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Risk, Usage, Funding and Pricing of Revolving Credit Lines

Risk, Usage, Funding and Pricing of Revolving Credit Lines. Vikrant Tyagi Loan Exposure Management Group. Introduction. Most bank loan portfolios consist mainly of revolvers which have uncertain usage and hence uncertain funding requirements

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Risk, Usage, Funding and Pricing of Revolving Credit Lines

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  1. Risk, Usage, Funding and Pricing of Revolving Credit Lines Vikrant Tyagi Loan Exposure Management Group

  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 · 9/22/2014 · page 2

  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. · 9/22/2014 · page 3

  4. Part – ITotal Risk of a Revolver Portfolio · 9/22/2014 · page 4

  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 · 9/22/2014 · page 5

  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 matrixΣ. 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). · 9/22/2014 · page 6

  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 periodis 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 obtainthe 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 · 9/22/2014 · page 7

  8. Graphical Illustration of Rating Migrations Asset returns simulated by simulating systematic and idiosyncratic factors Initial Rating Asset Return New Rating Time Distribution of Asset Returns · 9/22/2014 · page 8

  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 · 9/22/2014 · page 9

  10. % change due to other reasons Total % hazard rate change % change due to rating migrations Hazard Rate Changes • Percentage change in hazard rates between time t and t+Δt is assumed to be given by · 9/22/2014 · page 10

  11. % change due to rating migrations % hazard rate change between generic curves* of old and new rating 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 · 9/22/2014 · page 11

  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 andσhare estimated from historical spread data and εh is given by where ωandεi are macro and firm-specificfactors respectively with a standard normal distribution N(0,1) andβis a correlation parameter estimated from the history of credit spreads and index spreads · 9/22/2014 · page 12

  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. · 9/22/2014 · page 13

  14. Part – IIUsage of a Revolver Portfolio · 9/22/2014 · page 14

  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 · 9/22/2014 · page 15

  16. Relation between spreads and usage • Suppose that usage of the revolver depends on its expected usage and spreads as follows: • Various examples include · 9/22/2014 · page 16

  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 · 9/22/2014 · page 17

  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 · 9/22/2014 · page 18

  19. Part – IIIFunding of a Revolver Portfolio · 9/22/2014 · page 19

  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 · 9/22/2014 · page 20

  21. Graphical illustration: Static term funding Time 3 utilization at 99% CI given U0 U3 • One Possible Actual Utilization Path U2 • 3 Year Static Unexpected Funding at 99% CI given U0 U1 • Utilization UE • Time 3 Expected Utilization given U0 • U0 Static Term Funding for 3 Year Maturity given U0 Time 3 usage distribution as of time 0 given time 0 usage is U0 Time 0 Actual Utilization Time 1 Time 2 Time 3 · 9/22/2014 · page 21

  22. 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 Static Term Funding: Summary · 9/22/2014 · page 22

  23. Graphical illustration: Dynamic term funding Time 3 utilization at 99% CI given U0 3 Year Static Unexpected Funding at 99% CI given U0 Time 1 utilization at 99% CI given U0 U1 Utilization Dynamic Unexpected One Year Funding at 99% CI Mean Time 3 Utilization given U0 Mean Time 1 Utilization given U0 • U0 Time 1 usage distribution as of time 0 given time 0 usage is U0 Static Term Funding for 3 Year Maturity given U0 Time 3 usage distribution as of time 0 given time 0 usage is U0 Dynamic Term Funding for 3 Year Maturity Given U0 Time 1 Time 2 Time 3 · 9/22/2014 · page 23

  24. Graphical illustration: Dynamic term funding Time 3 utilization at 99% CI given U0 Time 2 utilization at 99% CI given U1 3 Year Static Unexpected Funding at 99% CI given U0 U2 Dynamic Unexpected One Year Funding at 99% CI Mean Time 2 Utilization given U1 U1 Utilization Dynamic Term Funding Adjustment for 2 Yr Maturity given U1 Mean Time 3 Utilization given U0 Mean Time 1 Utilization given U0 U0 Time 2 usage distribution as of time 1 given time 1 usage is U1 Static Term Funding for 3 Year Maturity given U0 Time 3 usage distribution as of time 0 given time 0 usage is U0 Dynamic Term Funding for 3 Year Maturity Given U0 Time 1 Time 2 Time 3 · 9/22/2014 · page 24

  25. Graphical illustration: Dynamic term funding Time 3 utilization at 99% CI given U0 Time 3 utilization at 99% CI given U2 U3 Dynamic Unexpected 1Yr Funding at 99% CI Mean Time 3 Utilization given U2 3 Year Static Unexpected Funding at 99% CI given U0 U2 Dynamic Term Funding for 1 Yr Maturity at time 2 given U2 Mean Time 2 Utilization given U1 Utilization U1 Dynamic Term Funding for 2 Yr Maturity at time 1 given U1 Mean Time 3 Utilization given U0 Time 3 usage distribution as of time 2 given time 0 usage is U2 U0 Mean Time 1 Utilization given U0 Static Term Funding for 3 Year Maturity given U0 Time 3 usage distribution as of time 0 given time 0 usage is U0 Dynamic Term Funding for 3 Yr Maturity at time 0 given U0 Time 1 Time 2 Time 3 · 9/22/2014 · page 25

  26. Comparison between the two alternatives · 9/22/2014 · page 26

  27. Part – IVPricing of a Revolver · 9/22/2014 · page 27

  28. Borrower Funding Counterparty Bank Loan Pricer Funding Model Bank charges funding cost upfront and pays it over time Liquidity Premium Charged Calibration Liquidity Premium Paid Loan Price including upfront funding cost Funding term structure to pay funding cost Relation Between Pricing and Funding Loans sponsored by Businesses · 9/22/2014 · page 28

  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 · 9/22/2014 · page 29

  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 · 9/22/2014 · page 30

  31. Appendix · 9/22/2014 · page 31

  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 · 9/22/2014 · page 32

  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 · 9/22/2014 · page 33

  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 · 9/22/2014 · page 34

  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) · 9/22/2014 · page 35

  36. Dynamic funding offers more protection against unexpected draws than static funding during a high volatility environment U4 U3 Funding Shortfall in Static Case U2 Utilization U1 Unexpected Funding in Static Case U0 Unexpected Funding in Dynamic Case Time 3 usage distribution as of time 0 given time 0 usage is U0 Time 1 Time 2 Time 3 Time 4 · 9/22/2014 · page 36

  37. Dynamic funding has less model risk than static funding U4 U3 Funding shortfall in static case is exacerbated with model risk U2 U1 Utilization U0 Unexpected Funding in Dynamic Case Unexpected Funding in Static Case Time 3 usage distribution as of time 0 given time 0 usage is U0 Time 1 Time 2 Time 3 Time 4 · 9/22/2014 · page 37

  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 · 9/22/2014 · page 38

  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: • 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 model • we model both industry and country correlations using the DB’s Economic Capital methodology whereas Merrill Lynch uses only industry correlations • 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. · 9/22/2014 · page 39

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