Risk, Usage, Funding and Pricing of Revolving Credit Lines Vikrant Tyagi Loan Exposure Management...
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Transcript of Risk, Usage, Funding and Pricing of Revolving Credit Lines Vikrant Tyagi Loan Exposure Management...
Risk, Usage, Funding and Pricing of Revolving Credit Lines
Vikrant TyagiLoan Exposure Management Group
· 04/19/23 · 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
· 04/19/23 · 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.
· 04/19/23 · 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
· 04/19/23 · 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
· 04/19/23 · 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
· 04/19/23 · 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
· 04/19/23 · 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
,
· 04/19/23 · 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
· 04/19/23 · 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
· 04/19/23 · 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
· 04/19/23 · 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.
· 04/19/23 · 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
· 04/19/23 · 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
· 04/19/23 · 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
· 04/19/23 · 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
· 04/19/23 · 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
· 04/19/23 · 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
· 04/19/23 · 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
· 04/19/23 · page 23
Time 1 utilization at 99% CI given U0
Time 3 utilization at 99% CI given U0
Mean Time 1 Utilization given U0
Mean Time 3 Utilization given U0
Time 3 usage distribution as of time 0 given time
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
Time 1 Time 2 Time 3
Dynamic Term Funding for 3 Year Maturity
Given U0
3 Year Static Unexpected Funding at 99% CI
given U0
Static Term Funding for 3 Year Maturity
given U0
•
· 04/19/23 · page 24
Time 2 utilization at 99% CI given U1
Time 3 utilization at 99% CI given U0
Mean Time 1 Utilization given U0
Mean Time 3 Utilization given U0
Time 3 usage distribution as of time 0 given time
0 usage is U0
Time 2 usage distribution as of
time 1 given time 1 usage is U1
U0
U1
Util
izat
ion
Graphical illustration: Dynamic term funding
Time 1 Time 2 Time 3
Dynamic Term Funding for 3 Year Maturity
Given U0
3 Year Static Unexpected Funding at 99% CI
given U0
Static Term Funding for 3 Year Maturity
given U0
Mean Time 2 Utilization given U1
Dynamic Unexpected One Year Funding at 99% CI
Dynamic Term Funding Adjustment for 2 Yr Maturity given U1
U2
· 04/19/23 · page 25
Time 3 utilization at 99% CI given U2
Mean Time 1 Utilization given U0
Mean Time 2 Utilization given U1
Mean Time 3 Utilization given U2
Mean Time 3 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
U0Util
izat
ion
Graphical illustration: Dynamic term funding
Time 1 Time 2 Time 3
Time 3 usage distribution as of
time 2 given time 0 usage is U2
U2
U1
3 Year Static Unexpected Funding at 99% CI
given U0
Static Term Funding for 3 Year Maturity
given U0
Dynamic Term Funding for 3 Yr Maturity at time 0 given U0
U3
Dynamic Term Funding for 2 Yr Maturity at time 1 given U1
Dynamic Term Funding for 1 Yr Maturity at time 2 given U2
Dynamic Unexpected 1Yr Funding at 99% CI
· 04/19/23 · 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
· 04/19/23 · 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
· 04/19/23 · 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
· 04/19/23 · 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
· 04/19/23 · 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
· 04/19/23 · 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
· 04/19/23 · 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
· 04/19/23 · 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
· 04/19/23 · page 36
Time 3 usage distribution as of time 0 given time
0 usage is U0
U0Util
izat
ion
Dynamic funding offers more protection against unexpected draws than static funding during a high volatility environment
Time 1 Time 2 Time 3
U2
U1
U3
Time 4
U4
Unexpected Funding in Dynamic Case
Unexpected Funding in Static Case
Funding Shortfall in Static Case
· 04/19/23 · page 37
Time 3 usage distribution as of time 0 given time
0 usage is U0
U0Util
izat
ion
Dynamic funding has less model risk than static funding
Time 1 Time 2 Time 3
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
· 04/19/23 · 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
· 04/19/23 · 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.