mau mortgage refinancing paper

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  1. 1. 1 SINGLE CALCULATION APPROACH FOR DETERMINING BENEFIT OF MORTGAGE REFINANCING Abstract Assessing the benefit of residential mortgage refinancing currently requires performing numerous computations. This study streamlines these calculations to provide a simpler method to assess mortgage refinancing. For the net present value calculation of the savings of refinancing, this work substitutes calculus integrations for discrete summations and derives a single, explicit, analytical expression for fixed-to-fixed rate mortgage refinancing. The developed expression precisely calculates the net present value of mortgage refinancing with errors of less than 0.5% when compared to accepted models that sum discrete values. INTRODUCTION This paper emphasizes static, in-the-money mortgage refinancing models mortgage refinancing that offers savings to the borrower based on specified, not stochastic, refinancing interest rates. This paper does not review the dynamic portion of models that deal with uncertain future interest rates. This work streamlines the computations of mortgage refinancing through a single calculation for a fixed-to-fixed rate mortgage refinancing rather than relying on possibly extensive spreadsheet computations and summations of per period savings. To establish the single calculation, this paper substitutes continuous integrations for discrete summations. This approach provides a simple, one-step calculation to summarize the possible savings associated with mortgage refinancing for a typical homeowner. LITERATURE REVIEW
  2. 2. 2 This review illustrates how static, in-the-money models have progressed beyond traditional, simple rules to the point of endeavoring to precisely calculate savings resulting from mortgage refinancing. This progression in calculation complexity requires a correct formulation of mortgage refinancing, leading to the need to account for the complete mortgage refinancing transaction. Review and Evaluation of Refinancing Models To summarize the review of several studies, Table 1 highlights the in-the-money components included in the respective models. This summary table shows that most mortgage refinancing models are incomplete in their methodologies or capabilities for evaluating the complete mortgage refinancing transaction. This paper suggests that the Tai and Przasnyski (1996) model incorporates all of the components of the mortgage refinancing transaction and includes means for evaluating all combinations of fixed-to-fixed rate mortgages (their model also includes acceptable expressions for adjustable rate loan refinancing). They implement their model in a spreadsheet platform. This paper chooses this model as the basis for further developing mortgage refinancing expressions. Detailed Review of Tai and Przasnyski Model This model consists of two main components per period savings (Sm) and net present value (NPV) as follows: The per period savings expression, Sm, applicable for any type of combination of fixed or adjustable rate mortgages, is: 1 11 2 22 2121 D TLC D TLC TIIPPS iiii ttttm where, the following defines the various variables: Sm Per Period Savings in Period m
  3. 3. 3 iitP Loan i Total Payment in Period ti iitI Loan i Interest Payment in Period ti T Borrowers Marginal Tax Rate Ci Points Paid on Loan i Li Loan i Amount Di Duration (Amortization Period) for Loan i Delta function for Loan 1 = 1 if Loan 1 was obtained through refinancing = 0 if Loan 1 was the original loan and the following defines the subscripts: m variable period with respect to Loan 2 i for Loan i ti for time period ti Period m = 1 refers to the first period when Loan 2 would take effect. The net present value expression, NPV, applicable for a Loan 2 that is fixed rate, is: j njj n j m m m j rT BB BLTlQLCF rT S NPV 2 112 11222 1 2 )1(1 )( )( )1(1 where, the following defines the various variables: j Holding Period or Period of Interest of Loan 2 r2 Loan 2 Interest Rate for Fixed Rate Loan F Sum of Fees Paid in Refinancing Loan 2 Q Prepayment Penalty on Early Repayment of Loan 1 iitB Outstanding Balance on Loan i at time ti B1n-1 Outstanding Balance on Loan 1 at the time of refinancing (time period of Loan 1 = n-1) n Refinancing Period (of Loan 1) For double subscripts, the first refers to the loan (Loan 1 or Loan 2), while the second refers to the time period. The current study applies the following throughout: 12 Annualr r (all subsequent expressions only use r) and Di expressed as periods (Di = Duration (in years) x 12)
  4. 4. 4 These substitutions assume all mortgage payments are monthly and require expressing n, m and any other timing designation as periods. For example, if refinancing is considered in year 5, the value for n is 60 (periods). By inspection of Sm and NPV, the two expressions account for the tax deductibility of interest payments, points costs amortization, and prepayment penalties. Also, the discount rate is the new loan interest rate and is after-tax. FRAMEWORK FOR MORTGAGE REFINANCING The framework for this papers approach for assessing mortgage refinancing includes establishing a common timeline for the original and new loan, defining (and accounting for) both loan durations, providing scaling to generalize results, and developing explicit equations that have similar timescales for the various terms in the per period savings and net present value expressions. Synchronizing Loan Timescales Each loan has its own timeframe and periods the first period of a refinanced loan is the nth period of the original loan. Providing synchronized loan timescales establishes a common or real time for all loan transactions. For two fixed rate loans, Figure 1 shows the timing of payments and relationship between the first and second loan. Illustrated in this figure, both period ti = n for Loan 1 and period m = 1 for Loan 2 refer to the first period for mortgage payments based on Loan 2 conditions, i.e., the time period when mortgage refinancing would take effect. Thus, to reflect real time and to synchronize the two loan time scales, the following defines the absolute time periods of Loans 1 and 2, t1 and t2, respectively: mt nmt 2 1 1
  5. 5. 5 As required for derivations shown later in this paper, this time scale reduces time for both loans to a single, synchronized variable, m, with the following limits: At the point of refinancing: m = 0 (t1 = n - 1 & t2 = 0) At the holding period: m = j (t1 = j + n 1 & t2 = j) Loan Durations As shown in Figure 1, the duration for Loan 2 does not need to be the same or match the same end date as the first loan. For the Loan 2, typical loan durations are: duration)standard(anyD DD 2 12 Scaling This paper factors L1 from all per period and net present value terms to provide general expressions. A simple example of this scaling occurs between L1 and L2: 12 LL L This scaling establishes expressions that are independent of initial loan values (and apply universally to all mortgage refinancing scenarios without loss of generality). This paper shows this factoring or scaling throughout, while later presenting specific expressions for the loan scale factor, L, and other scaling factors. Explicit, Analytical Expressions For Per Period Savings Terms For fixed-to-fixed rate loan refinancing, this work develops explicit expressions, using the synchronized timescale and loan value scaling, for each component of the per period savings expression, Sm: iititit CIPB iii ,,, .
  6. 6. 6 Loan Balance, iitB . The general, explicit expression for iitB , at any time ti, is: i i i i D i t it iiit r r rLB 11 11 1 The remaining balance for Loan 1, to any holding period: ti = t1 = m + n 1, is: 111 1 1 1 11 1111 11 11 1 nmBD nm nm nm L r r rLB The remaining balance for Loan 2, to any holding period: ti = t2 = m, is: mBLD m m m L r r rLB 22 1 2 2 222 11 11 1 Loan Payment, iitP . The loan payment is constant over the entire duration of a loan with a fixed interest rate; therefore, we do not need a second, or time, subscript, as follows: ii PiD i i ii L r r LP 11 where: iiti PP Loan i Total Payment in any period For Loans 1 and 2, direct implementation of this general equation only requires substitution of the appropriate ri and Di (and express both as a function of L1 and factor L1). Interest Payment, iitI . By identity, the interest payment depends directly on the interest rate and remaining loan balance: 1 ii itiit BrI Rewriting this general expression by substituting for the respective remaining loan balance gives: i i i i D i t it iiiit r r rLrI 11 11 1 1 1
  7. 7. 7 Factoring and rearranging this expression provides: 11 1111 11 ii i ii ii tD iPi tD iD i i iit rLr r r LI The interest payment expression for Loan 1, to any holding period: ti = t1 = m+n-1, is: 2 11 2 1 1 1 111 1 1 1 1 1111 11 nmD P nmD Dnm rLr r r LI while, the interest payment expression for Loan 2, to any holding period: ti = t2 = m, is: 1 21 1 2 2 2 22 2 2 2 2 1111 11 mD PL mD Dm rLr r r LI Points, Ci. The expression defining the relationship between points and loan values is: 1 1 2 2 1 1 11 2 12 1 11 2 22 D C D C TL D TLC D TLC D TLC D TLC LL In all cases, this expression is constant in time. EXPLICIT, ANALYTICAL EXPRESSION FOR PER PERIOD SAVINGS, Sm Updating the previous per period savings expression to reflect the appropriate subscripting for synchronized time and loan values (to match the basis for the individual terms completed above) gives: 1 11 2 22 211211 D TLC D TLC TIIPPS mnmmnmm This form of the Sm expression allows the direct substitution of the explicit, analytical expressions for the individual terms as developed above, giving: 1 1 2 2 1 1 2 2 11 1 2 2 1 1 21 1111 D C D C TL T