MATHS 2 - Terrible Risk of Government Bonds

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This section is about the safety or otherwise of government bonds.

Readers are invited to have a look at this report on the British Banks. Much the same applies to banks around the world. I Quote from The Economic Consequences of the Vickers Commission - Laurence J. Kotlikoff - Professor at Boston, June 2012.pdf.

"...Indeed, today’s safest assets are, according to the market, UK gilts and U.S. Treasuries. But based on long‐term fiscal gap analysis, they are among the riskiest assets in the world. Yet, the Commission would allow good, ring­‐fenced banks, to borrow 25 pounds for every pound of equity and invest it all in gilts. In this case, the Commission’s ring­‐fenced banks would fail if gilt prices dropped by just 4 percent..."

If these assets are so safe then why do they behave with such extreme volatility? See FIG 2.8 below for USA Bonds as traded on a year on year basis:

FIG 2.8 - we will return to this later. The Blue line represents year on year market value changes in USA Treasuries.
Source: Morgan Stanley Research c/o Money Game Chart of the day.

As already explained on other pages, this graph is telling us that as America reduced inflation over the period from 1980 to 2010 (the period illustrated), this was costing USA tax payers 9% p.a. of GDP in interest per GDP of government debt at first and declining over time - see the trendline. And for the markets it is/was a massive wealth re-distribution machine. The re-distribution happens when people and institutions deal in these bonds. Looking at the BLUE line which represents the true rate of return on an investment in these bonds on a year by year basis we see as follows:

A fictitious investor that bought and sold these bonds every year and who owned a GDP's worth of them, would see a small loss in the first year, and a massive 27% of GDP increase in value in the second year. The loser would be the one that sold the bonds at the end of the first year. The winner, the one that bought and sold in the second year.
Having such kind of re-distribution of wealth both from tax payers and to and from dealers in the market place on such a huge scale in any economy is without question the source of a lack of confidence in the wealth of the nation, (neither wealth invested nor the cost of wealth borrowed is safe), never mind in the health of its banking sector that is forced to invest in such bonds as a 'risk free' asset!

The need to reform the structure of government debt is urgent and essential to the future stability of the economy.

Here is the maths of Fixed Interest and GDP-Linked Bonds:

In 'MATHS straight in' we derived Ingram's Safe Entry Cost Equation:

P% = C% + D% + I%

The margins (C% +D%) must be allowed to fall (by the risk manager) if I% rises, or else P% will have to jump up, threatening arrears and repossessions.

FIG 2.1

If C% falls the total repayment period increases.

If C% falls to zero, the ‘X’ falls onto the standing loan line. The loan is never repaid if it stays there. 

If C% stays positive but D% falls to zero, the ‘X’ moves to the right and hits the AEG% p.a. line. The % of income needed remains the same throughout the repayment period.

If both fall to zero, the ‘X’ moves onto the standing loan line and the AEG% line – where the two intersect: at the top of the lower small triangle. The % of income needed never alters and the loan never gets repaid. This is a Wealth Bond.

This format, with the 'X' firmly attached to that intersection, can be used by a government. It issues a bond paying a fixed level of true interest only and index-linking the capital to AEG% p.a. or Nominal GDP - NGDP.

Indexation has the same effect on the capital value of a bond as re-investing the same percentage of interest. For example, if a bond values at 100 is indexed up by 5% it becomes worth 105 in money terms. If 5% tax free interest is added instead, it still becomes worth 105 in money terms.

A government borrower does not need any D% - it does not get payments fatigue like ordinary people do, and it does not need to plan for retirement. Its revenues are likely to correlate fairly well to NGDP. So an index-link to NGDP is likely to suit a government's budgeting plans quite well. The servicing cost will be a fixed percentage of revenues and the amount of wealth that was borrowed will remain constant until maturity.

That is what happens if the 'X' is placed at the top of the lower triangle. The payment becomes an interest coupon which rises at AEG% p.a. because e% = AEG% p.a. and AEG% p.a. should correlate quite well to GDP growth (NGDP) and government revenues. The rest of the interest, (the core interest of AEG%) increases (indexes up) the capital value of the debt - the bond. See FIG 2.2 as an example:

FIG 2.2 - A wealth bond offering 1% true interest for 10 years.
Figures in billions
Source: E C D Ingram Spreadsheet
The bond is repaid on the maturity date. It is called a Wealth Bond because by index-linking to AEG it preserves the wealth (the amount of average income invested in the bond) for the lender / investor. 

Typically a wealth bond with a 1% interest coupon might be issued by a government to get cheap borrowing. Doing that instead of offering fixed interest bonds also adds to the nation's economic stability, removes the 'wealth redistribution machine' and underpins the reserves and the stability of the banking sector. It enables a government to reduce the level of inflation at no additional cost, and it protects the nation's wealth if inflation rises.

In short, it creates economic and financial stability in a major part of the economy.

Economic stability is increased because not only is the government's call on tax-payers both low and pre-defined, but the wealth invested in the bond is also safe. Even if the bond is traded on the open market its value will not vary wildly as current fixed interest bonds are prone to do.

For simplicity the above illustration shows that the government borrows 10% of annual revenue and repays a total of 11% of annual revenue over 10 years, which is 1% p.a. of the wealth borrowed plus the return of the wealth borrowed at maturity.

If the government prefers to create an index of total national income which is normally thought of as GDP, and if that figure is well defined in that way and is not manipulated,  then a wealth bond can be created with an index link to GDP. It should be about the same as a link to AEG% p.a. There is a discussion on this topic on the MATHS 3 page.

With this index the share of the GDP that a person has earned can be stored in such a wealth bond. The cost to tax payers is the true interest payable of maybe just 1% p.a. as illustrated above in FIG 2.2. The true cost is the 10% (1% p.a.) true interest paid and is the same as the wealth added to / earned by the lender. At maturity the amount of GDP that was borrowed has to be repaid. That costs no additional wealth / share of GDP than was borrowed.

Here is the bar chart for that kind of bond. It compares the wealth bond (ILS) with a fixed interest (LP) bond paying the same 1% true interest. For the LP bond this is a 5% interest coupon in the same AEG = 4% p.a. environment.

FIG 2.3  - Index-Linked Wealth Bond payments of 1% compared with 5% Fixed Interest Bond payments. 

Paying 5% p.a. fixed interest means paying more p.a. but results in a maturity value which is less. The maturity value payable is 7.09% of revenues compared to the 10% of revenues originally borrowed / lent using the fixed interest bond whereas the maturity value of the wealth bond is the full 10% of the revenues / all of the wealth lent. This shows that the fixed interest payments were eroding the stock of wealth owed at maturity.

Here is an illustration of the above Fixed Interest Bond when AEG% = 4% p.a. in which the 7.09% figure is shown - right hand column, final payment.

FIG 2.4 - Fixed Interest 10 year bond paying 5% at AEG% = 4%
Source: E C D Ingram Spreadsheet
10% of government revenue is borrowed and 10.81% is paid back but the final installment paid on maturity is only 7.09% of revenue  including the interest coupon of 5%.

In the case of a fixed interest bond issued when AEG% p.a. is higher than 4% p.a. and repayable when AEG% p.a. is low or negative, the government has to pay out a great deal more of its revenues / wealth. The same thing happens when an economy is in trouble and AEG% p.a. falls as illustrated next.

But the Wealth Bond pays out the same 1% p.a. at any level of AEG% p.a. 

Here is how it would look if the same bond found that AEG% p.a. suddenly fell to -2% p.a. on account of austerity:

FIG 2.5 - Fixed Interest bond after AEG% p.a. has fallen
Source E C D Ingram Spreadsheet
Now that same bond is costing tax payers 17.84% of revenues instead of 11%, (7.84% net cost instead of 1% net cost), even though the government borrowed only 10% of revenues. 

The final payment is 12.85% of revenues to repay a debt of 10% of revenues, and the cost of the interest coupon was rising every yer as well. See right hand column.

The mathematical reason for this is that based on the equation:
I% = r% - AEG%

The true interest cost is 5% nominal interest + 2% (for -2% AEG% p.a.) = 7% true cost p.a.

But the same wealth bond is largely unaffected in these terms. The true cost remains at 1% p.a:

FIG 2.6 - Wealth Bond cost / benefit is almost unchanged
Source: E C D Ingram Spreadsheet
In FIG 2.2 with AEG% p.a. = 4% the wealth bond cost tax payers 11% to service a 10% of revenue bond - exactly the same figure as here in FIG 2.6 with AEG% p.a. = -2%.

The money repaid is less but the investor still gets back the same proportion of GDP as was lent. The government has enough trouble with revenues falling, so the fixed interest bond would be adding a significant additional cost when the economy can least afford it. 

With fixed interest bonds during recession or austerity, wealth moves from the poorer tax payers to the wealthier investment institutions and others, and economic recovery takes longer and is more costly.

Here is the Bar Chart comparison:

FIG 2.7 - % of revenues needed to repay - comparison of fixed interest (LP) and wealth bonds (ILS)

Source: The above spreadsheets
Now as shown, (blue bars), the government is paying out five times as much every year and still has to repay more from its revenues on maturity.

In fact in FIG 2.5 it shows that the cost-to-revenue is rising every year with the fixed interest bond.

This is the kind of thing that happened in the 1980s to American tax payers after the great inflation was brought under control. Remember, the following chart shows the return to investors who were trading in government bonds at that time:

FIG 2.8 Cost / Returns on USA Treasuries 1980 - 2010. This needs a little explaining - see below the graph

Source: Morgan Stanley Research c/o Money Game Chart of the day.

It may be implied that the trend-line represents the true cost (the rate of wealth transfer) from USA tax-payers to investors in government bonds. This is not small money. It is a percentage of the national debt invested in these bonds. The volatility of the blue line shows the return to investors that trade in these securities. The volatility is caused by changing perceptions of the value of the bonds and is partly related to the expected rate of inflation going forward. 

To inject this kind of uncertainty into the wealth of an economy and to create such a huge cost on tax payers is to make a real mess of the economy in many ways mostly discussed already.

Any bond that is index-linked to AEG% or to GDP as defined here, can be called a wealth bond. It does not matter whether the capital is repaid in installments or at maturity, the indexation simply preserves the share of the national income or of National Average Earnings (NAE) that was invested. 


There is more related reading on these issues in the GLOSSARY page and in the WEALTH BONDS page. Click on those links.

The Glossary discusses true interest rate v real interest rate in more detail.

The Wealth Bond page has some amazing illustrations of wealth bonds as applied to mortgage finance.

More reading of interest can be found in the NEW PRODUCTS RESULTING page. Click that link to read it. It is full of fascinating practical uses for the knowledge that you have gained already.

I will also be adding a lot of back-testing and discussions for the ILS Mortgage model in later pages of this Blog as explained on the Home page.

Wealth bonds can provide lenders to business and home buyers with a fixed true interest rate and so reduce or eliminate the variability of the true interest rate mix (of funds to lend) on offer to borrowers. In fact the true rate can be fixed for both the lender and the borrower, with certain provisos written into both contracts - investors and borrowers, to protect cash flows and to match the lender's assets and liabilities. The outcome would be a Defined Cost Mortgage, funded by a Wealth (protection) bond.

The Defined Cost Mortgage has a fixed I% and a fixed D% and a fixed N, repayment period.

P% = C% + a constant.

Every year end C% rises taking P% up with it until the loan is repaid.

Governments can also raise funds in the same way, offering wealth bonds that repay capital in installments. This is a cheap and safe way for governments to borrow, especially during times of recession when AEG% p.a. may be falling as already illustrated.


As long as C% is positive, the capital content of the repayments P% rises at year end and that takes P% up with it. This means that if the mortgage is a variable interest rate model, there is more scope for I% to rise as time passes, without causing any damage:

As P% rises so F% (C% + D%) rises for a given I%.

An example of P% rising every year can be seen in the far right hand column of FIG 2.2 below.

FIG 2.2 - P% rises every year as does C%. This is normal.
P% rises of its own accord because P% = P / L x 100%, all else being equal. 

P% is the ratio of the annual payments P compared to the debt / Loan, L. 

As L reduces the payments become a higher proportion of what is left to repay. This happens in any repayments model as long as C% is positive.

As P% rises it gives more safety margin to the borrowers and lenders because as P% rises F% also rises if I% remains the same

F% = C% + D% which is the total safety margin.

This means that the first five or so years are the most risky.

What we need to avoid is having the payments jump around in a disorderly fashion, especially during those years.

Please note that the spreadsheets can also show a steady % of average income is paid for a rental. You can set any figure but in all the spreadsheets used by me it is set at 70% of the entry cost for the 100% mortgage on the same property. This comes to a flat 21% of income.

I like to depict this with a sketch as in FIG below:

FIG 2.3 - ILS compared to rental. The rental costs more than the mortgage after around 12 years If D% = 4% p.a.


Because, for the standard Level Payments Mortgage, D% = AEG% the risk manager has no control over this D% function. This is why this Mortgage Model gets into difficulties when AEG% is low or negative: the downwards slope of the ‘% of income’ can become level or turn upwards as AEG% reaches zero or below zero. The cost may go above and outside of the 30% of income 'box'.

With the LP Model, this means that if I% rises, and D% is determined by AEG% p.a. at all times, then the value of C% is all that remains for the risk manager to mange in order to protect P% from rising or jumping upwards.

FIG 2.5 - This is a bad case of the LP Model going out of control. The blue bars are the LP payments and the other bars are how ILS Model payments might behave with the same interest and AEG rate data.

In practice, the Level Payments (LP) Model nearly always jumps the 'X' up and down the Y-Axis so as to keep C% where it needs to be to repay the loan on schedule.

This is a requirement of the regulators who will treat any deviation with suspicion.

Another way to write the equation for the LP Model is this:
P% = C% + AEG% + (r% - AEG%)
because I% = r% - AEG% by definition. r% being the nominal rate of interest.

This then reduces to:
P% = C% + r%

So if r% rises P% rises, making the payments jump upwards. If C% reduces to zero as r% rises, then we have an interest only payment:
P% = r%

This arrangement is far too inflexible. It is unable to ride out many changing conditions.

Furthermore, when r% falls, P% also reduces and then lenders increase the amount of wealth that they lend.

When (not if)  r% rises again P% rises, but by then too much wealth has been lent and costs rise out of control.

The Level Payments (LP) Model has neither proper control over the slope of the '% of income' (cost) bars nor any proper control over the amount of wealth that may be lent. The interest rate and inflation rate risks and the risk to the value of the collateral security, (property values generally), are simply out of control.

If the mortgage model is released from this straight jacket of always paying all of the nominal interest r% and P% is made to start high enough (by the risk manager) to ensure that F% has a useful value (F% = C% + D%) then both D% and C% can be reduced to help to protect the value of the payments going forward, if I% rises.

If the payments are not enough to prevent the mortgage balance from rising lenders can offer a rent-to-buy contract in place of a mortgage, and the arithmetic remains basically the same except for additional costs which may be small.

In Turkey they have introduced a wages-linked mortgage where D% = 0%, and I% is fixed. It is not very user friendly and the risk of arrears and losses by the lender is made significant by this omission.  The mortgages run for only 15 years.

Do a Google Search on "Kanak Patel" and "Turkey Mortgage"
Various commentaries can be found there.

In fact the Turkish model was first designed by me in 1974 and it was published in the Building Societies Gazette in October of that year.

But on 11th April 1975 my letter to The Times was published asking for funds needed to refine that model. I needed to find the Safe Entry Cost Equation. I needed to add D%.

Click on FIGs to enlarge and read then click back page to return.

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