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The chart below is the central topic of all of the remaining lessons.

P% is the percentage of the loan that is paid in the current year meaning that:
P% p.a. = P / L x 100% where P = the total annual payment, and
L is the amount of the mortgage or Loan.

FIG 3.1

To repay the mortgage the value of P% has to be greater than r%, so if we were to place an 'X1' on the Y-axis, above r%, this would represent the current level of payments needed to repay a mortgage at the current rate of interest, r%.

If a variable rate is used then the value of r% will alter and this will cause the point 'X' to move if the repayment period is to remain the same.

It turns out that if r% moves then the 'X' has to move about the same percentage (distance) upwards so as to keep the repayments on schedule.

FIG 3.2 below:

As long as the 'X' is above the standing loan line, which  slopes down at 45 degrees from left to right, the mortgage will be repaid.

The higher the 'X' is above this line the faster the mortgage will be repaid.

It turns out that to stay on schedule, the 'X' always needs to be about the same distance 'C%' above this standing loan line.

As the rate of interest changes the standing loan line rises and falls. It always crosses both axes at position r%.

To stay above this standing loan line the 'X' must either move up and down or from to the left and right.

A vertical movement is NOT the only option, as we have become accustomed to believe.

The problem with the vertical motion is that this interferes with the pricing of mortgages, as explained in LESSON 1. And it also creates property price bubbles and crashes and it creates potentially catastrophic arrears and funding problems for lenders.

The effect of moving to the right when the standing loan rises is to keep the 'X' the same distance above the standing loan line and to repay the debt on schedule without having to immediately escalate (jump up) the payments. The payment increases are smoothed out over the remaining period. But there is a condition attached. There is a boundary to be observed.

FIG 3.3 Readers will notice a vertical dashed line at position AEG% on the X-axis.

AEG stands for Average Earnings (or Incomes) Growth and the position of the line is the current rate of AEG% p.a. The 'X' has to be placed to the left of this line so that the cost of the payments rises less quickly than incomes.

There may be some guesswork about this AEG% p.a. value, and even how to define it. Not all of the borrowers will have incomes growing at the AEG% p.a. rate so we will need to have some margins.

When we get into looking at government debt we may want to change over to using the rate of growth of net tax revenues - or more likely, we may use the rate of growth of GDP as an approximation to the rate of increase of government revenues and thus what they can afford to pay for borrowing.

Quite a lot of debate surrounds the selection of this index.

Whereas a government may not need to place their 'X' to the left of the GDP% p.a. line, (their equivalent of the AEG% p.a. line) when it comes to mortgages, the idea is that the 'X' should always be a decent distance to the left of this AEG% p.a. line. 

In part, this is to allow for any error or misrepresentation in the value of AEG% p.a. but the main idea is to ensure that for the typical borrower the cost of borrowing in terms of the impact on their income will reduce every year. If this is not done then what is known as 'Payments Fatigue' sets in.

It is thought that the distance to the left of AEG% p.a. should be about 4% p.a. to get a high percentage of borrowers into the safety net.

At 4% p.a. this means that on average this provides about a 12% reduction in the cost-to-income after three years no matter whether incomes are rising or falling. 

FIG 3.4 shows a condition in which average incomes / earnings (AEG% p.a.) are not rising. The AEG% p.a. line is on the Y-Axis.

Now the 'X' is to the left of the Y- Axis so the payments are falling every year. Readers may be curious to know how that would work out in practice. Click here and look at FIG 1 for a tabulation

Remember that the 'X' must always be above the standing loan line, but also remember that at times of austerity when incomes may be falling, the interest rate and the standing loan line will be low, as shown. So it should not be a problem to move the 'X' to the left as long as the 'X' started high enough in the first place - like where it would normally be in normal conditions.

In FIG 3.4 the monthly repayment cost stays the same but the borrower's income is shown as static so the AEG% p.a. line moves onto the Y-Axis.

SPREADSHEET TESTS - to be added on new pages soon.

Tests using spreadsheets seem to confirm this 4% figure will work in most conditions, provided that the entry cost, (the INITIAL height of the 'X') is set high enough to cope with normal or mid-cycle conditions. It is suggested that readers go back to that mid-cycle link later. 

This places the 'X' at around P% = 8.5% p.a. for an economy like the UK and maybe the USA and some of Europe.

In order to take the mathematics a step further we will need to define the distances from the 'X' in relation to the standing loan line and the AEG% p.a. line.

As stated, we need the 'X' to be always above the standing loan line and always to the left of the AEG% line. We will come to that in LESSON 4.

One of the problems with the traditional mortgage is when the 'X' is stuck on the Y-Axis and is not permitted to move to the left of it if incomes are rising slowly or not rising at all. The AEG% p.a. line can even move to the left of the Y-axis at times of austerity and so the idea of maintaining the 'X' on the Y-axis, which is what the traditional fixed interest and variable interest rate mortgages attempt to do, is very hazardous.


The same problem happens with a government fixed interest rate bond which will be at a fixed rate point on the Y-axis somewhere similar in height to the 'X' shown here. The market rate of interest may fall but the 'X' on the Government's chart stays in the same place - far above market rates of interest in these conditions.


Government bonds should not do that. By doing that they increase the risk of a government default and this means that re-funding the debt, or borrowing more, will be done at a higher interest rate than the dangerously high original cost. A 'default-risk and interest rate' spiral develops:

As the risk of default rises so the interest rate at which governments or mortgagees must pay to cover the risk rises.

This raises the standing loan line and makes default much more likely which then raises the risk and causes the interest rate to rise - the system is likely to fail completely and disastrously.

We will need to have another think and find another model for both debt types, allowing them freedom to move away from the Y-Axis in both cases if market rates are not to be flouted,and financial stability is to be maintained.

This appendix is for those who want proof that the standing loan line falls 1% for every 1% it moves to the right.

There are two proofs - one algebraic in LESSON 1 and  the other is a simple proof done here by careful illustration:

There is an algebraic proof based on this starting point:

P/L = constant   or P% p.a. = constant.

The condition for this to be happening is the condition whereby the payments are escalating at e% p.a. and the amount of the debt L is also escalating at e% p.a. This is a race which neither P nor L side can win. The ratio P / L is a constant.

If it was a government debt then we would ask:

How much is the debt 'L' and what is the GDP? If that ratio of debt / GDP never changes then we might conclude that this was also a standing loan, provided that the net tax revenues were always a constant percentage of GDP.

But here is a simple way to think it through:

Imagine a loan of 100 and an interest rate r% = 10%. Add 10% interest and you get 110 owed.

Subtract a payment of 9 (P% = 9% this year) and we get a balance of 101.

So the loan L has risen by 1% from 100 to 101.

The next year's payment will be 1% higher because the loan is now 1% more.

This means two things:

a. To catch up, the payments must rise by 1% at the end of the year, and every year thereafter as long as only 9% is paid and 10% interest is added. The escalation rate e% = 1% for this year and every year.
b. The loan and the payments are both rising at 1% p.a.

We can repeat this arithmetic every year and we will always find that P% = P / L remains the same at year end.

We can also repeat this for a loan L = 100 and a payment P% = 8%.

This time we get a year-end balance of 
100 + 10 (interest of 10%) - 8 (payment of 8%) = 102.

So to catch up the payment P must rise by 2 % next year and every year.

Then the payment will be 2% more than 8 and the loan will be 2% more than 100. So the same race to nowhere is on. P/L is a constant once more.

Hence by doing this for all the different values of P% we can find this standing loan line falls by 1% for every 1% it moves to the right.

When there is time a series of spreadsheet examples that were done in the original paper will be added to this page.

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