MARCO LEGAL

Previous sections have shown how bootstrapped estimates of default probabilities capture statistical noise; Table 4.8 shows that the 99% upper bound for the default probability of BBB is 1.36 times the best estimate from the benchmark matrix. Empirical evidence has been presented that shows that downward momentum is (statistically) significant, with probabilities of default of excited states higher than

the non-excited states in the same rating category. Lastly, approaches to capturing time-varying effects of the rating process show how real-world default probabilities are linked to the real economy.

Based on these observations it is straightforward to hypothesize that VaR esti- mates using bootstrapped matrices will have higher VaR than the non-bootstrapped counterparts, that the presence of downward momentum leads to higher VaR and that attempts to capture time-inhomogeneity lead to higher VaR estimates. Ul-

timately, the interest lies with the extent to which VaR estimates differ from the

benchmark.

The Monte Carlo model is parametrized as follows; simulating a portfolio using

100 bonds (N = 100) of 100 notional amount, a maturity date in 6 years (T = 6), a

coupon rate of 5% paid annually and a fixed annualized risk-free rate of 2% for all maturities. An asset correlation of 0.1998 is used, as estimated by Zeng and Zhang

(2001). Both D¨ullmann et al. (2007) and Lopez (2004) arrive at similar estimates

for asset correlations for medium to large US corporations. Recovery rates are

simulated from a beta distribution with mean of 0.476 and standard deviation of 0.229 (Altman and Kishore, 1996) in the event of a default.

For the benchmark matrix, the bootstrapped estimates of the benchmark ma- trix and the migration matrix with an extended state space, the CIR processes are projected forward one year from their September 2015 historical value in monthly

time steps5_{. For the simulations challenging the time-homogeneity assumptions, no}

stochastic scalar is used and instead the focus lies with illustrating how the distri- bution of rating classes varies from the benchmark using the real world transition dynamics.

Based on 25,000 simulations, the 1-year VaRα=99% of the benchmark matrix

(Table 4.3) is -4.01%, which will be the value against which the VaR values of competing models based of different migration matrices are compared in Figure 4.12.

5_{Please note that two CIR processes have been calibrated; one based on the benchmark matrix} (used in the benchmark en bootstrapped benchmark case) and a process calibrated using the matrix with an extended state space.

Benchmark Statistical Uncertainty Downward Momentum 100 110 120 130 140 150 160 % VaR of Benchmark Model

Figure 4.12: VaR differences from benchmark matrix for simulation with standard param- eters.

Figure 4.12 illustrates clearly how ignoring statistical uncertainty of estimated migration process impacts the simulated risk of the credit risky portfolio. With levels

ofV aRα=99%21% higher than the benchmark, this result is economically significant.

Since the statistical uncertainty is straightforward to quantify given the underlying ratings events, and given that running simulations using varying realisations of the same underlying process is straightforward, the results are easily obtained which makes constructing a simulation that accounts for statistical uncertainty a realistic option on a real world commercial setting. The difference in VaR levels is as hy- pothesized for the rating process that attempts to incorporate rating history under a new Markov process; the VaR of the ‘momentum-aware’ transition model is 49% higher than the benchmark. Using a model based on bootstrapped samples of the ‘momentum-aware’ matrix, thereby accounting for statistical uncertainty in the es- timation of the transition matrix with downward momentum, yields, as expected, an ever larger difference in Value-at-Risk of 83%.

For the matrices that have been conditioned on the state of the economy (GDP growth), the regime is allowed to switch on a monthly basis, projecting forward 20 years using the transition matrix in Table 4.9, estimated from the event data in

Figure 4.6 (right).

From / To High Growth Normal Growth Low Growth

High Growth 0.285 0.653 0.062

Normal Growth 0.064 0.847 0.092

Low Growth 0.043 0.736 0.221

Table 4.9: Quarterly transition matrix for three economic regimes, defined by GDP growth.

In each of the simulated months, ratings are projected forward according to the rating migration matrix that corresponds to the regime. Rather than look at Value- at-Risk measures, the focus of the investigation is on the real-world distribution of simulated ratings and how it varies between a model where the benchmark matrix is used and a model where the transition matrix depends on the projected state of the economy. Both simulations start with 1000 bonds of credit quality AA and for simulation years 1, 5, 10 and 20, rating membership is observed for each of the 25,000 simulations. Recording the percentage of bonds in each rating category, for each simulation, gives, at each point in time, a distribution of rating membership for each of the ratings. For instance, using the benchmark model, it could turn out that in year five, a mean percentage of 70% of bonds still labelled A-rated, and, for instance, the 5th quantile is 45% and the 95th quantile is 84%. These numbers are

only for illustrative purposes6_{, ultimately the interest lies in thedifferences between}

the values in the benchmark model and the economy-dependent model, rather than the observed values. Figure 4.13 shows how the difference in 5th quantile and 95th quantile of the rating distributions for the economy-dependent simulation compare to those of the benchmark model, expressed in terms of the benchmark model.

6_{Note that the expectation of the resulting distribution could easily obtained by taking powers} of the 1-year migration matrices.

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 50 75 100 125 150 5 10 15 20 Years ahead % of Benchmar k Model Quantile ● ● 5th 95th Rating AAA B BBB

Figure 4.13: Relative distribution of rating membership at various years ahead, for the benchmark model and the economy dependent model

Figure 4.13 shows how the 5th and 95th quantile of the rating distributions (AAA, BBB and B) are different, at various years ahead, for the benchmark model and the economy dependent model. The results are expressed as a percentage of the benchmark model. For instance, simulation 5 years ahead, the 5th quantile for AAA is only 74% of the benchmark model indicating a lower 5th quantile, which indicates there is the potential for fewer AAA-rated bonds using the economic-dependent model. Figure 4.13 shows that relative to the benchmark, in general, the distribution

of rating membership is wider for the economy-dependent model. There is both

the potential for (far) more, or (far) less rating changes in the economy-dependent model; a result of the randomness in the economic growth that is simulated which can be very different from the average case. Table 4.9 illustrates how GDP growth appears to be highly mean reverting with the highest transition probabilities always towards the medium growth. Evidence of this tendency for mean reversion can be seen in Figure 4.13 as well. The difference between the benchmark model and the economy-dependent model appears to decrease with simulation time, indicating that over longer time period it is rather unlikely that one would experience, say, twenty

years of low growth7.