LOAN DEBT ANALYSIS (Developed, Composed & Typeset by: J B Barksdale Jr / )

Transcription

1 LOAN DEBT ANALYSIS (Developed, Composed & Typeset by: J B Barksdale Jr / ) Article 01.: Loan Debt Modelling. Instances of a Loan Debt arise whenever a Borrower arranges to receive a Loan from a Lender. Lenders assess a Lending Fee charged to the Borrower in payment for the use and privilege of the advanced Loan Amount. The Loan Industry has evolved over a period of centuries and has arrived at its modern-day state along with its present: definitions, structures, processes and practices. The current practices regarding loan debt modelling are described by the following principles: (i) A Sum of Money-- the Loan Debt (L)--is borrowed from a lender. (ii) The Borrower then repays the loan debt (L) by submitting a a specified number (n) of periodic payments each of a payment amount (p). (iii) The lender loan fees and the loan debt principle (L) are paid to the lender via the stream of (n) periodic payment amounts of (p) over the Term of the Loan. (iv) The Term of the Loan is defined to be the number (n) times the interval of time between two periodic payments; that time of common duration between periodic payments is called a conversion period. Hence, the Term of the Loan is the time duration of (n) conversion periods. (v) The Sum of lender loan fees plus the loan debt principal (L) is paid with the stream of (n) periodic payments of the payment amount (p). (vi) The periodic payment amount (p) is numerically related to: the Loan Debt principal (L); the number of conversion periods (n); a Lender fee specified interest rate (i) for each conversion period; and, the Remaining Loan Balance (B ) still owed after submitting periodic payment number. (vii) As a consequence of the above described Loan Debt Model, it can be demonstrated that the quantities: Pß :ß 3ß and F are all numerically related by the following Loan Debt Equation. (01) Ð"3Ñ Ð3P:Ñ œ Ð3F :Ñ (Loan Equation). Article 02.: Loan Debt Equation. By applying the above described principles of the Loan Debt Model presented in Article 01, equation-(01) can be established. In order to demonstrate this claim, consider the below detailed development. Imagine that a Lender loans an amount ÐPÑ to a borrower. Suppose, also, that the Lender specifies a payment amount ÐpÑbe paid at the end of each conversion period; further, imagine that the Lender charges the Borrower a Loan Fee of (i BÑß where Ð3Ñis a Lender specified, common interest rate for each conversion period, and ÐFÑ denotes the remaining loan balance owed. Then, applying the principles of the above described model, it follows that after the first payment (p) the remaining balance ÐF Ñ is thus (02a) F œ P 3P : œ Ð"3ÑP : ß " since the Borrower then owes: the Loan amount (L), plus the Loan Fee (i L), " - 1 -

2 minus the first payment amount (p); these debits and credits occur at the end of the first conversion period. In the spirit of convenience in notation, the symbol (B!Ñ shall hereby be used to denote the Loan Amount principal L; hence, F! œ Pß indicating the original Loan Principal after NO (i.e., zero) payments. Hence, Item-(02a) becomes: F" œ Ð"3ÑF! :Þ It similarly follows that, (02b) F œ F 3F : œ Ð"3ÑF : ß # " " " calculates the remaining balance after the second payment of (p). Continuing with each successive payment, it logically follows that, (02c) F œ Ð"3ÑF : ß $ # (02d) F œ Ð"3ÑF : ß % $ (02e) F& œ Ð"3ÑF% : ß Þ Þ Þ And so, after arbitrary payment number R, the Remaining Loan Balance (B Ñ is clearly specified by (03) F œ Ð"3ÑF : " (Balances Recursion Relation). OBSERATIONS. Note that the behavior of the Sequence of Remaining Balances, F! ß F" ß ÞÞÞ Fß ÞÞ., is determined by the relative values of: the loan amount (B!Ñ ; the conversion period interest rate (i) ; and the conversion period payment (p). Suppose that, 3F! :àthen, F" œf! 3F! : œ F! Ð3F! :Ñ F! Þ Moreover, in such case as this, it also follows that, F# œ F" Ð3F" :Ñ F" ß since Ð3F" :Ñ Ð3F! :Ñ!Þ Thus, F! F" F# à continuing with such subsequent comparisons, it follows that the sequence of Recursive Balances: F! F" F# ÞÞÞ is a strictly increasing sequence; therefore, the Loan Balance could never reach zero! Similarly, if 3F! œ :ß then every Recurring Loan Balance would equal B! Þ Thus, in this case also, the Loan could never be repaid! However, in the instance that, 3F! :ßthen: F" œf! 3F! : Fß! since, Ð3F:Ñ!! ; so that, F! FÞ " Continuing with such comparisons, it follows that the recursive sequence: F! F" F# ÞÞÞ F5 ÞÞÞ is a strictly decreasing sequence; and so, the sequence of Loan Principal credit amounts: Ð:3F! Ñ Ð:3F" Ñ Ð:3F# Ñ ÞÞÞ Ð:3F5 Ñ ÞÞÞ is strictly increasing. F! F! Now, creating integer, Aœ œ (least integer) ß it then follows that Ð:3F Ñ A" A"!!! 5 5œ! 5œ! Ð:3F Ñ!! F Ÿ A Ð: 3F Ñ Ÿ Ð:3F Ñ Ÿ Ð:3F Ñ à therefore, the total Loan Debt could be repaid with no more than A payments of the amount (p). Hence, the recursive balances must successively diminish to a non-positive value, since it follows that, A" A" F F œ ÐF F Ñ œ Ð:3F Ñ Ÿ F implies: F Ÿ!Þ A! 5" 5 5! A 5œ! 5œ! - 2 -

3 Article 03.: Equivalence of: Loan Equation & Balances Recursion. By applying the above principles of Loan Debt Modelling as described in Article 01, it was observed that the Balances Recursion Relation of remaining balance amounts did logically emerge as displayed in equation-(03) of Article 02. What is somewhat surprising (and, perhaps unexpected) is the logical equivalence of the Loan Equation-(01) and the Balances Recursion Relation-(03). Hence, if either equation is assumed, then the other equation must also follow. The below presented developmental details establish the following biconditional: [eqn (01)] Í [eqn (03)]. PROOF that: [eqn (01)] Í [eqn (03)]. First, suppose the hypothesis that [eqn (01)] is satisfied for each integer value, where denotes the set of positive integers. Then, for each ß it follows that " (04) Ð"3Ñ Ð3P:Ñ œ Ð3F :Ñ and Ð"3Ñ Ð3P:Ñ œ Ð3F :Ñ à hence, Ð3F :Ñ œ Ð"3Ñ Ð3P:Ñ œ Ð" 3ÑÐ"3Ñ Ð3P:Ñ œ Ð" 3ÑÐ3F :Ñ. Then, 3F : œ Ð"3Ñ3F : 3: œ 3 Ð"3ÑF : : à now, to each side of " " this last, preceding equation, add :, and then divide by 3 in order to render, (05) F œ Ð"3ÑF : ; therefore, the implication: [eqn (01)] Ê [eqn (03)] " has been established. In order to complete the proof of the biconditional, now suppose the hypothesis that [eqn (03)] is satisfied for each integer and F! œ P. Thus, by hypothesis, F5 œ Ð"3ÑF5" : for each 5 Þ To each side this last, preceding equation: multiply by 3 and then add Ð:Ñ in order to render, Ð3F5 :Ñ œ 3ÒÐ"3ÑF5" :Ó : œ Ð"3Ñ3F5" 3: : à hence, Ð3F5 :Ñ œ ("3Ñ3F5" Ð"3Ñ: ; simplifying the right-hand side of this very last equality yields, Ð3F5 :Ñ œ Ð"3Ñ Ð3F5" :Ñ Þ Consequently, Ð3F5 :Ñ (06) Ð"3Ñ œ, for each integer 5 Þ Ð3F :Ñ 5" In order to establish the conclusion of this claim, let be given arbitrarily. Now, apply Item-(06) to render, " " Ð"3Ñ œ 5 Ð"3Ñ œ Ð3F5:Ñ à so, " 5œ" 5œ" Ð3F5:Ñ Ð3F:Ñ Ð3F5:Ñ Ð3F :Ñ Ð3F :Ñ 5œ" 5œ" Ð"3Ñ œ œ " œ œ Þ Ð3F :Ñ Ð3P :Ñ It now follows that Ð3F :Ñ Ð3F :Ñ Ð3F :Ñ 5"! 5 5œ" 5œ" (07) Ð"3Ñ Ð3P:Ñ œ Ð3F :Ñ ; therefore, the second of the implications, [eqn (03)] Ê [eqn (01)], has thus ß also ß been established. Consequently, a proof of the asserted biconditional has been completed and achieved.! Ð3F 5" :Ñ " - 3 -

4 Article 04.: Loan Equation Analysis. The Loan Debt Model described by the six defining principles as displayed in Article 01 did logically render the Balances Recursion Relation-(03); the development presented in Article 03 then established that the Loan Equation-(01) is a logical consequence of equation-(03), and vise-versa. Therefore, equation-(01) is, in fact, a Mathematical Model for the Loan Debt Concepts as described in Article (01). An analysis of the equation-(01) shall now be investigated. Observation (4.1). Suppose that the quantities: Pß 3ß : are numerically specified for the loan debt model, equation-(01). Now, let 8 denote the final payment-number so that: œ8 Ê F œf8 œ!þ Hence, with these specifications, equation-(01) becomes 8 (08) Ð"3Ñ Ð3P :Ñ œ :. Therefore, equation-(08) declares the numerical relationship among: (a)--the loan debt Principal (L); (b)--the loan conversion period Interest (i); (c)--the loan Term of n-conversion periods; and, (d)--the loan Payment amount (p) per conversion period. Observation (4.2). Appealing to equation-(08), computational formulas for the payment amount (p) and the loan principal (L) can be easily created as follows. From (08), Ð"3Ñ 3P Ð"3Ñ : œ : implies that Ð"3Ñ 3P œ ÒÐ"3Ñ "Ó : à hence, 8 8 Ð"3Ñ 3P ÒÐ"3Ñ "Ó : (09) (a)-- :œ and (b)-- Pœ. ÒÐ"3Ñ8"Ó 3Ð"3Ñ8 Observation (4.3). Appealing to equation-(08), a computational formula for the (least required) number of conversion periods (n) can be formulated as : ln : (10) 8œ :3P ( ln"3 œ ln:3p least integer) Þ ln"3 Observation (4.4). A formula to compute the Interest rate (i) appears to be somewhat elusive. Clearly, however, with (i) specified, then given any two of the quantities: :ß Pß or n, then, the remaining third quantities could be determined by implementing the appropriate selection from among the list of: (09a), (09b) and (10)

5 Article 05.: Loan Analysis Illustrations. The following loan debt scenarios illustrate how to analyze the loan debt quantities by implementing the formulas cited in Observation (4.4), above. The Excel File: LoanDebtWrkbk.xlsx companion file, which accompanies the Loan Debt Analysis file, displays a variety of scenarios regarding loan debt analysis. The following presented illustration scenarios appear as rows 6, 7 and 8, respectively, on sheet numbers 1, 2 and 3 of the Excel Workbook companion file printouts displayed on pages 6, 7, and 8 of this composition. ( NOTE: Forget not that: [APR] = "#3 à so that, 3 œ [APR] ƒ "# ). Scenario #01. Imagine the loan of a borrower with the following, GIEN: L = 140,000 ; i = ; n = 180 (months; i.e., 15 years). (NOTE: Forget not that: [APR] = "#3 à so that, 3 œ [APR] ƒ "#). FIND: p = monthly payment amount. (here: 3œ[(3.123%) ƒ12]. SOLUTION: By appealing to the given data and using formula (09a), the calculation yields, ")! Ð"Þ!!#'!#& Ñ Ð!Þ!!#'!#& Ñ Ð"%!!!!Ñ :œ œ *(&Þ"# Þ ÒÐ"Þ!!#'!#& Ñ")! "Ó See Page-6, below: [ Sheet-1/Row-6: Input Columns {D, E, F}; Output Column {G} ] Scenario #0#. Imagine the loan of a borrower with the following, GIEN: i = ; n = 180 (months; i.e., 15 years) ; p = *(&Þ"# Þ FIND: L = Loan Principal. SOLUTION: By appealing to the given data and using formula (09b), the calculation yields, ")! ÒÐ"Þ!!#'!#& Ñ "Ó Ð*(&Þ"#Ñ Pœ œ "%!,!!! Þ Ð!Þ!!#'!#& Ñ Ð"Þ!!#'!#& Ñ")! See Page-7, below: [ Sheet-2/Row-7: Input Columns {E, F, L}; Output Column {M} ] Scenario #03. Imagine the loan of a borrower with the following, GIEN: L = 140,000 ; i = ; p = *(&Þ"# Þ FIND: n = number of conversion periods. SOLUTION: By appealing to the given data and using formula (10), the calculation yields, *(&Þ"# ln *(&Þ"# Ð!Þ!!#'!#& Ñ Ð"%!!!!Ñ 8œ œ lnð"þ!!#'!#& Ñ See Page-8, below: [ Sheet-3/Row-8: Input Columns {D, E, L}; Output Column {M} ] From the above Formula applications of (09a), (09b) and (10) to the given data of Scenarios (#01), (#02) and ((#03), the calculations reveal that the numerical results of L, n and p all agree for each one of the three scenarios; and, each scenario computes a different loan model quantity. (iew pages 6, 7 and 8 and, respectively, and examine columns 6, 7 and 8 of the Excel Workbook Sheet Printouts: Sheet-1, Sheet-2 & Sheet-3)

6 LOAN DEBT Excel File Workbook Sheets-1, -2 & -3 PRINTOUTS SHEET 1 Monthly Loan Amortization Calculations for INPUT ALUES: Formula A = ((1+E/12)^F)*D*E/12 ; Formula B = (1+E/12)^F 1 ; Formula C = ((1+E/12)^J)*D*((1+E/12)^(F J) 1) Monthly Pmts = A/B ; R = LN( L+(1 L)*(1+E/12)^F )/LN(1+E/12) ; Loan DEBT = input ; Loan APR = input ; Term of Loan (in months) = input ; # of Monthly Pymts PAID = Input. Formula A Formula B Formula C INPUT INPUT INPUT Calculate Calculate Calculate INPUT Calculate INPUT: 0 < r < 1 R = # Pmts req'd A B C LoanAmt D Loan APR Term of Loan Monthly Pymts TOTAL COST Total Interest # of Pmts PAID LoanAmt UNPAID r = decimal portion so Loan Bal = rd (Input D,E,F) (Input E,F) (Input D,E,F,J) (Decimal) (Months) (A / B) (F * G) (H D) (C / B) of the LoanAmt D (R alues Below) NOTES regarding this SPREADSHEET:: (01) ROW numbers 6 through 24 contain both INPUT data & CALCULATED results; (02) COLUMNS A, B and C are FORMULA COLUMNS; the ENTRIES ARE CALULCATED IA THE Formulas A, B, and C; (03) COLUMNS D, E, F, J and L are data INPUT ALUES; hence,... these NUMERICAL INPUT values CAN BE CHANGED AS DESIRED; (04) COLUMNS G, H, I, K and M are ARE OCCUPIED by FORMULAS WHICH CALCULATE THE DISPLAYED Cell Entries FROM THE INPUT DATA AULES of Columns D, E, F, J and L; (05) An entry of Column L is an Input Decimal alue, 0 < r < 1, which declares a portion of the loan Debt, rd, denoting an unpaid, reduced Loan Debt Balance; and, Column M is the calculated number of Loan Payments required to reduce the Loan Debt Balance D to that reduced portion value rd. Therefore: When this Excel file is opened,... The USER may ENTER OTHER data INPUT ALUES in order to Calculate & Analyze a variety of different Loan Debt Scenarios

7 SHEET 2 Monthly Loan Amortization Calculations for INPUT ALUES: Formula A = ((1+E/12)^F)*D*E/12 ; Formula B = (1+E/12)^F 1 ; Formula C = ((1+E/12)^J)*D*((1+E/12)^(F J) 1) Monthly Pmts = A/B ; R = LN( L+(1 L)*(1+E/12)^F )/LN(1+E/12) ; Loan DEBT = input ; Loan APR = input ; Term of Loan (in months) = input ; # of Monthly Pymts PAID = Input. Formula A Formula B Formula C INPUT INPUT INPUT Calculate Calculate Calculate INPUT Calculate INPUT: Declare p Recalc Loan Amt A B C Loan Amt Loan APR Term of Loan Monthly Pymts TOTAL COST Total Interest # of Pmts PAID Loan Bal Unpaid New Monthly Pmt New LOAN Amt (Input E,F) (Input D,E,F,J) (Decimal) (A / B) (F * G) (H D) (C / B) (Input New p value) (Input E,F,L) NOTES regarding this SPREADSHEET:: (01) ROW numbers 6 through 24 contain both INPUT data & CALCULATED results; (02) COLUMNS A, B and C are FORMULA COLUMNS; the ENTRIES ARE CALCULCATED IA THE Formulas A, B, and C; (03) COLUMNS D, E, F, J and L are data INPUT ALUES; hence,... these NUMERICAL INPUT values CAN BE CHANGED AS DESIRED; (04) COLUMNS G, H, I, K and M are ARE OCCUPIED by FORMULAS WHICH CALCULATE THE DISPLAYED Cell Entries FROM THE INPUT DATA AULES of Columns D, E, F, L and M; For this Sheet #3 of this Spreadsheet File,... NOTE THAT:... M = ( 12*L/E)*(1 (1+E/12)^( F) ),... AND... (05) An entry of Column L is NOW ASSIGNED (at will) as an Input Declared (New) Monthly Payment Amount; and, Column M NOW DISPLAYS the Calculated New Loan Amounts Allowed by that Declared New Assigned Monthy Payment Amount while retaining both the same: (i) the Loan APR (Column E) AND (ii) the Term of Loan (Column F). Therefore: When this Excel file is opened,... The USER may ENTER OTHER data INPUT ALUES in order to Calculate & Analyze a variety of different Loan Debt Scenarios

8 SHEET 3 Monthly Loan Amortization Calculations for INPUT ALUES: Formula A = ((1+E/12)^F)*D*E/12 ; Formula B = (1+E/12)^F 1 ; Formula C = ((1+E/12)^J)*D*((1+E/12)^(F J) 1) Monthly Pmts = A/B ; R = LN( L+(1 L)*(1+E/12)^F )/LN(1+E/12) ; Loan DEBT = input ; Loan APR = input ; Term of Loan (in months) = input ; # of Monthly Pymts PAID = Input. Formula A Formula B Formula C INPUT INPUT INPUT Calculate Calculate Calculate INPUT Calculate INPUT: Declare p Recalc Loan Term A B C Loan Amt Loan APR Term of Loan Monthly Pymts TOTAL COST Total Interest # of Pmts PAID Loan Bal Unpaid New Monthly Pmt New LOAN Term (Input E,F) (Input D,E,F,J) (Decimal) (Months) (A / B) (F * G) (H D) (C / B) (Input New p value) (Input D,E,L) NOTES regarding this SPREADSHEET:: (01) ROW numbers 6 through 24 contain both INPUT data & CALCULATED results; (02) COLUMNS A, B and C are FORMULA COLUMNS; the ENTRIES ARE CALCULCATED IA THE Formulas A, B, and C; (03) COLUMNS D, E, F and J are data INPUT ALUES; hence,... these NUMERICAL INPUT values CAN BE CHANGED AS DESIRED; (04) COLUMNS G, H, I, K and M are ARE OCCUPIED by FORMULAS WHICH CALCULATE THE DISPLAYED Cell Entries FROM THE INPUT DATA AULES of Columns D, E, F, J and L. For this Sheet #2 of this Spreadsheet File,... NOTE THAT:... (Col M) = (LN(L6/(L6 (E6/12)*D6))/LN(1+E6/12)),... AND... (05) An entry of Column L is NOW ASSIGNED an Input Declared (New) Monthly Payment Amount; and, Column M NOW DISPLAYS the Calculated New Loan Term (in months) by that Declared New Assigned Monthy Payment Amount while retaining both the same: (i) the Loan Amt (Column D) AND (ii) the Loan APR (Column E). Therefore: When this Excel file is opened,... The USER may ENTER OTHER data INPUT ALUES in order to Calculate & Analyze a variety of different Loan Debt Scenarios