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## Credit Quality Thresholds

### Introduction

An equivalent way to represent transition probabilities is by transforming them into credit quality thresholds. These are critical values of a standard normal distribution that yield the same transition probabilities.

An M-by-N matrix of transition probabilities TRANS and the corresponding M-by-N matrix of credit quality thresholds THRESH are related as follows. The thresholds THRESH(i,j) are critical values of a standard normal distribution z, such that

TRANS(i,N) = P[z < THRESH(i,N)],

TRANS(i,j) = P[z < THRESH(i,j)] - P[z < THRESH(i,j+1)], for 1<=j<N

Financial Toolbox supports the transformation between transition probabilities and credit quality thresholds with the functions transprobtothresholds and transprobfromthresholds.

### Compute Credit Quality Thresholds

To compute credit quality thresholds, transition probabilities are required as input. Here is a transition matrix estimated from credit ratings data:

load Data_TransProb
trans = transprob(data)
trans =

93.1170    5.8428    0.8232    0.1763    0.0376    0.0012    0.0001    0.0017
1.6166   93.1518    4.3632    0.6602    0.1626    0.0055    0.0004    0.0396
0.1237    2.9003   92.2197    4.0756    0.5365    0.0661    0.0028    0.0753
0.0236    0.2312    5.0059   90.1846    3.7979    0.4733    0.0642    0.2193
0.0216    0.1134    0.6357    5.7960   88.9866    3.4497    0.2919    0.7050
0.0010    0.0062    0.1081    0.8697    7.3366   86.7215    2.5169    2.4399
0.0002    0.0011    0.0120    0.2582    1.4294    4.2898   81.2927   12.7167
0         0         0         0         0         0         0  100.0000

Convert the transition matrix to credit quality thresholds using transprobtothresholds:

thresh = transprobtothresholds(trans)

thresh =

Inf   -1.4846   -2.3115   -2.8523   -3.3480   -4.0083   -4.1276   -4.1413
Inf    2.1403   -1.6228   -2.3788   -2.8655   -3.3166   -3.3523   -3.3554
Inf    3.0264    1.8773   -1.6690   -2.4673   -2.9800   -3.1631   -3.1736
Inf    3.4963    2.8009    1.6201   -1.6897   -2.4291   -2.7663   -2.8490
Inf    3.5195    2.9999    2.4225    1.5089   -1.7010   -2.3275   -2.4547
Inf    4.2696    3.8015    3.0477    2.3320    1.3838   -1.6491   -1.9703
Inf    4.6241    4.2097    3.6472    2.7803    2.1199    1.5556   -1.1399
Inf       Inf       Inf       Inf       Inf       Inf       Inf       Inf

Conversely, given a matrix of thresholds, you can compute transition probabilities using transprobfromthresholds. For example, take the thresholds computed previously as input to recover the original transition probabilities:

trans1 = transprobfromthresholds(thresh)

trans1 =

93.1170    5.8428    0.8232    0.1763    0.0376    0.0012    0.0001    0.0017
1.6166   93.1518    4.3632    0.6602    0.1626    0.0055    0.0004    0.0396
0.1237    2.9003   92.2197    4.0756    0.5365    0.0661    0.0028    0.0753
0.0236    0.2312    5.0059   90.1846    3.7979    0.4733    0.0642    0.2193
0.0216    0.1134    0.6357    5.7960   88.9866    3.4497    0.2919    0.7050
0.0010    0.0062    0.1081    0.8697    7.3366   86.7215    2.5169    2.4399
0.0002    0.0011    0.0120    0.2582    1.4294    4.2898   81.2927   12.7167
0         0         0         0         0         0         0  100.0000

### Visualize Credit Quality Thresholds

You can graphically represent the relationship between credit quality thresholds and transition probabilities. Here, this example shows the relationship for the 'CCC' credit rating. In the plot, the thresholds are marked by the vertical lines and the transition probabilities are the area below the standard normal density curve:

load Data_TransProb
trans = transprob(data);
thresh = transprobtothresholds(trans);

xliml = -5;
xlimr = 5;
step = 0.1;
x=xliml:step:xlimr;
thresCCC = thresh(7,:);
labels = {'AAA','AA','A','BBB','BB','B','CCC','D'};

centersX = ([5 thresCCC(2:end)]+[thresCCC(2:end) -5])*0.5;
omag = round(log10(trans(7,:)));
omag(omag>0)=omag(omag>0).^2;
fs = 14+2*omag;

figure
plot(x,normpdf(x),'LineWidth',1.5)
text(centersX(1),0.2,labels{1},'FontSize',fs(1),...
'HorizontalAlignment','center')
for i=2:length(labels)
val = thresCCC(i);
line([val val],[0 0.4],'LineStyle',':')
text(centersX(i),0.2,labels{i},'FontSize',fs(i),...
'HorizontalAlignment','center')
end
xlabel('Credit Quality Thresholds')
ylabel('Probability Density Function')
title('{\bf Visualization of Credit Quality Thresholds}')
legend('Std Normal PDF','Location','S')


The second plot uses the cumulative density function instead. The thresholds are represented by vertical lines. The transition probabilities are given by the distance between horizontal lines.

figure
plot(x,normcdf(x),'m','LineWidth',1.5)
text(centersX(1),0.2,labels{1},'FontSize',fs(1),...
'HorizontalAlignment','center')
for i=2:length(labels)
val = thresCCC(i);
line([val val],[0 normcdf(val)],'LineStyle',':');
line([x(1) val],[normcdf(val) normcdf(val)],'LineStyle',':');
text(centersX(i),0.2,labels{i},'FontSize',fs(i),...
'HorizontalAlignment','center')
end
xlabel('Credit Quality Thresholds')
ylabel('Cumulative Probability')
title('{\bf Visualization of Credit Quality Thresholds}')
legend('Std Normal CDF','Location','W')


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