Mortgage Burnout and Selection Effects in Heterogeneous Cox Hazard Models

Mortgage Bur nout and Selection Effects in Heter ogeneous Cox Hazard Models Andrew Lesniewski Department of Mathematics Baruch College One Bernard Baruch W ay Ne w Y ork, NY 10010 USA March 20, 2026 Abstract W e study the aggregate hazard rate of a heterogeneous population whose indi vidual e v ent intensities are modeled as Cox (doubly stochastic) processes. In the deterministic hazard setting, the observ ed pool hazard is the surviv al weighted mean of the indi vidual hazards, and its time deri v ati ve equals the mean individual hazard drift minus a variance term. This yields a transparent structural explanation of b urnout in mortgage pools. W e extend this perspective to stochastic intensity models. The observed pool hazard remains a survi v al-weighted mean, b ut no w e volv es as an Ito process whose drift contains the mean drift of the individual hazards and a negativ e selection term driv en by cross-sectional dispersion, together with a diffusion term inherited from the common factor . W e formulate the general identity and discuss special cases relev ant to mortgage prepayment modeling. 1 2 A. Lesnie wski 1 Intr oduction A standard feature of mortgage prepayment modeling is the b urnout ef fect : conditional on a fa v orable re- financing environment, the observed prepayment rate of a mortgage pool tends to decline with seasoning because the most refinance sensiti ve borro wers exit first. This phenomenon is widely documented in the mortgage-backed securities literature [3], [5], [1]. Empirically , burnout is one of the most robust features of mortgage prepayment data. The standard explanation is borrower heterogeneity . Borrowers differ in trans- action costs, financial sophistication, mobility , credit constraints, and other factors af fecting their propensity to refinance. When interest rates decline, borrowers with the strongest refinancing incentiv es tend to prepay first, lea ving behind a population with progressi vely lo wer propensity to refinance. As a result, the observed pool prepayment rate declines ov er time e ven if the aggre gate refinancing incentiv e remains unchanged. In heterogeneous deterministic hazard models this mechanism admits a simple structural representation, d dt ¯ λ ( t ) = E t ( ˙ λ ( t, x )) − Va r t ( λ ( t, x )) , where ¯ λ ( t ) is the observed pool hazard and E t , Va r t are computed under the survi v al weighted cross- sectional distribution of borro wer types. The identity abov e has an interesting conceptual interpretation. The e volution of the pool hazard can be vie wed as a selection effect in a heterogeneous population. Borrowers with higher hazard rates tend to exit the pool earlier , so that the surviving population becomes progressi vely enriched in lower -hazard types. This mechanism is mathematically analogous to the Price equation from ev olutionary biology [4], which decomposes the change in the mean v alue of a trait into a selection component and a transmission compo- nent. In its simplest form, the Price equation states that the change in the population mean of a trait z i satisfies ∆ ¯ z = Cov ( w i , z i ) ¯ w + 1 ¯ w E ( w i ∆ z i ) , where w i denotes fitness. In the present context, the role of the trait is played by the borro wer hazard λ ( x ) , while the role of fitness is played by survi val. The negati v e variance term in the b urnout identity is precisely the continuous-time analog of the covariance selection term in the Price equation. Thus mortgage burnout can be interpreted as a form of natural selection within a heterogeneous population of borrowers: high-hazard borrowers prepay first, lea ving behind a population with progressi vely lo wer av erage hazard. Importantly , the burnout effect therefore arises generically in heterogeneous hazard models and does not require behavioral assumptions about borro wer refinancing decisions. The purpose of this note is to formulate a stochastic analog of this identity when each borrower -le vel hazard is itself a Cox process. 2 Mortgage P ool Heter ogeneity Let (Ω , F , ( F t ) t ≥ 0 , P ) be a filtered probability space describing the time ev olution of the pool of mortgages, and let ( X, X , F ) be a measurable type space representing the unobserved borrower heterogeneity . For each x ∈ X , let λ t ( x ) be a nonne gati ve F t -adapted intensity process. T ypically , λ t depends e xplicitly on a number of loan characteristics y , such as refinance incenti ve (the difference between the coupon on the loan and the current mortgage rate), size, age, loan to value ratio, FICO score, geographic region, which represent the observed heterogeneity . Unless stated otherwise, we will suppress this dependence for the sake of notational con venience. Mortgage Burnout and Selection Effects in Heterogeneous Hazard Models 3 Conditionally on the filtration, the survi v al process of a borro wer of type x is S t ( x ) = exp  − Z t 0 λ s ( x ) ds  . The aggregate survi val process of the pool is ¯ S t = Z X S t ( x ) F ( dx ) . Definition 2.1. The observed pool hazar d ¯ λ t is defined by ¯ S t = exp  − Z t 0 ¯ λ s ds  . Equivalently , ¯ λ t = − d dt log ¯ S t . Since dS t ( x ) = − λ t ( x ) S t ( x ) dt, it follo ws that ¯ λ t = R X λ t ( x ) S t ( x ) F ( dx ) R X S t ( x ) F ( dx ) . Definition 2.2. Define the survival weighted cr oss-sectional measur e F t ( dx ) = S t ( x ) F ( dx ) R X S t ( y ) F ( dy ) . F or any integr able function ϕ ( x ) , E t ( ϕ ) = Z X ϕ ( x ) F t ( dx ) , is the expected value with r espect to F t and V a r t ( ϕ ) is the variance with r espect to F t . T aking the time deriv ativ e of log ¯ S t , we find that ¯ λ t = E t ( λ t ( x )) . (1) The surviv al weighted measure F t e volv es ov er time because borro wers with higher hazard rates exit the pool earlier . This induces a selection effect in e xpectations taken under F t . Let ϕ t ( x ) be a semimartingale inde xed by borrower type x . A direct differentiation of E t ( ϕ t ) = R X ϕ t ( x ) S t ( x ) F ( dx ) R X S t ( x ) F ( dx ) yields the identity d E t ( ϕ t ) = E t ( dϕ t ) − Cov t ( λ t ( x ) , ϕ t ( x )) dt. (2) This identity may be viewed as a continuous-time analog of the Price equation from ev olutionary biol- ogy , which decomposes the change in the population mean of a trait into a direct evolution component and a selection component. In the present setting, borrower survi val plays the role of fitness, and the co variance term captures the selecti ve remo v al of high-hazard borrowers from the pool. 4 A. Lesnie wski 3 Deterministic Hazards If each borro wer hazard λ ( t, x ) is differentiable in time, differentiation yields the following deterministic burnout identity . Theorem 3.1. Suppose eac h borr ower hazard λ ( t, x ) is differ entiable in time . Then the pool hazard satisfies d dt ¯ λ ( t ) = E t ( ˙ λ ( t, x )) − Va r t ( λ ( t, x )) . (3) The second term represents a selection ef fect: borro wers with higher hazard rates exit the pool earlier, reducing the av erage hazard of the survi ving population. In particular , if individual hazards are constant in time, ˙ λ ( t, x ) = 0 , then d dt ¯ λ ( t ) = − V a r t ( λ ( t, x )) ≤ 0 . (4) Thus heterogeneity alone produces a monotone decline in the observed pool hazard. 4 Stochastic Hazards 4.1 Common F actor Hazard Dynamics Suppose each borro wer hazard follo ws an Ito process dλ t ( x ) = µ t ( x ) dt + σ t ( x ) dW t . (5) Idiosyncratic borro wer shocks diversify in large pools and therefore do not contrib ute to the aggregate hazard dynamics. Since the pool hazard satisfies ¯ λ t = E t ( λ t ( x )) , the selection identity yields the follo wing stochastic generalization of the burnout formula. Proposition 4.1. Under the common factor dynamics above, the pool hazar d evolves accor ding to d ¯ λ t =  E t ( µ t ( x )) − Va r t ( λ t ( x ))  dt + E t ( σ t ( x )) dW t . (6) The drift contains the same negati ve v ariance term − Va r t ( λ t ( x )) as in the deterministic b urnout iden- tity . Thus heterogeneity generates a structural downw ard pressure on the observed pool hazard e ven when indi vidual hazards follo w stochastic dynamics. Similar selection effects arise in heterogeneous credit hazard models, where dispersion in firm lev el default intensities leads to declining a verage hazard among survi ving firms. 4.2 Measure Change Inter pr etation The surviv al weighted distrib ution F t can also be interpreted as a change of measure induced by the survi val process. Define the weighting factor H t ( x ) = S t ( x ) ¯ S t . Mortgage Burnout and Selection Effects in Heterogeneous Hazard Models 5 Then expectations under the survi val weighted distrib ution satisfy E t ( ϕ ) = Z X ϕ ( x ) H t ( x ) F ( dx ) . Thus F t may be vie wed as a Radon-Nikodym tilt of the original type distribution F with density proportional to surviv al. From this perspectiv e the b urnout identity arises because the weighting factor H t ( x ) ev olves according to d dt log H t ( x ) = − λ t ( x ) + ¯ λ t . Consequently the change in expectations under the tilted measure contains a co v ariance term − Cov t ( λ t ( x ) , ϕ t ( x )) , which represents the selecti ve remo v al of high hazard borrowers from the population. This representation shows that b urnout can be interpreted as a measure change phenomenon: the ob- served pool dynamics correspond to expectations taken under a surviv al biased distribution of borrower types. In other words, the cross-sectional distribution of borro wer types observed in a seasoned mortgage pool is endogenously tilted tow ard lower hazard borrowers. Burnout therefore arises as a purely statistical selection ef fect rather than a behavioral change in borro wer refinancing incentiv es. 5 Frailty F actor Models A commonly used mortgage specification is the fr ailty model [6], [2]. In mortgage applications the frailty factor captures unobserved borrower characteristics such as refinancing costs, mobility , financial sophistica- tion, or credit constraints. It assumes that borrower hazards f actor into a borro wer specific frailty parameter and a common stochastic intensity λ 0 t ( y ) depending on the observ ed features y , λ t ( x, y ) = f ( x ) λ 0 t ( y ) , where f ( x ) > 0 represents borrower heterogeneity . Then ¯ λ t ( y ) = λ 0 t ( y ) E t ( f ( x )) , and V a r t ( λ t ( x, y )) = λ 0 t ( y ) 2 V a r t ( f ( x )) . The b urnout dynamics are therefore governed by the ev olution of the survi val weighted distrib ution of f ( x ) . W e now illustrate these ideas with sev eral explicit e xamples of the distribution F . In mortgage appli- cations the gamma and lognormal frailty models are particularly natural. The gamma model is analytically tractable and produces a hyperbolic b urnout profile, while the lognormal model arises naturally from multi- plicati ve borro wer characteristics such as refinancing incentiv e, credit quality , and mobility factors. 5.1 Gamma Frailty The gamma frailty model is widely used in surviv al analysis because it admits closed form expressions. Suppose the initial frailty distribution is f ∼ Gamma( k, θ ) , 6 A. Lesnie wski with mean k θ and variance kθ 2 . If λ 0 t ( y ) = λ is constant, then S t ( f ) = exp( − λf t ) . The survi v al weighted distribution remains g amma, f | survi val at t ∼ Gamma  k , θ 1 + θ λt  . Hence E t ( f ) = k θ 1 + θ λt . The observed hazard becomes ¯ λ t = λ k θ 1 + θ λt . Thus burnout produces a h yperbolic decay of the pool hazard. Proposition 5.1. Under gamma frailty with constant common factor λ , the heter ogeneous Cox population is observationally equivalent to a deterministic hazar d model with time-varying pool hazard ¯ λ t = ¯ λ 0 1 + θ λt . (7) 5.2 Lognormal Frailty Suppose f = exp( Y ) , Y ∼ N ( µ, σ 2 ) . Then the survi v al weighted distribution satisfies F t ( d f ) ∝ exp( − λf t ) 1 f σ √ 2 π exp  − (log f − µ ) 2 2 σ 2  d f . Closed-form expressions are not a vailable, b ut the pool hazard ¯ λ t = λ E t ( f ) can be e v aluated numerically . For small dispersion σ , a Laplace approximation yields ¯ λ t ≈ λ exp  µ + σ 2 2  exp  − σ 2 λt  , which produces approximately exponential b urnout. 5.3 Normal Frailty Strictly speaking, frailty v ariables in surviv al models are required to be nonnegati ve, since they scale the baseline hazard. F or this reason the gamma and lognormal distrib utions are commonly used in applications. The normal distrib ution does not satisfy this positi vity constraint and is therefore not appropriate as a struc- tural frailty model. Nev ertheless, it is useful as a simple analytic benchmark illustrating ho w cross-sectional dispersion generates burnout. Mortgage Burnout and Selection Effects in Heterogeneous Hazard Models 7 For illustration we therefore also consider the normal distrib ution. f ∼ N ( m, s 2 ) , with f > 0 implicitly assumed. Then ¯ λ t = λ R f e − λf t ϕ ( f ) d f R e − λf t ϕ ( f ) d f . For small v ariance s 2 , a second-order expansion gi ves ¯ λ t ≈ λ  m − s 2 λt  , sho wing a linear burnout ef fect for short horizons. 5.4 Multivariate Frailty In man y mortgage applications borrower heterogeneity is driven by sev eral unobserv ed f actors rather than a single scalar frailty variable. A natural extension is therefore a multivariate frailty model [2], in which the borro wer hazard takes the form λ t ( x, y ) = f ( x ) ⊤ λ 0 t ( y ) , where f ( x ) = ( f 1 ( x ) , . . . , f k ( x )) ⊤ represents a v ector of borro wer-specific frailty f actors and λ 0 t ( y ) is a vector of common factor intensities that may depend on the observed loan characteristics y . Examples of such latent factors include borrower mobility , refinancing transaction costs, credit constraints, and other behavioral characteristics that are dif ficult to observe directly b ut influence the propensity to prepay . Under this specification the pool hazard becomes ¯ λ t ( y ) = E t ( f ( x )) ⊤ λ 0 t ( y ) , where E t ( f ( x )) denotes the surviv al weighted mean frailty vector . The burnout dynamics are therefore gov erned by the ev olution of the surviv al weighted distribution of the multiv ariate frailty factors. Applying the deterministic burnout identity yields d dt ¯ λ t ( y ) = E t ( ˙ λ t ( x, y )) − λ 0 t ( y ) ⊤ Cov t ( f ( x )) λ 0 t ( y ) , where Cov t ( f ( x )) is the survi v al weighted covariance matrix of the frailty v ector . Thus in the multiv ariate setting the b urnout effect is dri ven by the cross-sectional co v ariance structure of borro wer frailties. Directions in factor space with larger dispersion contrib ute more strongly to the decline of the pool hazard as high frailty borro wers exit the pool. This formulation sho ws that mortgage burnout can be interpreted as a selection effect operating on a multidimensional latent factor structure for borrower behavior . 5.5 Connection with Cox Proportional Hazard Models The representations considered abov e can be vie wed as a particular parameterization of Cox proportional hazard models. In the classical Cox framew ork, the hazard is written as λ t ( y ) = λ base t exp  β ⊤ y t  , where λ base t is a baseline hazard and y t denotes observ able cov ariates. 8 A. Lesnie wski In the present setting, the roles of baseline hazard and cov ariates may be interpreted dif ferently . The frailty component λ t ( x ) , or more generally f ( x ) in the factor specification, can be viewed as a borro wer- specific random modification of the baseline hazard, while the common factor λ 0 t ( y ) plays the role of a time-v arying systematic component dri ven by observ able characteristics. Under this interpretation, heterogeneous Cox intensity models correspond to Cox-type specifications with random, possibly multidimensional, baseline heterogeneity . The burnout ef fect then arises from the endogenous ev olution of the cross-sectional distribution of these latent baseline components under surviv al weighting. This perspectiv e highlights that burnout is not tied to a particular functional form of the hazard, but rather reflects a general selection mechanism operating in Cox-type models with latent heterogeneity . 6 Conclusion The observed hazard of a heterogeneous Cox population remains a survi val-weighted average of borro wer hazards. Under common-factor stochastic dynamics, the observed pool hazard satisfies d ¯ λ t =  E t ( µ t ( x )) − Va r t ( λ t ( x ))  dt + E t ( σ t ( x )) dW t . The negati ve v ariance term provides a structural stochastic analog of burnout: heterogeneity induces a sys- tematic downward drift in the observed hazard through selection ef fects. The resulting formula may be interpreted as a stochastic continuous time analog of the Price equation, showing that burnout arises from a uni versal selection mechanism in heterogeneous hazard populations. 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