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### 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

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

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:

Saunders & Allen Chapters 10 & 11

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

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 =

Saunders & Allen Chapters 10 & 11

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

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.

Saunders & Allen Chapters 10 & 11

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

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

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

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

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

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

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

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

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

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

Saunders & Allen Chapters 10 & 11

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.

Saunders & Allen Chapters 10 & 11

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

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

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

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

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

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:

Saunders & Allen Chapters 10 & 11

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.

Saunders & Allen Chapters 10 & 11

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

Saunders & Allen Chapters 10 & 11

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

- A0 , AH , AMvaluations. Stochastic process generating AH, AM:
- The random component = systematic portion f + firm-specific portion u. Each simulation draws another risk factor.
- Using AH andAMcan calculate EDFH and EDFM

Saunders & Allen Chapters 10 & 11

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:

Saunders & Allen Chapters 10 & 11

Step 3: Aggregation to Construct Portfolio

- Sum the expected values VHfor all loans in the portfolio.

Saunders & Allen Chapters 10 & 11

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.

Saunders & Allen Chapters 10 & 11

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

- Using Table 11.8:

Saunders & Allen Chapters 10 & 11

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

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.

Saunders & Allen Chapters 10 & 11

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