Loan Portfolio Selection and Risk Measurement

Loan Portfolio Selection and Risk Measurement

Chapters 10 and 11

Saunders & Allen Chapters 10 & 11 2

The Paradox of Credit

• Lending is not a “buy and hold”process.

• To move to the efficient frontier, maximize return for any given level of risk or equivalently, minimize risk for any given level of return.

• This may entail the selling of loans from the portfolio. “Paradox of Credit” – Fig. 10.1.


Return

The EfficientFrontier

A

B

C

Risk0

Figure 10.1 The paradox of credit.


Managing the Loan Portfolio According to the Tenets of Modern Portfolio Theory

• Improve the risk-return tradeoff by:– Calculating default correlations across assets.– Trade the loans in the portfolio (as conditions

change) rather than hold the loans to maturity.– This requires the existence of a low transaction

cost, liquid loan market.– Inputs to MPT model: Expected return, Risk

(standard deviation) and correlations


The Optimum Risky Loan Portfolio – Fig. 10.2

• Choose the point on the efficient frontier with the highest Sharpe ratio:– The Sharpe ratio is the excess return to risk

ratio calculated as:

p

fp rR


Return (Rp)rf

A

BD

C

Risk (p)

Figure 10.2 The optimum risky loan portfolio


Problems in Applying MPT to Untraded Loan Portfolios

• Mean-variance world only relevant if security returns are normal or if investors have quadratic utility functions.– Need 3rd moment (skewness) and 4th moment

(kurtosis) to represent loan return distributions.

• Unobservable returns– No historical price data.

• Unobservable correlations


KMV’s Portfolio Manager

• Returns for each loan I:– Rit = Spreadi + Feesi – (EDFi x LGDi) – rf

• Loan Risks=variability around EL=EGF x LGD = UL– LGD assumed fixed: ULi = – LGD variable, but independent across borrowers: ULi =

– VOL is the standard deviation of LGD. VVOL is valuation volatility of loan value under MTM model.

– MTM model with variable, indep LGD (mean LGD): ULi =

)1( EDFEDF

22)1( ii EDFiVOLLGDEDFiEDFi

222 )1()1( iii VVOLEDFiEDFiVVOLLGDEDFiEDFi


Valuation Under KMV PM

• Depends on the relationship between the loan’s maturity and the credit horizon date:

• Figure 11.1: DM if loan’s maturity is less than or equal to the credit horizon date (maturities M1 or M2).

• MTM if loan’s maturity is greater than credit horizon date (maturity M3). See Appendix 11.1 for valuation.


0

Figure 11.1 Loan maturity ( M) versus loan horizon ( H).

M1 M2 H M3Date


Correlations

• Figure 11.2 – joint PD is the shaded area.GF = GF/GF

GF =

• Correlations higher (lower) if isocircles are more elliptical (circular).

• If JDFGF = EDFGEDFF then correlation=0.

)1()1(

)(

FFGG

FGGF

EDFEDFEDFEDF

EDFEDFJDF


Firm F

Firm G

Firm F’sDebt Payoff

100

100(1-LGD)

Market Valueof Assets - Firm G

Market Valueof Assets - Firm F

Face Value of Debt

Figure 11.2 Value correlation.


Role of Correlations• Barnhill & Maxwell (2001): diversification can reduce

bond portfolio’s standard deviation from $23,433 to $8,102.

• KMV diversifies 54% of risk using 5 different BBB rated bonds.

• KMV uses asset (de-levered equity) correlations, CreditMetrics uses equity correlations.

• Correlation ranges:– KMV: .002 to .15– Credit Risk Plus: .01 to .05– CreditMetrics: .0013 to .033


Calculating Correlations using KMV PM• Construct asset returns using OPM.• Estimate 3-level multifactor model. Estimate coefficients and then

evaluated asset variance and correlation coefficients using:• First level decomposition:

– Single index model – composite market factor constructed for each firm.

• Second level decomposition:– Two factors: country and industry indices.

• Third level decomposition:– Three sets of factors: (1) 2 global factors (market-weighted index of

returns for all firms and return index weighted by the log of MV); (2) 5 regional factors (Europe, No. America, Japan, SE Asia, Australia/NZ); (3) 7 sector factors (interest sensitive, extraction, consumer durables, consumer nondurables, technology, medical services, other).


CreditMetrics Portfolio VAR

• Two approaches:– Assuming normally distributed asset values.– Using actual (fat-tailed and negatively skewed)

asset distributions.

• For the 2 Loan Case, Calculate:– Joint migration probabilities– Joint payoffs or loan values– To obtain portfolio value distribution.


The 2-Loan Case Under the Normal Distribution

• Joint Migration Probabilities = the product of each loan’s migration probability only if the correlation coefficient=0.– From Table 10.1, the probability that obligor 1

retains its BBB rating and obligor 2 retains it’s a rating would be 0.8693 x 0.9105 = 79.15% if the loans were uncorrelated. The entry of 79.69% suggests a positive correlation of 0.3.


Mapping Ratings Transitions to Asset Value Distributions

• Assume that assets are normally distributed.• Compute historic transition matrix. Figure 11.3

uses the matrix for a BB rated loan.• Suppose that historically, there is a 1.06%

probability of transition to default. This corresponds to 2.3 standard deviations below the mean on the standard normal distribution.

• Similarly, if there is a 8.84% probability of downgrade from BB to B, this corresponds to 1.23 standard deviations below the mean.


Joint Transition Matrix

• Can draw a figure like Fig. 11.3 for the A rated obligor. There is a 0.06% PD, corresponding to 3.24 standard deviations below the mean; a 5.52% probability of downgrade from A to BBB, corresponding to 1.51 std dev below the mean.

• The joint probability of both borrowers retaining their BBB and A ratings is: the probability that obligor 1’s assets fluctuate between –1.23 to +1.37 and obligor 2’s assets between –1.51 to +1.98 with a correlation coefficient=0.2. Calculated to equal 73.65%.


Class:Transition Prob. (%):Asset ():

Def1.06 2.30

CCC1.00 2.04

B8.84 1.23

BB80.53

BBB7.731.37

A0.672.39

AA0.142.93

AAA0.033.43

Figure 11.3 The link between asset value volatility ( )and rating transition for a BB rated borrower.


Calculating Correlation Coefficients

• Estimate systematic risk of each loan – the relationship between equity returns and returns on market/industry indices.

• Estimate the correlation between each pair of market/industry indices.

• Calculate the correlation coefficient as the weighted average of the systematic risk factors x the index correlations.


Two Loan Example of Correlation Calculation

• Estimate the systematic risk of each company by regressing the stock returns for each company on the relevant market/industry indices.

• RA = .9RCHEM + UA

• RZ = .74RINS + .15RBANK + UZ

A,Z=(.9)(.74)CHEM,INS + (.9)(.15)CHEM,BANK

• Estimate the correlation between the indices.• If CHEM,INS =.16 and CHEM,BANK =.08, then

AZ=0.1174.


Joint Loan Values

• Table 11.1 shows the joint migration probabilities.• Calculate the portfolio’s value under each of the

64 possible credit migration possibilities (using methodology in Chap.6) to obtain the values in Table 11.3.

• Can draw the portfolio value distribution using the probabilities in Table 11.1 and the values in Table 11.3.


Credit VAR Measures

• Calculate the mean using the values in Table 11.3 and the probabilities in Tab 11.1.– Mean =

– Variance =

– Mean=$213.63 million – Standard deviation= $3.35 million

ii

iVp

64

12

64

1

)( MeanVp ii

i


Calculating the 99th percentile credit VAR under normal

distribution

• 2.33 x $3.35 = $7.81 million

• Benefits of diversification. The BBB loan’s credit VAR (alone) was $6.97million. Combining 2 loans with correlations=0.3, reduces portfolio risk considerably.


Calculating the Credit VAR Under the Actual Distribution

• Adding up the probabilities (from Table 11.1) in the lowest valuation region in Table 11.3, the 99th percentile credit VAR using the actual (not normal) distribution is $204.4 million.

• Unexpected Losses=$213.63m - $204.4m = $9.23 million (>$7.81m).

• If the current value of the portfolio = $215m, then Expected Losses=$215m - $213.63m = $1.37m.


CreditMetrics with More Than 2 Loans in the Portfolio

• Cannot calculate joint transition matrices for more than 2 loans because of computational difficulties: A 5 loan portfolio has over 32,000 joint transitions.

• Instead, calculate risk of each pair of loans, as well as standalone risk of each loan.

• Use Monte Carlo simulation to obtain 20,000 (or more) possible asset values.


Monte Carlo Simulation• First obtain correlation matrix (for each pair of loans)

using the systematic risk component of equity prices. Table 11.5

• Randomly draw a rating for each loan from that loan’s distribution (historic rating migration) using the asset correlations.

• Value the portfolio for each draw.• Repeat 20,000 times! New algorithms reduce some

of the computational requirements.• The 99th% VAR based on the actual distribution is

the 200th worst value out of the 20,000 portfolio values.


MPT Using CreditMetrics

• Calculate each loan’s marginal risk contribution = the change in the portfolio’s standard deviation due to the addition of the asset into the portfolio.

• Table 11.6 shows the marginal risk contribution of 20 loans – quite different from standalone risk.

• Calculate the total risk of a loan using the marginal contribution to risk = Marginal standard deviation x Credit Exposure. Shown in column (5) of Table 11.6.


Figure 11.4

• Plot total risk exposure using marginal risk contributions (column 6 of Table 11.6) against the credit exposure (column 5 of Table 11.4).

• Draw total risk isoquants using column 5 of Table 11.6.

• Find risk outliers such as asset 15 which have too much portfolio risk ($270,000) for the loan’s size ($3.3 million).

• This analysis is not a risk-return tradeoff. No returns.


0

9

8

7

6

5

4

1

2

3

0

Credit Exposure ($ Millions)

141210

“Isoquant” Curve ofEqual Total Risk

$70,000

864

15

7

14

13

6

16

5

12 10

9

20 1 188

2 16

Figure 11.4 Credit limits and loan selection in CreditMetrics.


Default Correlations Using Reduced Form Models

• Events induce simultaneous jumps in default intensities.• Duffie & Singleton (1998): Mean reverting correlated

Poisson arrivals of randomly sized jumps in default intensities.

• Each asset’s conditional PD is a function of 4 parameters: h (intensity of default process); (constant arrival prob.); k (mean reversion rate); (steady state constant default intensity).

• The jumps in intensity follow an exponential distribution with mean size of jump=J.

• So: probability of survival from time t to s:

p(t,s) = exp{(s-t)+(s-t)h(t)} where (t) = -(1 – e-kt)/k (t) = -[t + (t)] – [/(J+k)][Jt – ln(1 - (t)J)]


Numerical Example

• Suppose that =.002, k=.5, =.001, J=5, h(0)=.001 (corresponds to an initial rating of AA).

• Correlations across loan default probabilities:

• Vc=common factor; V=idiosyncratic factor. As v0, corr0 As v1, corr1.

• If v=.02, V=.001, Vc=.05: the probability that loani intensity jumps given that loanj has experienced a jump is = vVc/(Vc+V) = 2%. If v= .05 (instead of .02), then the probability increases to 5%.

• Figure 11.5 shows correlated jumps in default intensities.

• Figure 11.6 shows the impact of correlations on the portfolio’s risk.

= vVc + V


150

100

50

0

MarketwideCredit Event

Year

Source: Duffe and Singleton (1998), p.25.The figure shows a portion of a simulated sample path of total default arrivalintensity (exactly 1,000 firms). An X denotes a default event.

CalendarTime

3.43.2 3.83.632.82.62.42.2 4

Figure 11.5Correlated default intensities.


0

0.7

0.5

0.6

0.3

0.4

0.1

0.2

0

Time Windowm (Days)

70

High CorrelationMedium CorrelationLow Correlation

60 90805040302010 100

Source: Duffe and Singleton (1998), p.27.The figure shows the probabilty of anm-day interval within10 years having four or more defaults (base case).

Figure 11.6Portfolio default intended.


Appendix 11.1: Valuing a Loan that Matures after the Credit Horizon – KMV PM

• Maturity=M3 in Figure 11.1. Use MTM to value loans.• Four Step Process:

– 1. Valuation of an individual firm’s assets using random sampling of risk factors.

– 2. Loan valuation based on the EDFs implied by the firm’s asset valuation.

– 3. Aggregation of individual loan values to construct portfolio value.

– 4. Calculation of excess returns and losses for portfolio.

• Yields a single estimate for expected returns (losses) for each loan in the portfolio. Use Monte Carlo simulation (repeated 50,000 to 200,000 times) to trace out distribution


Step 1: Valuation of Firm Assets at 3 Time Horizons – Fig. 11.7

• A0 , AH , AM valuations. Stochastic process generating AH, AM:

• The random component = systematic portion f + firm-specific portion u. Each simulation draws another risk factor.

• Using AH and AM can calculate EDFH and EDFM

ln AH = ln A0 + (-.52)tH + HtH (11.21)

where AH = the asset value at the credit horizon date H, = the expected return (drift term) on the asset valuation,

= the volatility of asset returns, tH = the credit horizon time period,

H = a random risk term (assumed to follow a standard normal distribution).


Step 2: Loan Valuation Using Term Structure of EDFs

• Convert EDF into QDF by removing risk-adjusted ROR.

• Also value loan as of credit horizon date H:

V0 = PV0(1 – LGD) + PV0(1-QDF)LGD (11.22) where V0 = the loan’s present value,

PV0 = the present value factor using the riskfree rate to discount the loan’s cash flows to time t=0, QDF = the (cumulative) risk neutral quasi-EDF, LGD = the loss given default

VH|ND = CH + PVH(1 – LGD) + PVH(1-QDF)LGD (11.23)

where VH|ND = the loan’s expected value as of the credit horizon date given that default has not occurred,

CH = the cash flow on the credit horizon date, PVH = the present value factor using the riskfree rate as the discount factor to discount the loan’s cash flows to time t=H.

However, there is a possibility that the loan will default on or before the credit horizon date. The expected value of the loan given default is:

VH|D = (CH + PVH)LGD (11.24)

VH = (EDF) VH|D + (1-EDF) VH|ND (11.25)


Step 3: Aggregation to Construct Portfolio

• Sum the expected values VH for all loans in the portfolio.

PtV =

i

itV (11.26)

where PtV = the value of the loan portfolio at date t=0,H, itV = the value of each loan i at date t=0,H.


Step 4: Calculation of Excess Returns/Losses

• Excess Returns on the Portfolio:

• Expected Loss on the Portfolio:

• Repeat steps 1 through 4 from 50,000 to 200,000 times.

HR = FP

PPH RV

VV

0

0 (11.27)

where RH = the excess return on the loan portfolio from time period 0 to the credit horizon date H, P

HV = the expected value of the loan portfolio at the credit horizon date,

PV0 = the present value of the loan portfolio, RF = the riskfree rate.

0

|

V

VVEL HNDHH

(11.28)


A Case Study: KMV PM valuation of 5 yr maturity $1 loan paying a fixed rate of 10% p.a.

• Using Table 11.8:V0 = PV0(1 – LGD) + PV0(1-QDF)LGD = 1.2103(.50) + (1.0675)(.50) = $ 1.1389

Table 11.8 Valuing the Loan’s Present Value

Time Period

(1)

Cash flows per

period

(2)

Discount Factor

FtRe

(3)

Risk-free Present Value of

Cashflows (2) x (3) = (4)

EDFi cumulative

(5)

QDFi cumulative

(6)

Risky Present Value of

Cashflows (7)

1 .10 .9512 .0951 .0100 .0203 .0932 2 .10 .9048 .0905 .0199 .0471 .0862 3 .10 .8607 .0861 .0297 .0770 .0795 4 .10 .8187 .0819 .0394 .1088 .0730 5 1.10 .7788 .8567 .0490 .1414 .7356

Totals 1.2103 1.0675


Valuing the Loan at the Credit Horizon Date =1• Using Table 11.9:

VH|ND = CH + PVH(1 – LGD) + PVH(1-QDF)LGD = 0.10 + 1.1723(.50) + (1.0615)(.50) = $ 1.2169

VH|D = (CH + PVH)LGD = (0.10 + 1.1723)(.50) = $ 0.63615

VH = (EDF) VH|D + (1-EDF) VH|ND = (.01)(.63615) + (.99)(1.2169) = $ 1.2111

Time Period

(1)

Cash flows per

period

(2)

Discount Factor

FtRe

(3)

Risk-free Present Value of

Cashflows (2) x (3) = (4)

EDFi cumulative

(5)

QDFi cumulative

(6)

Risky Present Value of

Cashflows (7)

1 .10 1 0 2 .10 .9512 .0951 .0100 .0203 .0932 3 .10 .9048 .0905 .0199 .0471 .0862 4 .10 .8607 .0861 .0297 .0770 .0795 5 1.10 .8187 .9006 .0394 .1088 .8026

Totals 1.1723 1.0615


KMV’s Private Firm Model

• Calculate EBITDA for private firm j in industryj.• Calculate the average equity mulitple for industryi

by dividing the industry average MV of equity by the industry average EBITDA.

• Obtain an estimate of the MV of equity for firm j by multiplying the industry equity multiple by firm j’s EBITDA.

• Firm j’s assets = MV of equity + BV of debt• Then use valuation steps as in public firm model.


Credit Risk Plus Model 2 - Incorporating Systematic Linkages in Mean Default rates

• Mean default rate is a function of factor sensitivities to different independent sectors (industries or countries).

• Table 11.7 shows as example of 2 loans sensitive to a single factor (parameters reflect US national default rates). As credit quality declines (m gets larger), correlations get larger.

AB = (mAmB)

1/2

N

k1

AkBk(k/mk)2 (11.20)

where AB = default correlation between obligor A and B, mA = mean default rate for type A obligor, mB = mean default rate for type B obligor, A = allocation of obligor A's default rate volatility across N sectors, B = allocation of obligor B's default rate volatility across N sectors, (k/mk)2 = proportional default rate volatility in sector k.

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Loan Portfolio Selection and Risk Measurement

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