Credit Portfolio Management phần 2 pdf

Modern Portfolio Theory and Elements of the Portfolio Modeling Process The argument we made in Chapter 1 is that the credit function must form into a loan portfolio management function.

1 If the mark-to-market (MTM) model is used, then b(PD) is given by:

If a default mode (DM) model is used, then it is given by:

b(PD) = 7.6752 × PD2– 1.9211 × PD + 0.0774, for PD < 0.05 b(PD) = 0, for PD > 0.05

The Credit Portfolio Management Process

Modern Portfolio Theory

and Elements of the Portfolio Modeling Process

The argument we made in Chapter 1 is that the credit function must form into a loan portfolio management function Behaving like an assetmanager, the bank must maximize the risk-adjusted return to the loan port-folio by actively buying and selling credit exposures where possible, andotherwise managing new business and renewals of existing facilities Thisleads immediately to the realization that the principles of modern portfoliotheory (MPT)—which have proved so successful in the management of eq-uity portfolios—must be applied to credit portfolios

trans-What is modern portfolio theory and what makes it so desirable? Andhow can we apply modern portfolio theory to portfolios of credit assets?

MODERN PORTFOLIO THEORY

What we call modern portfolio theory arises from the work of HarryMarkowitz in the early 1950s (With that date, I’m not sure how modern it

is, but we are stuck with the name.)

As we will see, the payoff from applying modern portfolio theory is

that, by combining assets in a portfolio, you can have a higher expected turn for a given level of risk; or, alternatively, you can have less risk for a given level of expected return.

re-Modern portfolio theory was designed to deal with equities; sothroughout all of this first part, we are thinking about equities We switch

to loans and other credit assets in the next part

The Efficient Set Theorem and the Efficient Frontier

Modern portfolio theory is based on a deceptively simple theorem, calledthe Efficient Set Theorem:

27

An investor will choose her/his optimal portfolio from the set of folios that:

port-1 Offer maximum expected return for varying levels of risk.

2 Offer minimum risk for varying levels of expected return.

Exhibit 2.1 illustrates how this efficient set theorem leads to the cient frontier The dots in Exhibit 2.1 are the feasible portfolios Note that

effi-the different portfolios have different combinations of return and risk Theefficient frontier is the collection of portfolios that simultaneously maxi-mize expected return for a given level of risk and minimize risk for a givenlevel of expected return

The job of a portfolio manager is to move toward the efficient frontier

Expected Return and Risk

In Exhibit 2.1 the axes are simply “expected return” and “risk.” We need

to provide some specificity about those terms

EXHIBIT 2.1 The Efficient Set Theorem Leads to the Efficient Frontier

In modern portfolio theory, when we talk about return, we are

talk-ing about expected returns The expected return for equity i would be

written as

E[R i] = µiwhereµi is the mean of the return distribution for equity i.

In modern portfolio theory, risk is expressed as the standard deviation

of the returns for the security Remember that the standard deviation for

equity i is the square root of its variance, which measures the dispersion of

the return distribution as the expected value of squared deviations about

the mean The variance for equity i would be written as1

The Effect of Combining Assets in a

The variance for our two-equity portfolio is where things begin to getinteresting The variance of the portfolio depends not only on the variances

of the individual equities but also on the covariance between the returnsfor the two equities (σ1,2):

Since covariance is a term about which most of us do not have amental picture, we can alternatively write the variance for our two-equityportfolio in terms of the correlation between the returns for equities 1and 2 (ρ1,2):

σp2=w12σ12+w22σ22+2w w1 2 1 2 1 2ρ σ σ,

σp2 =w12σ12+w22σ22+2w w1 2 1 2σ ,

σi2 =E E R[( [ ]i −R i) ]2

This boring-looking equation turns out to be very powerful and haschanged the way that investors hold equities It says:

Unless the equities are perfectly positively correlated (i.e., unless ρ1,2= 1) the riskiness of the portfolio will be smaller than the weighted sum of the riskiness of the two equities that were used to create the portfolio.

That is, in every case except the extreme case where the equities areperfectly positively correlated, combining the equities into a portfolio willresult in a “diversification effect.”

This is probably easiest to see via an example

Example: The Impact of Correlation

Consider two equities—Bristol-Meyers Squibb and Ford Motor Company Using historical data on the share prices, we found that the mean return for Bristol-Meyers Squibb was 15% yearly and the mean return for Ford was 21% yearly Using the same data set, we calculated the standard deviation in Bristol-Myers Squibb’s return as 18.6% yearly and that for Ford as 28.0% yearly.

E(R BMS) = µBMS= 15% E(R F) = µF= 21%

The numbers make sense: Ford has a higher return, but it is also more risky.

Now let’s use these equities to create a portfolio with 60% of the portfolio invested in Bristol-Myers Squibb and the remaining 40% in Ford Motor Company The expected return for this portfolio is easy to calculate:

Expected Portfolio Return = (0.6)15 + (0.4)21 = 17.4%

The variance of the portfolio depends on the correlation of the returns on Bristol-Meyers Squibb’s equity with that of Ford ( ρBMS, F):

The riskiness of the portfolio is measured by the standard deviation of the portfolio turn—the square root of the variance.

re-The question we want to answer is whether the riskiness of the portfolio (the portfolio

standard deviation) is larger, equal to, or smaller than the weighted sum of the risks (the

standard deviations) of the two equities:

Weighted Sum of Risks = (0.6)18.6 + (0.4)28.0 = 22.4%

To answer this question, let’s look at three cases.

Variance of Portfolio Return = +

+

( ) ( ) ( ) ( ) ( )( )( )( , )( )( )

C ASE 1: T HE R ETURNS A RE U NCORRELATED (ρBMS,F= 0):

Variance of Portfolio Returns = (0.6) 2 (18.6) 2 + (0.4) 2 (28) 2 + 0 = 250.0

In this case, the riskiness of portfolio is less than the weighted sum of the risks of the two equities:

Standard Deviation of Portfolio = 15.8% yearly < 22.4%

If the returns are uncorrelated, combining the assets into a portfolio will generate a large diversification effect.

C ASE 2: T HE R ETURNS A RE P ERFECTLY P OSITIVELY C ORRELATED (ρBMS,F= 1):

In this extreme case, the riskiness of portfolio is equal to the weighted sum of the risks of the two equities:

Standard Deviation of Portfolio = 22.4% yearly

The only case in which there will be no diversification effect is when the returns are fectly positively correlated.

per-C ASE 3: T HE R ETURNS A RE P ERFECTLY N EGATIVELY C ORRELATED (ρBMS,F= –1):

In this extreme case, not only is the riskiness of portfolio less than the weighted sum of the

risks of the two equities, the portfolio is riskless:

Standard Deviation of Portfolio = 0% yearly

If the returns are perfectly negatively correlated, there will be a combination of the two sets that will result in a zero risk portfolio.

as-From Two Assets to N Assets

Previously we noted that, for a two-asset portfolio, the variance of theportfolio is

Variance of Portfolio Returns = +

( ) ( ) ( ) ( ) ( )( )( )( )( )( )

0 6 18 6 0 4 28

2 0 6 0 4 1 18 6 28 0

500 0

This two-asset portfolio variance is portrayed graphically in Exhibit 2.2.The term in the upper-left cell shows the degree to which equity 1varies with itself (the variance of the returns for equity 1); and the term

in the lower-right cell shows the degree to which equity 2 varies with self (the variance of the returns for equity 2) The term in the upper-right

it-shows the degree to which the returns for equity 1 covary with those for

equity 2, where the term ρ1,2σ1σ2is the covariance of the returns for uities 1 and 2 Likewise, the term in the upper-right shows the degree to

eq-which the returns for equity 2 covary with those for equity 1 (Note that

ρ1,2=ρ2,1.)

Exhibit 2.3 portrays the portfolio variance for a portfolio of N ties With our two-equity portfolio, the variance–covariance matrix con-

equi-tained 2 × 2 = 4 cells An N-equity portfolio will have N × N = N2cells in

its variance–covariance matrix.

In Exhibit 2.3, the shaded boxes on the diagonal are the variance

terms The other boxes are the covariance terms There are N variance terms and N2– N covariance terms.

If we sum up all the cells (i.e., we sum the i rows and the j columns) we

get the variance of the portfolio returns:

The Limit of Diversification—Covariance

We have seen that, if we combine equities in a portfolio, the riskiness of theportfolio is less than the weighted sum of the riskiness of the individual eq-uities (unless the equities are perfectly positively correlated) How far can

we take this? What is the limit of diversification?

j

N i

N

w w

2

1 1

EXHIBIT 2.2 Graphical Representation of

Variance for Two-Equity Portfolio

Equity 1 w1σ1 w1w2ρ1, 2σ1σ2

To answer this question, let’s consider a portfolio of N equities where all the equities are equally weighted That is, w i = 1/N.

We can express the portfolio variance in terms of the average

vari-ances and average covarivari-ances Remember that we have N variance terms and N2– N covariance terms Since the portfolio is equally weighted, each

of the average variance terms will be weighted by (1/N)2, and each of the

average covariance terms will be weighted by (1/N) × (1/N) = (1/N)2:

After doing a little algebra, we can simplify the preceding expression to:

What happens to the variance of the portfolio returns as the number of

eq-uities in the portfolio increases? As N gets large, 1/N goes to zero and (1– 1/N) goes to one So as the number of equities in the portfolio increases,

the variance of the portfolio returns approaches average covariance Thisrelation is depicted graphically in Exhibit 2.4

“Unique” risk (also called “diversifiable,” “residual,” or atic” risk) can be diversified away However, “systematic” risk (also called

“unsystem-“undiversifiable” or “market” risk) cannot be diversified away And, as wesaw previously, systematic risk is average covariance That means that thebedrock of risk—the risk you can’t diversify away—arises from the waythat the equities covary

For a portfolio of equities, you can achieve a “fully diversified” lio (i.e., one where total portfolio risk is approximately equal to averagecovariance) with about 30 equities

portfo-CHALLENGES IN APPLYING MODERN PORTFOLIO

THEORY TO PORTFOLIOS OF CREDIT ASSETS

In the preceding section, we saw that the application of modern portfolio

theory results in a higher expected return for a given level of risk or, natively, less risk for a given level of expected return.

alter-This is clearly an attractive proposition to investors in credit assets.However, there are some challenges that we face in applying modern port-folio theory—something that was developed for equities—to credit assets

Credit Assets Do Not Have Normally Distributed

Loss Distributions

Modern portfolio theory is based on two critical assumptions The first sumption is that investors are “risk averse.” Risk aversion just means that

EXHIBIT 2.4 As the Number of Equities Increases, Portfolio Risk Approaches Average Covariance

Portfolio

standard deviation

Number of securities

Unique risk

Systematic risk

if the investor is offered two baskets of assets—basket A and basket B—where both baskets have the same expected return but basket A had higherrisk than basket B, the investor will pick basket B, the basket with thelower risk And that assumption is not troublesome It is likely that in-vestors in credit assets are at least as risk averse as equity investors.The second assumption—the troublesome one—is that security returns

are jointly normally distributed This means that the expected return and

standard deviation completely describe the return distribution of each curity Moreover, this assumption means that if we combine securities intoportfolios, the portfolio returns are normally distributed

se-First, we have to do some mental switching of dimensions For ties, we are interested in returns For loans and other credit assets, we areinterested in expected losses So the question becomes: Can the loss dis-tributions for loans and other credit assets be characterized as normaldistributions? And, as long as we are here, we might as well look at thedistribution of equity returns

equi-Exhibit 2.5 examines these questions Panel A of equi-Exhibit 2.5 contains

a normal distribution and the histogram that results from actual daily pricechange data for IBM It turns out that the daily price changes for IBM arenot normally distributed: There is more probability at the mean thanwould be the case for a normal distribution; and there are more observa-tions in the tails of the histogram than would be predicted by a normal dis-tribution (The actual distribution has “fat tails.”) Indeed, if you look atequities, their returns are not, in general, normally distributed The returnsfor most equities don’t pass the test of being normally distributed

But wait a minute We said that a critical assumption behind modernportfolio theory is that returns are normally distributed; and now we havesaid that the returns to equities are not normally distributed That seems to

be a problem But in the case of equity portfolios, we simply ignore the viation from normality and go on In just a moment, we examine why this

de-is okay for equities (but not for credit assets)

Panel B of Exhibit 2.5 contains a stylized loss distribution for an inate-and-hold” portfolio of loans Clearly, the losses are not normally dis-tributed

“orig-Can we just ignore the deviation from normality as we do for equityportfolios? Unfortunately, we cannot and the reason is that credit portfoliomanagers are concerned with a different part of the distribution than arethe equity managers

Managers of equity portfolios are looking at areas around the mean.And it turns out that the errors you make by ignoring the deviations fromnormality are not very large In contrast, managers of credit portfolios fo-cus on areas in the tail of the distribution And out in the tail, very smallerrors in the specification of the distribution will have a very large impact

EXHIBIT 2.5 The Distribution of Equity Returns May Not Be Normal; but the Distribution of Losses for Loans Is Not Even Symmetric

So what does this mean? The preceding tells us that the mean and standarddeviation are not sufficient When we work with portfolios of credit assets,

we will have to collect some large data sets, or simulate the loss tions, or specify distributions that have long tails

distribu-Other Sources of Uncertainty

Working with portfolios of credit assets also leads to sources of uncertaintythat don’t occur in portfolios of equities

We noted previously that, for credit portfolios, we work with the tribution of losses rather than returns As is illustrated in Exhibit 2.6,

dis-WAYS THAT CREDIT MODELS INCORPORATE NONNORMALITY

In the discussion of Moody’s–KMV Credit Monitor®in Chapter 3 wesee that much of the technique is based on assuming a normal distrib-ution But we see that at the critical point where we need to go to adistribution to retrieve the probability of default, Credit Monitordoes not use normal distribution Instead, the probability of defaults

is obtained from a proprietary distribution created from actual lossdata; and this proprietary distribution is distinctly nonnormal

In Chapter 4, we see that Credit Risk+™ is based on a Poisson

distribution Why? Because a Poisson distribution will have the longright-hand tail that characterizes loss distributions for credit assets

In the other models we examine in Chapter 4, the loss tion is simulated By simulating the loss distribution, we can createdistributions that make sense for portfolios of credit assets

distribu-EXHIBIT 2.6 Additional Sources of Uncertainty

{Probability} {Expected}

Complicated function of firm, industry, and economy-wide variables

• Amount outstanding at time of default (“usage given default”)

• Expected loss given default (“severity” or “LGD”)

• Volatility of loss given default

losses are themselves dependent on two other variables Since probability

of default is a complex function of firm-specific, industry-wide, and omy level variables, this input will be measured with error In the case ofthe exposure at default, it depends on the amount outstanding at the time

econ-of default, the expected loss given default (or the inverse, recovery), andthe volatility of loss given default

Unlike equity portfolios, for portfolios of credit assets there is no rect way to estimate the covariance term—in this case, the covariance ofdefaults Because the vast majority of the obligors of interest have not de-faulted, we cannot simply collect data and calculate the correlation Conse-quently, much more subtle techniques will be required

di-Bad News and Good News about the Limit

of Diversification—Covariance

We have some bad news for you Look again at Exhibit 2.4 In the case ofequity portfolios, we note that a “fully diversified” portfolio can beachieved with a limited number of equities The number of assets needed

to create a “fully diversified” portfolio of loans or other credit assets ismuch larger It is certainly bigger than 100 assets and it may be largerthan 1,000 assets

But we have some good news for you as well The diversification fect for portfolios of loans or other credit assets will be larger than thediversification effect for portfolios of equities Remember that thebedrock risk—the risk that cannot be diversified away—is average co-variance As before, I find it easier to think about correlations than covariances, so, since both of them are telling me about the same thing,

ef-I switch and talk about correlation The typical correlation of equity returns is 20%–70% However, the typical correlation of defaults ismuch smaller—5% to 15% So the risk that cannot be diversified awaywill be smaller

The bad news is that it is going to take many more assets in the lio to achieve a “fully diversified” portfolio The good news is once youhave a “fully diversified” portfolio, you’re going to get a much larger di-versification effect

portfo-ELEMENTS OF THE CREDIT PORTFOLIO

Banks are currently the predominant users of credit portfolio ing The models are being used to accomplish a number of functions:

model-■ Calculation of economic capital

■ Allocation of credit risk capital to business lines

■ Supporting “active” management of credit portfolios through loansales, bond trading, credit derivatives, and securitization

■ Pricing transactions and defining hurdle rates

■ Evaluation of business units

■ Compensation of underwriters

Insurance companies are using credit portfolio models to:

■ Manage traditional sources of credit exposure

■ Guide the acquisition of new credit exposures—to date, mostly ment grade corporate credits—in order to provide diversification tothe core insurance business (Note: This has been accomplished pri-marily through credit derivatives subsidiaries.)

invest-Monoline insurers use credit portfolio models to:

■ Manage core credit exposure

■ Anticipate capital requirements imposed by ratings agencies

■ Price transactions and evaluate business units

Investors use credit portfolio models for:

■ Optimization of credit portfolios

■ Identification of mispriced credit assets

Exhibit 2.7 provides a way of thinking about the credit portfolio eling process Data gets loaded into a credit portfolio model, which out-puts expected loss, unexpected loss, capital, and the risk contributions forindividual transactions

mod-In Chapter 3, we describe the sources for the data We look at ways in which the probability of default for individual obligors andcounterparties can be estimated From the perspective of the facility, welook at sources for data on utilization and recovery in the event of de-fault And we examine the ways that the correlation of default is beingdealt with

To adapt the tenets of portfolio theory to loans, a variety of lio management models have come into existence Four of the most

portfo-widely discussed models are Moody’s–KMV Portfolio Manager™, the Metrics Group’s CreditManager™, CSFB’s Credit Risk+, and McKinsey’sCreditPortfolioView™ In Chapter 4, we describe the various creditportfolio models.

Risk-NOTE

1 The Statistics Appendix contains more detailed explanations of theseexpressions

EXHIBIT 2.7 Elements of the Credit Portfolio Modeling Process

Risk Cont

Exp Loss

Unexp