Loan Portfolio Selection and
Risk Measurement
Chapters 10 and 11
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.
Saunders & Allen Chapters 10 & 11
2
Figure 10.1
The paradox of credit.
B
The Efficient
Frontier
Return
C
0
A
Risk
Saunders & Allen Chapters 10 & 11
3
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
Saunders & Allen Chapters 10 & 11
4
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:
r

R
p
f

p
Saunders & Allen Chapters 10 & 11
5
Figure 10.2
The optimum risky loan portfolio
B
D
Return (Rp )
rf
A
C
Risk (p )
Saunders & Allen Chapters 10 & 11
6
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
Saunders & Allen Chapters 10 & 11
7
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 = EDF (1 EDF )
– LGD variable, but independent across borrowers: ULi =
EDFi(1  EDFi) LGDi2  EDFiVOL2i
– 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 =
EDFi(1  EDFi) LGDi2  EDFiVVOL2i  (1  EDFi)VVOL2i
Saunders & Allen Chapters 10 & 11
8
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.
Saunders & Allen Chapters 10 & 11
9
Figure 11.1 Loan maturity (
0
M1
M) versus loan horizon (
M2  H
Saunders & Allen Chapters 10 & 11
H).
M3
Date
10
Correlations
• Figure 11.2 – joint PD is the shaded area.
• GF = GF/GF
• GF =
JDFGF  ( EDFG EDFF )
EDFG (1  EDFG ) EDFF (1  EDFF )
• Correlations higher (lower) if isocircles are
more elliptical (circular).
• If JDFGF = EDFGEDFF then correlation=0.
Saunders & Allen Chapters 10 & 11
11
Figure 11.2 Value correlation.
Market Value
of Assets - Firm G
Firm G
Market Value
of Assets - Firm F
Firm F
100(1-LGD)
100
Face Value of Debt
Firm F ’s
Debt Payoff
Saunders & Allen Chapters 10 & 11
12
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
Saunders & Allen Chapters 10 & 11
13
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).
Saunders & Allen Chapters 10 & 11
14
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.
Saunders & Allen Chapters 10 & 11
15
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.
Saunders & Allen Chapters 10 & 11
16
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.
Saunders & Allen Chapters 10 & 11
17
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%.
Saunders & Allen Chapters 10 & 11
18
Figure 11.3 The link betw een asset value volatility (
and rating transition for a BB rated borrow er.
Class:
Def
CCC B
BB
BBB
Transition Prob. (%): 1.06 1.00 8.84 80.53 7.73
Asset ():
2.30 2.04 1.23
1.37
Saunders & Allen Chapters 10 & 11
)
A
AA AAA
0.67 0.14 0.03
2.39 2.93 3.43
19
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.
Saunders & Allen Chapters 10 & 11
20
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.
Saunders & Allen Chapters 10 & 11
21
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.
Saunders & Allen Chapters 10 & 11
22
Credit VAR Measures
• Calculate the mean using the values in
Table 11.3 and the probabilities in Tab 11.1.
– Mean =
64
 pV
i 1
– Variance =
i i
64
p
i 1
i
(Vi  Mean ) 2
– Mean=$213.63 million
– Standard deviation= $3.35 million
Saunders & Allen Chapters 10 & 11
23
th
99
Calculating the
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.
Saunders & Allen Chapters 10 & 11
24
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.
Saunders & Allen Chapters 10 & 11
25
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.
Saunders & Allen Chapters 10 & 11
26
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.
Saunders & Allen Chapters 10 & 11
27
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.
Saunders & Allen Chapters 10 & 11
28
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.
Saunders & Allen Chapters 10 & 11
29
Figure 11.4 Credit limits and loan selection in CreditMetrics.
9
15
8
7
7
6
5
14
4
3
“Isoquant” Curve of
Equal Total Risk
 $70,000
16
13
6
2
9
5
1
12 10
20
1
0
0
2
4
6
8
8
18
10
12
14
16
Credit Exposure ($ Millions)
Saunders & Allen Chapters 10 & 11
30
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)]
Saunders & Allen Chapters 10 & 11
31
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:  = vVc + V
• Vc=common factor; V=idiosyncratic factor. As v0, corr0 As v1,
corr1.
• 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.
Saunders & Allen Chapters 10 & 11
32
Figure 11.5 Correlated default intensities.
Source: Duf f e and Singleton (1998), p.25.
The f igure shows a portion of a simulated sample path of total def ault arriv al
intensity (exactly 1,000 f irms). An X denotes a def ault ev ent.
150
100
50
0
2.2
2.4
2.6
2.8
3
Year
3.2
3.4
Marketw ide
Credit Event
Saunders & Allen Chapters 10 & 11
3.6
3.8
4
Calendar
Time
33
Figure 11.6 Portfolio default intended.
Source: Duf f e and Singleton (1998), p.27.
The f igure shows the probabilty of anm-day interv al within
10 y ears hav ing f our or more def aults (base case).
0.7
0.6
0.5
0.4
0.3
High Correlation
Medium Correlation
Low Correlation
0.2
0.1
0
0
10
20
30
40
50
60
70
80
90
100
Time Window m (Days)
Saunders & Allen Chapters 10 & 11
34
Appendix 11.1: Valuing a Loan that Matures
after the Credit Horizon – KMV PM
• Maturity=M3 in Figure 11.1.
• Four Step Process:
Use MTM to value loans.
– 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
Saunders & Allen Chapters 10 & 11
35
Step 1: Valuation of Firm Assets at
3 Time Horizons – Fig. 11.7
• A0 , AH , AM valuations. Stochastic process generating AH, AM:
ln AH = ln A0 + (-.52)tH + HtH
(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).
• 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
Saunders & Allen Chapters 10 & 11
36
Step 2: Loan Valuation Using
Term Structure of EDFs
• Convert EDF into QDF by removing risk-adjusted ROR.
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
• Also value loan as of credit horizon date H:
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
Saunders & Allen Chapters 10 & 11
(11.25)
37
Step 3: Aggregation to Construct Portfolio
• Sum the expected values VH for all loans in
the portfolio.
Vt P
= Vt i
(11.26)
i
where Vt P = the value of the loan portfolio at date t=0,H,
Vt i = the value of each loan i at date t=0,H.
Saunders & Allen Chapters 10 & 11
38
Step 4: Calculation of Excess Returns/Losses
• Excess Returns on the Portfolio:
RH
=
V HP  V0P
 RF
V0P
(11.27)
where RH = the excess return on the loan portfolio from time period 0 to
the credit
horizon date H,
V HP = the expected value of the loan portfolio at the credit
horizon date,
V0P = the present value of the loan portfolio,
RF = the riskfree rate.
• Expected Loss on the Portfolio:
ELH 
VH | ND  VH
V0
(11.28)
• Repeat steps 1 through 4 from 50,000 to 200,000 times.
Saunders & Allen Chapters 10 & 11
39
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
Cash
flows
per
period
(1)
1
2
3
4
5
Totals
(2)
.10
.10
.10
.10
1.10
Discount
Factor
e  tRF
Risk-free
Present
Value of
Cashflows
EDFi
QDFi
cumulative
cumulative
(3)
(2) x (3) = (4)
(5)
(6)
.9512
.9048
.8607
.8187
.7788
.0951
.0905
.0861
.0819
.8567
1.2103
.0100
.0199
.0297
.0394
.0490
.0203
.0471
.0770
.1088
.1414
Saunders & Allen Chapters 10 & 11
Risky
Present
Value of
Cashflows
(7)
.0932
.0862
.0795
.0730
.7356
1.0675
40
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
Cash
flows
per
period
(1)
1
2
3
4
5
Totals
(2)
.10
.10
.10
.10
1.10
Discount
Factor
e
 tRF
Risk-free
Present
Value of
Cashflows
EDFi
QDFi
cumulative
cumulative
(3)
(2) x (3) = (4)
(5)
(6)
1
.9512
.9048
.8607
.8187
0
.0951
.0905
.0861
.9006
1.1723
.0100
.0199
.0297
.0394
.0203
.0471
.0770
.1088
Saunders & Allen Chapters 10 & 11
Risky
Present
Value of
Cashflows
(7)
.0932
.0862
.0795
.8026
1.0615
41
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.
Saunders & Allen Chapters 10 & 11
42
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).
N
AB = (mAmB)1/2  AkBk(k/mk)2
(11.20)
k 1
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.
•
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.
Saunders & Allen Chapters 10 & 11
43