Intensity Model

In­ten­sity mod­els (also called re­duced form mod­els) are a form of de­fault model. Lets start by con­sid­er­ing what are known as mor­tal­ity mod­els of de­fault. These are es­sen­tially a dis­crete form of in­ten­sity model. Once we have es­tab­lished them, we will take a limit as time in­ter­vals go to zero—and out will pop in­ten­sity mod­els.

Mor­tal­ity mod­els de­rive their name from their sim­i­lar­ity to ac­tu­ar­ial mod­els of human mor­tal­ity. De­fine a sur­vival func­tion s(t). It might in­di­cate the prob­a­bil­ity of a human sur­viv­ing until age t or the prob­a­bil­ity of a bond sur­viv­ing with­out de­fault for t years. For the rest of this ar­ti­cle, we shall use it to de­note the lat­ter.

The prob­a­bil­ity of a bond de­fault­ing in year t + 1 is given by

This is an un­con­di­tional prob­a­bil­ity. It re­flects the prob­a­bil­ity at time 0 (when the bond is is­sued) of de­fault be­tween time t and time t + 1. If we want the con­di­tional prob­a­bil­ity of de­fault—that is, the prob­a­bil­ity of de­fault be­tween time t and time t + 1, con­di­tional on sur­vival to time t—we apply Bayes’ the­o­rem to ob­tain

A sur­vival func­tion can be con­structed from his­tor­i­cal bond de­fault data. Con­structed in this man­ner, the sur­vival func­tion de­fines a mor­tal­ity model of de­fault. Ex­hibit 1 in­di­cates em­pir­i­cal sur­vival func­tions by orig­i­nal credit qual­ity. If the num­bers were smoothed, it could rea­son­ably be used to spec­ify a mor­tal­ity model for de­fault.

Ex­hibit 1

Now, in­stead of con­sid­er­ing a one-year time in­ter­val, let’s con­sider an ar­bi­trary time in­ter­val Δt. Gen­er­al­iz­ing [2], the prob­a­bil­ity of de­fault be­tween time t and time t + Δt, con­di­tional on their being no de­fault by time t, is

We can ex­press this as an av­er­age rate of de­fault by di­vid­ing by the time in­ter­val Δt:

Con­sider an ex­am­ple. Let t be three years. As­sume s(5) = 0.8921 and s(8) = 0.8609. Then the con­di­tional prob­a­bil­ity of de­fault be­tween years 5 and 8 is

This is a prob­a­bil­ity of de­fault over a three-year pe­riod. By [4], we con­vert it to an av­er­age an­nual rate of de­fault by di­vid­ing by 3. The re­sult is an av­er­age rate of .0117 de­faults per year over the three-year pe­riod.

To ob­tain an in­stan­ta­neous rate of de­fault f (t) at any time t, we take the limit as t goes to 0 in [4]:

where s’(t) is the first de­riv­a­tive of s with re­spect to t.

The in­stan­ta­neous rate of de­fault f (t) is called the de­fault in­ten­sity or, to bor­row a word from in­sur­ance, the haz­ard rate. In­ten­sity mod­els work by as­sum­ing some func­tional form for f (t) and then cal­i­brat­ing that to cur­rent in­ter­est rate spreads. f (t) can re­flect “real” prob­a­bil­i­ties to sup­port credit risk man­age­ment ap­pli­ca­tions. It can re­flect risk neu­tral prob­a­bil­i­ties to sup­port fi­nan­cial en­gi­neer­ing ap­pli­ca­tions. The sur­vival func­tion is re­cov­ered from f (t) by re­ar­rang­ing [8] and in­te­grat­ing:

From this, prob­a­bil­i­ties of de­fault can be ob­tained from [1] or [2] as ap­pro­pri­ate.

Note how f(t) plays a role sim­i­lar to that of a con­tin­u­ously com­pounded in­ter­est rate in [9]. Use of de­fault in­ten­si­ties tends to sim­plify math­e­mat­ics, which is one rea­son in­ten­sity mod­els are pop­u­lar with fi­nan­cial en­gi­neers.

Alt­man (1989), Asquith, Mullins and Wolff (1989) and Alt­man and Sug­gitt (2000) dis­cuss mor­tal­ity mod­els of de­fault. The first pub­lished in­ten­sity model ap­pears to be Jar­row and Turn­bull (1995). Sub­se­quent re­search in­cludes Duffie and Huang (1996), Jar­row, Lando and Turn­bill (1997) and Duffie and Sin­gle­ton (1997a, 1997b).

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