Class 15; Chap 12

 Purpose: Gain a working knowledge of how FIs measure and manage the risk of a loan portfolio
1.
a) b)
Simple models
Migration analysis
Concentration limits
2.
a) b)
Modern Portfolio Theory Models
Portfolio Diversification
KMV model
3.
a)
Loan Concentration Models
Loan volume-Based Model



Basic Idea:
FI managers want to know how the credit risk (Rating) of loans in their portfolio should change over time (that is, what can managers expect).
If the actual change is different from what managers expect they can ration credit or adjust premiums





For Example:
Suppose that on average over the last five years 11 out of 100 loans have been downgraded from a rating of BBB to a rating of BB each year.
If the FI has a portfolio of 500 BBB rated loans How many loans should the manager expect to be rating of BB by the end of the year?
55 loans should drop to a rating BB
Suppose that over the last year, on average, 15 out of 100 BBB loans were downgraded to a rating of BB, would the manager be concerned? Yes! More loans than expected were downgraded
This would suggest that BBB rated loans have become riskier – So what can the manager do?
The manager could then cut-back on lending to this rating category or increase the credit risk premium


Loan Rating Migration Matrix
The migration matrix tells us the percentage of bonds that start in one rating category and move to another
The column header tells us the rating category where bonds end the year
The row header tells us the rating category where loans begin the year
AAA-A
BBB-B
CCC-C
AAA-A BBB-B CCC-C
0.85
0.10
0.04
0.12
0.03
0.83
0.13
0.03
0.80
D*
0.01
0.02
0.04



We read the table by rows across columns DO NOT READ DOWN THE COLUMNS
The number inside the matrix are called transition probabilities. They tell us what portion of an FIs portfolio is expected to transition to a new rating each year
Example:
 Of the loans that started the year in AAA-A category 85% remained in AAA-A
10% transitioned to BBB-B
4% transitioned to CCC-C
1% Defaulted

Suppose an FI manager has 211 BB rated loans, 310 AA rated loans and 130 C rated loans what would the manager expect his portfolio to look like at the end of the year
Begin with the AA rated Loans
•
AAA-A = (310)(.91) = 282.1
•
BBB-B = (310)(.08) = 24.8
•
CCC-C = (310)(.01) = 3.1
•
D = (310)(0) = 0
AAA-A
BBB-B
CCC-C
AAA-A BBB-B CCC-C
0.91
0.08
0.01
0.08
0.03
0.83
0.10
0.05
0.80
D*
0.00
0.04
0.07
BB rated Loans
•
AAA-A = (211)(0.08) = 16.88
•
BBB-B = (211)(0.83) = 175.13
•
CCC-C = (211)(0.05) = 10.55
•
D = (211)(0.04) = 8.44
C rated Loans
•
AAA-A = (130)(0.03) = 3.9
•
BBB-B = (130)(0.10) = 13
•
CCC-C = (130)(0.80) = 104
•
D = (130)(0.07) = 9.1
Loan Portfolio at the end of the year
•
AAA-A = 302.88
•
BBB-B = 212.93
•
CCC-C = 117.65
•
D = 17.54
Sum each category

Suppose an FI has 100 BBB rated loans, 500 A rated loans, and 13 C rated loans. Calculate the expected loss of the portfolio if all loans have face value of $100,000. The average recovery rate for loans is 60%. The transition matrix is shown below
AAA-A BBB-B CCC-C D*
AAA-A
BBB-B
CCC-C
0.85
0.11
0.03
0.10
0.80
0.10
0.03
0.05
0.81
0.02
0.04
0.06
Step #1 – calculate the number of bonds in each category expected to fail
AAA-A 500*.02 = 10
BBB-B 100*.04=4
CCC-C 13*.06 = .78
Step #2 – calculate the loss on the portfolio
AAA-A 10*100,000*0.6 = 600,000
BBB-B 4*100,000*0.6 = 240,000
CCC-C 0.78*100,000*0.6 = 46,800
Total $886,800.00

Basic Idea:
The FI can set limits on the amount lent to an individual borrower or a sector to limit exposure to a particular borrower or sector
 If defaults of a borrower/sector are highly correlated the FI will usually set one combine limit
 Geographic limits are also commonly used
 Regulators have limited the amount lent to one borrower to 10% of equity capital

Suppose management has placed a limit on lending to the auto manufacturing sector such that the maximum potential loss is no greater than 10% of equity capital. The firm has total assets of 500 mill and liabilities of 450 mill. Historical recovery from the Auto manufacturing sector is 40%. How much can the FI lend to the auto manufacturing sector?
Step #1 calculate the maximum equity capital the firm can lose
( 0 .
10 )( 500 mill

450 mill )

5 mill
Step #2 Calculate the loan principal that can be lent to auto manufactures
If the full amount lent to auto manufactures (X) defaults the FI will lose 1-.4= 60% of the principal lent
( 1

.
40 ) X

5 mill
X
X

$ 5 , 000 , 000
.
60

$ 8 , 333 , 333 .
33
Loan concentration limit
The maximum amount the FI can lend to auto manufactures

Example: Managers at Manhattan bank have limited lending to the computer software sector.
No more than 5% of total loan value can be lent to this sector. The bank is currently at its limit. How much software loan value will the bank have to sell if the value of their loan portfolio decreases by 20% but the value of their software loans stays constant. Assume that the loan portfolio currently has a value of $50M.
Step #1 calculate the maximum equity capital the firm can lose
( capital )( 0 .
05 )
Step #2 Calculate the loan principal that can be lent to auto manufactures
( X )(.
15 )
 capital (.
05 )
X
X
 capital
0 .
05
0 .
15
 capital ( 0 .
3333333 )
X ≤33.333%

Basic Idea:
 FIs can reduce the level of loan portfolio risk by ensuring that the portfolio is well diversified – changes in default probabilities are not perfectly correlated.
 The portfolio diversification model also allows the manager to measure the level of risk in his/her portfolio

The portfolio diversification model also allows FI managers to measure how well their portfolio is performing relative to other allocations (portfolios) with the same risk level

Expected Return of a portfolio:


Given historical returns of the individual assets, how do you calculate the expected portfolio return?
It is just the weighted average of individual returns
E
 
P
 i
N 

1
 i
E
  i
 i
N 

1
 i
R i
Risk of a Portfolio:


Risk is measured by the volatility (standard deviation) of portfolio returns
This can be calculated using the variance of individual asset returns

2 p
 i
N 

1
 i
2
 i
2  i
N N 

1 j
 i
2
 i
 j
 i
 j
 i , j

2 Loan Portfolio:
Given the following statistics for two loans, find the risk and return of the portfolio
Loan Market Value R i
 i
 i
2
 i , j
1 40 mill 0.1
0.0857
0.007340
2 60 mill 0.12
0.0980
0.009604
-0.84
Step #1 Calculate the portfolio expected return
R p
 
1
R
1
 
2
R
2

40
40 mill mill

60 mill

0 .
1

40
60 mill mill

60 mill
R p

 

0 .
1

 

0 .
12

.
112

11 .
2 %

0 .
12
Step #2 Calculate the risk (volatility) of the portfolio

2 p
 
1
2

1
2  
2
2

2
2 
2

1

2

1

2

1 , 2

2 p

( 0 .
4
2
)(.
00734 )

( 0 .
6
2
)(.
009604 )

2 ( 0 .
4 )( 0 .
6 )( 0 .
0857 )( 0 .
098 )(

0 .
84 )

.
0012462
 p

0 .
0012462

.
0353

3 .
53 %



Portfolio Performance ( Side Note ):
The question here is can portfolio manager do better with a different allocation?
The answer is maybe – in the 2 asset case NO (if there are only 2 assets in the universe )

Example: Suppose an FI holds a bond portfolio that is fully invested in two sectors: $20 million to consumer durables and $50 million to the healthcare sector. The expected return and volatility for the consumer durables sector portfolio is .125 p.a. and .081 p.a. respectively. The expected return and volatility for the health care sector portfolio is .083 p.a. and .041 p.a. respectively.
Find the risk and return of the portfolio. Correlation between the two portfolios is .32

Basic Idea:

Diversification models are only appropriate for assets with normally distributed returns but bond & loan returns are not usually considered normal

Loans are also infrequently traded – this makes estimating the expected return and standard deviation from market data very difficult

The Moody’s KMV portfolio manager model addresses these shortcomings

Main contribution – they use proprietary probabilities of default (EDF) to directly estimate the return on the loan and use that return in the portfolio diversification model
 We add one more step

Loan Returns:
E
  i

AIS

EDF i

LGD i
All-in-spread AIS = (return on loan – funding cost) + annual fees k”
Expected default frequency EDF = probability of default
Loan Risk: Volatility of the company’s EDF
 i
  i

LGD

EDF i

1

EDF i


LGD
Correlation:
In the KMV model the correlation is the correlation between the systematic component of a bank asset returns (e.g. the systematic component of loan 1 and loan 2 returns)

Given the following portfolio of loans calculate the risk and return of the portfolio
Loan Value Return on loan
– funding cost
1 30 mill 5%
2 20 mill 4.5%
Annual Fees LGD EDF Corr
2%
1.5%
25%
20%
3%
2% -.25
Step #1 Calculate Expected Loan Returns for each loan
E
  i

AIS i

EDF i

LGD i
E
 
1
E
 
1


( 0 .
05

0 .
02 )
(.
07 )

0 .
0075

.
03

.
25

.
0625

6 .
25 %
E
 
2
E
 
2


( 0 .
045

0 .
015 )

.
02

.
2
(.
06 )

0 .
004

.
0056

5 .
6 %

Given the following portfolio of loans calculate the risk and return of the portfolio
Loan Value Return on loan
– funding cost
1 30 mill 5%
2 20 mill 4.5%
Annual Fees LGD EDF Corr
2%
1.5%
25%
20%
3%
2% -.25
Step #2 Calculate Loan Risk (volatility) - for each loan
 i

EDF i

1

EDF i


LGD

1


1

( 0 .
03 )( 1

0 .
03 )

0 .
25
.
0291

0 .
25

.
04265

4 .
265 %

2


2

( 0 .
02 )( 1

0 .
02 )

0 .
2
.
0196

0 .
2

0 .
028

2 .
8 %

Given the following portfolio of loans calculate the risk and return of the portfolio
Loan Value Return on loan Return Volatility Corr
1 30 mill 6.25% 0.04265
2 20 mill 5.6% .028
-.25
Step #3 Calculate the portfolio return
E
  p
 
1
E
 
1
 
2
E
 
2
E
  p

30
50

.
0625

20
50

.
056
E
  p

0 .
6

.
0625

0 .
4

.
056

5 .
99 %

Given the following portfolio of loans calculate the risk and return of the portfolio
Loan Value Return on loan Return Volatility Corr
1 30 mill 6.25% 0.04265
2 20 mill 5.6% .028
-.25
Step #4 Calculate the risk (volatility) of the portfolio

2 p
 
1
2

1
2  
2
2

2
2 
2

1

2

1

2

1 , 2

2 p

( 0 .
6
2
)(.
04265
2
)

( 0 .
4
2
)(.
028
2
)

2 ( 0 .
4 )( 0 .
6 )( 0 .
04265 )( 0 .
028 )(

0 .
25 )
 p
2 
0 .
0006369
Portfolio Risk = 2.52%
Portfolio return = 5.99%

2 p

0 .
0006369

0 .
252

2 .
52 %

Example: Suppose an FI has holds 2 loans in its portfolio with the given characteristics. Find the portfolio return if the correlation between the two firms assets is 0.45
Loan #1
EDF = 0.05
LGD = 0.30
AIS = 0.12
Principal = 25 mill
Loan #2
EDF = 0.07
LGD = 0.60
AIS = 0.20
Principal = 75 mill


Basic Idea:
Loan prices are difficult to obtain but volume (aggregate principal) is available

The volume based model measures how different a bank’s lending activity is from the average bank in a pier group

Given the concentration of lending to each of the four sectors for bank A and bank
B along with the national average below, which of the two bank differs more in its lending activity from the average national bank
Real Estate
C&I
Individual
Other
National Average Bank A Bank B
45% 65% 10%
30%
15%
10%
20%
10%
5%
25%
55%
10%
Step #1 Calculate the squared difference between the national average and each bank
Bank A
(.
45

.
65 )
2 
.
04
(.
30

.
20 )
2 
.
01
(.
15

.
10 )
2 
.
0025
(.
10

.
05 )
2 
.
0025
Bank B
(.
45

.
10 )
2 
.
1225
(.
30

.
25 )
2 
.
0025
(.
15

.
55 )
2 
.
16
(.
10

.
10 )
2 
0

Given the concentration of lending to each of the four sectors for bank A and bank
B along with the national average below, which of the two bank differs more in its lending activity from the average national bank
Real Estate
C&I
Individual
Other
National Average Bank A Bank B
45% 65% 10%
30%
15%
10%
20%
10%
5%
25%
55%
10%
Step #2 average the squared differences and take the square root
Bank A
0 .
04

0 .
01

.
0025

.
0025

0 .
01375
4
0 .
01375

11 .
73 %
Bank B
0 .
1225

0 .
0025

0 .
16

0

0 .
071
4
0 .
071

26 .
69 %

Given the concentration of lending to each of the four sectors for bank A and bank
B along with the national average below, which of the two bank differs more in its lending activity from the average national bank
Real Estate
C&I
Individual
Other
National Average Bank A Bank B
45% 65% 10%
30%
15%
10%
20%
10%
5%
25%
55%
10%


Is it bad to be different?
Banks may specialize in certain types of lending
Banks may be located in a region where certain types of lending is more prevalent


We saw 5 different ways to measure the credit risk of a portfolio of assets
1.
a)
Simple models
Migration analysis b)
Concentration limits
2.
a) b)
Modern Portfolio Theory Models
Portfolio Diversification
KMV model
3.
a)
Loan Concentration Models
Loan volume-Based Model