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We propose a real option framework to value distressed properties and restructure their loans. Our approach reconciles the interests of borrowers and lenders through a constrained optimization model yielding mutually beneficial restructure terms. Borrowers receive lower loan balances and payments, while lenders replace non-performing loans with performing loans that have higher market values. A numerical illustration shows that the market value of a restructured loan can exceed that of the original non-performing loan and the post-foreclosure cash flows when the lender repossesses the property.

Many studies have applied option theory to real estate investment, abandonment, and timing decisions over the past three decades. [

The precipitous drop in home loan market values coupled with increased reserve requirements has necessitated loan restructuring. However lenders are often unwilling to restructure loans. We suggest one reason lenders are apprehensive is that they are unaware of the related issues of current home valuation and optimum restructuring terms. [

To illustrate the potential benefits to borrowers and lenders, consider the average market-to-book values for performing and non-performing loans, 83% and 35%, respectively. The restructured loan market value will exceed the non-performing loan market value as long as the restructured principal balance is greater than

The lender can increase the market value of the non-performing loans 66% by restructuring into performing loans. Of course there are many other choice variables for the lender such as the loan rate and term. The trio of principal reduction percentage, loan rate, and loan term are used as the lender’s choice variables in our model that follows.

For the purposes of this study, a “distressed” property is a property where a homeowner has ceased repayment due an “underwater” condition: the current loan balance exceeds the current market value of the home. Our

model first implementsthe underwater condition by setting the real option value to the borrower to zero, and then solving for the current market value of the home S_{0} (“stock price” in standard options nomenclature). Given the current distressed property market value S_{0} and existing loan amount K_{0}, the lender chooses the principal reduction amount a, new interest rate r, and new loan term T that maximizes the value of the now-performing loan

We make several simplifying assumptions for model tractability. These assumptions are intentionally restrictive in that we do not intend for this model to apply to all non-performing loans. However, loans that meet the criteria identified in the following assumptions do represent a subset of all non-performing loans.

Assumption 1: The non-performing loan represents a negative equity scenario where the real option value on the home is equal to zero. Admittedly, it is possible that a loan is non-performing for reasons other than negative equity. The borrower may have experienced unemployment or under-employment recently or an investor may walk away from a property in which mortgage payments exceed rental income. In either case, borrowers and investors can avoid default by selling unless the home has negative equity. Hence, this study focuses on negative equity distressed properties.

Assumption 2: The loan will become performing after restructuring. The restructured loan, by construction of our model, has a lower balance, lower monthly payments, and a positive real option value. Assuredly, not all home loans can be restructured. However, we consider successful convergence to a solution, subject to several constraints, an indication of potential applicability. We deem the loan a restructure candidate when the model converges to a solution. We do not consider the loan a restructure candidate when the model does not converge.

Assumption 3: The home price process follows a lognormal distribution. We make this assumption to apply the Black-Scholes pricing valuation to our model. This is a standard assumption with option pricing made in [

Assumption 4: The The borrower’s time horizon is t years. We acknowledge that the same agreement was made in the non-performing original loan. However, since the output of our model is a restructured loan beneficial to both the borrower and the lender, we consider t, which is distinct from the loan term T, a required commitment from the borrower for restructuring to occur.

Additional simplifying assumptions.We further assume the market-to-book-value of non-performing loans and performing loans remains constant. In line with option pricing practice, we also assume that price volatility and the risk free rate are stationary, there are no transaction costs or taxes, and the stock (home) does not pay a dividend. It is true that these assumptions do not always hold in the real world. However, the transaction costs and taxes will be minimal for a distressed property due to a major decline in value. In addition, although rental revenue may be positive, it is likely offset by costs of maintaining the property. Finally, [

where

where

The real option value of the restructured loan to the borrower is computed in the same fashion as under distress with changes to the interest rate

where

Should the lender reposses the home and sell on the open market (referred to Real Estate Owned or REO sale), the foreclosed home will sell at a discount to its current market value. This discount can be attributed to ex- pected repairs associated with a property that has been vacant for six months or more (RealtyTrac). In a REO sale both the lender and third party buyer incur repair expenses [

where

Note that the cash flow to the lender in the REO sale is not the sale price

The lender chooses the loan balance discount a, the new loan interest rate

where

Constraint C1: The restructured loan must be smaller than the original loan. This constraint follows Assump- tion 1 and the suggestion by [

Constraint C2: The borrower and lender evenly split potential gains. This constraint guides our model to a solution that is mutually beneficial to borrowers and lenders. Simply reducing a principal balance just to bring a borrower current may be insufficient incentive for a lender. We incorporate the suggestion of shared gains [

Constraint C3: The market value of the restructured performing loan must exceed the market value of the current non-performing loan. This constraint is included to ensure that restructuring is beneficial to the lender. Without this constraint it would be possible to restructure into a performing loan whose market value is lower than the current non-performing loan.

Constraint C4: The new loan rate must be greater than or equal to the market rate for performing loans.A lender may be unable to fund loans at rates below the current market rate. Therefore, we require the restructured rate tobe at least the average market rate for performing 30 year loans.

Constraint C5: The new loan term is between the current loan term and 30 years. Given the finite lives of borrowers and customary industry practice we restrict the maximum new loan term to 30 years. Also, setting the minimum term to the remaining term on the existing loan follows Assumption 2 (the loan will become perform- ing due to lower balance and payment).

Again, we utilize numerical optimization methods to solve the constrained optimization problem in Equation (13) to produce optimum values of the lender choice variables

We obtain the average market rate for 30 year conforming loans ^{®} (PMMS) and the 30 year risk free rate

Symbol | Description | Value | Source |
---|---|---|---|

US average market rate of 30 year conforming loan | 4.20% | Freddie Mac PMMS, September 2014 | |

Risk free rate. | 3.21% | US Treasury, September 2014 | |

Annual US home price volatility | 4.77% | FHFA HPI 1991Q1 to 2014Q3 | |

Non-performing loan market to book ratio | 0.3379 | FDIC 2014Q3 home loan sales data | |

Performing loan market to book ratio | 0.6877 | FDIC 2014Q3 home loan sales data | |

Average length of home ownership | 6 years | National Association of Realtors | |

Average down payment requirement | 10% | mortgageqna.com | |

Average REO discount from market value | 35.90% | RealtyTrac | |

Average buyer REO repair expense | 7663 | La Jeunesse [ | |

Average lender REO repair expense | 2252 | La Jeunesse [ |

correction from 2006 to mid-2011, and the post-2011 rebound are evident.

We begin our illustration with the median values of negative equity loans from [

Next, we perform a sensitivity analysis to investigate factors that impact distressed property values. The results of the sensitivity analysis are presented in

lower value. This could reveal that additional years of payments were unable to bring the loan balance of an extremely inflatedpurchase price in line with the fundamental value. Third, home values are relatively insensitive to the original interest rate

We determine optimum loan restructuring terms for the base case using our real options-based distressed property value estimate. We employ numerical methods [

to the lender post foreclosure sale

We now examine optimality conditions for different non-performing loan balances.

The restructured value

We present a method to establish the market value of distressed properties and a model to reconcile the interests

251,300 | 132,599 | 0.3994 | 0.0420 | 28.5 | 23,825 | 47,649 | 84,914 | 82,744 | 103,797 |

287,200 | 151,542 | 0.4297 | 0.0420 | 28.5 | 33,296 | 66,592 | 97,045 | 94,886 | 112,644 |

323,100 | 170,485 | 0.4527 | 0.0420 | 28.5 | 42,767 | 85,535 | 109,175 | 107,029 | 121,604 |

359,000 | 189,427 | 0.4710 | 0.0420 | 28.5 | 52,239 | 104,477 | 121,306 | 119,171 | 130,599 |

394,900 | 208,370 | 0.4859 | 0.0420 | 28.5 | 61,710 | 123,420 | 133,437 | 131,313 | 139,606 |

430,800 | 227,313 | 0.4984 | 0.0420 | 28.5 | 71,181 | 142,363 | 145,567 | 143,455 | 148,617 |

466,700 | 246,255 | 0.5089 | 0.0420 | 28.6 | 80,563 | 161,306 | 157,698 | 155,598 | 157,608 |

of borrowers and lenders. Our approach applies real option theory using the Black-Scholes option pricing formula, a constrained maximization framework, and numerical methods. Sensitivity analysis reveals distressed property values are most sensitive to the non-performing loan balance and least sensitive to market volatility. The non-performing loan balance sensitivity indicates information remains in the non-performing balance. The volatility insensitivity suggests the resulting negative equity position of distressed homes is driven more by inflated purchase prices than overall market volatility. We present evidence that restructuring is the highest value alternative among the lender’s choice to sell the loan, foreclose, or restructure. Borrowers also benefit from prin- cipal and interest rate reductions. We show that the real option under restructuring is a significant improvement over the zero-value option during non-performance. Overall, we believe our approach can be used to arrive at mutually beneficial loan terms thereby relieving current and future negative equity positions.

David J.Moore,NuriddinIkromov, (2015) A Real Options Approach to Distressed Property Borrower-Lender Reconciliation. Journal of Mathematical Finance,05,73-81. doi: 10.4236/jmf.2015.51007