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Economic Capital

  • The required capital reserve – which acts as a buffer against insolvency for the bank in the event of the default by an obligor –  is known as economic capital
  • With the fluctuation in the unexpected loss the economic capital will also fluctuate
  • Economic Capital required to cover UL
  • Pricing, Loan Reserves take care of EL

Problem

ABC bank has extended a line of credit of US$120mn to XYZ company. The company has already drawn US$50mn of the line. The company is assumed to have a Usage Given Default of 40% and companies with similar ratings are known to have a default probability of 20%. The assets of the company are such that it will not be possible to recover more than 50% of the value. The Variance observed in LGD has been 20% and that in PD has been 10%.How much economic capital must the bank have to cover the unexpected loss on this asset?
 

Solution

OS = $50mn
COM = $120mn
PD = 20%
LGD = 50%
σ2PD = 10%
σ2LGD = 20%
 
Undrawn Portion                 = $120mn – $50mn = $70mn
UGD                                    = 40%
Adjusted Exposure (EA)     = $50mn + 40% * ($70mn) = $50mn + $28mn = $78mn
Unexpected Loss (UL)       = $ 78mn *  √[PD* σ2LGD +LGD2* σ2PD]
                                           = $78mn *  √[0.2*0.2 +0.52*0.1]
                                           = $78mn *  √[0.04 +0.025]
                                           = $78mn * 0.255
                                           = $19.88mn

 



 
Example (Expected Loss )

Assume the original commitment is $20mn, of which $10mn has been drawn by the borrower
(i.e., $10mn is outstanding and $10mn is the unused commitment)
(BB rated: UGD = 50% ), PD/EDF = 1%, LGD = 50%
What is the expected loss (EL)?
How does EL relate (vary with) PD and LGD (e.g., linear, nonlinearly)?
What is the meaning of UGD? What is the difference between UGD and LGD?
we have everything we need to calculate unexpected loss (UL) except for which one value. Which value do we require that’s difficult?

 

Solution

AE = 10 + 10 * 50% = 15; EL = AE * LGD * PD = 15 * 50% * 1% = 75,000
Positively, linearly
Explained earlier
Variance of LGD and variance of PD

 
 
Question (Expected Loss )

 

Total COM $10,000,000
OS $5,000,000
Risk rating 3(1year non secured)
UGD 65%
AE $8,250,000
EDF 0.15%
LGD 50%
EL $6,188
 
 Question (Unexpected Loss )

 
Total COM $10,000,000
OS $5,000,000
RC 3IN 1 year non secured
UGD 65%
AE $8,250,000
EDF 0.15%
Sigma EDF 3.87%
LGD 50%
Sigma LGD 25%
UL $178,511


Question (Portfolio , Unexpected & Expected Loss )
Assume a two-asset portfolio. The first exposure is as above ($10mn outstanding; $10mn unused commitment; usage given default (UGD) = 50%; probability of default (PD) = 1%; loss given default (LGD) = 50%; standard deviation of LGD = 50%; standard deviation of EDF = 30%. The second exposure is the same EXCEPT its
LGD = 25% and the standard deviation of its LGD = 25%. Default correlation between the two exposures is 10%
What is the portfolio’s expected loss (EL)?
What is the portfolio’s unexpected loss (UL)?
Why are the sum of unexpected losses (UL + UL) greater than portfolio UL?
Under what condition would (UL) + (UL) = portfolio UL?

 

Solution

ELA = 15 * 1% * 50% = 75,000; ELB = 15 * 1% * 25% = 37,500; total = 112,500
ULA = 15 * [1% * 50%2 + 50%2 * 30%2]0.5 = 0.61 ; ULB = 15 * [1% * 25%2 + 25%2 * 30%2]0.5 = 0.31 (millions)
ULTotal = [(0.5ULA)2 + (0.5ULB)2 + 2 * 0.5 * 0.5 * ULA * ULB * 0.10]2 = $3,55,756 million
Because correlation is 10% i.e., diversified
When correlation is 100%

 



 




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