LESSON 1 - AN INTRODUCTION AND AN APPENDIX TO LESSON 3

SCIENCE OF DEBT PRICING

INTRODUCTION

Everyone please read down to FIG 1.3

THE BASIC CHART
The basic framework for the charts is this:

FIG 1.1

The charts display the payments P% p.a. as a percentage of the amount borrowed L.

The interest rate r% can also be shown on the same axis.

When P% = r% there is no repayment of capital. This is known as a standing loan. The loan remains the same size and so do the payments.

If an 'X' is placed on the Y-Axis at a value of r% as shown, this is a standing loan with Level Payments. If the 'X' is above r% then the mortgage is being repaid.

There is also an X-Axis in case the lender should want to increase or decrease the amount of the payments P p.a. (not P% p.a.) at a rate of e% p.a.

For example, the first year payments might be 100 per month, and the next year's payments might be 105 per month. In this case e% p.a. = 5% p.a.

PROBLEMS FACED BY LENDERS

Normally lenders use the value e% = 0% p.a. They do not alter the payments on a regular basis. But the problem with this Level Payment (LP) mortgage model is that incomes are not always rising. Sometimes they may be falling. That can create serious problems.

Another problem is that raising the funds using fixed interest rates can be expensive because the owners of the funds may be worried about inflation and there is no adjustment made for that. That can also present serious problems.

Sometimes lenders find it is easier or cheaper to use a variable interest rate. The problem with that is that the payments can jump upwards if the interest rate increases, and the jump can be many times greater than the increase in the rate of interest or the rate at which incomes are rising.

Here is the annual cost sensitivity table to nominal interest rates which lenders use.

FIG 1.2 - Sensitivity to nominal interest rates

THE PRICING OF MORTGAGES

This pricing model is also used for fixed interest mortgages. Whereas the price of goods has a relation ship with aggregate demand, and so as incomes rise so may the price of goods, ceteris paribus, It is like fixing the 'price' for the monthly repayments in a way that has no connection with the level of supply or the level of demand. It is an economic nonsense and it gives rise to shortages and surpluses of money to lend. It also gives rise to property price bubbles and crashes and it gives rise to very high rates of arrears at times, making the raising of interest rates to the level where the cash inflows to the lenders at a sufficient level to continue lending at the desired rate, impossible.

THE FUNCTION OF LENDERS

Lenders have a function in an economy:

To tap into the funds needed in order to satisfy the level of demand at the current price of money, and to do so without creating a high level of arrears.

The mathematics starts here:

THE INITIAL STEPS IN THE GENERAL MATHEMATICS OF LENDING

FIG 1.3

Here an additional line is shown, called the standing loan line.

If the 'X' is placed anywhere on this line the debt will never be repaid. The ratio of the payments / the debt, P% p.a. never changes.

There is an algebraic way of finding this line and there is another way to do it which non-mathematicians may find easier.  That is given in LESSON 3.

Non mathematicians may like to move now straight to LESSON 3

APPENDIX TO LESSON 3

Here is the algebraic derivation.

FINDING   STANDING   LOANS   WHERE   e%   IS   NON-ZERO

In this section we will find out where on this chart standing loans can be found.

It is called finding the locus of standing loans.

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 the factor in { } brackets.

                          100

                                          

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 the factor in { } brackets.

                       100

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 multiplying through

                             L₁

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.