# NPV of a mortage loan

I need to model the expected NPV of a mortage loan over his whole life-time. Assume that only the prepayment and default risk matters and that these events can occour at only discrete time-points. I'm not shure whethere to use conditional probabilities - cunditioned on having survived up to the payment date or "unconditional" ones.

Option 1: $E(NPV)=\sum_t \frac{1}{(1+r)^t} ( P(T=t,prepayment) CF^{pre}+ P(T=t,default) CF^{def}+ P(T=t,regular) CF^{reg})$

Option 2:

$E(NPV)=\sum_t \frac{1}{(1+r)^t} ( P(T=t,prepayment|T>=t) CF^{pre}+ P(T=t,default|T>=t) CF^{def}+ P(T=t,regular|T>=t) CF^{reg})$

Where T denotes the stochastic variable, modelling the discrete timepoints at which the payments are expected.

Do you have a reference/textbook on which option to choose?

Or do I even miss an option?

A few initial observations but the quick answer is your Option 2.

(1) Assuming both prepayment and default can occur only at discrete time-points is not strictly correct since a borrower has the right to payoff the loan in full at any time between payment dates. (2) Default is a nebulous concept - are you referring to the act of missing a payment itself, or when the loan is ultimately declared as charged-off with some recovery depending on net proceeds after liquidation costs.

Industry practice is to model the conditional probability of a missed payment with recovery proceeds received after a pre-specified liquidation lag. It is also convention to assume that NO cures for missed payments occur. The conditional probability of prepayment is applied to the prior month's balance (after scheduled amortization). Combining these terms with the contractual cash flows (and adjusting the "performing" balance appropriately every period) should allow you to compute actual principal and actual interest which discounted back using the appropriate discount rate should give you the NPV.

Empirically, hazard models estimated on actual data on performance, i.e., panel data sets for each loan where you can estimate multinomial models for a loan being in one of 3 mutually exclusive states (alive = 0, prepaid = 1, charged-off = 2) [these can also be estimated using pair-wise binomial logits] are used as inputs into the cash flow formulas laid out earlier.

• Thanks for the quick and detailed reply! Concering your (1) point - since I only do have strictily monthly updates, I think a discrete model is appropriate. Concerning (2): Sorry for the "simplification" I know that I should introduce the LGD and the EAD, since I mean the ultimately declared charged-off status when using the term "default". So you would propose to choose option (2)? Do you have any reference for that? – Jogi Sep 4 '17 at 19:20
• Regarding discrete data - you are correct in that empirically the 'event' is observed only at discrete times, the correct framework for accounting for this is using grouped-hazard models which allow you to recover continuous-time estimates. This is useful if you ever want to scale up (or down) the time periods and need a new hazard rate for that. As far as a reference, please refer to the following for additional details regarding cash flows: sifma.org/wp-content/uploads/2017/08/chsf.pdf – HookahBoy Sep 4 '17 at 19:25
• @HookahBoay Thanks for the reference and the clarification. I recently read this paper [link]onlinelibrary.wiley.com/doi/10.1111/j.1540-6229.2006.00166.x/… which showed that a multinomial logit model with a random intercept outperforms hazard models. – Jogi Sep 4 '17 at 20:05