Technical Paper Series

Congressional Budget Office

Washington, DC

VALUING FEDERAL LOANS AND LOAN

 GUARANTEES USING OPTIONS-PRICING METHODS

Deborah Lucas

Northwestern University and NBER

Evanston, IL

E-mail: d-lucas@Kellogg.Northwestern.edu

Marvin Phaup

Congressional Budget Office

Washington, DC

E-mail: Marvin.Phaup@CBO.gov

Ravi Prasad

formerly Consultant to Congressional Budget Office

Bank of America Securities

E-mail: OhmsatyaP.Ravi@bankofamerica.com

August 2004

2004-12

Technical papers in this series are preliminary and are circulated to stimulate discussion and

critical comment. These papers are not subject to CBO’s formal review and editing processes.

The analysis and conclusions expressed in them are those of the authors and should not be

interpreted as those of the Congressional Budget Office. References in publications should be

cleared with the authors. Papers in this series can be obtained at www.cbo.gov/tech.cfm. 
ABSTRACT

This paper provides more detail about the methods used in the Congressional Budget

Office (CBO) study, “Estimating Subsidies for Federal Loans and Guarantees,”

(August 2004). That study estimates the market value of loans and loan guarantees

under both credit reform and risk-adjusted approaches. The purpose of this paper is

to increase the transparency of those methods for analysts who may wish to adopt

them. Although this paper includes a brief overview of the basics, it is not a

substitute for a textbook on options pricing. Rather, the discussion focuses on

instances where the standard models were modified to better reflect the terms of the

AWA and Chrysler guarantees.
Table of Contents

Abstract

Introduction 1

Basics of Valuing Loan Guarantees Using Options-Pricing Methods 3

Estimating the Value of Guarantees and Guarantee Fees 

Using Binomial Asset Pricing Models 5

Estimating Asset Values and Volatilities 8

Merton’s Derivative Pricing Model 9

An Alternative Approach 13

Valuation of Warrants 15

Dilution Adjusted Black-Scholes 16

Valuation Under Credit Reform 19

Appendix I: The Effect of Market Risk on the Price

of Loans and Loan Guarantees 22

Appendix II: Credit Rating-Based Alternatives 

for Estimating Expected Losses 25
1. The authors thank Robert McDonald (Northwestern University), David Torregrosa (CBO), and Susan

Woodward (Sandhill Econometrics) for useful discussions and Wendy Kiska (CBO) for technical

assistance.

1

1. Introduction1

The CBO study, “Estimating Subsidies for Federal Loans and Guarantees,” (August

2004) finds that including the market price of risk in cost estimates of federal loans

and loan guarantees is necessary to measure cost comprehensively and, thereby,

improves the comparability of those estimates with the cost of other federal

programs in the budget. That study also estimates the costs of loan guarantees made

to Chrysler and to America West Airlines with and without risk adjustment, and

reports considerably higher costs with market risk.

Incorporating the market price of risk into subsidy cost estimates for loans and loan

guarantees is accomplished most efficiently using options pricing techniques, which

is the approach followed in “Estimating Subsidies for Federal Loans and

Guarantees.” That approach is useful because guarantees are a type of option and

because the exposure of options to market risk varies over time. The latter feature

of guarantees makes options pricing more accurate than traditional discounted

present value estimates.

Interested readers of the CBO study may find the overview provided here of 

options-pricing models and the additional detail about the use of binomial models

useful. This paper also explains the use of options pricing models to derive two key
2

parameters for valuing corporate loan guarantees–the value of the firm’s underlying

assets, and the volatility of the value of those assets. Since many government

guarantees involve the receipt of warrants, the paper also explains how standard

options-pricing methods are used to value warrants, and how their valuation differs

under Credit Reform. Appendix I is an extended example of the effect of market risk

on the price of guarantees that includes an explanation of why market risk leads to

a discount rate that is different than the risk-free rate. Appendix II describes an

alternative approach to estimating expected losses based on credit ratings.
2. As explained in the CBO study, the credit subsidy cost of a loan or a guarantee on a loan with the same

terms is equivalent. Hence, the discussion refers only to loan guarantees, but applies equally well to

direct loans.

3. For a more complete description of binomial option pricing methods see McDonald, Robert,

Derivatives Markets, Addison-Wesley, 2003.

3

2. Basics of Valuing Loan Guarantees Using OptionsPricing Methods

The pricing methods described in this paper rely on the observation that the payments

on a loan guarantee are equivalent to those on a put option written by the government

and held by the borrower. Thus, the cost of the loan guarantee provided by the 2

government may be estimated using either the Black-Scholes formula for put options

or a binomial option-pricing model. The former is easier to use when it is applicable,

but the latter can be applied much more generally. The Black-Scholes formula is

appropriate when the government has first lien over all the assets of the borrower and

the borrower cannot prepay the loan. In more complicated situations, for instance, if

the loan can be prepaid or the government does not have a senior claim over all the

assets of the borrower, the binomial option-pricing model is more appropriate. In

addition, the default point, (i.e., the asset value at which the borrower chooses to

default on its obligations or the lenders force the borrower to default), tends to change

over the life of the loan guarantee, which also precludes using the standard BlackScholes formula. Consequently, the binomial option pricing approach, which offers

more flexibility, is used in computing the Chrysler and AWA loan guarantee values.

The main model is constructed using an Excel spreadsheet and Visual Basic.3
4

When the binomial option-pricing model is used, the information required to compute

the cost of the loan guarantee includes the following: 

1) Current market value of assets (A)

2) Current book/market value of liabilities (L)

3) Market value of assets that triggers bankruptcy (Default Point, D, may vary with

time)

4) Market value of assets that triggers prepayment (Prepayment Point, P, may vary

with time)

5) Original time to expiration of loan guarantee (T)

6) Vo Alatility of return on assets ( )

7) Ri fsk-free interest rate (r )

8) Dividend y Aield on assets ( )
5

3. Estimating the Value of Guarantees and Guarantee

Fees Using Binomial Asset Pricing Models

The value of loan guarantees and of associated guarantee fees were found using

similar binomial asset pricing models. The binomial model approximates the future

distribution of asset value over time, based on the initial asset value, asset volatility,

and expected returns. The results also depend on the conditions that trigger default

and prepayment.

Estimates of guarantee value using a binomial tree are based on the following steps:

1. A binomial tree is created, where each node in the tree represents the market

value of the guaranteed firm’s assets on a future date, and in a particular “state

of the world” that reflects one possible history of asset returns. Going forward

any one step in the binomial tree, the market value of assets takes two possible

values (hence the name “binomial”). The two outcomes are selected to

capture the assumed mean and variance of asset returns over the next short

time period. A probability also is associated with an up or a down move (in

such a way that the target mean return and variance are matched, and so that

the tree structure is tractable). These probabilities are constant throughout the

tree. 

2. The assumed rules for prepayment and default are applied at each node to

determine whether a default or prepayment will occur. Given these rules and

the value of assets, the size of any loss to the government at each node is 
6

calculated.

3. The insight exploited by options pricing theory is that the cost to the

government at each node is equivalent to the payoff on a portfolio consisting

of a fraction of the firm’s assets and risk-free borrowing or lending. This

“replicating portfolio” is different at different nodes, reflecting that the

amount of market risk implicit in the guarantee changes over time. The

replicating portfolio is easily identified and calculated, using standard

methods. Since the model provides the value of the replicating portfolio at

each node, and since its value is equal to the value of the guarantee payments,

the calculation yields the value of the guarantee. The value of guarantee fees

received is found similarly.

While the procedure is computationally straightforward, it is not immediately obvious

how risk adjustment is being incorporated. The answer is that the risk-adjusted

market rate enters the calculation of market value via the portion of the replicating

portfolio invested in the firm’s assets. The larger the effective position in the firm’s

assets, the higher the risk. The market risk is also magnified by the fact that the

guarantee is equivalent to a highly levered position in the assets. This effect on the

position of the guarantor means that the guarantee is like financing a purchase of the

firm’s risky assets with risk-free debt.

Credit reform cost estimates and cost estimates using market prices may both be

obtained from the same binomial tree. However, to obtain credit reform cost
7

estimates, the present value of the cash flows from the government is determined by

discounting back to the present at a risk-free rate rather than determining the value of

a replicating portfolio. This method is used in the CBO study. 
8

4. Estimating Asset Values and Volatilities

To value the cost of government loan guarantees using option-pricing models, it is

necessary first to estimate the market value and volatilities of the assets of the firm

receiving the guarantee. The market value of assets of a firm is equal to the sum of

the market value of its liabilities, including its long and short-term debt and other

fixed obligations, and the market value of its equity. (The market value of assets is

not equal to the book value of assets, in general, since book value reflects historical

costs and market values reflect investor expectations about future value.) The

volatility of asset values depends on the volatilities of liabilities and equity, and on

the correlations between them. In general, the market value and volatility of the

equity of publicly traded firms can be calculated from recent share price data. If the

market value and volatilities of all the liabilities of the firm and the correlation

coefficient between the equity and debt returns is known, it is possible to calculate the

market value and volatilities of the assets directly. Although the data required for

estimating the market value and volatilities of the publicly traded debt of a firm may

be available, the corresponding parameters for other liabilities (“non-trading

liabilities”) of the firm are difficult to estimate, but often significant. An alternative

approach is thus needed for estimating the market value of the assets and the asset

volatility.

For healthy firms, the simplest approach to estimating the market value of assets is

to set it equal to the sum of the book value of liabilities and the market value of
9

equity. The inherent assumption in these calculations is that the market value of

liabilities is closely approximated by their book value. This approximation can be

expected to give good estimates of the market value of the assets of companies with

strong balance sheets, e.g., for firms with negligible probability of default on their

debt as evidenced by their very high debt ratings. However, this procedure can give

incorrect results for companies in distress, as was the case for Chrysler and AWA,

because the market value of their liabilities will be significantly lower than the book

value of the liabilities. An alternative is to estimate these parameters using one of the

two procedures described below. Both involve approximations with known

limitations but they produce similar estimates for both Chrysler and America West

Airlines. 

4.1 Merton’s Derivative Pricing Model 

Over the past several years, some firms (most notably, KMV Corporation, a

subsidiary of the rating agency, Moody’s) have proposed using Merton’s general

derivative pricing model for understanding the link between the market value of the

firm’s assets and the market value of its equity. Equity has the residual claim on the

assets after all other obligations have been met. It also has limited liability. A call

option on the underlying assets has the same properties as equity. Thus, equity can

be thought of as a call option on the assets of the firm. The holder of a call option on

the assets has a claim on the assets after meeting the strike price of the option. When

equity is valued as a call option, the strike price of the call option is equal to the book
10

value of the firm's liabilities. If the value of the assets is insufficient to meet the

liabilities of the firm, then the shareholders, as holders of the call option, will not

exercise their option and will leave the firm to its creditors.

In Merton’s approach, the option nature of equity is exploited to derive the asset value

and asset volatility of the firm’s underlying assets implied by the equity’s market

value and equity volatility. Both the equity value and equity volatility can be

observed from share price data. In particular, the option price (equity value) and

option price volatility (equity volatility) are used for determining the implied asset

value and asset volatility.

Valuing equity as a call option on the firm’s assets using the Black-Scholes option

pricing formula for a non-dividend paying stock:

(1)

 (2)

(3)

In the Black-Scholes framework, the volatility of equity and the volatility of assets
11

are related by:

(4)

where E is the equity value, A is the asset value, X is the book value of total

E

liabilities, T is time to expiration of call option, is the volatility of common

A f stock, is the volatility assets, N(.) is the cumulative normal distribution, and r is

the risk-free interest rate.

The Black-Scholes formula given by equation (1) and the relation between equity

volatility and asset volatility provided by equation (4) provide two equations, which

can be solved simultaneously to obtain the two unknowns, asset value (A) and asset

vo Alatility ( ).

As an example, consider a firm with a market capitalization (E) of $3 billion, an

annual equity volatil Eity ( ) of 40 percent and total liabilities (X) of $10 billion.

Further assume that all the liabilities are due within one year (T=1 year) and that the

f

risk free rate (r ) is 5 percent. The asset value and asset volatility implied by the

equity value, equity volatility and liabilities are calculated by solving the call price

and volatility equations, (1) and (4), simultaneously. In this case, the implied market

value of the firm’s assets is $12.51 billion, and the implied asset volatility is 9.6%.

(This result implies that the market value of debt is $9.51 billion, $.49 billion lower
12

than its book value due to the possibility of default.)

The Black-Scholes model described above, while limited in certain dimensions, is

widely understood and provides a useful framework to review the issues and to

generate rough estimates. In practice, however, it can be important to use a more

general option-pricing relationship that allows for a detailed specification of the

firm’s liabilities. In Merton’s approach, the market value and volatilities of assets

are calculated by simultaneously solving two equations (1 and 4). KMV extends this

procedure to handle firms with more complex capital structures, and their approach

is one that could be adopted by budget analysts.

The KMV approach is likely to give reasonable estimates of the market value and

volatilities of assets of healthy firms. For firms in distressed conditions, the book

value of the liabilities is likely to be above both the market value of the assets and

the market value of the liabilities. In addition, the market value of equity will only

be a small fraction of the book value of total liabilities for a distressed firm. (For

instance, the market value of equity of AWA was about $138 million, while the book

value of liabilities was more than $1,500 million, when the government guaranteed

$420 million in new loans.) Consequently, the equity of distressed firms is similar

to a deep out-of-money call option. It is well known that the implied volatility

estimated from deep out-of-money options may not accurately reflect the actual

volatility of the asset on which the option is written. Consequently, an alternative

approach, described below, can be used instead. It also involves error because it
13

relies on statistical estimates of asset betas which are difficult to measure. 

4.2 An Alternative Approach 

An alternative model for estimating the volatility of assets is based on the

observation that the option and the underlying asset on which the option is written

will have same "Sharpe Ratios." The Sharpe ratio (S) is defined as: ,

where e f r is the expected return, r is the risk free rate and is volatility. Using an

asset pricing model known as the capital asset pricing model (the “CAPM”) returns

are m f given by: where (r - r ) is the market risk premium.

In the case of loan guarantees, asset refers to the assets of the firm and the option

refers to the equity of the firm. Therefore,

(Beta of Asset*Market Risk Premium)/Volatility of Asset = 

(Beta of Equity*Market Risk Premium)/Volatility of Equity

It follows that:

Volatility of Asset = Volatility of Equity*(Beta of Assets/Beta of Equity) (5)

It is possible to estimate the volatility of equity, the beta of assets and the beta of

equity for any publicly traded firm. These data can be used for calculating the
14

volatility of the assets using equation (5). 

This method may be more robust for firms in financial distress than the KMV

approach because the beta of the assets of a firm should depend primarily on its

business or the industry in which the firm operates. On the other hand, the difficulty

of estimating betas precisely might lead some analysts to prefer the KMV approach,

which does not require estimating industry betas. 

After estimating the volatility of assets using equation (5), the market value of assets

can be calculated using the Black-Scholes formula for call options. The market

value of equity is the value of the call option on the firm’s assets. The value of the

call option depends on asset value, strike price, risk-free rate, asset volatility, time

to maturity of the option and the dividend yield. The market value of the assets is the

only unknown among these variables.
15

5. Valuation of Warrants

The government sometimes accepts warrants as partial compensation for granting

loan guarantees to distressed corporations. This was the case for both America West

Airlines and Chrysler. To estimate the net cost to the government on a market value

basis, the market value of the warrants also must be determined. 

A warrant to purchase stock is analytically similar to a call option–the right but not

the obligation to purchase shares at a specified price for a specified period. A call

option on a stock allows the holder to buy the stock at a pre-specified price (strike

price) on or before a pre-specified date, regardless of its current price. At expiration,

the value of a stock option is the maximum of the difference between the market

price and the strike price or zero and the holder will exercise the option only if it is

to her advantage to do so. A warrant differs from a traded call option in that the

number of shares outstanding increases when warrants are exercised (this effect is

called dilution).

Warrants often have much longer maturities than traded call options. The value of

either a warrant or a call option generally increases with maturity for two reasons.

First, stocks tend to increase in value over time, and more time provides greater

chance for movement in a stock price. At the time a warrant or call option expires,

its value is worth the difference between the current stock price and the exercise

price, or zero (when the current stock price is below the exercise price). Because of
16

the time value of a warrant, a rational investor would not exercise it prior to the

expiration date. An investor who wanted to lock in gains if the stock price had risen

above the exercise price would generally get a bigger return by selling the option on

the open market.

The value of a warrant or call option also increases with volatility, because this

increases the probability that the future stock price will exceed the exercise price.

Finally, the value of a warrant or call option increases with the level of the risk-free

interest rate because a call option permits the holder to defer purchase and thereby

save the financing, or carrying, cost of the asset.

5.1 Dilution Adjusted Black-Scholes

Market participants commonly use the dilution adjusted Black-Scholes model to

value warrants. This model assumes that stock returns are normally distributed and

that stock prices follow a random walk (diffusion process). The Black-Scholes

model values “European” options–those options that can be exercised only at

maturity. American options can be exercised at any time, though it is seldom

optimal to do so before maturity. (The exception is when a stock pays a dividend.)

For this reason, the value of an American call option is same as the value of a

European call option when the stock pays no dividend, as is expected to be the case

for Chrysler and AWA. Thus, both Chrysler and AWA warrants may be treated as

European warrants and valued using the Black-Scholes model.
17

Accordingly, the market value of a warrant share (W) is given by:

(6)

(7)

(8)

where S = S + (M/N)W, which is substituted for stock price, S, in the standard *

Black-Scholes formula to account for the effect of the increase in the number of

shares on exercise of the warrant, or dilution. There are eight variables in equations

(6)-(8). They are: share price at the time the warrant is issued (S), strike price or the

price at which shares can be purchased using the warrant (X), the number of regular

shares outstanding (N), the number of shares that the warrant holder can buy (M),

time to expiration of warrant (T), volatility of common stock return ( ), risk-free

f

interest rate (r ) and dividend yield ( ). An iterative procedure is used for solving

equation (6) since warrant price (W) is needed as an input to calculate S which is *

needed for calculating the warrant price (W) in turn.
18

The inputs into this formula are for the most part the same as for the binomial model

used to value loan guarantees and associated fees. There are, however, a few

considerations specific to warrant valuation.

The share price (S) used in the warrant price calculations should be the share price

immediately after the government loan guarantee has been publicly announced.

Taking an example from the CBO study, AWA’s share price jumped from $2.75 to

$4 following the announcement of the loan guarantee. If the share price is set equal

to $2.75 instead of $4, the warrant share price is in the range of $1.49 to $1.82,

which implies that the total market value of the warrant for 18.8 million common

shares is $28.01 million to $34.22 million. However, warrants are to buy stock in

the post-guarantee AWA. Therefore, the correct stock price to use in warrant

calculations is the post-guarantee price. Chrysler’s share price immediately after the

government agreed to guarantee the loan was $7.50. 

The value of a warrant or stock option is very sensitive to the assumed volatility of

the share price, as measured by the standard deviation of the average percentage

change in stock prices. Other things equal, the price of a warrant will increase with

the expected variability of a firm’s share price. Warrants to purchase stock in a firm

whose share price shows little variability over time will be priced below those for a

firm subject to large swings in the market value of its equity. 

For a distressed company issuing a long-lived warrant, the question is how to
4. Options-pricing methods, including the Black-Scholes model use an analytical technique known as

“risk-neutral pricing” to estimate the market value of risky options. The use of risk-free rates with this

technique does not imply that investors are risk-neutral or that markets use risk-free rates to discount

risky future cash flows. Risk-neutral pricing is simply a computationally efficient means of replicating

risk-adjusted market prices. For details, see McDonald, Derivatives Markets, pp. 336-40 and 358-63.

19

estimate the average volatility over the life of the warrant, taking into account that

the current volatility is likely to be higher than long-run volatility. When publicly

traded option prices are available, an estimate of volatility can be backed out from

the option prices. This is known as the “implied volatility,” and it can be used as an

input to value other options or warrants. For instance, when Chrysler issued the

warrant, the volatility of the common stock was extremely high--101 percent. The

estimated value of the warrants using a volatility of 101 percent is $6.67 a share.

Alternatively, the implied volatility of the 5-year warrants that were publicly traded

was 70.5 percent on May 12, 1980. Substituting a volatility of 70.5 percent for 101

percent in the Black-Scholes options pricing model generates a price of $5.60 for the

warrants. Thus, the total value of the warrants for 14.4 million shares ranged from

$80.6 million to $96.5 million, depending on which volatility assumption is chosen.

The number reported in the CBO study reflects the lower implied volatility, which

seems more appropriate for the government’s long-lived warrants.

5.2 Valuation Under Credit Reform

The difference between the estimated market value of warrants and the credit-reform

estimates is that credit reform uses a risk-free rate for discounting cash flows, rather

than a risk-adjusted rate. As with market valuation, it is assumed that the warrants 4

can be exercised only at the end of the maturity period, so their terminal value
20

depends on the stock price at that time relative to the strike price. The distribution

of possible stock prices at the end of the maturity period is simulated and the warrant

payoffs corresponding to these simulated stock prices are calculated. The expected

value of warrant payoff at maturity is discounted using a risk-free interest rate to

obtain the warrant value at the time it was issued.

For the AWA and Chrysler estimates under Credit Reform, and consistent with the

assumptions underlying the Black-Scholes formula, a log normal distribution of

stock prices at the end of the warrant maturity period is assumed. With a log normal

distribution, the stock prices can be represented by:

(9)

where T 0 S is the share price at the end of the warrant maturity period, S is the initial

share price, is the CAPM-based expected return on common stock, is the

dividend yield, T is the time to maturity, Z is the standard normal random variable

and is the volatility of the common stock. 

Fo Tr any share price at time T, denoted by S , the payoff for each warrant share that

can be exercised only at warrant maturity is given by:

Warrant Payoff at time T = max (0.0, (NST + MX)/(N+M) - X) (10)

where N is the number of common shares, M is the number of warrant shares and X
21

is the strike price for warrants. By averaging warrant share payoffs at several share

prices (i.e., at different values of the standard normal random variable Z), the

expected payoff to the warrant share at maturity is calculated. The expected payoff

of each warrant share at its maturity is discounted at a risk-free interest rate to

determine the warrant share price when the warrant is originally issued. This

procedure effectively replicates the assumptions of the market-based warrant

valuation, except for the use a risk-free discount rate.
5. This higher value of payment when income is low is also an implication of diminishing marginal utility

of wealth.

22

Appendix I: The Effect of Market Risk on the Price of

Loans and Loan Guarantees

This appendix consists of an example illustrating how an aversion to market risk

translates into an effect on discount rates and asset prices.

Market risk is costly because people would pay more today to receive of $100 next

year if the economy is in a downturn, than they would to receive $100 next year if

the economy is in a boom. This is equivalent to saying that people are willing to pay

for insurance against bad economic times.5

To make this reasoning precise, assume that the booms and busts have equal

probabilities of 50 percent. Also suppose that investors are willing to pay $45 today

to receive a safe promise of $100 next year if the economy is in a boom, and $49

today to receive $100 next year if the economy is in a downturn (so-called “state

payoffs”). Then the price today of $100 for sure next year is $94.00 ($49 + $45), the

price of buying a claim for $100 in both the good and bad states. This implies a riskfree rate of 6.38%, since $100/(1.068) = $94.

Consider a risky government loan with an expected return of 6.38%, the risk-free

rate. For instance, assume that the government lends out $100 at a rate of 12.76%,

and that the government anticipates the loan will be repaid in full if the economy is
23

in a boom, but that the loan will only repay principal in a downturn. Using credit

reform assumptions, the loan would appear to be unsubsidized, because its value is

. Private investors, however, would assign a value to the loan

of less than $100, because payments are relatively low when they are most valuable.

Investors would value the loan at , which implicitly

discounts the loan at the higher rate of 6.66% since . The

market would assign a subsidy value (or loss) of $0.26 per $100 of funds loaned. If

the government attempted to re-insure the guarantee with private investors, it would

appear to lose $0.13 per $100 guaranteed.

Alternatively, suppose the government guarantees a similarly risky loan. With the

government guarantee, the borrower could obtain a rate only slightly above the riskfree rate of 6.38%, since the private lender would set a rate on the loan taking into

account the government guarantee. Assume again that the loan is repaid in full in a

boom, including interest of 6.45%, but that it only repays principal in a downturn.

In a boom there is no cost, but in a downturn the cost is $6.45. Under credit reform,

the budget would report a present value cost of . Private
24

guarantors would assign the higher cost of to the

guarantee, reflecting that they anticipate paying out money when it is especially

valuable.
25

Appendix II: Credit Rating-Based Alternatives for

Estimating Expected Losses

The rules of credit reform permit a variety of approaches to estimating the

probability and severity of losses on loan guarantees, and the value of expected fees

paid. For comparability, and because it is arguably the most reliable approach, the

credit-reform estimates reported in the CBO study are based on the same model of

future asset values, defaults, and prepayments as the market-based estimates. As a

robustness check, and to demonstrate an alternative that may be easier to use in some

circumstances, the credit reform value of guarantee fees is also estimated using

default probabilities based on credit ratings. This approach relies on historical

default probabilities and recovery rates associated with debt of a particular rating.

The weakness of this approach is that within a rating class, the probability and

especially the severity of default can vary widely, making the estimates imprecise.

The expected cash flows from guarantee fee payments in i year are given by: th

CFi i p i = (1 - CPD )*(1 - P ) *Fi

where CPDi p = Cumulative probability of default by year “i”, P = annual prepayment

probability i, F = Promised guarantee fee payments in the i year. The default th

probability is based on the historical default performance of the debt with the same

credit rating as that of the firm whose debt the government is guaranteeing. The

prepayment probability is based on the annual probability of credit upgrade by two
26

notches. For example, consider the prepayment probability of a CCC rated debt.

Historical statistics available from CreditMetrics indicate that a CCC rated debt has

41.53% probability of getting upgraded one category to B and 6.8% probability of

an upgrade to BB or higher within the first year of issuance. In this situation, the

prepayment probability is set to 6.8%. The expected guarantee fee payments are

discounted at a risk-free rate to determine the value of guarantee fees at the time

government extends the guarantee. Following this approach, the expected value of

loan guarantee fee payments to the government is $69.49 million in AWA case and

$38.55 million in Chrysler case. Both numbers are somewhat higher than those

estimated using the model of cash flows corresponding to the options calculations

($52.5 million for AWA and $28.9 million for Chrysler).