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Measuring risk-adjusted performanceMichel Crouhy and Stuart M. Turnbull

Market Risk Management, Canadian Imperial Bank of Commerce, 161 Bay Street,

Level 11, Toronto, Ontario, Canada M5J 2S8

Lee M. Wakeman

Risk Analysis & Control, 2 Weed's Landing, Darien, Connecticut 06820, USA

Many banks follow the dictum of maximizing the risk-adjusted return on economiccapital, subject to constraints imposed by regulatory requirements. The authors show

that commonly employed methods may result in decisions that adversely affect

shareholder value. They present an alternative methodology, adjusted RAROC, that

corrects the inherent limitations of the existing methods.

1. INTRODUCTION

Many large US and foreign banks have introduced performance evaluation

methods designed to allocate capital among their dierent businesses. Tradi-

tional performance measures such as return on capital have long beenrecognized as failing to take into account the risk of the underlying business

and the value of future cash ows. The capital asset pricing model (CAPM)

provides a framework for determining the net present value of a business.

Expected future cash ows are discounted using a risk-adjusted expected rate of

return. Risk is dened in terms of the covariance of changes in the market value

of the business with changes in the value of the market portfolio. This form of

risk measure is usually referred to as the beta coecient. For an introduction to

the CAPM and discussion of the beta coecient, see Brealey and Myers (2000).

For a portfolio of assets, risk in the CAPM is dened in terms of the standard

deviation of the portfolio's return. The Sharpe ratio is a measure of the relative

attractiveness of dierent portfolios. It maps the expected return and the riskinto a single measure. It is argued that the larger the Sharpe ratio, the better the

portfolio, and hence the objective of portfolio manager's is to maximize the

Sharpe ratio or, equivalently, the expected return on risk-adjusted capital.

For many businesses, measuring the rm's beta and the beta of an invest-

ment presents a formidable challenge given the absence of market data.

Practitioners have also expressed the concern that measures of risk such as

beta, do not consider the risk and costs associated with default. Consequently,

other forms of expected return/risk measures have been introduced (see Froot

and Stein 1996).

The stated objective of these risk-based capital allocation systems, which are

often grouped together under the acronym RAPM (risk-adjusted performance

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measurement), is to provide a uniform measure of performance that manage-

ment can use to compare the economic protability of businesses with dierentsources of risk and dierent capital requirements.

A generic RAPM takes the form

RAPM Expected Revenues Costs Expected LossesEconomic Capital

Y 1

where Èxpected Revenues' is the expected revenues assuming no losses,

Èxpected Losses' is the expected losses from default, and Èconomic Capital'

is usually dened as the capital necessary to cushion against unexpected losses,

operating risks, and market risks, and is often referred to as value-at-risk (see

Ong 1999, p. 218). The magnitude of the economic capital is usually determined

so that the probability of unexpected losses is below some specied level. Matten

(1996, p. 59) describes four dierent risk-adjusted performance measurement

models, the two commonly used models being RAROC (risk-adjusted return on

capital) and RORAC (return on risk-adjusted capital).1 The RAROC approach

adjusts the numerator of expression (1) to take account of various risks, and the

RORAC approach adjusts the denominator to the account for the same risks.

Matten correctly observes that there is considerable confusion between the two

approaches. In this paper we will simply use the term RAROC.

Bankers Trust developed the RAROC methodology in the late 1970s.

According to Zaik et al. (1996), their original intent was to measure the riskof the bank's credit portfolio and the amount of equity capital necessary to limit

the bank to a specied probability of loss. The method of measuring the risk

and the time horizon are issues that require specication. The denition of risk

has moved away from a market-driven denition of risk to a measure of risk that

is purely rm specic. It is implicitly assumed that RAROC correctly compen-

sates for changes in risk. Consequently, it is assumed that it can be used to

measure the performance of dierent types of businesses and that decisions

based on RAROC are consistent with maximizing the wealth of existing

shareholders.

Zaik et al. (1996) state that Bank of America's policy is to capitalize each of

its business units in a manner consistent with the bank's desired credit rating onthe unit's stand-alone risk. If a business unit's RAROC is higher than the cost of

the bank's equitythe minimum rate of return required by shareholdersthen

the unit is deemed to be adding value to shareholders. This assumes that the risk

of the economic capital of the stand-alone business is the same as that of the

bank's equity (see footnote 8 for further discussion).

1RAROC makes a risk adjustment to the numerator by subtracting a risk factor from the return:

RAROC Return Risk AdjustmentaCapital. RORAC makes a risk adjustment to thedenominator by selecting economic capital based on the shape of the gain/loss distribution:

RORAC ReturnaCapital Risk Adjustment.This list is not exhaustive. Many consulting rms, such as Coopers and Lybrand, have introduced

variants of RAPM such as RARORAC (risk-adjusted return on risk-adjusted capital):

RARORAC Return Risk AdjustmentaCapital Risk Adjustment.

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The implicit assumption is that the RAROC measure adjusts the risk of a

business to that of a rm's equity. Consequently, if a rm is consideringinvesting in a business (or closing a business down), it can compute the RAROC

for the business and compare it with the rm's cost of equity capital. If the

RAROC number is greater than the rm's cost of the equity capital, the business

will add value to the rm. The RAROC methodology avoids having to calculate

the beta for the business. It measures risk by using the current probability of

default for the rm.

In Section 2 we examine the underlying premise of RAROC: that it is possible

to construct a risk-adjusted rate of return such that it can be compared with a

rm's cost of equity capital. We do this using the contingent claims framework

of Merton (1974). We consider a business that is nanced in such a way that theexpected rate of return on equity equals some prespecied value. We then

compute the probability of default. We show, not surprisingly, that while it is

possible to pick a capital structure so as to achieve a required rate of return on

equity, the probability of default will change as the volatility of the rate of return

on the rm's assets changes.

We also show that while it is possible to pick a capital structure so that the

probability of default equals some prespecied level, the expected rate of return

on equity will change as the volatility of the rate of return on the rm's assets

changes.

In Section 3 we consider an all equity rm undertaking a risky investment.

The rm sets up a reserveeconomic capitalso that the probability of defaultremains constant at some prespecied level. We show that risk-adjusted

performance measures such as RAROC change as the risk of the business

changes, even though the probability of default is kept constant. We then

introduce an alternative measure that corrects the problem: adjusted RAROC.

In Section 4 we consider the RAROC of a bank's loan portfolio. We

demonstrate that RAROC varies as the volatility of the assets underlying the

loan varies, due to changes in the level of the rm's economic capital, even

though the probability of the rm defaulting on its loan is kept constant.

Businesses within a rm are typically correlated with each other. In the

RAROC methodology, economic capital for each business is determined bytreating each business in isolation. The economic capital for the rm takes into

account the covariances between the dierent businesses. The sum of the

economic capital for each business is typically greater than the economic capital

for the rm. One common method of addressing this problem is to introduce a

weighting scheme, where it is assumed that the economic capital for a stand-

alone business is proportional to the economic capital of the rm. We show in

Section 5 that such a method can introduce major errors.

Section 6 presents some closing comments. We consider some of the

incentives generated by a RAROC methodology. If compensation decisions

are based on RAROC, senior management may benet at the expense of

shareholders.

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2. PURE FINANCING DECISION

A premise of RAROC is that it keeps the probability of default constant and

can be directly compared with the rm's cost of equity capital. In this section

we show that this premise is awed: maintaining the probability of default

constant is inconsistent with a constant expected rate of return on equity and

vice versa.

In the following we consider a rm that undertakes a project and adjusts its

capital structure so that the probability of default is set at some prespecied level

and we then compute the expected rate of return on equity capital for two cases.

First, we alter the volatility of the rm's risky assets, adjust the rm's capital

structure so that the probability of default is kept constant, and compute the

expected rate of return on equity capital. We show that keeping the probabilityof default constant does not imply that the cost of equity capital is invariant to

changes in the risk of the rm's assets. In the second case the rm undertakes a

pure nancing decision to achieve a degree of leverage so that the expected rate

of return on equity equals some prespecied level. We compute the probability

of default. We alter the volatility of the rm's risky assets and adjust the rm's

capital structure so that the expected rate of return on equity is kept constant.

We show that the probability of default is not invariant to changes in the risk of

the rm's assets even if the expected rate of return on equity is kept constant.

2.1 The Model

We consider a rm that undertakes a pure nancing decision to alter its degree

of leverage so that the probability of default is kept constant at some

prespecied level. To start we assume that the rm is totally nanced by equity.

The rm issues zero-coupon debt and with the proceeds repurchases equity.

Capital markets are assumed to be frictionless and for simplicity there is no

form of corporate or personal taxation. Corporate debt is risky, as default can

occur. If default occurs, bondholders take over the rm. It is assumed that there

are no transaction costs associated with default and consequently capital

structure is irrelevant (see Merton 1977).

Let At denote the market value of the rm's assets at time t. The rm issueszero-coupon debt with face value F. The debt matures at date T. Let DtYTdenote the market value of the rm's debt at time t. At maturity the payo to

bondholders is

DTYT F if ATb F,AT if AT ` F,

@2

Looking at the right-hand side of expression (2), the payo to bondholders is

composed of two terms. The bondholders receive the face value of debt, F, and

a put option is written. This put option will be exercised against the bondholders

if default occurs. Let St denote the market value of equity at time t. The valueJournal of Risk

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of equity at date T is

ST AT F if ATb F,0 if AT ` F,

&3

which is the payo to a European call option on the rm's assets with strike

price F. The total market value of the rm equals the sum of the market value of

equity plus the market value of debt:

At St DtYT for 0T tT TX 4

The expected rate of return on equity is

"RS fEST S0gaS0X 5

To compute this expected rate of return, we need to impose some assumptions

about the specication of changes in the market value of the assets.

2.2 Valuation

Following Merton (1974), it is assumed that

dAtA

t

"A dt 'A dWtY 6a

where "A is the instantaneous expected rate of return, 'A is the volatility, and

Wt is a Brownian motion. It is assumed that "A and 'A are constant, implyingthat At is lognormally distributed. Hence, we can write

At A0 exp"A 12 '2At 'AWtY 6b

where Wt is normally distributed with zero mean and variance t. FollowingMerton (1974), assuming that the default free rate of interest is deterministic

over the period 0Y T and using expression (6) implies that the market value ofdebt and equity at time 0 is given by

D0YT FB0Y T pA0Y TYF 7and

S0 cA0Y TYFY 8

where B0YT is the value of a Treasury bill that pays one dollar for sure at dateT; pA0Y TYF and cA0Y TYF respectively represent the value of a Europeanput and call option described by

pA0Y TYF FB0YTNd2 A0Nd1Y 9with

d1 flnA0aFB0Y T 12 '2ATga'A Tp and d2 d1 'A Tp XVolume 2/Number 1, Fall 1999


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Here N

is the cumulative normal distribution function, and

cA0Y TYF A0Nd1 FB0Y TNd2X 10

The expected value of equity is

EST A0exp "TNd1 FNd2Y 11awhere

d1 flnA0aF "A 12 '2ATg

'A

T

pand d2 d1 'A

T

pX 11b

Note that, in contrast to the denitions of d1 and d2 in expression (9), d1 and d2are dened in terms of the instantaneous expected rate of return "A on the

rm's risky assets. In calculating the expected value of equity, we use the natural

(or objective) probability distribution and not the equivalent martingale

distribution.

The expected rate of return on equity is dened by expression (5). By altering

the capital structure of the rm and thus the face value F of the zero-coupon

debt, we can vary the expected rate of return on equity.

2.3 Probability of Default

The probability that default occurs is

p PrAT T F j A0X

Using expression (6b), we can write

p Pr

ZT

lnFaA0 "A 12 '2AT'A

Tp

!Y

where Z is normally distributed with zero mean and unit variance. Hence,

p Nd2Y 12

where d2 is dened in expression (11b).Two observations can be made about expression (12). First, the probability of

default is calculated using the distribution that describes the changes in the

market value of the assets, as described by expression (6). This is dierent from

the equivalent martingale distribution used to calculate the option prices in

expressions (9) and (10). Second, because the ordinary probability distribution is

used, the expected rate of return on the assets appears in expression (12), and

consequently there will be a trade-o between the expected rate of return and

volatility in determining the probability of default.

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keeping the probability of default constant is inconsistent with a constant

expected rate of return on equity for projects with dierent volatilities and

correlations with the market portfolio.

Keeping the expected rate of return on equity constant

The capital structure is altered so that the expected rate of return on equity

equals the target expected rate of return. In this example, the target expected rate

of return is set at 17%. For a xed value of the correlation coecient &, we varythe volatility 'A of the rate of return on the risky asset. Increasing the volatility

of the rate of return on the risky asset increases the expected rate of return on the

rm's assets, and this will decrease the required degree of leverage to achieve the

target expected rate of return on equity. The results are shown in Table 2.

The striking dierence between the results in Part A and Part B is that the

probabilities of default are an order of magnitude higher in Part A than in Part

B. An increase in the correlation coecient, keeping volatility constant,

increases the expected rate of return on the risky asset and decreases the degree

of leverage necessary to reach the target return on equity. This lowers the

probability of default, which is inversely related to the asset's expected rate of

return and directly related to the face value of debt.

TABLE 1. Altering the capital structure to keep the probability of default constant:

p 17.Standarddeviation,

'A (%)

Expected rateof return on

the risky assets,"RA (%)

Face valueof debt,

F

Market values Expected rateof return on

equity,"RS (%)

Debt, D0YT Equity, S0

Part A: Correlation coecient & 0X255 5.70 939.8 893.7 106.3 10.46

10 6.27 838.0 796.8 203.2 10.71

20 7.42 661.2 628.4 371.6 11.25

40 9.71 399.4 379.3 620.7 12.48

Part B: Correlation coecient & 0X505 6.27 944.9 898.5 101.5 16.28

10 7.42 847.0 805.2 194.8 16.77

20 9.71 675.3 641.7 358.3 17.81

40 14.29 416.1 394.9 605.1 20.21

Probability of default: p Nd2 1%Default free rate of interest,a Rf 5.13%

Expected rate of return on the market portfolio,a "RM 12.0%

Volatility of the return on the market portfolio, 'M 15.0%

Market value of assets 1000

Maturity of debt 1 year

a Expressed as a discretely compounded rate of return.

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3. PROJECT EVALUATION

We now examine the use of risk-adjusted performance measures in determining

whether to accept or reject an investment project. We start by considering an all

equity nanced rm. The price per share is denoted by s0 and the number of

shares by n0. Initially, the only asset of the rm is cash g, where g n0s0. Werelax this assumption in Section 5.

The rm is assumed to undertake a risky investment. The cost of theinvestment is denoted by I. The market value of the risky assets is denoted by

A0. The stochastic process describing changes in the market value of the riskyasset is given by expression (6). The rm also sets up a reserve fund G of cash so

that the probability of default after it undertakes the investment is equal to some

prespecied level p. We refer to G as the economic capital of the rm. Tonance the investment, the rm raises I G g in cash by issuing zero-coupon debt and additional equity. After undertaking the investment, the rm's

assets are A0 and G. The reserve fund is invested in a default free asset,generating a rate of return RfX

The market value of the zero-coupon debt is denoted by D

0YT

. The debt

matures at date T. The number of new shares of equity is denoted by m and the

TABLE 2. Altering the capital structure to keep the expected rate of return on equity

constant:"

RS 177.Standarddeviation,

'A (%)

Expected rateof return on

the risky assets,"RA (%)

Face valueof debt,

F

Market values Probabilityof default,

p Nd2(%)

Debt, D0YT Equity, S0

Part A: Correlation coecient & 0X255 5.70 1024.7 965.1 34.9 27.56

10 6.27 993.8 927.9 72.1 26.73

20 7.42 919.6 846.7 153.3 24.92

40 9.71 728.2 660.7 339.3 20.48

Part B: Correlation coecient & 0X505 6.27 952.1 905.2 94.8 1.48

10 7.42 851.4 809.3 190.7 1.15

20 9.71 647.7 615.8 384.2 0.56

40 14.29 239.8 228.1 771.9 0.01

Default free rate of interest,a Rf 5.13%



Market value of assets 1000

Maturity of debt 1 year




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new equilibrium price per share by s

0

.4 The total market value of equity is

denoted by S0, whereS0 n0 ms0X 14

The total amount of nancing is

D0Y T ms0 I G gX 15

3.1 Valuation of Equity and the Credit-Risky Debt

The market value of the risky investment is described by expression (6).5 The

payo to the debt holders is

DTYT F if AT G1 Rfb F,AT G1 Rf if AT G1 Rf ` F.

@16

The above expression can be written in the form

DTY T F0 if ATbK,K AT if AT ` K,

@17

where K F G1 Rf and can be interpreted as the net indebtness of therm, measured at time T. The rm has issued risky debt with a face value F and

can use its economic capital G to help pay o bondholders in the event that the

value AT of the risky assets is less than the face value F of the debt. Using theresults from expression (7), the market value of the risky debt is given by

D0Y T FB0Y T pA0Y TYKX 18

The promised rate of return RP on the credit risky debt is dened by

D0YT Fa1 RP 19using discrete compounding.

Following from expression (8), the market value of equity is given by

S0 cA0Y TYKX 20

3.2 The Probability of Default

The level of economic capital G is set such that, given debt has a face value F,

the probability of default equals some prespecied value p:

p PrAT G1 RfT F PrAT TKY 21

4 If m is negative, this means that shares are repurchased. Note that n0 m b 0.5 Note that A

0

I only for a zero net present value investment. Otherwise A0

is greater or less

than I, depending on whether the investment has a positive or negative net present value.

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where K

F

G

1

Rf

. If G

1

Rf

b F, the probability of default is zero,

since the net indebtness of the rm is negative. We assume that the rm is a netborrower, so that there is a positive probability of default. Using expression (12),

we get

p Nd2Y 22

where d2 flnA0aK "A 12 '2ATga'A

Tp

. The value of p is specied bysenior management. Given p, expression (21) implies the value of K, that is,

K A0 exp"A 12 '2AT 'A

Tp

N1pY 23

where N1 is the inverse of the cumulative normal distribution function.Given K, we can immediately determine the market value of equity, usingequation (20), and the default put option pA0Y TYK in expression (18). Todetermine the market value of debt, we must make some assumption about the

debt/equity nancing of the investment I G g. Note that K and F and Gare related by the expression

G F KB0YTX 24

3.3 Calculating RAROC

The market value of equity at time T is given by

ST AT G1 Rf DTY TXHence, the expected value of equity is

EST EAT G1 Rf EDTY T EAT G1 Rf D0YT1 "RDY 25

where "RD is the expected rate of return on debt. The expected rate of return on

equity to new shareholders is

"RE

E

S

T

aS

0

1X

26

The expected rate of return to existing shareholders is

"RHE

ESTan0 ms0

1X 27

Up to this point we have made no assumption about the debt equity mix used

in the nancing of the project. In calculating RAROC a common assumption is

that the risky asset is totally nanced by debt, implying that

I D0YTY 28

with economic capital G being nanced by the initial cash position g and new



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equity (see expression (15)). RAROC is then dened as the expected return on

economic capital:6

RAROC fEAT G1 Rf D0YT1 "RDgaG 1 ESTaG 1X 29

Comparing expression (29) with the expected rate of return to new shareholders

(expression (26)) and the expected rate of return to existing shareholders

(expression (27)), it is seen that it is generally dierent from the expected rate

of return to equity shareholders, as rst observed by Wilson (1992).

According to Zaik et al. (1996), Bank of America uses a xed hurdle rate

the bank's cost of equity capital. It is well known that this produces two types of

error: accepting `high-risk' projects that will decrease the value of the rm and

rejecting `low-risk' projects that will increase the value of the rm (see Brealey

and Myers 1996). Zaik et al., while recognizing the errors that may occur, argue

that the costs of estimating individual betas for dierent businesses outweigh the

benets. This raises two questions. First, how is the expected rate of return to

existing shareholders related to RAROC? Second, how sensitive is the RAROC

methodology to changes in the risk of the underlying business?

Appendix A proves the following proposition relating RAROC to the

expected rate of return to existing and new shareholders.

Proposition 1 Assuming that the risky asset is totally nanced by debt

(expression (28)) and that the market value of debt is less than the market valueof the risky assets, D0Y T ` A0, we have:(1) for a positive net present value project,

"RHE b RAROC b "REY

(2) for a zero net present value project,

"RHE RAROC "REY

(3) for a negative net present value project,

"RE b RAROC b "RHEX

where "RE is the expected rate of return to new shareholders and "RHE is the expected

rate of return to existing shareholders.

Proof. See Appendix A. &

The interesting aspect of Proposition 1 is that we have not used any type of

valuation model in deriving the relative ordering of the expected rates of return.

The result is simply a consequence of the denition of RAROC. The proposition

6 This expression is completely consistent with the expression in footnote 1, recognizing that the risk

adjustment is the economic capital G.

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shows that, for zero net present value projects, the expected rate of return on

equity for both the new and the existing shareholders and RAROC are equal.This provides some credence for comparing RAROC to the rm's cost of equity

capital for stand-alone projects (see Zaik et al. 1996).

Proposition 1 suggests an appropriate hurdle rate for accepting or rejecting a

project. In equilibrium the expected rate of return to equity shareholders is given

by the capital asset pricing model (see expression (13)):

"RE Rf E "RM RfY

where E is the beta of equity. From Proposition 1, a project will increase the

wealth of existing shareholders if

"RHE Rf

Eb

RAROC RfE

b RM RfY 30

assuming E b 0. We dene the adjusted RAROC measure to be

Adjusted RAROC RAROC RfE

X 31

3.4 Properties of RAROC

To examine some of the properties of RAROC, we consider two examples that

are extensions of the example described in Section 2.4. First, we consider a zeronet present value investment. The volatility of the return on the risky asset is

varied, keeping xed the correlation coecient between the return on the risky

assets and the return on the market portfolio.7 This aects the systematic risk of

the project A and its expected rate of return. The probability of default is kept

constant, implying that the level of the economic capital must be adjusted as the

volatility of the risky asset and its expected rate of return change.

The RAROC value is calculated using (29) and the adjusted RAROC

measure using expression (31). The beta for equity is estimated using the capital

asset pricing model and the fact that equity is a call option on the assets of the

rm (see expression (20)):

E A0S0 Nd1AY 32

where A is the beta coecient of the risky assets. The use of expression (32)

represents an approximation. It is an instantaneous beta, and not a discrete time

beta. This introduces a small error.

The sensitivity of the RAROC measure to changes in the volatility of the

risky asset are shown in Table 3. The project is a marginal project with zero net

present value, as the market value A0 of the risky asset equals the cost I of theinvestment. Therefore, we should be indierent to the project. We also observe

that, as expected from Proposition 1, RAROC and the expected rate of return to

7 Beta is related to the correlation coecient via &AM'Aa'MX



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equity are equal. The adjusted RAROC is just equal to the expected excess rate

of return on the market portfolio, that is, "RM Rf 6X877.Four points are worth noting. First, RAROC is sensitive to the level of the

standard deviation of the risky asset. This is to be expected, given our results

from the last section. Second, RAROC is sensitive to the correlation of the

return on the underlying asset and the market portfolio. Third, the adjusted

RAROC measure is relatively insensitive to changes in volatility and correlation.Fourth, if, as described by Zaik et al. (1996), we follow Bank of America's use of

a xed hurdle rate, we would pick high volatility and high correlation projects.

The results in Table 3 show that RAROC depends upon the level of volatility.

This suggests that, for large volatility, RAROC may be suciently large that it is

above the required hurdle rate. Hence a project, with a negative net present value

may well be accepted. We consider a minor variation of the example used in

Table 3 by altering the cost I of the investment. In the rst case we set I 1050,implying a negative net present value: the share price declines to $9.50 per share

from $10 per share. In the second case we set I 1000, implying a zero netpresent value: the share price remains unchanged at $10 per share. This case was

already considered in Table 3. In the third case we set I 950, implying a

TABLE 3. Sensitivity of RAROC to volatility for a zero net present value investment.

Standarddeviation,

'A (%)

Expected rate of return (%) Economiccapital,

G

RAROC(%)

AdjustedRAROC

(%)Risky asset,"RA Equity, "RE

Part A: Correlation coecient 0X25

5 5.70 10.46 106.3 10.46 6.88

10 6.27 10.71 203.2 10.71 6.88

20 7.42 11.25 371.6 11.25 6.88

40 9.71 12.48 620.7 12.48 6.88

Part B: Correlation coecient 0.50

5 6.27 16.28 101.5 16.28 6.90

10 7.42 16.77 194.8 16.77 6.90

20 9.71 17.81 358.3 17.81 6.89

40 14.29 20.21 605.1 20.21 6.88




Market value of the risky assets, A0 1000Cost of investment, I 1000

Initial value of the rm, g 1000

Initial number of shares, n0 100

Maturity of debt, T 1 year

Probability of default, p 1%


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positive net present value: the share price increases to $10.50 per share from $10

per share. The results are given in Table 4.

Looking along the rows, it is seen that the variation in RAROC depends

upon the level of volatility. For low volatility, RAROC is relatively sensitive as to

whether the project has positive or negative net present value. The level of

sensitivity decreases as the level of volatility increases. RAROC is an increasing

function of net present value. Suppose the net present value decreases. This

implies that the numerator in RAROC decreases and the denominator increases,

and hence RAROC decreases.

The variation of RAROC with volatility depends upon whether the project

has positive or negative net present value. If the project has positive net present

value, the variation of RAROC with volatility is U-shaped. For zero andnegative net present value projects, RAROC increases as volatility increases.

RAROC depends upon the variation of EATaG and I1 "RDaG withvolatility. The level of reserves G declines as the net present value increases.

For zero and negative net present value projects, the change in EATaG islarger than the change in I1 "RDaG for a given increase in volatility. Thisexplains why RAROC increases as volatility increases. This is also the case for

positive net present value projects when volatility is large. For low volatility, the

reverse holds.

The results in Table 4 show that, for a project with high volatility and negative

net present value, the RAROC value may be suciently large so as to be above an

investment hurdle rate, i.e. 17% in our example. The adjusted RAROC provides

TABLE 4. Net present value and RAROC.

Standarddeviation

(%)

Negative Zero Positive

RAROC AdjustedRAROC

RAROC AdjustedRAROC

RAROC AdjustedRAROC

5 10.82 3.52 16.18 6.90 22.31 10.62

10 13.84 5.16 16.77 6.90 19.84 8.72

20 16.19 6.01 17.81 6.89 19.48 7.80

40 19.22 6.43 20.21 6.88 21.21 7.34

Price per share,s0

9.50 10.00 10.50




Original price per share, s0 10.00

Market value of assets, A0 1000Maturity of debt, T 1 year

Correlation coecient, & 0.50




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the correct investment decision. For negative net present value investment

projects it is below the hurdle rate, and for positive net present value projectsit is above the hurdle rate, where adopting adjusted RAROC the corresponding

hurdle rate becomes the expected excess return on the market portfolio, "RM Rf.The adjusted RAROC is equal, below, or above the expected excess return on the

market portfolio, that is, 6.87%, when the net present value of the project is zero,

negative, or positive respectively. This is true for all levels of volatility.

3.5 Estimating the Adjusted RAROC

The adjusted RAROC dened by expression (31) requires that we measure the

beta of the equity. It is commonly argued that the whole reason for RAROC is

that it adjusts for risk without requiring that we measure beta. However, as wehave demonstrated, RAROC does not adjust for risk and consequently we are

faced with the problem of estimating beta.

Expression (32) provides us with a mechanism to estimate beta. First,

experience typically allows us to estimate the value A0 of the assets, theirvolatility 'A, and beta A. We must estimate the term Nd1, where

d1 flnA0aK r 12 '2ATg

'A

Tp

X

The probability of default is given by expression (22). If we know the beta of the

assets, then we know the required rate of return "A (see expression (13)). Given

an estimate of the probability of default, we can use expression (23) to infer theparameter K. Hence we are in a position to estimate d1 and S0, usingexpression (20). Consequently, we can estimate all the terms on the right-hand

side of expression (32).

3.6 Value-at-Risk and Economic Capital

We end this section with a brief discussion of the denition of economic capital

and the relationship between economic capital and value-at-risk. We have

dened the rm's economic capital as being equal to the reserves G that the

rm keeps so that the probability of default is equal to some prespecied level

p. This is not the denition of economic capital that is used in practice.

Normally, it is dened as the dierence between the expected future value of the

rm's assets and the value-at-risk. The value-at-risk v for the rm is dened as

the critical value for which the probability of the rm's risky assets being less

than or equal to v is equal to some specied value p. We can write this in theform

p PrATT vX

The standard practice is to set p equal to the desired probability of default p.This implies that, by comparison with expression (21), v K.

The standard denition of economic capital is

EC EAT v A0 exp"AT KX

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Substituting expression (24), we have

EC A0 exp"AT F G1 Rf 1 RffA0 exp"A rfT FB0YT GgY

where rf is the continuously compounded default free interest rate. The above

expression implies that the traditional denition of economic capital may

overestimate or underestimate the correct economic capital.8

4. A BANK'S LOAN PORTFOLIO

Bankers Trust originally introduced the RAROC methodology to measure the

performance of its loan portfolio. In this section we examine the performance of

a loan portfolio by extending the model described in the last section. It is

assumed that the bank purchases the credit-risky debt issued by a rm. The

bank nances the loan by borrowing. The bank requires a reserve fund GB of

cash so that the probability of default is equal to some prespecied level pB. Thereserve fund is referred to as the bank's economic capital.

To simplify the analysis, we assume that before undertaking this investment

the bank is all equity nanced. The only asset of the bank is cash gB. Let IBdenote the cost to the bank of purchasing the bond from the rm or alternatively

advancing a loan to the rm. The total cost of the investment is IB GB gB,which is nanced by borrowing D0YT and issuing additional equity. Weassume that the loan advanced to the rm is nanced by the bank borrowing

the amount IB. At time T, the bank must pay the amount IB1 RB, where RBis the bank's cost of borrowing.

The probability of default is

pB PrDTY T GB1 Rf T IB1 RBY 33

where DTYT is described by expression (16). Let

KB IB1 RB GB1 Rf 348 A slightly more sophisticated denition denes economic capital as the present value of the

dierence between the expected future value of the rm's assets and the value-at-risk:

EC PVAT F G1 RfXIf the risk of default is ignored, then the promised rate on debt RP (see expression (19)) is equal to

the expected rate "RD (see expression (25)). Hence

F D0Y T1 RP D0Y T1 "RDXUsing this expression, the economic capital is

EC PVST S0XHence, for a zero net present value project, EC G. Furthermore, G is less (greater) than EC fornegative (positive) net present value projects. While this is an interesting result, it ignores default and

hence is a rst-order approximation at best.



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denote the net payment the bank must make at time T. Expression (33) can then

be written asp

B PrDTYTTKBX 35

If KB b F, then default by the bank is certain. The maximum the bank will

receive from the bond payment is F, which is less than the bank's obligation,

implying default will occur with certainty. We will assume that the bank follows

a strategy where default does not occur with certainty, implying KB T F. Let DBdenote the event of default by the bank and D the event of default by the rm.

The probability of default by the bank is

Pr

DB

Pr

DB

jD

Pr

D

Y

36

the proof being given in Appendix B.

The rm will default if the value of its assets is less than K, that is, if the event

AT ` K occurs (see expression (21)). Conditional on default by the rm, therm pays the bank the amount AT G1 Rf. The bank will default on itsobligations if

AT G1 Rf ` KBX 37

The probability of default by the bank conditional on default by the rm is

PrDB j D PrAT G1 Rf ` KB j AT ` KXLet

KHB KB G1 RfY 38

so that

PrDB j D PrAT ` KHB j AT ` K 1 if KHB b K,

PrAT ` KHB if KHB TK.

@

39The unconditional probability of default by the bank is

PrDB Pr

D

if KHB b K,

PrD PrAT ` KHB if KHB TK.@ 40

The case KHB b K occurs when the rm is more conservative than the bank.Default by the rm triggers default by the bank. The probability of default for

the bank is thus determined by the rm.


The expected payment at date T to the bank from the rm can be written, using

expression (17), in the form

F EK AT j AT ` KY 41Journal of Risk

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where the second term is the expected value of the loss, given default. Using

expression (29), the value of RAROC is

RAROC fF EK AT j AT ` K GB1 Rf IB1 RBgaGB 1Y42

assuming GB b 0.

4.2 Example

The risky debt issued by the rm, described in Section 3.4, is purchased by the

bank. The bank nances the purchase by debt. We assume that the bank sets its

reserves so that the probability of default over the one year period is 0.01%,

which is consistent with Moody's expected default frequency for an A credit-

rated company. We compute the RAROC of the loan to the bank using

expression (42). For simplicity, we assume that cost of borrowing for the bank

is the risk-free rate of interest.9 While this introduces a slight error, it is common

practice to use the risk-free rate. The results are shown in Table 5.

In Part A, the probability of default for the rm is assumed to be 1.79%,

TABLE 5. RAROC for a bank's loan portfolio

Standarddeviation,

'A (%)

Expected rate of returnfor the rm's:

Bank'seconomiccapital, G

RAROC(%)

Risky assets, "RA Debt,a "RD

Part A: Probability of default for the rm 1.79%

5 6.27 5.15 19 6.37

10 7.42 5.15 34 6.35

20 9.71 5.19 55 6.32

40 14.29 5.20 68 6.26

Part B: Probability of default for the rm 8.31%

5 6.27 5.23 72 6.6110 7.42 5.32 130 6.62

20 9.71 5.45 210 6.66

40 14.29 5.55 267 6.73

Default free rate of interest, Rf 5.13%

Expected rate of return on the market portfolio, "RM 12.0%


Correlation coecient, & 0.50

Market value of assets, A0 1000Maturity of debt, T 1 year

Probability of default for the bank, p 0.01%


9 In practice, the LIBOR interest rate is used and it is implicitly assumed that there is no chance of

default by the bank.



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which is consistent with Moody's Ba credit rating. In Part B, the probability of

default is assumed to be 8.31%, which is consistent with Moody's B creditrating.

Two interesting observations can be made about the results in Table 5. First,

RAROC varies as the volatility of the rm's risky assets change, even though the

probability of default for the rm and the bank are held constant. Second,

RAROC can be an increasing or decreasing function of the volatility of the

rm's risky assets depending upon its credit rating. For low credits, RAROC

increases as volatility increases.

5. DIVERSIFICATION AND ECONOMIC CAPITAL

For a rm with dierent businesses, the economic capital for a particular

business can be determined by viewing the business on a stand-alone basis.

The stand-alone economic capital is often used for measuring the performance

of dierent businesses within the rm. The economic capital for the rm as a

whole is normally less than the sum of the economic capital of the individual

businesses determined on a stand-alone basis, because the businesses are not

perfectly correlated. For entry/exit decisions, it is necessary to determine the

marginal economic capital, not the stand-alone economic capital. This raises the

practical diculty of how to measure the marginal economic capital.

One common method of addressing this problem is to introduce a weighting

scheme, where it is assumed that the economic capital for a stand-alone business

is proportional to the economic capital of the rm. Let ECj denote the stand-

alone economic capital of the jth business within the rm. Let ECB denote the

economic capital for the rm as a whole. Dene

w ECB0n

i1ECiY 43

where n is the number of businesses within the rm. Note that the right-hand

side does not depend explicitly upon the particular business j, a nd 1 wrepresents the portfolio diversication eect. The marginal economic capital

for the jth business is given by

w ECjX 44

The sum of the marginal economic capitals for the businesses is

nj1

w ECj ECBY 45

as expected.

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5.1 Valuation of Equity and the Credit-Risky Debt

We start by describing the rm and then the investment project to be undertaken

by the rm.

The market value of the rm's assets is described by expression (6). The rm

has zero-coupon debt with a face value F. The debt matures at date T. The rm

also has a reserve fund G of cash so that the probability of default over the

period 0Y T is equal to some prespecied level p. The value of the rm's debt isdescribed by expression (17), with K, the strike price of the put option, being

given by expression (23). The level of the rm's economic capital is described by

expression (24).

The rm undertakes an investment project. The cost of the investment is I1.

The market value of the investment project is A10. The stochastic processdescribing changes in the market value of the investment project is

dA1tA1t

"1 dt '1 dW1tY 46

where "1 is the instantaneous expected rate of return, '1 is the volatility, and

W1t is a Brownian motion. In general, changes in the market value of theproject are correlated with changes in the value of the rm's existing assets. The

project is assumed to be value additive, so that the value of the rm after it

undertakes the project is

Vt At A1tX 47

The stochastic process describing the value of the rm is, in general, not

lognormal, as the sum of lognormal distributions is not lognormal. For

computational ease, we will, however, assume that the distribution can be

approximated by a lognormal distribution by matching the rst two moments

"VY 'V of the distribution. The details are described in Appendix C. Theaccuracy of this assumption is examined in Levy and Turnbull (1992).

The project is assumed to be totally nanced with debt. The rm also wants

to alter the amount of cash reserves to the level G1 so that the probability of

default remains unchanged at the prespecied level p. The total cost of theinvestment is I1 G1 G and it is nanced by issuing zero-coupon debt andequity.

The number of new shares of equity is denoted by m and the new equilibrium

price per share by s0. The total market value of equity is denoted by S0,where

S0 n0 ms0Y 48

where n0 is the number of shares in the rm before the project. The investment I1is assumed to be nanced by issuing zero-coupon debt that matures at date T.

The rm already has zero-coupon debt outstanding with a face value F that

matures at the same date. After the rm has nanced the investment project, it



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will have debt outstanding with a total face value F1 and market value D1

0YT

.

LetD10Y T F1d10YTY 49

where d10Y T is the market value of a zero-coupon bond with a face value of 1,given that the rm has F1 such bonds. Given that the rm has announced it will

undertake the investment, the market value of the rm's original debt is

Fd10Y T, so that the market value of the new debt issues is F1 Fd10YT,where

F1 Fd10Y T I1X 50

Substituting expression (49) into expression (50) gives

D10YT1 FaF1 I1X 51Using expression (23), the probability of default p

implies

K1 V0 exp"V 12 '2VT 'V

Tp

N1pY 52

where "V is the instantaneous expected rate of return for the rm and 'V the

volatility of the return. The above expression determines K1. Given expression

(50) and K1, the value of the zero-coupon can be determined as

D10Y T F1B0YT pV0Y TYK1Y 'VY 53

using expression (17). The total value of equity is given by

S10 cV0Y TYK1Y 'VY 54

using expression (20). The level of reserves is given by

G1 F1 K1B0Y TX 55


In general, the eects of correlation between the investment project and the risky

assets of the rm are nonlinear. Correlation aects the value of the put option

that bondholders have implicitly written, and hence the value of existing debt,the economic capital, and RAROC. One case that allows a simple analysis is

that of a scale-expanding project. To be precise, we dene a scale-expanding

project to be of the formA1t AtY 56

where is a positive constant. The project is nanced so that the capital

structure of the rm remains unchanged. Expression (56) implies that the return

on the project is perfectly positively correlated with the original risky assets of

the rm.

Proposition 2 For a scale-expanding project that is nanced such that the

capital structure remains unchanged, the economic capital is also scale expanding.

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That is,

G1 G GX 57Proof. See Appendix D. &

This proposition implies that the marginal economic capital is equal to the

stand-alone economic capital. In general, this is not the case for non-scale-

expanding projects. To explore the eects of correlation, we consider a

numerical example using the theory described in Section 5.1.

5.3 Example

The initial conditions of the rm are given in Table 6. These gures are taken

from Table 3, Part B. The rm has issued zero-coupon debt with a currentmarket value of 1000. The debt matures in one year's time. The rm undertakes

an investment project. The market value A10 of the project is assumed tobe 1000.

In Table 7, Part A, the cost I1 of the project is assumed to equal 1000,

implying the project has zero net present value. We alter the correlation

coecient &IM of the project with the market portfolio, keeping the volatility

'1 constant. Once the correlation coecient &AM of the return on the rm's

existing risky assets with the market portfolio is specied, then the correlation

coecient &AI between the return on the project and the rm's existing assets is

determined as

&AI &AM&IMY 58

assuming a one-factor model of asset returns.10

The project is rst considered in isolation. We increase the correlation

coecient of the project's rate of return with the return on the market portfolio.

This increases the project's beta coecient and hence its expected rate of return.

The volatility of the rate of return of the project is kept constant. The probability

10 Let the return on the market portfolio be described by

dM

M "M dt 'M dWMX

The return on the existing assets of the rm is described by

dA

A " dt 'AM dWM 'AI dWAY

where WM and WA are independent Brownian motions. The return on the investment project is

described bydAI

AI "1 dt 'IM dWM 'II dWIY

where WM, WA, and WI are independent Brownian motions.

The correlation coecient of the existing assets of the rm with the market is

&AM 'AMa'Aand, for the investment project,

&IM 'IMa'IXThe correlation coecient of the existing risky assets of the rm and the investment project is

&AI 'AM'IMa'A'I &AM&IMX



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of default is also kept constant, so that the strike price K increases as the

correlation coecient increases. From equation (24), this implies that the

economic capital decreases. The stand-alone RAROC value increases. The

results are similar to those in Table 3.

Next, the project and the rm's existing risky assets are considered together.

As we increase the correlation coecient &IM of the project's rate of return with

the return on the market portfolio, the correlation &AI between the project and

the rm's existing risky assets also increases, given expression (58).

For the economic capital of the project and rm together, two opposing

eects must be considered. An increase in the correlation coecient of the risky

project with the market increases the project's expected rate of return. This will

have the eect of increasing the strike price K1. However, an increase in the

correlation coecient of the risky project with the market increases the

correlation coecient &AI of the project with the existing risky assets of the

rm. This has the eect of increasing the volatility 'V and lowering the strike

TABLE 6. Initial state of the firm.

Market value of debt, D0YT 1000Market value of equity, S0 358Economic capital, G 358

Volatility, 'A 20%

Correlation coecient with the market, & 0.5

Beta of the risk assets, A 0.667

Probability of default, p 1%Maturity of debt, T 1 year

RAROC 17.81%

TABLE 7. Investment project with present value A10 1000 and volatility '1 207.

Project Correlationwith market,

&IM

Correlationwith rm,

&AI

Projectstand-alone basis

Projectand the rm

Economic

capital, G

RAROC Economic

capital, G

RAROC

Part A: Zero net present value

1 0.25 0.125 372 11.25 562 17.21

2 0.50 0.250 358 17.81 575 20.89

3 0.75 0.375 345 24.85 588 24.44

4 1.00 0.500 332 32.43 599 27.87

Part B: Negative net present value

1 0.25 0.125 377 9.77 567 16.18

2 0.50 0.250 363 16.19 581 19.85

3 0.75 0.375 350 23.07 593 23.39

4 1.00 0.500 337 30.46 604 26.81

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price K1. In Table 7 the increase in volatility eect dominates and hence the

economic capital increases. For RAROC, the decrease in the strike price and the

increase in the expected rate of return increase the expected future value of

equity and hence RAROC. Osetting this increase is the increase in economic

capital that, everything else being equal, will cause RAROC to decrease. In this

case, the net eect causes RAROC to increase.

For the valuation of debt, the associated put option is deep-out-of-the-money, because the prespecied probability of default is set at 1%. Con-

sequently, the change in the value of the put option is small when the rm

undertakes the risky investment, implying that wealth transfer eects between

bond and equity holders can be ignored.

For an investment project with negative net present value, the results are very

similar, as shown in Table 7, Part B.

Marginal economic capital

Tables 6 and 7 allow us to compute the marginal economic capital and to

compare it with the method described by expression (44). The results are shown

in Table 8. We have computed the marginal economic capital using the results inTable 7.11 It is seen in Table 8 that there is a wide discrepancy between the two

sets of values. The weighting scheme is insensitive to the project's systematic risk

(beta) with the market.

Projects with constant beta coefcient

In Table 9, we vary the correlation of the project with the market such that the

project's beta (systematic risk) remains constant. This implies that the expected

rate of return on the project remains unchanged. If the correlation of the project

TABLE 8. Marginal economic capital of the investment

project.

Project Marginalcapital

Weightingscheme

Part A

1 204 286

2 217 288

3 230 289

4 241 288

Part B

1 209 291

2 223 293

3 235 293

4 246 293

11 For project 1, the marginal economic capital is 562 358 204. For the weighting scheme,2

j1ECj

358

372

730, implying wj

562a730

0X7699. The economic capital for the

project is 0X7699 372 286.



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with the market increases, then the volatility of the project must decrease, given

the assumption of a constant beta, and the correlation with the rm will increase

(see expression (58)). Increasing the correlation of the project with the market

with the resulting decrease in the project's volatility implies that the project's

stand-alone economic capital decreases and also the economic capital of the

project and the rm together decreases. Consequently RAROC increases. For

estimating the marginal economic capital, the weighting scheme can produce

large errors.

6. CLOSING COMMENTS

To maximize the risk-adjusted return on economic capital, subject to constraints

imposed by regulatory requirements, is a dictum followed by many banks. We

have shown that commonly employed methods may result in decisions that

adversely aect shareholder value. We have described an alternative methodo-

logy, adjusted RAROC, that corrects the inherent limitations of the existing

methods.

6.1 Games with RAROC

We end this paper by examining the incentives that a RAROC methodology

TABLE 9. Beta of project constant.

Part A

Project Standarddeviation

Correlationwith

market

Correlationwithrm

Projectstand-alone basis

Projectand the rm

Economiccapital

RAROC Economiccapital

RAROC

1 60 0.25 0.125 785 13.13 1114 14.88

2 30 0.50 0.250 499 17.76 712 20.35

3 20 0.75 0.375 350 23.07 593 23.39

4 15 1.00 0.500 263 28.94 543 25.04

Market of project A10 1000.Cost of investment I1 1005.

Part B

Project Marginalcapital

Weightingscheme

1 756 765

2 354 414

3 235 293

4 185 230

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generates for senior management. The initial state of the rm is as described in

Table 6. The current RAROC value for the rm is 17.81%.

Game 1

In Table 10, Part A, we consider four mutually exclusive investment projects.

The marginal economic capital, as estimated by the weighting scheme is

approximately the same for all the projects. On a stand-alone basis, project 4

has the highest RAROC. When we consider the RAROC value of the rm andproject together, project 4 again comes out top. Advocates of RAROC would

pick project 4. Unfortunately, there is a problem. Project 4 has negative net

present value, and will decrease the wealth of existing shareholders.

Game 2

In Table 10, Part B, we consider four mutually exclusive investment projects.

The rm's RAROC before undertaking any new investment projects is 17.81%.

We see that project 1 will lower the rm's RAROC, while the remaining projects

will increase the rm's RAROC value. Advocates of RAROC would reject

project 1, and accept one of the remaining projects. Unfortunately, there is a

problem. The remaining projects have negative net present value, and will

TABLE 10. Games with RAROC.

Part A: Game 1

Project Projectstand-alone basis

Marginaleconomic

capital

Weightedeconomic

capital

RAROCrm and

projectEconomic capital RAROC

1a 358 17.81 217 288 20.89

2a 345 24.85 230 289 24.44

3b 350 23.07 235 293 23.39

4b 337 30.46 246 293 26.81

a From Table 7, Part A.

b From Table 7, Part B.

Part B: Game 2

Project Firm and project together

RAROC of projectand the rm

Economiccapital

1a 17.21 562

2b 19.85 581

3b 23.39 593

4b 26.81 604

a From Table 7, Part A.

b From Table 7, Part B.



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decrease the wealth of existing shareholders. Project 1, which was rejected, has

zero net present value.

If compensation for senior management is based on RAROC, then the

methodology provides incentives for management to undertake investments

that can increase RAROC, but decrease the wealth of existing shareholders.

APPENDIX

A. Proof of Proposition 1

We rst consider the expected rate of return to new shareholders. Given the

assumption that I D0YT, expression (15) can be rewritten in the formms0 G gY

so that G ms0 n0s0. For a nonnegative net present value project s0b s0,implying that

GT m n0s0 S0XHence,

1aGb 1aS0X

Given limited liability, then ESTb 0, so that

ESTaG 1b ESTaS0 1XHence,

RAROC b "REX A1

For a negative net present value project,

RAROC ` "REX A2

Next we consider the expected rate of return to original shareholders. For a

nonnegative net present value project, s0 b s0, we have

Gb m n0s0Yimplying that

"RHE bRAROCX A3

For a negative net present value project

"RHE ` RAROCX &

B. The Bank's Loan Portfolio

Let "D denote the event the rm does not default on its bond and D the event

that the rm defaults. Suppose that the rm does not default. Then the bank

receives the payment F. The conditional probability that the bank will default

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is

PrDB j "DA 1 if F ` KB Bank defaults,0 if Fb KB Bank does not default.

&B1

If KB b F, then default will occur with certainty, implying that

PrDB 1X B2

Suppose that the rm defaults on its bond payment. The bank receives the

amount AT G1 Rf. The bank will default if AT G1 Rf ` KB.Hence,

PrDB j D PrAT G1 Rf ` KB j DX B3

NowPrDB PrDB j "D Pr "D PrDB j D PrDX B4

For KB T F, using equation (B1), we obtain

PrDB PrDB j D PrDX B5

C. Two Risky Assets

By assumption,

AiT Ai0 exp"i 12 '2i T 'i Tp

Zi i 1Y 2Y

where Z1 and Z2 are bivariate normally distributed with correlation coecient &;

Zi $ N0Y 1. Let Vt A1t A2t. It is assumed that the distribution ofVT can be approximated by a lognormal distribution:

Vt V0 exp"V 12 '2VT 'V

Tp

ZVY

where ZV $ N0Y 1. The moments "V and '2V are chosen to match the rst twomoments of A1T A2T:

EA1T A2T

2

i

1

A10 exp"iT V0 exp"VTX C1

Expression (C1) denes "V.For the second moment,

EfA1T A2Tg2 EA1T2 2EA1TA2T EA2T2 A102 exp"1T '21T

2A10A20 exp"1 "2T '1'2&T A202 exp2"1T '22 T

V02 exp2"VT '2VTX C2

Expression (C2) denes '2V.



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D. Proof of Proposition 2

By assumption, A10 A0 b 0, so that

V0 1 A0X D1

In expression (52), recognizing that "V " and 'V ', we have, by comparisonwith expression (23),

K1 1 KX D2

Given the assumption of lognormality, options are homogeneous of degree 1 in

A0 and K. Hence,

pA10Y TY

K1Y ' 1 pA0Y TY

KY 'X D3andcA10Y TYK1Y ' 1 cA0Y TYKY 'X D4

SinceV0 D10Y T S10Y D5

it follows that (D1) and (D4) imply that

D10Y T 1 D0Y TX D6

Expressions (D3) and (D6) imply that

F1

1

FX

D7

Expressions (D2) and (D7) imply that

G1 1 GX D8

Acknowledgements

The views expressed in this paper are those of the authors and do not necessarily

reect the position of the Canadian Imperial Bank of Commerce. The rst draft

of this paper was written when Lee Wakeman was a consultant for CIBC. The

authors gratefully acknowledge comments from Robert Mark and the Editor,

Philippe Jorion.

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