity index of the bank
asset portfolio?
ZENG Yuhao SEEM 3580: Risk Analysis for Financial Engineering
Measurement of Liquidity Risk: Liquidity Index (cont’d)
Example (Liquidity Index cont’d)
Note that the DI receives 100% of the fair market values of cash
under fire sale. Let w1, w2, and w3 be the weight of Cash, Loans
and Securities, respectively.
w1 =
$30M
$30 + $20M + $50M
= 0.3;
w2 =
$20M
$30 + $20M + $50M
= 0.2;
w3 =
$50M
$30 + $20M + $50M
= 0.5;
From definition, the liquidity index is given by:
I = 0.3× 1.0 + 0.2× 0.80 + 0.5× 0.90 = 0.91.
ZENG Yuhao SEEM 3580: Risk Analysis for Financial Engineering
Measurement of Liquidity Risk: Financing Gap
The Daily Financing Gap ID is defined as:
ID = (OS + OU + OSD)− (IS + IU + ISD),
where
OS : daily scheduled cash outflow,
OU : daily unscheduled cash outflow,
OSD : daily semidiscretionary cash outflow,
IS : daily scheduled cash inflow,
IU : daily unscheduled cash inflow,
ISD : daily semidiscretionary cash inflow.
In fact, ID is the daily amount of the discretionary fund to be
raised in order to balance the net daily cash outflow.
ZENG Yuhao SEEM 3580: Risk Analysis for Financial Engineering
Measurement of Liquidity Risk: Financing Gap
Equation (2) can be rewritten as
ID = OS − IS + R
where
R = (OU + OSD)− (IU + ISD)
R is a random variable, since the unscheduled and
semidiscretionary flows evolve randomly according to the
behavior of customers and the bank’s normal operation.
Assume R ∼ N (R, σ2R), then
ID ∼ N (OS − IS + R, σ2R).
R and σR can be estimated by collecting historical data for
OU , OSD , IU , and ISD to calculate R for each past day.
The larger σR is, the greater is liquidity risk.
ZENG Yuhao SEEM 3580: Risk Analysis for Financial Engineering
Measurement of Liquidity Risk: Financing Gap
Define Iα% as the α-th percentile of ID , i.e.
P(ID ≤ Iα%) = α%,
Iα% is the level of the discretionary fund to be raised daily in
order to make the DI have α% (confidence level) chance to
meet the daily net cash outflow.
With the assumption of ID in (4), we have
Iα% = OS − IS + R + zα% · σR ,
where zα% is the α-th percentile of the standard normal
distribution.
ZENG Yuhao SEEM 3580: Risk Analysis for Financial Engineering
Measurement of Liquidity Risk: Financing Gap
How about a period of T days?
Assume that R over the subsequent days are independent and
all follow N (R, σ2R).
The required level of discretionary fund over a period of T
days for the confidence level α% is given by:
Iα%,T = OS ,T − IS ,T + R · T + zα% ·
√
T · σR .
where OS,T and IS,T are the scheduled cash outflow and
inflow over a period of T days.
ZENG Yuhao SEEM 3580: Risk Analysis for Financial Engineering
Scaling a Normal Random Variable
If a random variable X is normally distributed as
X ∼ N (µ, σ2), then any linear transformation of X , given by
Y = aX + b (where a and b are constants), will also be
normally distributed:
Y ∼ N (aµ+ b, (aσ)2).
The mean of the new variable Y is a times the mean of X
plus b, and the variance is a2 times the variance of X .
ZENG Yuhao SEEM 3580: Risk Analysis for Financial Engineering
Sum of Correlated Normal Random Variables
For two correlated normal random variables X1 ∼ N (µ1, σ21)
and X2 ∼ N (µ2, σ22) with correlation coefficient ρ, the sum
S = X1 + X2 has the following distribution:
S ∼ N (µ1 + µ2, σ21 + σ22 + 2ρσ1σ2).
Here, ρ is the Pearson correlation coefficient which measures
the linear correlation between X1 and X2.
The variance of the sum S includes the covariance term
2ρσ1σ2 reflecting the correlation between X1 and X2.
ZENG Yuhao SEEM 3580: Risk Analysis for Financial Engineering
Thanks for Your Attention!
ZENG Yuhao SEEM 3580: Risk Analysis for Financial Engineering