The Math of Modification NPV Analysis

The purpose of this post is to illustrate, with a simple example, the mathematics behind loan modification decisions. Also, it takes a first crack at filling a void in the practical mortgage mathematics literature. (See the Calculated Risk "Ubernerd" posts for more on the mechanics of modifications.) The overall message of this post is that there are numerous “moving parts” and assumptions that undermine the often assumed conclusion that modifying delinquent loans is essentially a no-brainer.

The basic mathematics of loan modifications

The expected present value (PV) of an unmodified loan (PVNoMod) is a default probability-weighted average of its recovery values in foreclosure (FCRecovery), and the discounted value of the payments on mortgages that do not foreclose (PVOld):

PVNoMod = FCProb x FCRecovery + (1 – FCProb) x PVOld

Typical loan loss models assume that, without meaningful modifications, about 70 percent of loans that are seriously delinquent will end up foreclosed (FCProb = 70 percent).

After accounting for distress sale discounts and the various expenses of the foreclosure process, including legal fees, property maintenance and real estate commissions, and payment arrears (interest, insurance and taxes), the FCRecovery value might be expected to be between 60 and 70 percent. (According to the FDIC “Mod in a Box” spreadsheet, before considering home price depreciation during the foreclosure process, recovery values range from about 60 percent for homes valued at around $150,000 to 70 to 75 percent for homes worth $200,000 and over.)

The expected present value of a modified loan (PVMod) is a redefault probability-weighted average of its recovery value if it redefaults (RedefProb), and the discounted value of the payments on the new loan (PVNew):

PVMod = RedefProb x FCRecovery + (1 – RDefProb) x PVNew

Historically, redefault rates have ranged from 40 to 50 percent, although some evidence suggests the redefault rate on modifications that result in significant (e.g., 10 percent) payment reductions are lower (e.g., 30 percent). Also, homeowners with positive equity in their homes (i.e., the value of the house exceeds outstanding mortgage loans) are less likely to default. Hence the RDefProb variable might be expected to vary from 30 to 50 percent, with more aggressive modifications being at the lower end of the range.

Complicating things somewhat is the choice of the appropriate discount rate, and accounting for the impact of prepayments. The FDIC “Mod in a Box” spreadsheet uses the Freddie Mac Weekly Rate, which seems lower than what servicers should be using on behalf of their clients. The FDIC spreadsheet also ignores prepayment risk by taking all PV calculations out to the loan’s ultimate 30 or 40 year amortization term.

Example of the modification math in action

Suppose that a homeowner earning $3,669 per month is funding a $250,000 home with a $250,000 8.00 percent 30-year mortgage. The monthly mortgage payment of $1,834 implies a debt-to-income (DTI) ratio of 50 percent, which would have to be reduced to $1,137 per month to get down to a 31 percent DTI target. A “traditional” modification might consider reducing the interest rate from 8.00 to 3.61 percent.

If the house value has declined to $150,000, the recovery value on an immediate foreclosure (FCRecovery) is $105,000, and using an 8.00 percent discount rate, the value of the mortgage if it remains current (PVOld) is $250,000. A foreclosure probability of 70 percent implies an expected PV (PVNoMod) of $148,500, in this case, regardless of the horizon over which it is calculated.

The expected PV of the interest rate modification (PVMod) over a 30-year horizon is $130,000 with a redefault probability (ReDefProb) of 50 percent and the 8.00 percent discount rate, implying that the modification is not worth doing from the lender’s viewpoint. However, over a five-year horizon, PVMod is $155,938, implying that the modification is worthwhile.

If the discount rate is increased to 14 percent, all of the PV components associated with loans that cure decrease, and even the five-year horizon analysis becomes a closer call. PVNoMod falls from $148,500 to $132,703, and PVMod from $155,938 to $132,944.

High discount rates also favor Hope for Homeowners (H4H) modifications in extreme home price depreciation scenarios such as this one, because it locks in the PV, in this case at $140,408 – 96.5 percent of the $150,000 home value, minus the three percent up-front insurance.

Of course, this is vastly simplified, as a close look at the FDIC “Mod in a Box” spreadsheet would reveal. For example, it contains a much deeper analysis of recovery values, and incorporates a much richer array of modifications. On the other hand, it hopefully shows that the modification decision is by no means a simple one.