GRADUATE SCHOOL OF BUSINESS Global Risk Management: A Quantitative Guide
Credit Risk Management Ren-Raw Chen Fordham University
Types of Credit Risk • Bankruptcy • Rating migration • Spread change
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Bankruptcy • Equity investors – Almost nothing
• Bond investors – Recovery – Secured – 80~90% (due to drop in value) – Senior unsecured – 40% – Junior unsecured – 15%
• Jump to default risk (now popular) 3
Migration • One step before bankruptcy • Rating announcements carry information • BBB or higher vs. BB or lower – buy and sell pressure – due to pension fund regulation
• Large literature (details later)
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Migration
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Spreads • Day to day risk – market risk – hedgeable
• CVA – CDS – Correlation risk
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The Market
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The Market
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The Market
Source: http://viableopposition.blogspot.com/
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The Market
Source: Federal Reserve Board
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The Market
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The Market
Source: FDIC
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The Market
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Model • For bankruptcy – accounting • Altman Z • Ohlson O
– finance • reduced-form – Jarrow-Turnbull 1995, Duffie-Singleton 1997
• structural – Black-Scholes-Merton, Geske 1977, Leland 1990
• Hybrid – Black-Cox 1976, CreditGrades 14
Model • For migration – Markov chain • Jarrow-Lando-Turnbull • weak link to default • no link to spread
• For spread – Black-Scholes • no link to default • no link to migration 15
Measures of Credit Risk • Two building blocks – PD (probability of default) – LGD (loss given default) • = 1 – recovery • = exposure at default (note: exposure = notional)
• Measures – EL, UL, EAD, JTD, etc.
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Instruments to Transfer Credit Risk • • • •
CDS FTD/NTD CDO etc.
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Credit Default Swap (CDS)
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Credit Default Swap (CDS)
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Credit Default Swap (CDS) • source BBA
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Credit Default Swap (CDS) • source BBA
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Credit Default Swap (CDS)
Seller
Principal+accrued interest-recovery
Default occurs
0
T Spread
Buyer
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Credit Default Swap (CDS) • Back-of-the-envelope formula receive 1 - R if default
p
p(1 − R) = (1 − p)s ≈ s p=
1− p
s 1−R
pay spread (s) if survive
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Credit Default Swap (CDS) • pi: default prob; Qi: survival prob 1 - recovery 1 - recovery
p1
1 - recovery
p2
1 - recovery
p3
1 − p1
spread
spread
Q1
p4
1 − p2 1 − p3
spread Q2
1 − p4
spread
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Credit Default Swap (CDS) • Quotes (Disney 12/23/2005) term 1 2 3 5 7 10
sprd 9 13 20 33 47 61
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Credit Default Swap (CDS)
1 - recovery rate = 0.6 if default
0.6 × (1 − Q1 ) 0.0009 ×Q1 = 1.05 1.05 Q1 = 0.9985 = e −λ1
p1
Q1 = 1 − p1
λ1 = 14.99 basis points spread = 0.0009 if survive
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Credit Default Swap (CDS) 1 - recovery = 0.6 p1 = 0.0015
p2 Q1 = 0.9985 spread = 0.0013 Q2
1 - recovery = 0.6 prem leg =
1 − p2
spread = 0.0013
0.0013 × 0.9985 0.0013 ×Q2 + = 2 1.05 1.05 0.0015 × 0.6 0.9985 × p2 × 0.6 + = 1.05 1.052 def leg sub in p2 = 1 −
Q2 Q2 = 1− Q1 0.9985
hence, Q2 = 0.9956 = Q1e −λ2 hence, λ1 = 28.65 basis points 27
Credit Default Swap (CDS) CDS Term 1 2 3 4 5
Bootstrapping Market Risk-free Fwd. Surv.Pr. Def.Pr. Spread P(t) lambda(t) Q(t) -dQ(t) 0.0009 0.9512 0.0015 0.9985 0.0015 0.0013 0.9048 0.0029 0.9956 0.0029 0.002 0.8607 0.0059 0.9898 0.0058 0.7788 0.0092 0.9808 0.0091 0.0033 0.7788 0.0092 0.9718 0.009
smoothing
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Credit Default Swap (CDS)
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Credit Default Swap (CDS) • Bilateral financial contracts • Allow the transfer of credit risk from one party to another. Using these products, investors may hedge themselves against credit risk • Related to some risk or volatility • Do not require initial investment • Credit event: bankruptcy, failure to pay, restructuring etc. 30
Credit Default Swap (CDS) • Two parties: protection buyer, protection seller • Do not require either of the parties to actually hold the reference asset • Two ways of settlement: cash and physical • An over-the-counter contract that provides insurance against credit risk. 31
Credit Default Swap (CDS) • The protection buyer pays a fixed fee or premium, often termed as the “spread” to the seller for a period of time. • When a credit event occur at some point before the contract's maturity, the protection seller pay compensation to the buyer of protection, thus insulating the buyer from a financial loss. 32
Credit Default Swap (CDS) • CDS can be viewed as a put option: if one default event occurs, the bond can be put back to the seller at the principal. • CDS is similar to an insurance contract, providing buyers with protection against specific risks.
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Credit Default Swap (CDS) • CDS benefits – a short positioning vehicle not available in the cash market – access to maturity exposures not available in the cash market – does not require an initial cash outlay – access to credit risk not available in the cash market due to a limited supply of the underlying bonds 34
Credit Default Swap (CDS) – ability to effectively “exit” credit positions in periods of low liquidity – off-balance sheet instruments which offer flexibility in terms of leverage – provide important anonymity when shorting an underlying credit
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The CDS Big Bang: an example Consider a CDS contract on Bank of America with notional value of 10M dollars. The quoted 5Y CDS spread is 326 basis points. A protection buyer needs to • Before the Big Bang: pay $326,000 per year
… 1Y
•
2Y
After the Big Bang: – fixed 100: pay $990,254 upfront plus $100,000 per year
… 1Y
– fixed 500: receive $756,788 upfront and pay $500,000
2Y
… 1Y
2Y 36
Structural Models • KMV (Black-Scholes-Merton) method 400 350 300 250
AT > K
200
⇓
150 100
ET = AT − K DT = K
DD
At
50
PD
AT < K
0 0
4
8 12 16 20 24 28 32 36 40 44 48 52
ET = 0 DT = AT
Weeks t
⇓
T
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Structural Models • Equity is a call option (Black-Scholes) E (t ) = A(t )N (d1 ) − e −r (T −t )KN (d 2 ) σE = [A(t )/ E (t )]σAN (d1 )
• where ln A(t ) − ln K + (r − ½σA2 )(T − t ) d2 = σA T − t d1 = d2 + σA T − t
• PD 1 − N (d2 ) 38
Structural Models • K 1-year debt (=STD+0.5×LTD); σA asset vol; σE equity vol; r risk-free rate; T-t time horizon; E equity value; A asset value; N(.) normal probability; • Note −r (T −t ) D(t ) = A(t ) − E (t ) = A ( t )[1 − N ( d )] + e KN (d 2) 1 Recovery value Survival value
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Structural Models • KMV (Black-Scholes-Merton) method Risk-free Debt Value
Risky Debt Value
Asset Value
Asset Value
Loss Given Default
Asset Value
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Structural Models • Multiple debts – Geske • discrete time • flexible capital structure
– Leland • continuous time • steady state capital structure
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Structural Models • Geske E 0 = A0M (h1+ , h2+ ; ρ) − e −r (T2 −t )K 2M (h1− , h2− ; ρ) − e −r (T1 −t )K 1N (h1− )
D0,1 = A0N (−y1+ ) + e −r (T1 −t )K1N (y1− ) D0,2 = A0 − D0,1 − E 0 = A0 (N (y1+ ) − M (h1+ , h2+ ; ρ)) − e −r (T1 −t )K1 (N (y1− ) − N (h1− )) + e −r (T2 −t )K 2M (h1− , h2− ; ρ) D0,1 + D0,2 = A0 [1 − M (h1+ , h2+ ; ρ)] Recovery −r (T1 −t ) −r (T2 −t ) + e K N ( h ) + e K 2M (h1− , h2− ; ρ) 1 1 − 1st Yr Survival 2nd Yr Survival
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Structural Models • Geske – default barrier is A sum of all debts: (K1 + D12)
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First to Default (FTD) • Joint default
A 0
B
0 1
80% 10% 90%
1 0 10% 10%
80% 20% 100%
p(A | B )p(B ) or p(A ∩ B ) = p(B | A)p(A) p(A ∩ B ) = p(B | A)p(A) = p(A) = 10% 44
First to Default (FTD)
p(A ∩ B ) = p(B | A)p(A) = p(A) = 10% p(A ∩ B ) 10% p(B | A) = = = 100% p(A) 10%
p(A | B ) =
p(A ∩ B ) 10% = = 50% p(B ) 20%
B completely depends on A A only 50% depends on B
Default correlation = 0.6667 (highest possible) 45
First to Default (FTD) A
p(A ∩ B ) = p(B | A)p(A)
0
B
= p(A)
0 1
70% 20% 90%
1 10% 0% 10%
80% 20% 100%
= 0% p(A ∩ B ) 0% p(B | A) = = = 0% p(A) 10%
B opposite of A
p(A ∩ B ) 0% = = 0% p(B ) 20%
A opposite of B
p(A | B ) =
Default correlation = -0.1667 (lowest possible) 46
First to Default (FTD) • Default correlation reaches 1 as pA=pB • Default correlation reaches –1 as pA+pB=1 • Multi-party becomes more complex
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First to Default (FTD) • In the event of perfect dependency, i.e. p(B|A)=1, the basket valuation is: 1 [p(A) + p(B ) − p(B | A)p(A)] 1+r 1 p(B ) = 1+r
V =
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First to Default (FTD) • In the event of perfectly negative dependency, i.e. p(B|A)=0, the basket valuation becomes: 1 [p(A) + p(B ) − p(B | A)p(A)] 1+r 1 = [p(A) + p(B )] 1+r
V =
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Collateral Debt Obligation (CDO)
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Collateral Debt Obligation (CDO) • Waterfall Tranche loss Senior tranche
Equity tranche
Mezzanine tranche Total loss K0
K1
K2
K3 51
Collateral Debt Obligation (CDO) • Types – cash CDO (real bonds) – synthetic CDO (CDS)
• by action – cash-flow CDO (boxed) – market-value CDO (non-boxed)
• by sponsor – arbitrage CDO (active) – balance-sheet CDO (passive) 52
Collateral Debt Obligation (CDO)
– http://thismatter.com/money/bonds/types/cdo.htm 53
Synthetic CDO • Risky bond + CDS = risk-free bond – hence, risky bond = risk-free bond - CDS – i.e. risky bond = long risk-free bond and short CDS (provide protection) – e.g. $100 mil risky bonds = $100 mil Treasury and $100 mil CDS (which has no value) – Treasury is collateral • if no collateral, then no treasury 54
default
Synthetic CDO CDS
A
CDS CDS
B POOL Z
$1,250 million
$1,250 million
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$4 million
WDFA
$1,250 million
$6 million
LOSS
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Collateral Debt Obligation (CDO) • CDX – a CDX CDO is a CDO with 125 credit default swaps (8% each) with US$10 million notional – 0-3%, 3-7%, 7-10%, 10-15%, and 15-30%. – very liquid (more liquid than single name CDS)
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Collateral Debt Obligation (CDO) • Copula (how to correlate defaults) – Gaussian copula (solve the dependency problem) – Key equations x i = ρWˆM + 1 − ρWˆi (x < K | W = f ) = Pr ( ρ f + 1 − ρW < K ) pˆi|f = Pr i i M i i
( =N(
Wi < = Pr Ki − ρ f
=N
(
1−ρ
ρ f −K i 1−ρ
)
)
N −1 ( pˆi )− ρ f 1−ρ
) 58
Collateral Debt Obligation (CDO) – Loss distribution • Fourier inversion • Recursive algorithm
prob(rho=0.9) 0.7 0.6 0.5 0.4 0.3 0.2 0.1
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39
42
45
48
51
54
57
60
36
39
42
45
48
51
54
57
60
33
30
27
24
21
18
15
9
12
6
0
3
0 -0.1
Loss prob(rho=0.5)
prob(rho=0) 0.16 0.14 0.12 0.1 0.08 0.06 0.04
Loss
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30
27
24
21
18
15
12
9
6
3
-0.02
0
60
57
54
51
48
45
42
39
36
33
30
27
24
21
18
15
9
12
3
0.02 0
6
0
0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0
Loss
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Collateral Debt Obligation (CDO) • Problems with such a loss distribution – thin tranches (100 tranche CDO) – CDO^2, CDO^3, ... – mezzanine tranches difficult to price
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Collateral Debt Obligation (CDO) • Tranche loss/spread plots here
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Collateral Debt Obligation (CDO) • A Cat analogy – Cats have nine lives (JPM)
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Default Prediction • • • • •
Early warning signal Quantitative rating KMV-Moodys Altman’s Z Olhson’s O
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EL and UL • Basic intuition (single name)
p
Default => 1 − R
EL = p(1 − R) UL = p(1 − p)(1 − R)
1− p
No default => 0
UL highest when p = 0.5 64
EL and UL • Portfolios – difficult to measure accurately – use standard deviation
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EAD (exposure at default) • Expected recoveries from counterparties • Correlations among defaults • Similar to FTD/NTD
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CVA • • • •
Became important after the crisis quantifies counterparty risk trading desk (transfer pricing) DVA – CVA to the counterparty – worsening credit (DVA falls) helps NI – correlation of credit and other products
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CVA • An old method: exposure (call option) Exposure
Moneyness of Deal
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CVA • Valuation – = CDS protection value
• An example (IRS) • A pays fixed 4% • B is BBB rated; CDS spread is 200 bps • A needs to hedge for B’s default • A books the trade at 6% -- true cost • IRS matches with CDS
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CVA • A different example (bond) • A buys a Treasury bond from B ($100 face) • B CDS spread is 200 basis points • PV 5 years of 200 bps is, say 6.2% ($6.2) • cost to A is $106.2
• Complexities – correlation among counter parties – correlation between assets and counter parties 70
CVA
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CVA • CVA - DVA
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CVA • Wrong Way Risk (WWR) – In general, the exposure with a CP is not independent of the CP’s credit quality – Wrong Way Risk is cases where the exposure increases when the credit quality of the CP deteriorates – i.e. exposure tend to be high when PDs are high
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CVA • WWR – Negative correlation between PD and LDG (source: Altman)
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CVA • Two types of WWR – General WWR: the CP’s credit quality is for correlated with macroeconomic factors which also affect the value of the derivatives (e.g. correlation between declining corporate credit quality and high (or low) interest rates causing higher exposures – Specific WWR: CP’s exposure is highly correlated with CP’s PD. (e.g. a company writing put options on its own stock, derivatives collateralized by own shares)
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CVA • WWR quantification is still an open challenge other than the self referencing specific WWR due to: – Difficulty to separate statistical noise from systematic correlation – Challenge of dynamic forward looking adjustment to historical calibration
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CVA Sensitivity and Hedging • CVA Sensitivity is straight forward by definition:
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CVA • Meaningful hedging and P&L explain: a long way to go – Common hedges on the street • Counterparty credit spread delta, FX delta, IR delta, FX vega
– In progress and outstanding • IR Vega, FX/IR Gamma, Cross Gamma - WWR
– Additional note • CVA hedging and P&L explain gap, Debt/Liability DVA management • Direction of CVA desk’s roles and performance factors – real PnL, VaR capital mitigation, counterparty risk capital mitigation • Difficulty to draw clear line between hedging to limit risk and hedging for profit
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CVA • Practical challenges – Path dependent impact: early exercise, Barrier, Bermudan – Recalibration noise: bucketed FX Vega/IR Delta/IR Vega – IR Vega and cross gamma for meaningful hedging/P&L explain.
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Credit VaR • Skewed loss distribution • Term structure of CVaR (Basel II) – no more Gaussian, no more scalability – highly dependent on models
• Difficulties to use models on banks – Case study of Lehman (CCIS 2014)
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Lehman Case Study • Balance sheet (look at capital 1.6%) as of 2002 Assets
Cash Securities Coll Ag’mt Receivables Real Estate
Liabilities
2,265 70,881 101,149 21,191 138
Total 196,219 million $
Short-term Debt Other Securities Coll ST Financing Payables Long-Term Debt Equity Total
123 50,352 121,844 12,758 7,990 3,152 196,219
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Lehman Case Study • 2006 (capital 3.8%) Cash and Short Term Investments Accounts Receivable - Trade, Net Receivables - Other Total Receivables, Net Total Inventory Prepaid Expenses Other Current Assets, Total Total Current Assets Property/Plant/Equipment, Total - Gross Accumulated Depreciation, Total Goodwill, Net Intangibles, Net Long Term Investments Other Long Term Assets, Total
$ $ $ $ $ $ $ $ -
Total Assets
$
12,078.00 Accounts Payable 25,919.00 Accrued Expenses Notes Payable/Short Term Debt 27,971.00 Current Port. of LT Debt/Capital Leases Other Current liabilities, Total Total Current Liabilities Long Term Debt Capital Lease Obligations 5,194.00 Total Long Term Debt (1,925.00) Total Debt 2,417.00 Deferred Income Tax 945.00 Minority Interest 451,752.00 Other Liabilities, Total Total Liabilities Redeemable Preferred Stock, Total Preferred Stock - Non Redeemable, Net Common Stock, Total Additional Paid-In Capital Retained Earnings (Accumulated Deficit) Treasury Stock - Common Other Equity, Total Total Equity 503,545.00 Total Liabilities & Shareholders' Equity
$ 43,912.00 $ 14,697.00 $ 16,596.00 $ 12,878.00 $ 81,178.00 $ 81,178.00 $ 110,652.00 $ 315,093.00 $ 484,354.00 $ 1,095.00 $ 61.00 $ 8,727.00 $ 15,857.00 $ (4,822.00) $ (1,727.00) $ 19,191.00 $ 503,545.00
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Lehman Case Study • Liability term structure 30000.00
25000.00
20000.00
15000.00
10000.00
5000.00
2036
2034
2032
2030
2028
2026
2024
2022
2020
8 00 .2 8 ug 00 A 2 n. 8 Ju 00 .2 pr 08 A 20 b. 7 Fe 00 .2 ec D
2018
2016
2014
2012
2010
2008
0.00
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Lehman Case Study • Time line of events
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Lehman Case Study • Assets & Liabs
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Lehman Case Study
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Lehman Case Study • PD curves
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Lehman Case Study • PD over time
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Lehman Case Study • Equity value
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Lehman Case Study • Liquid • Semi • Illiquid
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Lehman Case Study
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Lehman Case Study
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Lehman Case Study
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Lehman Case Study
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Conclusion • Next topic: Liquidity
THANK YOU
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