who wish to know how to maintain a rating; consequently the CAR is the value that drives the analysis.
Then the natural assumption is to say that the chance of the portfolio defaulting is simply the product of the individual probabilities, that is
Unfortunately this is wrong. It is wrong because it ignores correl- ation, which is very important because without redress we tend to over- estimate the chances of default and thus the amount of capital to be set aside. How then do we investigate correlation effects?
We employ a derivative of the Merton methodology which has wide- spread use among loan portfolio managers. An alternative would be to directly utilize default data or use secondary bond market spreads and imply the default rates. Either way proceed according to the discus- sion in Section 2.16. We offer a critique of these approaches and do not wish to re-peddle here.
The essence of the Merton type approach is to map the firm’s asset value onto the rating. The motivation behind this is that the correlation Probability of default chance AA defaults chance BBB defaults.
Figure 2.11 The possible states for a two loan portfolio.
Table 2.8 Migration rates.
Rating Outcome (%)
Default Value loss Unchanged Value gain
AA 0.01 11.66 87.76 0.57
BBB 0.37 NA* 95.24 4.39
Source:Standard and Poors, 2002.
* We have aggregate probabilities between investment grade and default partially for pedagogical purposes and further due to the lack of spread data on Euro denominated speculative grade credit.
between equities is easily measured and further based on a liquid marketplace. Figure 2.12 shows how the mapping occurs.
We know the transition probability from the agencies. However we also know the distribution of asset returns – it is a normal distribu- tion. Thus we just need to calibrate the distribution to the probabil- ities. We show the case of default below:
This will enable us to back out the equivalent threshold default return. You can use Excel to do this by calling
we also require this for the next step of isolating the loss of value probability:
Continuing on in this manner we end up with details as shown in Table 2.9.
For a one-dimensional normal distribution the returns scale by the volatility, consequently in order to get the actual return, we need to multiply by the volatility as displayed in the last column of Table 2.9.
We are not through because we must do the same for the AA borrower – this is not just handle grinding because we need to introduce correl- ation. This is again conceptually simple – you are probably happy with the one-dimensional distribution. Just think of this in two dimen- sions, Figure 2.13 illustrates this.
The total area under the graph is one and correlation determines the exact ‘shift’ away from the uniform hump about the origin. This is math- ematically easy to determine, simply by counting the area under the
Probability of decrease cumulative normal(return ) cumulative normal(return ).
decrease default
returndefault NORMSINV (0.37%),
Probability of default cumulative normal(return default).
Rating
Asset value
Figure 2.12 Mapping asset values to changes in state.
graph between the relevant states labelled on the axis – mathematically this comes down to determining the definite integral below:
where f(return1, return2, covariance) is the binormal density function.
This can be evaluated from a standard mathematical library, such as Mathematica™ for example. The bounds are set according to the state for which we need to derive the transition probability. After evalu- ating all the states we end up with the joint default probability table (Table 2.10).
Probability both loans default
(return , return , covariance) return return d(return d(return
1 2
return return
1 2 1 2
2 default2
1 default1
f
•
•
Ú
Ú
) ),
Table 2.9 Returns from probabilities (mapping a distribution to a rating BBB borrower).
State Probability from transition matrix (%) Return (%)
Value gain 4.39 1.71
Unchanged 95.24 1.71
Value loss NA NA
Default 0.37 2.67
⫺2
⫺2 0
0
2
2 0
0.1 0.15
Probability
Return AA (#stds.)
Return BBB (#s
tds.) 0.05
Figure 2.13 The joint-distribution probabilities.
Loan revaluation
We need to complete the analysis by valuing the loan for each of the states.
We can revalue using the spread from the simulated pricing curve for the appropriate rating. Drawn loans will be treated in the same manner as a floating-rate note, that is revaluing it in each future state by dis- counting the cash flows consisting of the loan margin at the appropri- ate discount rate implied from the pricing curve. We are also going to neglect the effect of any accrual on the loan for reasons of transparency and furthermore the standard methodology for evaluating risk is to take a ‘snapshot’ of market rates, either historical or projected, and see the effect on the portfolio while maintaining its maturity profile.
The loan commitment is a facility (which gives the obligor an option of borrowing up to the face amount). If the loan is not fully drawn it must be factored when attempting to revalue the loan. This character- istic introduces the need for further analysis because the borrower pays interest on the drawn amount and a fee which is different from the margin on the undrawn portion. Consequently when revaluing the loan in future states we have to know the amount currently drawn and the change due to the obligor’s rating migration.
During an adverse credit environment the borrower will usually draw more and vice versa in a more conducive market. (You would also expect the covenants to have some bearing on the loan valuation. In particular if the interest spread is related to the obligor’s performance then the effect would be to keep the loan near par and the only volatil- ity remaining would be that due to a potential default.) In our examples we assume that such covenants are not in existence and consequently that the loan can be characterized as having an FRN like sensitivity to credit risk. If the practice is ‘marking to market’ we deploy the VAR approach outlined in Section 1.12 on fixed income credit but substitut- ing the credit sensitivity from Section 5.3 on the credit risk of libor instruments, which we directly derived for the instrument.
Table 2.10 Joint migration probabilities.
BBB BBB BBB BBB
higher value maintained lower value defaults
AA higher value 0.07% 0.47% NA 0.00%
AA maintained 4.17% 83.50% NA 0.28%
AA lower value 0.22% 11.17% NA 0.10%
AA defaults 0.00% 0.01% NA 0.00%
An estimate of changes of drawdown given possible rating changes is taken from Arsarnow and Marker. They provide both the amount of commitment at a given credit rating and further the utilization in the event of default. Given this information it is possible to revalue the drawn portions. As the borrower changes in rating we require the fee structure to revalue the undrawn portion in the new credit state.
This is provided in Table 2.12. Table 2.11 illustrates the changes in the drawn portion of the loan for each of the states that the borrower may migrate to at the end of the period. In particular if the state is unchanged then the draw remains at 20 per cent. For the other states we need to determine the increment of the currently undrawn portion, that is 80 per cent that will be utilized. This is supplied from Table 2.12, where the last column reveals the utilization rate. Given this applies to the notional of a previously undrawn facility. We need to actually translate to a percentage on the currently undrawn portion of 80 per cent, the relevant percentage is 153.2/80 33.5 on a BBB–BB transition.
Subsequently multiply the undrawn percentage by this result to get the extra-utilization of 26.8 per cent.
We then go on to revalue the commitment for each of the year-end ratings.
Table 2.13 shows the final result for both changes in value due to changes in drawdown and the resulting differences in fee on the Table 2.11 Changes in drawdown for the BBB obligation.
Horizon state Current draw (%) Change (%) New draw (%)
AAA 20 19.9 0.1
AA 20 18.4 1.6
A 20 15.4 4.6
BBB 20 0.0 20.0
BB 20 26.8 46.8
CCC 20 55.0 75.0
D 20 60.0 80.0
Table 2.12 Fee and utilization structure.
Rating Fee Utilization (%)
AAA 3 0.1
AA 4 1.6
A 6 4.6
BBB 9 20.0
BB 18 46.8
B 40 63.7
CCC 120 75.0
remaining commitment. The relative spread is the difference in yield as determined from the relevant Bloomberg fair market curves as of 30 December 2002. The second column captures the impact of the revaluation on the new drawn amount (using a credit spread sensi- tivity of 0.035 on a €100 M notional). Finally we need to work out the fees on the new drawn and undrawn portion. We collect together the individual valuations and display these in the table. To determine the CAR on the portfolio we simply weight by the joint default probabilities.
We subsequently repeat the analysis for the AA and then collect the revaluation in each of the states in Table 2.14. To determine the expected value on the portfolio we simply weight the sum of the combined valuations in each final state by the joint probabilities of the portfolio ending in that state. This will also enable us to determine the standard deviation necessary for the CAR.
CAR for the portfolio
All we require having got the loan values and the probabilities of tran- sitioning is to calculate the unexpected loss. This will be given by Table 2.13 Revaluation for the BBB obligation.
State Relative Drawdown Fee change Fee change Total change spread (bps) revaluation (bps) (€in M) (€in M)
(€in M)
AAA 148 0.05 6 0.06 0.11
AA 132 0.74 5 0.05 0.79
A 95 1.53 3 0.03 1.56
BBB 0 0 0 0 0
D 200* (56.00) 111 (222.00) (56.22)
* Default is at 200 bps to the BBB curve, assume fees are representative of the CCC grade.
Table 2.14 Portfolio revaluation.
BBB BBB BBB BBB
higher value maintained lower value defaults
AA higher value* 1.53 M 0.02 M NA (56.21 M)
AA maintained 1.51 M 0 NA (56.22 M)
AA lower value* (2.34 M) (3.85 M) NA (60.08 M)
AA defaults (91.68 M) (93.19 M) NA (149.41 M)
* We consider the higher and lower states to be a weighted combination of the rating states and consequently have used the marginal transition probabilities for the relevant outset rating.
a scaling of the volatility, which is given by the standard formula:
For a 95 per cent loss the CAR is determined as 3.11 per cent.
The 100 asset portfolio
Enough of the toy model. Let us analyse a proper commercial loan portfolio.
The profile of the loans is displayed above. The curve is comprised of the payouts consisting of both interest and redemption amounts, as the loans are medium to long dated, initially the bulk of the payments will be comprised of interest and then as we move towards maturity, redemption will become dominant. The majority of the loans were based on a floating-coupon structure; this means every year the obligor has to pay the relevant index rate – typically libor and a fixed margin which is dependent on the credit worthiness of the obligor (Figure 2.14).
There are approximately 100 loans under consideration with a nom- inal of 10 million. They were denominated in euros predominantly within the financial sector. This is the same example we have discussed in Section 4.9.
We require again the volatility of the losses. This was determined through a Monte Carlo simulation of the asset return. We assume that each loan is driven, as in the Merton type approach encountered in our two asset portfolio, by a purely firm-dependent factor but also a systematic component which represents the general economic envir- onment in which the borrower operates. This is a type of beta model.
We worked through these points in the example in Section 4.10. To apply this model to evaluate regulatory capital we determined the
Unexpected loss 1.65
1.65 probability (return average return) .2
states
Â
0 50 000 000 100 000 000 150 000 000
0 5 10 15 20
Payout (euros)
Maturity (months)
Commercial loan profile
Figure 2.14 The profile of the 100 loan portfolio.
number of defaults per face value of the portfolio for a given level of cor- relation at the 1-year horizon. The economic capital is proportional to the average rating of the collateral within the overall portfolio. We took the default probabilities from the transition matrix in Table 1.19 on fixed-income credit, together with a global recovery rate of 50 per cent.
The economic capital was determined to a 99 per cent confidence.
Figure 2.15 illustrates the outcome of various concentration scen- arios vs. the quality of the loan portfolio. As you can see it is very sensitive to the level of correlation within the portfolio.