Risk, Usage, Funding and Pricing of Revolving Credit Lines

Risk, Usage, Funding and Pricing of Revolving Credit Lines

Vikrant TyagiLoan Exposure Management Group

· 04/21/23 · page 2

Introduction

Most bank loan portfolios consist mainly of revolvers which have uncertain usage and hence uncertain funding requirements

Prior to the current liquidity crisis commercial bank loan portfolios were largely funded in the short-term/overnight market

As a result of the sharp drop in liquidity in the short-term money markets since the second half of 2007, commercial banks intend to reduce their reliance on short-term financing by increasing the term funding of their loan portfolio

· 04/21/23 · page 3

Banking industry is developing new funding practices

From UBS 2007 Shareholder’s Letter . . .

Until recently, the Investment Bank funded the majority of its trading assets on a short-term basis and therefore at short-term rates……Now, in order to encourage more disciplined use of UBS’s balance sheet, the Investment Bank will fund its positions at terms that match the liquidity of its assets as assessed by Treasury.

· 04/21/23 · page 4

Part – I

Total Risk of a Revolver Portfolio

· 04/21/23 · page 5

Introduction

Mark-to-Market (MtM) Loans incur both market and default risk: besides realized losses from defaults, the MtM loans incur unrealized PnL changes from spread movements

Need a model that captures default, rating migration and spread risks

Combine the structural and reduced form approaches for credit risk to capture all the above risks

Use this model in the subsequent sections to simulate spreads to determine the usage and funding of revolvers

· 04/21/23 · page 6

Asset Value Process

The returns r from a firm’s assets A are assumed to be given by

which can be normalized as

This residual term is assumed to be given by

where are N systematic (industry and country) factors and is an idiosyncratic factor. The systematic factors are assumed to be normally distributed with covariance matrix Σ. The term is the proportion of the residual return explained by systematic factors and are the weights on systematic factors. The term ensures that has standard normal distribution N(0,1).

11

1~

tt

t σ

μ r r

11

1

tt

ttt A

AA r

12

111

~1 ~

ti

N

iitt Rxwr

ix 1~

t

iw rt 1

~

2R

· 04/21/23 · page 7

Modeling of Defaults and Rating Migrations

Assume there are n+1 ratings with default rating d. For a given time horizon, let pij be the probability of migrating from rating i to rating j calculated using historical data. Then the cumulative probability of a credit with rating i being between rating 1 and rating k next period is given by

Estimate parameters and for each credit and simulate the independent normally distributed idiosyncratic factor and simulate the systematic (industry and country) factors from a normal distribution with covariance matrix Σ to obtain the residual term as per the expression on previous page. A credit with rating i migrates to rating k if

where N is the cumulative standard normal distribution.

I

d},..,n,{ ,21

k

jijki p

1,

iw

rt 1~

1~

t2R

ki ,

kitki rN ,11, )~(

ix

· 04/21/23 · page 8

Graphical Illustration of Rating Migrations

Asset returns simulated by simulating systematic and idiosyncratic factors

Ass

et R

etur

n

Time

Initial Rating

Distribution of Asset Returns

New Rating

· 04/21/23 · page 9

Hazard Rates

Hazard rates ht,T(s) for time interval (t,T) at time s are obtained for each borrower and generic curves using the risk-neutral survival probabilities q(t)

where risk-neutral survival probabilities qt are bootstrapped from the current spread data using the CDS pricing equation

which assumes constant recovery and independence between interest rate and default probabilities

tT

tq

Tq

h Tt

)(

)(ln

,

· 04/21/23 · page 10

Hazard Rate Changes

Percentage change in hazard rates between time t and t+Δt is assumed to be given by

% change due to rating migrations

Total % hazard rate change

% change due to other reasons

· 04/21/23 · page 11

Hazard Rate Change Due to Rating Migrations

The change in hazard rates due to rating migrations is given by

* Generic curve for a given rating is obtained from median spreads for that rating after exclusion of outliers and other adjustments

% change due to rating migrations

% hazard rate change between generic curves* of

old and new rating

· 04/21/23 · page 12

Hazard Rate Change Due to Other Reasons

Hazard rate change due to reasons other than rating migrations is given by a mean reverting process

where b0, b1 and σh are estimated from historical spread data and εh is given by

where ω and εi are macro and firm-specific factors respectively with a standard normal distribution N(0,1) and β is a correlation parameter estimated from the history of credit spreads and index spreads

iss

hs 1

211 1

htthss ttttmthbb )(,(ln

21 ,10

· 04/21/23 · page 13

Simulated Spreads and Portfolio Risk

After simulating the next period hazard rates using the previous expressions

the next-period risk-neutral survival probabilities are bootstrapped using the relation between hazard rates and survival probabilities given in a previous slide

the next period value of loan or CDS is calculated using the next-period survival probabilities

The loss distribution of the portfolio can be used to calculate various risk measures such as VAR, expected shortfall etc.

· 04/21/23 · page 14

Part – II

Usage of a Revolver Portfolio

· 04/21/23 · page 15

Introduction

The usage of a revolver is stochastic

The variation in loan usage is due to corporate financial decisions which are not observed by us

In this model usage is assumed to depend on the borrower’s credit spread and expected utilization of the loan At high spreads it is cheaper to draw on the revolver than borrow with some other instrument.

Other variables can be included if required such as borrower accounting variables

· 04/21/23 · page 16

Relation between spreads and usage

Suppose that usage of the revolver depends on its expected usage and spreads as follows:

Various examples include

· 04/21/23 · page 17

Statistical Distribution of Utilization

CDS spread for each credit is simulated 10,000 times at various times in the future using the model described in previous section

The simulated spreads at a future date and the mapping between spreads and utilization are used to obtain the usage on that date for each loan for each simulation.

The portfolio utilization is calculated for each simulation at each point in the future

A sample histogram for 1 year in the future is shown below

Distribution of 1 Year ahead Notional Weighted Percentage Utilization for FV Facilities with 0% Expected Utilization

0

200

400

600

800

1000

1200

3.00

%3.

75%

4.50

%5.

25%

6.00

%6.

75%

7.50

%8.

25%

9.00

%9.

75%

10.50

%

11.25

%

12.00

%

12.75

%

13.50

%

14.25

%

15.00

%

Utilization as % of Notional

Fre

qu

ency

· 04/21/23 · page 18

Expected and Unexpected Utilization

From the histogram for a given maturity bucket and a given future date, obtain the mean utilization and 95 percentile utilization for that maturity bucket and future date

The mean utilization represents the expected utilization for that maturity at that future date

The difference between the 95 percentile utilization and the mean utilization represents the unexpected utilization for that maturity and future date at the 95 percentile confidence interval

An illustrative output for a given future date is included below

Expected Utilization

12.5%

13.0%

13.5%

14.0%

14.5%

15.0%

15.5%

16.0%

16.5%

17.0%

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Maturity

Per

cen

tag

e o

f N

oti

on

al

Expected Utilization

Unexpected Utilization

0.0%

2.0%

4.0%

6.0%

8.0%

10.0%

12.0%

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Maturity

Per

cen

tag

e o

f N

oti

on

al

95% CI 99% CI

· 04/21/23 · page 19

Part – III

Funding of a Revolver Portfolio

· 04/21/23 · page 20

Introduction

The dramatic rise in short-term funding rates and decline in liquidity over the past six months require that commercial banks

reduce reliance on short-term funding

obtain long term funding for expected utilization of revolvers

keep a cushion for unexpected funding of revolvers

This section discusses two possible alternatives for term funding a revolver portfolio

Static Term Funding

Conditional on initial spreads with no subsequent funding adjustment

Dynamic Term Funding

Conditional on current spreads with regular funding adjustment

· 04/21/23 · page 21

Time 1 Time 2 Time 3

3 Year Static Unexpected Funding at 99% CI

given U0

Time 3 Expected Utilization given U0

Time 3 utilization at 99% CI given U0

Time 3 usage distribution as of time 0 given time

0 usage is U0

Static Term Funding for 3 Year Maturity

given U0

U0Util

izat

ion

Graphical illustration: Static term funding

•

UE •

U1

U2

U3 •

••

Time 0 Actual Utilization

One Possible Actual Utilization Path

· 04/21/23 · page 22

Static Term Funding: Summary

In static funding the CDS spreads are simulated till the loan maturity date using information as of loan inception date

Therefore, the usage distribution corresponds to the loan maturity date and is conditional on information at the loan inception date

The unexpected and unexpected funding remain fixed through the life of the loan

The cost of funding is fixed over the life of the loan

· 04/21/23 · page 23



Mean Time 1 Utilization given U0



0 usage is U0

Dynamic Unexpected One Year Funding at 99% CI

Time 1 usage distribution as of

time 0 given time 0 usage is U0

U0

U1

Util

izat

ion

Graphical illustration: Dynamic term funding


Dynamic Term Funding for 3 Year Maturity

Given U0


given U0


given U0

•

· 04/21/23 · page 24






0 usage is U0



U0

U1

Util

izat

ion



Dynamic Term Funding for 3 Year Maturity

Given U0


given U0


given U0


Dynamic Unexpected One Year Funding at 99% CI

Dynamic Term Funding Adjustment for 2 Yr Maturity given U1

U2

· 04/21/23 · page 25








0 usage is U0

U0Util

izat

ion





U2

U1


given U0


given U0

Dynamic Term Funding for 3 Yr Maturity at time 0 given U0

U3



Dynamic Unexpected 1Yr Funding at 99% CI

· 04/21/23 · page 26

Comparison between the two alternatives

Static Term Funding Dynamic Term Funding

Expected funding needs are conditional on

Initial spreads Current spreads

Funding projections provided for

All loans initially

New loans thereafter

All loans initially and later

Term Funding Adjustments over the life of loan

None Every quarter

Unexpected funding cushion used for

Unexpected usage needs over loan maturity Unexpected usage needs till next term funding adjustment

Unexpected Funding Cushion is

High Much lower

Reliance on expected funding is

Low High

Funding cost for a loan is Constant Stochastic

Model risk is High Low

Pricing is Simple Complicated

· 04/21/23 · page 27

Part – IV

Pricing of a Revolver

· 04/21/23 · page 28

Relation Between Pricing and Funding

Loans sponsored by Businesses

Loan Pricer

Loan Price including upfront

funding cost

Funding Counterparty

Bank

Bank charges funding cost upfront and pays it

over time

LiquidityPremium Charged

Funding Model

Funding term

structure to pay funding

cost

BorrowerLiquidityPremium

Paid

Calibration

· 04/21/23 · page 29

Pricing Issues

The price of revolver must incorporate variable usage and funding cost

Incorporating variable usage is straight-forward in a reduced form risk-neutral framework once the relation between usage and spread is decided

Incorporating funding cost depends on whether static or dynamic funding is used Static funding can be easily incorporated since the cost is fixed for the life of loan Dynamic funding requires incorporating stochastic funding cost

The stochastic funding cost in dynamic funding depends on stochastic funding spread of the bank and the usage (and hence credit spread) of the borrower This makes pricing with dynamic funding very complicated

Moreover since the pricing model and funding model are based on different assumptions, the two models need to be calibrated to ensure the bank is charging at least as much liquidity premium as it is paying

· 04/21/23 · page 30

Conclusion

The current liquidity crisis has highlighted the need to manage the liquidity risk of a bank loan portfolio

The presentation provides a framework to simulate future spreads and estimate future usage distribution of revolving credit lines

The future usage distribution is used to obtain expected and unexpected usage of the portfolio which can be term funded in two ways Static term funding is fixed over life of the loan and is conditioned on initial spreads Dynamic term funding changes over the life of the loan and is conditioned on spreads

at future adjustment dates

Dynamic funding has less model risk and has low reliance on unexpected funding than static funding but has stochastic rather than constant funding cost

Dynamic funding makes revolver pricing very complicated

· 04/21/23 · page 31

Appendix

· 04/21/23 · page 32

Example: Dynamic term funding

For simplicity, assume the following: There are two time points – Time 0 and Time 1 There are only loans with 1 year, 2 year and 3 Year maturities Term-matching is adjusted once a year

The example also includes Non-LEMG funding within DB to illustrate how it effects LEMG through the weighted average cost of funding

· 04/21/23 · page 33

Time 0 At time 0, assume that the expected LEMG and non-LEMG funding needs and the cost of

funding these requirements are as per graph below

Treasury term matches the expected funding requirements. Treasury is paid 1 year funding spread for loans with 1 year maturity and so on

ExampleTime 0 Expected Funding Needs and Funding Costs

120

180

200

100

150

180

40

65

80

0

50

100

150

200

1 2 3

Time to Maturity

Exp

ecte

d F

un

din

g

(MN

EU

R)

0

20

40

60

80

100

120

Sp

read

(B

PS

)

Expected LEMG Funding Expected Non-LEMG Funding CDS Spread

· 04/21/23 · page 34

Time 1 At time 1, new 3 year loans will come in and the previous 3 year (2 year) loans will become 2 year (1

year) loans. Based on these changes and time 1 spreads, the expected funding projections will be provided to treasury at time 1

Assume that the expected funding needs and the cost of funding these requirements at time 1 are as per graph below

ExampleTime 1 Expected Funding Needs and Funding Costs

150

250260

160170

250

45

75

90

0

50

100

150

200

250

300

1 2 3

Time to Maturity

Exp

ecte

d F

un

din

g

(MN

EU

R)

0

20

40

60

80

100

120

140

160

Sp

read

(B

PS

)

Expected LEMG Funding Expected Non-LEMG Funding CDS Spread

· 04/21/23 · page 35

Incremental Funding Need and Weighted Cost of Funding

At time 1, the incremental funding needs of LEMG and non-LEMG are netted and Treasury raises or unwinds this incremental funding requirement at the time 1 spread

The weighted average cost of funding for each maturity is obtained using the funding cost and the funded amounts at time 0 and time 1. Treasury is paid this funding cost

For example, a 3 year loan at time 0 is charged 80 bps at time 0 (see page 7) and is charged 79.5 bps at time 1 (see below)

ExampleTime 1 Incremental Funding Needs and Weghted Funding Costs

-30

50

260

10

-10

250

-20

40

510

66.3

79.590

-150

-50

50

150

250

350

450

550

1 2 3Time to Maturity

Incr

emen

tal

Fu

nd

ing

(M

N E

UR

)

0.0

20.0

40.0

60.0

80.0

100.0

120.0

Wei

gh

ted

Sp

read

(B

PS

)

Incremental LEMG Funding Incremental Non-LEMG Funding

Total Incremental Funding Weighted Spread

· 04/21/23 · page 36


0 usage is U0

U0Util

izat

ion

Dynamic funding offers more protection against unexpected draws than static funding during a high volatility environment


U2

U1

U3

Time 4

U4

Unexpected Funding in Dynamic Case

Unexpected Funding in Static Case

Funding Shortfall in Static Case

· 04/21/23 · page 37


0 usage is U0

U0Util

izat

ion

Dynamic funding has less model risk than static funding


U2

U1

U3

Time 4

U4

Unexpected Funding in Static Case

Unexpected Funding in Dynamic Case

Funding shortfall in static case is

exacerbated with model risk

· 04/21/23 · page 38

Expected usage is more reliable in dynamic funding

In the previous slide, we assumed model risk in estimating the standard deviation (unexpected usage) of the usage distribution. There was no model risk in the expected usage component

In reality, the expected usage is also subject to model risk

The further we look into the future, the more uncertainty we have in estimating defaults, rating migrations and spread movements which are the drivers of expected usage in this model

Since dynamic funding will look at shorter horizons than static funding, the expected usage will be more reliable in the case of dynamic funding

In short, dynamic funding will give more precise expected and unexpected funding estimates and better protection against unexpected funding draws

· 04/21/23 · page 39

Related Research

Merill Lynch uses a similar model to manage its liquidity requirements

The paper detailing the Merill model is as follows Tom Duffy, Manos Hatzakis, Wenyue Hsu, Russ Labe, Bonnie Liao, Xiangdong Luo,

Je Oh, Adeesh Setya, Lihua Yang, 2005, “Merrill Lynch Improves Liquidity Risk Management for Revolving Credit Lines”, Interfaces, Vol. 35, No. 5, September–October 2005, pp. 353–369

This model is an improvement over the Merrill Lynch model on three counts:1) usage may change in this model even if there is no rating migration (due to spread

changes) which is not the case in the Merrill Lynch model2) we model both industry and country correlations using the DB’s Economic Capital

methodology whereas Merrill Lynch uses only industry correlations3) We model the relation between usage and spreads whereas Merrill Lynch models the

relation between usage and ratings. Analysis of the historical data suggests that the mapping between spreads to utilization is more stable across time than the mapping between spreads and rating.

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Risk, Usage, Funding and Pricing of Revolving Credit Lines

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