## 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') 