# conditional probability of default

I would like to ask the following question.

I would appreciate if someone could help me out.

On what argument is based that states that conditional default rates ( loans of corporate borrowers) tend to decrease as time passes. are there any statistical research done on given issue.

$Variant 1 \qquad \qquad \qquad \qquad \quad Y1 \qquad Y2 \qquad Y3 \qquad Y4 \qquad Y5 \\ Cumulative \quad PDs \quad \quad \qquad 20 \% \qquad 30\% \qquad 38\% \quad 42\% \qquad 44\% \\ Conditional \quad PDs \qquad \qquad 20\% \qquad 13\% \qquad 9\% \qquad 4\% \qquad 2\% \quad \\ Formula=\frac{CumPD_{i}-CumPD_{i-1}}{1-CumPD_{i-1}}$

$Variant 2 \qquad \qquad \qquad \qquad \quad Y1 \qquad Y2 \qquad Y3 \qquad Y4 \qquad Y5 \\ Cumulative \quad PDs \quad \quad \qquad 20 \% \qquad 30\% \qquad 42\% \quad 55\% \qquad 70\% \\ Conditional \quad PDs \qquad \qquad 20\% \qquad 13\% \qquad 14\% \quad 15\% \qquad 18\% \quad \\ Formula=\frac{CumPD_{i}-CumPD_{i-1}}{1-CumPD_{i-1}}$

I presented two variants. The first one is with decreasing conditional probabilities. The second one is with increasing conditional probabilities. So the question was why the the first variant is in compliance with properties of conditional default rates where the second is not.

• Welcome, Oliver! It would be easier to answer your question if you provide some referene or even formulas ... – Ric Feb 26 '18 at 16:26
• Thank you very much Richard. Sorry for delay in response. – Oliver Mar 16 '18 at 12:35
• you should put this into the question (press edit) and use latex ... :) – Ric Mar 16 '18 at 12:54
• please: you have to edit the question. I guess noone is willing to follow these comments ... edit the queston and make it clear ... ok? – Ric Mar 19 '18 at 9:01
• thank you very much for your advise. I appreciate it. right now i am trying to do it properly but it will take a while. – Oliver Mar 19 '18 at 9:10

Let's assume a firm that maximizes returns for its shareholders. The firm can distribute dividends $D_t$, invest $I_t$, and borrow long term debt $B_{t}$.

Let's assume its production function depends on how much capital $k_t$ and labor $l_t$ and on the current productivity level $A_t$. Further let $w_t$ be the wage bill.

This firm solves the following problem:

$$Max_{k_{t+1}, l_t, B_{t+1}} \sum^\infty_{t=0} F(A_t, K_t, L_t) - (K_{t+1}-(1-\delta)K_t) - w_t l_t - M_t(B_t) + B_{t+1}$$

Where $M_t(B_t)$ is the payment due on the current outstanding long-term debt $B_t$ and $B_{t+1}$ is new issued debt. Also assume that equity issuances are prohibitively costly and that the firm defaults if $D_t < 0$.

This is an extremely complex problem to solve, but intuitively if you solve the problem you will have conditional probabilities of default that decrease with the level of leverage $B_t$. The reason is straighforward, as debt matures, leverage is lower for the firm and less likely it is to default (the cummulative PD is still increasing trivially).

There are a few papers with similar models. One that comes to my mind, is Gomes, Jermann and Schmid (2016). Take a look at figure 2 of that paper.

In Panel H you see that as time goes by the firm is deleveraging. In Panel G you see that default probability decreases as the firm deleverages. The cumulative PD is still increasing.

Empirically this is a pattern that is hard to see, because in practice as time goes by firms are decreasing some of their corporate loans but taking new loans at the same time. I do not know of any empirical paper that isolates the effect of time to maturity and default. The reason for this is probably that no one questions that the decreasing pattern of defaults is true as time goes by. Any structural model with long-term debt such as the one I outlined above will deliver that result.

• I would like to ask a question regarding presented equation: it does not contain $D_{t}$ and $I_{t}$. Should they be incorporated as well? In addition for what states $K_{t+1}-(1-\delta)*K_{t}$ (i mean intuition). – Oliver Mar 20 '18 at 6:50
• it turns out that $I_{t} = K_{t+1} - (1-\delta) * K_{t}$. – Oliver Mar 20 '18 at 9:55
• @Oliver, that is correct. You can either maximize the above or replace with $I_t$ and add a lagrange multiplier. Dividends are given by $D_t = F(A_t, K_t, L_t) - (K_{t+1}-(1-\delta)K_t) - w_t l_t - M_t(B_t) + B_{t+1}$ as there is no way in the model above for firms to save cash other than investing in capital. – phdstudent Mar 20 '18 at 10:01