Mortgage Loan NPV with Prepayment and Default Hazards
Article Quant Q&A · Author: Jogi
Summary
The document explains how to project a mortgage’s expected cash flows when prepayment and default can end or alter scheduled payments. It favors using conditional event probabilities at each period, conditional on the loan remaining active up to that point. Those probabilities are then combined with scheduled amortization, prepayment cash flows, and default recoveries to derive period cash flows, which are discounted to calculate net present value.
Key ideas
- Conditional prepayment and default probabilities apply to loans still active at the relevant period.
- Prepayment can occur between scheduled payment dates, so discrete event timing is a modeling simplification.
- Default modeling should distinguish a missed payment from a later charge-off and recovery.
- A common framework models missed payments with a liquidation lag and assumes no cures.
- Hazard models fitted to loan performance data can estimate probabilities of remaining active, prepaying, or charging off.
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Full text
# NPV of a mortage loan
# 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?
## Answer by HookahBoy (score 1, accepted)
https://quant.stackexchange.com/a/35909
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.