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Why Are the VaR and Put Approaches So Different: Self-Insurance vs. Third-Party Insurance

Why Are the VaR and Put Approaches So Different: Self-Insurance vs. Third-Party Insurance

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PORTFOLIO STRATEGY AND RISK MANAGEMENT



passenger on the Titanic? The put/insurance policy can still be valued and purchased.

We do a credit risk “CAT scan” that looks through all the assets and liabilities of the

firm and all of the counterparties to see how they are affected by key macro factors

like interest rates, home prices, foreign exchange rates, and stock indices. This is

exactly what we describe below. We then buy the appropriate insurance protection

on these macro factors. Almost all of them have exchange traded futures and (in some

cases) options, so we don’t have a Titanic problem.

In short, the difference between a VaR approach to capital needs and a put

option or insurance approach to capital needs is the difference between self-insurance

or insurance in an efficient market that recognizes diversification. In most cases, the

latter assumption will be a much more accurate indicator of how much capital

protection is necessary.

As we explained previously, a put option or insurance policy that insures the firm

against the same loss profile is the probability-weighted present value of all scenarios.

Given identical loss projections, this normally means that the “true” VaR will be

much higher than the put price (unless the losses are the same in all scenarios) because

the put is probability weighted and the VaR is not. Then what about this quote from

Bloomberg.com on January 28, 2008?

Merrill’s highest one-day value-at-risk in the third quarter was $92 million,

indicating that the firm’s maximum expected cost during the 63-trading day

period would be $5.8 billion. In fact, the firm wrote down $8.4 billion from the

value of collateralized debt obligations, subprime mortgages and leveraged

finance commitments, 45 percent more than the worst-case scenario. (www.

bloomberg.com/apps/news?pid¼newsarchive&sid¼axo1oswvqx4s.)

How could that happen?

What was reported in the Bloomberg story was “false” VaR, a gross underestimate of risk that has a host of problems that we listed in the previous section

“Is Your Value-at-Risk from Value-at-Risk?” The Bloomberg story goes on to state

that most of the large U.S. securities firms were using “false” VaR based on one to

four years of historical data. Given the history of U.S. home prices, a four-year

historical false VaR would have predicted minimal losses (as it did) because over the

four-year period ending December 31, 2007, home prices were strongly up. Even

though home prices peaked in Los Angeles in September 2006, they did not begin to

have a powerful impact on mortgage defaults until the second half of 2007. False

VaR is based on an average of history. True VaR, which is what we assumed in our

previous example, is a complete and accurate listing of all possible future outcomes

and their probabilities, Nobel laureate Kenneth Arrow’s famous “state space.” If we

base true VaR on that complete listing of outcomes and probabilities, the put price or

cost of insuring risks via a third party that can diversify will almost always be less

than the high percentile (say ninety-ninth) true VaR number. That true VaR number

is the amount of self-insurance you need to survive with 99 percent probability.

Wasn’t the miss with respect to VaR just a black swan as Nassim Taleb argued?

No, it was just bad math. It was an assumption that the world was flat. As we

explained previously, historical VaR contains an implicit assumption that neither

Lehman nor Bear Stearns could fail, because (based on historical equity returns) the



Value-at-Risk and Risk Management Objectives Revisited at the Portfolio and Company Level



731



implied probability of À100 percent stock returns within a month for each of them

was 0.000000 percent. Similarly, if the monthly changes in home prices have been

strongly positive over the measurement period, the historical VaR approach will

implicitly assume away the probability of a decline. The “true” VaR calculation and

a rationally priced put will consider all possible future scenarios, not just those that

have actually come about in the past 12 or 48 months.



CALCULATING THE JARROW-MERTON PUT OPTION VALUE AND

ANSWERING THE KEY 4 ỵ 26 QUESTIONS

“What is the hedge?” In other chapters in this book, we posed this question as the

best single-sentence test of risk management technology. For that reason, the JarrowMerton put option is a very powerful concept because the put option they propose, if

properly structured, is the hedge. What is the equivalent of the Merton and Jarrow

put option in the interest rate risk context? It is the value of an option to buy the

entire portfolio of the financial institution’s assets and liabilities at a fixed price at a

specific point in time.

Here are some examples of the Jarrow-Merton put option as a practical risk

management concept:

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Instead of the 10-day value-at-risk of a trading portfolio, what is the value of a

10-day put option on my current portfolio with an exercise price equal to the

portfolio’s current market value? The price of the put option will increase

sharply with the risk of my portfolio, and the put option’s price will reflect all

possible losses and their probability, not just the ninety-ninth percentile loss as is

traditional in value-at-risk analysis

Instead of stress testing the 12-month net income of the financial institution to

see if net income will go below $100 million for the year, what is the price of a

put option in month 12, which will produce a gain in net income exactly equal to

the shortfall of net income versus the $100 million target? The more interest rate

risk in the balance sheet of the financial institution, the more expensive this put

option will be. The put option will reflect all levels of net income shortfall and

their probability, not just the shortfalls detected by specific stress tests

Instead of the Basel II, Basel III, or Solvency II risk-weighted capital ratio for

the institution, what is the price of the put option that insures solvency of the institution in one year’s time? This put option measures all potential losses embedded in

the financial institution’s balance sheet and their probability of occurrence, including

both interest rate risk and credit risk, as we discuss at the end of this chapter.

Instead of expected losses on a collateralized debt obligation tranche’s B tranche,

what is the price of a put option on the value of the tranche at par at maturity?

This put option reflects all losses on the tranche, not just the average loss, along

with their probability of occurrence.

Instead of expected losses on the Bank Insurance Fund in the United States, the

Federal Deposit Insurance Corporation (FDIC) has valued the put option of

retail bank deposits at their par value as discussed in the FDIC’s loss distribution

model announced on December 10, 2003.



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PORTFOLIO STRATEGY AND RISK MANAGEMENT



The Jarrow-Merton put option concept helps to reconcile what many regard as

conflicting objectives to be managed from a risk management perspective:

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Net interest income (or net income), which is a multiperiod financial accountingbased exhibit that includes the influence of both instruments that the financial

institution owns today and those that it will own in the future.

Market value of portfolio equity, which is the market value of the assets a

financial institution owns today less the market value of its liabilities.

Market-based equity ratio, which is the ratio of the mark-to-market value of

the equity of the portfolio (“market value of portfolio equity” in bank jargon)

divided by the market value of assets. This is most closely related to the capital

ratio formulas of the primary capital era, Basel I, Basel II, Basel III, and Solvency II.

Default probability of the institution, which is another strong candidate as a

single measure of risk.



We next discuss how the valuation of these options can be done using the

technology in the first 38 chapters of this book.



VALUING AND SIMULATING THE JARROW-MERTON PUT OPTION

How do we value the Jarrow-Merton put option? In some of the cases mentioned

above, we already have known valuation approaches from Chapters 20 through 35

using an N-factor Heath, Jarrow, and Morton (HJM) interest rate model. In those

chapters, we used the three-factor HJM model from Chapter 9 for exposition purposes. For the full on- and off-balance sheet portfolio of a financial institution, we

simply repeat the process with finer time granularity and more transactions. This is

what computers are for. We need to calculate the risk-neutral expected values of the

cash flows that would be paid on the Jarrow-Merton put option that we are

analyzing.

We outlined the steps in this simulation in Chapter 19 in the context of a

portfolio of risky bonds:

For the reduced form model, the steps are as follows:

1. Simulate the risk-free term structure using an N-factor HJM approach.

2. Choose a formula and the risk drivers for the liquidity component of credit

spread for all of the relevant asset classes as discussed in Chapter 17.

3. Choose a formula and the risk drivers for the default-intensity process in the

Jarrow model as discussed in Chapter 16 for all asset classes.

4. Simulate the random values of the drivers of liquidity risk and the default

intensity for M time periods over N scenarios, consistent with the risk-free term

structure.

5. Calculate the default intensity and the liquidity component at each of the M time

steps and N scenarios for each counterparty, from retail to small business to

corporate to sovereign. Note that these will be changing randomly over time

because interest rates and other key macro factors (like home prices) are

changing as well. This is essential to capturing cyclicality in traded asset prices

and defaults.



Value-at-Risk and Risk Management Objectives Revisited at the Portfolio and Company Level



733



6. Apply the Jarrow model as we have done in Chapters 16 and 17 to get the

zero-coupon bond prices for each maturity for each transaction for each

counterparty.

7. Calculate the dates and cash flow amounts for the Jarrow-Merton put option

structure that is most relevant.

8. Calculate the put option’s value by discounting by the appropriate risk-free zerocoupon bond price for each of the M time steps and N scenarios.

This is a general valuation procedure that we can apply for all instruments

discussed in Chapters 20 through 26.



WHAT’S THE HEDGE?

Let’s say management has become comfortable with the Jarrow-Merton put option

as a comprehensive measure of risk. How do we address the key test of risk management technology: “What’s the hedge?”

If we have done a comprehensive job of fitting our default probability models

as in Chapters 16–17 and a thorough job of fitting our interest rate models as discussed in Chapters 5 through 10, we can stress test the put option value that

we derived in the previous section. We pick a macroeconomic risk factor to stress

test. If the financial institution is a lender in the United States, the S&P 500 is a

commonly used macro factor that can be proven to be a statistically significant driver

of default probabilities for a large range of counterparties. We can calculate what

position in S&P 500 futures is needed to control changes in the Jarrow-Merton put

option as a measure of risk in exactly the same way van Deventer and Imai (2003)

discuss macro hedging of a portfolio of risky credits:

1. Select the base case value X for the S&P 500 and all other macro factors.

2. Value the appropriate Jarrow-Merton put option using the approach of the

previous section.

3. Select the stress test value Y for the S&P 500 and use all of the same macro

factor values as in step 1. The same Monte Carlo “seed value” as step 1 also has

to be used.

4. Get the stress-tested value of the Jarrow-Merton put option.

5. The change in the Jarrow-Merton put option is the Value in Case 1 À Value in

Case 2.

6. The change in the S&P 500 is X – Y.

7. The proper hedge depends on what type of S&P 500 hedging instruments are

being used (futures contracts or not, maturity of futures contracts, etc.). The

number of contracts to be used for the hedge is the number such that the change

in the value of the futures contracts if the S&P 500 moves from X to Y will

exactly offset the change in the value of the Jarrow-Merton put option from its

Case 1 value to its Case 2 value.

Using this approach, we always know the proper integrated hedge of interest rate

risk and credit risk. Our objectives have been achieved.



PORTFOLIO STRATEGY AND RISK MANAGEMENT



734



LIQUIDITY, PERFORMANCE, CAPITAL ALLOCATION, AND

OWN DEFAULT RISK

Using the Jarrow-Merton put option approach that we have built in this chapter on

the foundation of Chapters 1 through 35, we can now definitively address key risk

management issues like those in our previous 4 ỵ 26 questions:

What is the liquidity risk of my organization?

What amount of capital should I have in the institution as a whole?

What amount of capital should I have in each business unit?

What is the performance of each business unit?

We turn to that task in the next three chapters.



NOTE

1. www.bloomberg.com/apps/news?pid¼newsarchive&sid¼axo1oswvqx4s.



CHAPTER



37



Liquidity Analysis and Management

Examples from the Credit Crisis



I



n Chapter 36, we applied the Jarrow-Merton put option concept as a comprehensive

measure of integrated interest rate, market risk, foreign exchange risk, and credit

risk. We listed four key questions and 26 supplementary questions that can be

answered as a result of the risk management process that are outlined in the first 36

chapters of this book. We reviewed the observable market data on the cost of the

Jarrow-Merton put option for Citigroup at various time horizons. Finally, we showed

the multiperiod simulation process that allows us to value the Jarrow-Merton put

option, giving us an alternative measure of the dollar amount of money that would be

necessary to eliminate the risk we face. We also noted in that chapter that the put

option concept can be applied to risk management defined as shortfalls in net income,

capital ratios, or provisions for loan losses.

In this chapter, we apply the same concept to shortfalls in cash—liquidity risk—

which is tightly linked with the credit risk of the institution. We take great care to

avoid the single greatest mistake in liquidity risk analysis—basing the analysis solely

on the cash flow history of a firm that has never had a “near death” experience from

liquidity risk. Indeed, our focus in this chapter is primarily on institutions that have

had such near death experiences. We start with a review of the five biggest liquidity

risk problems in North America during the 2006–2011 credit crisis.



LIQUIDITY RISK CASE STUDIES FROM THE CREDIT CRISIS

In the introduction to this book, we discussed the six biggest fallacies in risk management. As an introduction to five case studies of liquidity shortfalls, we survey each

of the risk fallacies that contributed to the 2006–2011 credit crisis, which, in turn,

caused the liquidity shortfalls that are outlined in this section:

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If it hasn’t happened to me yet, it won’t happen to me, even if it has happened to

someone else. Many market participants believed that home prices in the United

States would never go down, but they did.

Silo risk management allows my firm to choose the “best of breed” risk model

for our silo. Risk management was so fragmented at the largest U.S. and

European institutions that the pervasive impact of home price falls on a wide

range of counterparty risks and security values was not modeled realistically.



735



PORTFOLIO STRATEGY AND RISK MANAGEMENT



736

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I don’t care what’s wrong with the model. Everyone else is using it. The copula

approach to CDO valuation was hopelessly flawed and confined to a largely

desktop review of traded products. A macroeconomic factor–driven total balance sheet analysis of credit risk was rarely performed.

I don’t care what’s wrong with the assumptions. Everyone else is using them.

The assumption that home prices and other macro factors had returns that were

independent from period to period was dramatically wrong. There is a strong

persistence in trends in returns, causing macro factor swings to be much larger

than what was modeled.

Mathematical models are superior to computer simulations. Few banks had the

systems capability to realistically model the impact of a large number of random

macro factors. What analysis was done was dominated by closed form solutions

that could only be obtained by making unrealistic assumptions (like the independence assumption above) about the nature of macro factor movements.

Big North American and European banks are more sophisticated than other

banks around the world and we want to manage risk like they do. The evidence

in this section is proof positive that, sadly, their example is not one that should

be followed.

Goldman says they do it this way and that must be right. Michael Lewis’ (2011)

instant classic, The Big Short: Inside the Doomsday Machine, thoroughly

documents how a few securities dealers were able to whipsaw smaller market

participants with contrarian (and more accurate) views for an extended period of

time with arbitrary mark-to-market positions imposed by those dealers. This

aggravated the crisis, delayed recognition of its magnitude, and triggered some of

the biggest flows of bailout funds.1

We now turn to five case studies in liquidity risk during the 2006–2011 credit crisis.



CASE STUDIES IN LIQUIDITY RISK

Largest Funding Shortfalls

In this section, we review the data used in the Kamakura analysis of 21 cases studied

in liquidity risk and then review each of the top five funding shortfalls.2 Under the

Dodd-Frank Act of 2010, the Board of Governors of the Federal Reserve was

required to disclose the identities and relevant amounts for borrowers under various

credit facilities during the 2006–2011 financial crisis. These credit facilities provide,

perhaps, the best source of data about liquidity risk and funding shortfalls of the last

century. We use this data to determine to what extent there was a funding shortfall at

the largest institutions active in U.S. financial markets during the credit crisis.

The data used in the study consist of every transaction reported by the Federal

Reserve as constituting a “primary, secondary, or other extension of credit.”

Included in this definition are normal borrowings from the Fed, the primary dealer

credit facility, and the asset-backed commercial paper program. Capital injections

under the Troubled Asset Relief Program (TARP) and purchases of commercial paper

under the Commercial Paper Funding Facility are not included in this definition put

forth by the Federal Reserve. We analyze borrowings under the Commercial Paper

Funding Facility separately.3



Liquidity Analysis and Management



737



Kamakura took the following steps to consolidate the primary, secondary, and

other extensions of credit:

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From www.twitter.com/zerohedge, Kamakura downloaded the daily reports, in

PDF format, from the Federal Reserve on primary, secondary, and other extensions of credit from February 8, 2008, until March 16, 2009, approximately 250

reports in total.

Kamakura converted each report to spreadsheet form.

These spreadsheets were aggregated into a single database giving the origination

date of the borrowing, the name of the borrower, the Federal Reserve District of

the borrower, the nature of the borrowing (ABCP, PDCF, or normal), the

maturity date of the borrowing, and (in the case of Primary Dealer Credit

Facility) the name of the institution holding the collateral.

Consistency in naming conventions was imposed; that is, while the Fed listed two

firms as “Morgan Stanley” and “M S Co,” Kamakura recognized to the maximum

extent possible that they are the same institution and used a consistent name.

To the maximum extent possible, the name of the ultimate parent was used in order

to best understand the consolidated extension of credit by the Fed to that firm.4



Kamakura calculated the daily borrowings of a large number of prominent

international financial institutions during the period covered by the Federal Reserve

data. For each of these selected institutions, Kamakura calculated these statistics:

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Maximum Outstanding Borrowing

Average Outstanding Borrowing

First Borrowing Date

Average Borrowing on Nonzero Borrowing Days

Number of Borrowing Days



Exhibit 37.1 ranks these selected institutions by maximum borrowings during

the period covered. AIG leads the list with a maximum borrowing of $208 billion

during this period.

Readers should be under no illusion that this was just a temporary facility and

that these institutions could have survived without government aid. Nothing could be

further from the truth, as nicely summarized by the report by the Office of the Special

Inspector General of the Troubled Asset Relief Program entitled “Emergency Capital

Injections Provided to Support the Viability of Bank of America, Other Major Banks,

and the U.S. Financial System,” October 5, 2009. For each of the top five borrowers

in Exhibit 37.1, we discuss their borrowings in detail below. Exhibit 37.2 ranks

the same institutions by the average amount of borrowing on those days during the

period when borrowings were outstanding.

We now turn to a name-by-name summary of the top five liquidity crises in

North America during the period covered by the Fed data.



American International Group (AIG)

Excellence in risk management and good corporate governance require that financial

institutions analyze their own probability of default. The proposed Basel III



EXHIBIT 37.1 Kamakura Corporation, Analysis of Primary, Secondary, and Other Borrowings from the Federal Reserve Ranked by

Maximum Borrowing

First

Borrowing

Date



Average Of NonZero Borrowing

Days



Number of

Borrowing

Days



AIG

208,616,483,142 72,793,773,121 20080916

Consolidated JPMorgan, Bear Stearns,

and WaMu

101,125,000,000 17,728,196,729 20080306

State Street

77,802,450,046 9,000,920,496 20080220

Morgan Stanley

61,292,078,000 7,223,467,296 20080317

Dexia New York Branch

50,000,000,000 12,875,000,000 20080417

Consolidated BAC, Countrywide, and

Merrill Lynch

48,141,416,451 10,091,065,527 20080310

Barclays

47,942,000,000 2,452,411,911 20080317

Depfa Bank PLC New York Branch

47,800,000,000 9,716,203,474 20080930

BNY Mellon

41,588,201,426 2,495,094,570 20080922

Merrill Lynch

39,956,500,000 7,620,802,284 20080310

Wachovia Bank NA

36,000,000,000 5,962,593,052 20080416

Lehman Brothers

28,000,000,000

195,555,571 20080318

Citigroup

24,200,000,000 6,712,770,766 20080318

Goldman Sachs

24,200,000,000 2,155,973,945 20080318

Royal Bank of Scotland Group, including

CP*

20,460,500,000 5,531,579,799 20080229

Bank of America

13,000,000,000 1,895,449,347 20080619

Bank of Scotland PLC New York Branch 12,000,000,000 2,363,275,434 20080917

Royal Bank of Scotland PLC New York &

RBS Citizens

8,400,000,000

49,284,697 20080229

Societe Generale New York Branch

8,000,000,000 2,584,880,893 20080325

Countrywide Financial

6,295,000,000

574,813,896 20080317

HSH Nordbank AG New York Branch

5,250,000,000 1,662,622,829 20080327



161,186,211,912



182



23,666,720,690

19,821,699,234

16,172,540,667

15,534,805,389



290

183

180

334



14,120,484,053

6,177,012,500

24,022,269,939

5,880,252,114

16,164,122,739

23,791,336,634

5,253,926,333

12,524,289,901

11,584,766,667



288

160

163

171

190

101

15

211

75



9,604,281,409

4,106,806,919

8,140,170,940



364

186

117



2,837,390,429

3,960,863,118

2,271,078,431

3,004,650,224



7

263

102

223



Rank Institution

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21



*Average calculated through October 31, 2009.



738



Maximum

Outstanding



Average

Outstanding



EXHIBIT 37.2 Kamakura Corporation, Analysis of Primary, Secondary, and Other Borrowings from the Federal Reserve Ranked by Average of

Non-Zero Borrowing Days

First

Borrowing

Date



Average Of NonZero Borrowing

Days



Number of

Borrowing

Days



AIG

208,616,483,142 72,793,773,121 20080916

Depfa Bank PLC New York Branch

47,800,000,000 9,716,203,474 20080930

Wachovia Bank NA

36,000,000,000 5,962,593,052 20080416

Consolidated JPMorgan, Bear Stearns,

and WaMu

101,125,000,000 17,728,196,729 20080306

State Street

77,802,450,046 9,000,920,496 20080220

Morgan Stanley

61,292,078,000 7,223,467,296 20080317

Merrill Lynch

39,956,500,000 7,620,802,284 20080310

Dexia New York Branch

50,000,000,000 12,875,000,000 20080417

Consolidated BAC, Countrywide, and

Merrill Lynch

48,141,416,451 10,091,065,527 20080310

Citigroup

24,200,000,000 6,712,770,766 20080318

Goldman Sachs

24,200,000,000 2,155,973,945 20080318

Royal Bank of Scotland Group,

including CP*

20,460,500,000 5,531,579,799 20080229

Bank of Scotland PLC New York

Branch

12,000,000,000 2,363,275,434 20080917

Barclays

47,942,000,000 2,452,411,911 20080317

BNY Mellon

41,588,201,426 2,495,094,570 20080922

Lehman Brothers

28,000,000,000

195,555,571 20080318

Bank of America

13,000,000,000 1,895,449,347 20080619

Societe Generale New York Branch

8,000,000,000 2,584,880,893 20080325

HSH Nordbank AG New York Branch

5,250,000,000 1,662,622,829 20080327

Royal Bank of Scotland PLC New York

& RBS Citizens

8,400,000,000

49,284,697 20080229

Countrywide Financial

6,295,000,000

574,813,896 20080317



161,186,211,912

24,022,269,939

23,791,336,634



182

163

101



23,666,720,690

19,821,699,234

16,172,540,667

16,164,122,739

15,534,805,389



290

183

180

190

334



14,120,484,053

12,524,289,901

11,584,766,667



288

211

75



9,604,281,409



364



8,140,170,940

6,177,012,500

5,880,252,114

5,253,926,333

4,106,806,919

3,960,863,118

3,004,650,224



117

160

171

15

186

263

223



2,837,390,429

2,271,078,431



7

102



Rank Institution

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21



Maximum

Outstanding



Average

Outstanding



*Average calculated through October 31, 2009.



739



740



PORTFOLIO STRATEGY AND RISK MANAGEMENT



liquidity risk ratios are concrete symbols of the regulators’ focus on default risk.

Liquidity risk is the symptom that a firm has some other severe disease, be it credit

risk, market risk, interest rate risk, fraud, or something else. Default becomes inevitable when it becomes apparent that an institution cannot liquidate its assets

with sufficient speed or volume to meet cash needs from liabilities that have been

withdrawn (in the case of deposits) or that will not be rolled over (commercial paper,

bank lines, bonds, canceled insurance policies, and so on). For an institution that has

not failed, data from institutions that have failed or have had “near death experiences” usually provide more insights on liquidity risk than the institution’s own

history of liability amounts and costs. This section features the funding shortfalls at

AIG during the credit crisis for exactly that reason.

The primary, secondary, or other extensions of credit by the Federal Reserve

to AIG during the period February 8, 2008 to March 16, 2009 can be summarized

as follows:

First borrowing date:

Average from 2/8/2008 to 3/16/2009

Average when drawn

Maximum drawn



Continuous from September 16, 2008

$72.8 billion

$161.2 billion

$208.6 billion



Kamakura Risk Information Services (KRIS) version 5.0 Jarrow-Chava default

risk models showed a cumulative five-year default probability for AIG of 65.34 percent

on the date of its first Fed borrowing, September 16, 2008, as shown in Exhibit 37.3.

The pattern of total outstanding borrowings from the Federal Reserve shows a

steady increase from the first borrowing date on September 16, 2008, as shown in

Exhibit 37.4.

In addition to these borrowings (“primary, secondary, or other extensions of

credit”) from the Federal Reserve, AIG-related entities were substantial beneficiaries

of support from the Commercial Paper Funding Facility run by the Federal Reserve,

as shown in Exhibit 37.5.



EXHIBIT 37.3 American International Group



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