Students - David O Forfar

2012 MSc Operation Research (Finance) Dissertation
Supervisor : David Forfar (see http://www.davidforfar.com )
Title : Counterparty Credit Risk
‘Counterparty Credit Risk’ (CCR) is the risk of the other party to a transaction (whether
Government or a company) not being able to meet its obligation to pay interest or to repay
the capital on money it has borrowed or both.
In the case of Greek bonds, the Greek Government was unable to pay the redemption value
of its bonds. In order to remain part of the Euro, the Greek Government had to be relieved by
investors in Greek bonds of most of its obligation to repay the capital and interest due under
these bonds (and investors in Greek bonds suffered correspondingly!).
However, 'Counterparty Credit Risk' (CCR) usually refers to corporate entities (commercial
banks, investment banks and other companies) and not sovereign entities (like the Greek
Government).
Counterparty Credit Risk (CCR) became a very serious issue in the credit crunch of 2008-09
because if anyone had said, in 2006, that a huge bank like Lehman Brothers, full of 'quants'
(experts in quantitative finance), would fail, they would have laughed at you. However,
Lehman Brothers went bankrupt!
In the UK, Northern Rock, Royal Bank of Scotland and Bank of Scotland had to be bailed out
by the UK Government and UK taxpayer, costing them a very large sum of money.
In the USA, Fannie Mae and Freddie Mac were given best quality credit ratings - AAA - by
the rating agents (like Standard and Poors) - but still failed and (as well as American
International Group, AIG and others) had to be bailed out by the US Government and the US
taxpayer, costing them a very large sum of money.
So credit risk became a major issue.
Now all banks and other financial institutions are much more cautious and make an
adjustment (CVA, ‘credit valuation adjustment’) for their own assessment of the
creditworthiness of their counterparties and the determine whether a ‘trade’ is likely to be
profitable, taking account of the possibility that the counterparty might fail.
It would be useful for students to develop some familiarity (prior to the MSc beginning) of the
undernoted book by John Gregory and his spreadsheets, see his website www.oftraining.com
Students
(1) will cover the size of credit market, defining CCR, quantifying CCR, pricing CCR
(including the incorporation of ‘wrong-way’ risk) and hedging CCR,
(2) will know about credit ratings, AAA, AA etc.
(3) will calculate the time to default of one loan (using the exponential distribution and
depending on the hazard rate) and multiple loans,
(4) will use EXCEL spreadsheets to determine the price of financial instruments like
Interest Rate Swaps (IRS) and Credit Default Swaps (CDS),
(5) will program, in the computer language VBA (which comes with EXCEL),
(6) will learn how to mitigate CCR through use of the ISDA Master Agreement, netting,
collateral, hedging, creating an exchange etc.
(7) will learn how to make the calculation of CVA (‘credit valuation adjustment’) using
EXCEL and VBA both with no ‘wrong way’ risk and with ‘wrong-way’ risk,
(8) will determine if a ‘trade’ is profitable i.e. if the ‘spread’ exceeds the ‘CVA’,
(9) will recognise the failure of American International Group (AIG), one of the largest
insurance companies in the world, to use quantitative risk management. [AIG insured,
against default, certain parts of loans backed by sub-prime mortgages. AIG thought
there was no risk to them of writing this business! But the US Government had to bail
out AIG to tune of about 145 billion dollars as AIG was "too big to fail" etc....],
(10) Will learn something about BASEL II and III and the amount of regulatory capital that
a bank must hold,
(11) Will learn something about IFRS 9 and the accounting principle of measuring ‘fair
value’.
(12) the future of CCR.
Books
1. Gregory J. (2010), Counterparty Credit Risk, the new challenge for global financial
markets, John Wiley.
2. Löffler G. and Posch P. N., (2007), Credit Risk modeling using EXCEL and VBA, John
Wiley.
3. For interest rates swaps, currency swaps and options on swaps see Hull J. C. (2003)
Options, Futures and other Derivatives (for 5th Edition it is pages 134-147 and 520524)
Terminology
(1) Spot interest rates to-day, varying with the term of the loan (e.g. for a loan of
one year the spot interest rate may be 5.0% per annum but for a loan of two
years the 2-year spot interest rate may be 5.5% per annum….)
(2) Interest rate per annum convertible annually/yearly, half-yearly, quarterly…
(3) Force of interest to-day (continuously compounded)
(4) Yield curve to-day i.e. the term structure of the spot interest rates (how the
spot interest rates vary with the term of the loan)
(5) Forward interest rates to-day (nothing to do with interest rates in a year or in
two years, three years ….)
(6) Determination of to-day’s forward rates of interest from to-day’s yield curve of
to-days spot interest rates
(7) Basis point (an interest rate of 1% divided into 100 parts so a ‘basis point’
equals 0.01%)
(8) Zero Coupon Bonds
(9) Pricing of Zero Coupon Bonds,
1
P(0, N ) (1 the-spot-interest-rate-per-annum-converible-yearly-for-the-term-of-N-years)
N
(10) Coupon bonds
(11) Nominal/notional/face value
(12) Par value
(13) Coupon rate ( c )
(14) Pricing coupon bonds P(0, N ) c * a N
100v N
(15) Mark-to-Market (MtM)
(16) Gilt-edged securities (sovereign or government securities)
(17) Fixed interest rate
(18) Floating interest rate
(19) London Inter-Bank Offer Rate (LIBOR) – a rate of interest
(20) Interest Rate SWAP (IRS)
(21) Notional amount of a SWAP (notional=not exchanged)
(22) Fixed rate payer
(23) Fixed rate receiver
(24) Floating rate payer
(25) Floating rate payer
(26) Valuation of a SWAP at start (i.e. t=0)
(27) Valuation of a SWAP late (i.e. at t=t)
(28) Foreign Exchange (FX)
(29) Cross-Currency Swap (CCS)
(30) Valuation of a Cross-Currency SWAP at any time
(31) Actual Exchange of Principal (i.e. not notional) in a Cross-Currency Swap
(32) Over-the-Counter (OTC) derivative
(33) Bi-lateral transaction
(34) Gaussian distribution (Normal distribution)
(35) ‘Exposure’ of the first party to the second party - counterparty risk only if the
MtM is positive
(36) Calculation of ‘exposure’ (E)
(37) Exposure at default (EAD)
(38) Expected Exposure (EE)
(39) Expected Positive Exposure (EPE)
(40) Credit risk
(41) Counterparty-credit-risk (CCR)
(42) Credit quality/grade (AAA, AA,A, BBB….)
(43) Investment quality/grade (credit quality BBB or better)
(44) Speculative/Junk Bond (credit quality BB or worse)
(45) Migration of Credit Quality
(46) Default
(47) Probability of default
(48) Hazard rate (force of default)
(49) Loss given default (LGD)
(50) Recovery rates (1-LGD)
(51) Credit-Value-Adjustment (CVA)
(52) Wrong-way risk
(53) CVA=Exposure*probability-of-default*loss-given-default (if no ‘wrong-way’ risk)
(54) International Swaps and Derivatives Association (ISDA)
(55) ISDA Master Agreement
(56) Netting (close-out netting and payment netting)
(57) Netting Set (NS)
(58) Collateral
(59) Credit Default Swap (CDS)
(60) Pricing of a CDS, spread=insurance premium
(61) Hedging with a CDS
(62) Physical Settlement of a CDS
(63) Cash settlement of a CDS
(64) Option on a swap (swaption)
(65) Remargin period
(66) Threshold (uncollaterised within the threshold)
(67) Minimum Transfer Amount (MTA)
(68) Independent Amount (IA)
(69) Rounding
(70) Internal ratings-based approach (IRB)
(71) Gap risk (market value of a transaction may ‘gap’ substantially in a short
period of time)
(72) Credit spread (difference between ‘risky’ and risk-free yield) but, in the market,
bigger than required to compensate for actual historical rates of default as also
compensate for illiquidity, systematic risk etc.
(73) Credit derivative (designed to transfer ‘credit risk’ to a counterparty e.g. a
CDS)
(74) Collaterised Debt Obligation (CDO)
(75) Tranches of a CDO (Super Senior, Senior, Junior Senior, Senior Mezzanine,
Mezzanine, Junior Mezzanine, Equity )
(76) iTraxx index
(77) Insuring the Super Senior tranche of a CDO
(78) Leverage is a measure of how much debt is used to purchase assets; e.g. a
leverage ratio of 20:1 means that £20 of assets were purchased with £19 of
debt and £1 of equity capital
(79) Value at Risk (VaR) the amount of money that could be lost with, say, 95%
probability over, say, a 10 day period.
(80) Centralised Counterparties (CCPs)
(81) Commercial paper (CP) e.g. a loam/bond from a company
(82) Debt valuation adjustment (DVA) the opposite of CVA i.e. the counterparty’s
CVA
(83) Internal Model Method (IMM)
(84) Internal ratings-based (IRB)
Settlement of a CDS
Under a CDS the protection buyer (A) insures with the protection seller (B) a
nominal amount of the reference entity (C) bond.
Physical Settlement
On a default, (A) then delivers to (B) the bond of (C) (buying it if (A) has not got it,
it still trades - at a very low price - in the market even although (C) has defaulted and if a lot of people may need to buy it it to deliver to (B) because they have a
CDS, there may be a so-called 'squeeze')
alternatively, Cash Settlement (more popular)
(A) compensates (B) in cash for the estimated value of the bond - which (A) does
not need to hold(estimated by a market consensus among the traders).
Thus the risk of mis-estimating the value of bond (C) (through mis-estimating
recovery rate) is taken by (B) - the protection seller.
Bilateral Credit Value Adjustment (BCVA)
Although CVA is always positive, it should be noted that BCVA can be negative (if
the second term in (7.5) is larger than the first term).
BCVA differs from CVA in that it also includes (unlike CVA) the probability of the
other Party surviving (i.e. not defaulting).
BOTH Party and the Counterparty are assumed risky
If the Exposure (remember: must be positive) of the Party (i.e. E -the Trade viewed
from the first side) is positive but then some time later the Exposure of the
Counterparty (i.e. NE-the Trade viewed from the other side) is positive, we say the
Exposure swings between the Party and the Counterparty (e.g. on a SWAP).
The value of the Trade is then the (risk-free value of the Trade) minus BCVA (NOT
CVA).
BCVA has two terms and if there is no wrong-way risk (and to be precisely accurate
and also no simultaneous defaults) the BCVA is described as the formula on
Gregory p.182.
Marginal exposure
If the total exposure of all the trades in the Netting Set is E (note : E must be
positive), then the marginal exposure of each of the trades is the contribution
(normally %) of each trade to E.
It is ExpectedValue(E1|E>0)/ExpectedValue(E|E>0) and similarly For E2, E3, E4
.......
Credit Valuation Adjustment (CVA)
Stand Alone CVA
The CVA of the trade when it stands on its own and no other trade is considered.
Incremental CVA
When the trade is part of a Netting Set (NS) and a new trade is added to the
existing Netting Set.
ECVA is the CVA for the existing Netting Set.
The new CVA (NCVA) is calculated for the (existing Netting Set plus the new trade).
IncrementalCVA=NCVA-ECVA.
Also IncrementalCVA =SUM{ (1- recovery)* IncrementalExposure each
period*prob. of default in that period*discount factor for that period}
Marginal CVA
If E is positive there is exposure.
MarginalCVA =SUM{ (1- recovery)* Marginal Exposure each period*prob. of
default in that period*discount factor for that period}.
VALUATION OF A SWAP (Assume a 3 year swap)
Assume the 3 spot-rates are spot1, spot2, spot3
P(0,0) 1
P(0,1)
1
(1 spot1)1
P(0, 2)
1
(1 spot 2)2
P(0,3)
1
(1 spot 3)3
Forward rates are forward1, forward2, forward3
(0,1)
forward1 ( PP(0,0)
1) forward 2 ( PP(0,2)
1)
(0,1)
forward 3 ( PP(0,2)
1)
(0,3)
Valuation of the Fixed Leg
swap-rate*[ P(0,1) P(0, 2) P(0,3)]
Valuation of Floating Leg
forward1* P(0,1)
forward 2* P(0, 2)
forward 3* P(0,3)
(0,1)
( PP(0,0)
1) * P(0,1) ( PP(0,2)
1) * P(0, 2) ( PP(0,2)
1) * P(0,3)
(0.1)
(0,3)
P(0, 0) P(0,1) P(0,1) P(0, 2) P(0, 2) P(0,3)
1 P(0,3)
Jon Gregory Formula
Initially (i.e. at t=0) but NOT thereafter, the fixed and the floating legs are made equal
and this determines the swap-rate.
The Mark-to-Market (MtM) therefore starts at zero but becomes positive or negative
and is given (for a payer SWAP i.e. pay fixed) by Mark-to-Market (MtM) = (Value of
floating leg) – (Value of fixed leg).
The Exposure is the value of MtM BUT ONLY if this is positive. Exposure means
exposed to the risk (this is a credit risk) of the other party (the counterparty) owing
you money. This arises where you are the fixed rate payer of a SWAP (i.e. you pay
the fixed leg to the counterparty) and the counterparty pays you the floating rate (i.e.
the counterparty pay you the floating leg) and these floating rates have increased
since the start (i.e. t=0) so the counterparty owes you money as the value of the
floating leg minus the value of the fixed leg is positive.
MARGINAL EXPOSURE (WHERE THERE ARE TWO TRADES WHICH CAN BE
NETTED OFF AGAINST EACH OTHER)
Lemma: E[Z1 | Z2 k ] = (k ) where Z1 and Z2 are correlated standard normal
variables with correlation coefficient
and where ( x ) is the standard normal
1
2
probability density function i.e. ( x)
e
1 2
x
2
.
Proof:
1
k]
2
xe
2
1
x
1
2
1
x
y
1 2
y
2
y
2
2
1
1 2
y
2
e
e
2
y k
1
ue
2
))
dx
1
dy
xe
2(1
2
)
y )2
(x
dx
1
dy
(u
2(1
2
)
y)e
2
2(1
1
u
u2
du
y
u
)
u2
du
e
2
2(1
)
u2
du ] where u
x
y
u
y
1
2
y )2 y 2 (1
(( x
x
u
dy [
dxdy
x
y k
1
( x2 y 2 2 xy )
x
y
1
1
)
x
y k
2
2
2(1
xe
1 2
y
2
e
2
1
)
y k
1
2
1
dy
2
2(1
y k
y
x
2
1
y
x
E[ Z1 | Z 2
e
2
1
1 2
y
2
dy [0
y 2
1
2
]
y k
y
ye
2
1 2
y
2
y
dy ]
y k
e
2
y k
1 2
y
2
y2
d ( )] =
2
2
[ e
1 2
y
2
]k
(k )
( k)
□
The marginal exposures of X1 and X2 (assuming these two exposures can be netted
off against each other) are defined as E[ X1 | X1 X 2 0] and E[ X 2 | X1 X 2 0]
Where X1
Z and X 2
2
2 Z 2 and the correlation between X1 and X1+X2
is the same as the correlation between Z1 and the standard normal variable
1
*
2
2
Z3
( 1Z1
2 1 2 and is the correlation between X1
2 Z 2 ) where
1
2
*
1
1 1
and X2 (the same as the correlation between Z1 and Z2).
( 1
corr ( X 1 , X 1 X 2 ) corr ( Z1 , Z3 )
2
)
*
Hence
Where k
Marginal exposure of X1 E[ X1 | X1 X 2 0] E[ 1 1Z1 | Z3 k ]
( 1
2)
and Z3 is a standard normal variable correlated to Z1 with
*
correlation coefficient
Therefore, using the Lemma, the marginal exposures of X1 and X2 are, respectively
1
( k)
(
1
2
1
)
( k ) and
*
2
( k)
(
2
2
1
*
)
( k)
For percentage contributions divide by the sum, which is
(
1
2
) ( k)
( k)
*
Which is the same as
E[ X1 X 2 | X1 X 2
Which also equals (
1
2
) ( k)
*
0]
( k ) (see Jon Gregory book, p40)
BASEL I
Under BASEL I, for every pound lent to a corporation, regardless of whether it was
rated AAA, CCC or unrated, banks were required to hold 8% (of every pound lent)
in equity.
BASEL III and III
Pillar 1 is Minimum Capital Requirements (for credit risk, market risk, operational risk
etc...)
Pillar 2 is Regulation
Pillar 3 is Disclosure requirements (form of public documents, e.g. Profit & Loss
account, Balance Sheet etc.)
Under BASEL II banks could use:a) the so-called ‘standardised approach’ or
b) the so-called ‘internal ratings approach’ (IRB).
Under the IRB, an artificial variable Ai is constructed for each loan and it is
assumed:Ai
Z
1
Ei
where Z and Ei are standard normal N (0,1) variable which are independent and
all the Ei ' s are independent; therefore Ai is also N (0,1) . The same factor Z is
shared by all the loans and therefore the common Z means the loans are all
correlated with common correlation coefficient between the loans equal to the
common correlation between the Ai’s i.e. .
The default probability for the loans is PDi and the loans are assumed correlated.
1
( PDi ) .
The link to default is that default is deemed to occur if Ai
Given a value of Z that means default occurs if :1
Ai
( PDi )
Z
1
1
Ei
Ei
( PDi )
1
1
Z
( PDi )
1
So the probability of loan i defaulting is
[
( PDi )
Z
1
A conservative value of Z is a very negative value like
]
1
(0.999) .
So a conservative probability of default for loan i is
1
1
( PDi )
(0.999)
[
]
1
Ignoring the adjustment for maturity, the capital, under BASEL II, required to be
held by the bank against loan i thus equals:1
EADi *(1
i
)* [
1
( PDi )
1
(0.999)
]
Where EAD is the exposure at default and is the recovery rate. The correlation
coefficient is set (rather artificially) to be between 0.12 and 0.24 by the formula
*0.12 (1
where
) *.24
1 e 50*PD
1 e 50
It should be noted that PDi is the probability of default up until the time horizon
chosen (which might be ten years) and not the probability of default each year as
in the CVA calculation. Also there is no discounting factor (in contrast to the CVA
calculation).
IFRS 9
Works on the basis of ‘fair value’ which is based on the market value of the financial
instrument where there is a market and ‘best estimate’ for the value of the financial
instrument (reflecting market assumptions where possible) where there is not.