APPENDIX for Standing Loan Line

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STANDING   LOANS   WHERE   e%   IS   NON-ZERO

In a standing loan condition both the payments ‘P’ and the Loan ‘L’ will rise or fall at the same rate of e% p.a. This has nothing to do with the borrower’s ability to pay. This is just a mathematical definition of a standing loan – one that never gets nearer to being repaid and never gets further from being repaid. This condition can be written mathematically in the form of equation (i) below.  In the equation the suffixes represent the year numbers. This definition of a standing loan holds true for all values of e%, including when it is an LP Loan with e% = 0%: -

  P₂                P_{1         Pn}

-----   =   ------  =  -------   = Constant ………Standing Loan condition (i)

  L₂                L_{1         Ln}

Because payments are escalating at the specified rate of e% every year, we can write:  -

                             e

P₂  =  P₁  . { 1 + ------ }  =  P₁ * E  …………... (ii)   where ‘E’ is 

                          100

                                          

the factor in { } brackets.

Similarly, because for a standing loan the loan size of the debt is increasing at the rate at which interest r% is added, so we can also write: -

                          r

L_{2 =} L₁  . { 1 +  ---- }  - P₁ =  L₁ * R  - P₁………   (iii) where ‘R’ is 

                       100

the factor in { } brackets.

So for a standing loan to occur, re-arranging the first part of equation (i) we have : -

            P₁                                                                P₁

P₂ =  ------  *  L_{2  which taking} L_{2 from (iii)}  =    ----  * { L₁ . R  -  P₁ }

            L₁                                                               L₁

Replacing P₂ from (ii) : -

                P₁

P₁ . E  = ---- *{ L₁ . R  -  P₁ }  P_{1 and one} L₁ cancel in next line

                L₁

                             P₁

So  E  =  R  -  ----  and then, expanding the factors ‘E’ and ‘R’ and 

                             L₁

multiplying through by 100 : -

100 + e% = 100 + r%  - P%,   giving: -

P% = - e% + r% ………….    (iv)

What this means is that a standing loan occurs when (iv) is true.

We may re-write this as: -

 P%  =  - 1.e%  +  r%  ………….    (iv)

In this form this equation has the same format as the straight line graph, y = m.x +c where ‘P%’ is the Y-axis, ‘e%’ is the X-axis, ‘m’ is the slope of the straight line graph and ‘c’ is the X-axis intersect.  In which case the slope is ‘-1’ or 45^o down from left to right, and ‘r%’ is the value of ‘e%’ where the line intersects with the X-axis.