Debt is a Vector

Your Debt is Someone’s Wealth

Statistics showing rising debt always trigger concerns of impending economic crisis. We are living beyond our means and will soon have to pay the piper. In January 2019 the Institute of International Finance said that world debt was $244 trillion, which is more than 3 times global GDP and around $35,000 for every human on the planet.

What is seldom mentioned is that one person’s debt is another’s asset. Debt is a liability of the borrower, but wealth and income for the creditor. The world is a closed system - as such the balance sheet of the world is square from a purely financial perspective (environmental externalities aside).

Holding the debts of others seems to be the single largest source of wealth in the world. A 2016 report by Savills estimated the total value of all commercial and residential real estate at $217 trillion (a significant proportion of which is mortgaged). What about equity? It is the poor relation of debt - the total market capitalization of stock markets is only $77 trillion. The global stock of broad money is in the same ballpark as equity wealth – around $80 trillion by some estimates. Gold is a minor source of wealth – the total value of all the gold ever mined is only around $7.5 trillion.

(It is possible that the $244 trillion number should be netted down. If I owe you $10 and you owe me $10 then total debt is $20 but net debt is zero.)

Wealth arising from holding the debts of others has likely increased as a proportion of the total through a long bond market rally driven by extraordinary stimulus provided by central banks in support of the world economy. As interest rates go down, the price of bonds goes up. This is likely a major driver of growing inequality, since poor people don’t hold the debts of others. They provide the raw material for other’s wealth.

We don’t celebrate the logical increase in global wealth that goes hand in hand with an increase in debt, it wouldn’t feel right to so because we have a natural sympathy for the debtor’s side of the equation. We instinctively know that any increase in wealth through increase in global debt is to the benefit of an already wealthy section of society. 

When it comes to debt’s ability to crash the financial system, it is not inability of the debtor to pay that is important, but the ability of the lender to absorb losses. Even then, individual lenders may collapse with limited consequences so long as their default does not cascade through a network of interlinked obligations. It is not the stock of debt that is important, it is its direction.

Debt is a Vector

Debt isn’t a dimensionless quantity – it isn’t a scalar, but a vector with both magnitude and direction. Thinking of debts as vectors has the potential to significantly improve our analysis of this most important driver of the global economy.

Debt vectors flow in the direction from the borrower to the lender. The direction of the vector changes through the secondary debt markets as bonds are bought and sold. The magnitude of the debt vector changes through time as a result of repayments, restructuring, re-financing, interest rates and foreign exchange rates.

These debt vector arrows can pass through several aggregation points as we map out the patterns of indebtedness across the economy. For example, arrows from millions of credit card holders will point to a bank. From the bank arrows will point to its debt holders, e.g. institutional asset managers acting on behalf of pension funds. What we are trying to visualize is the global graph of debt so that we can see the choke points and gain an understanding of the ultimate beneficiaries of the world’s debt mountain.

Debt Vector Notation

To help the visualization we can introduce a notation to represent debt vectors. This is more of an aid to building a mental model than a serious attempt to force fit physics into finance (which doesn’t have a great track record).

• The direction of the vector follows the direction of the obligation, from the Borrower to the Lender.
• The magnitude of the vector defined as the current Net Present Value (NPV) of the obligation ($100 in the example given above). NPV is chosen in preference to the amount outstanding as it is a truer representation of the magnitude of the obligation.
• The NPV is a dynamic quantity that changes with respect to time, e.g. repayments.
• The NPV is negative for the Borrower and positive for the Lender, representing the respective liability and asset.
• In order to harmonize across different credit scoring notations, the credit risk of the borrower is given by a simple probability of repayment (80% in the example).
• The NPV for the Lender is adjusted for the Borrower probability of default and is therefore an expected NPV ($80 above).

It is hard to imagine even an ideal world in which all debt obligations could be captured in a single system with the data updated in real time. The value of this is to help visualize features of larger debt networks. It is through these networks that financial contagion is transmitted.

Debt Vector Directions

Thinking of debt as a vector helps identify large scale features of debt networks including aggregation points, magnitudes of flows and directions.

Some broad hypotheses can be made about the direction of the vectors that may make us stop and think. For example, it is probably reasonable to assume that the vectors tend to point from the poor to the already rich, as the poor need credit and not in a position to grant it. The vectors point from young to old, e.g. student debts, for the same reasons. Global debt vectors flow from countries with high savings to those with low savings, e.g. Asian ownership of US Treasuries.

Debt also flows to productive investment projects and is a major driver of economic growth – it is debt financing that builds factories, bridges, railways and roads. Through the maturity transformation that is at the heart of banking, short term customer deposits are turned into longer term loans to businesses and individuals.

Debt Vector Weather Forecasting

This dynamic system of debt vectors may be considered as a type of weather system with areas of high and low pressure, jet streams and other features that are stable over long periods of time. There are also seasons, or debt cycles, and of course storms of varying magnitude. The debt weather system may be shown to exhibit non-linear, chaotic behavior as butterflies fly through hedge fund windows.

Debt Vector Tracking

Our understanding of global debt vectors will only be as good as our models and measurements. In an ideal world, global debt obligations could be recorded and subject to analysis. Perhaps a Distributed Ledger could record all the obligations in anonymous form, with the entities only visible to the regulators in each country. After all, we are more concerned with the shape of the network and the potential transmission lines of contagion than we are in knowing who owes what, to whom.

Such a system would be fabulously expensive, difficult to implement and face incredible obstacles, but undoubtedly justified if another financial crisis could be avoided through the early warnings provided by such a system.

Conclusion

This short article makes the simple observation that debt is better understood as a vector quantity and that all the debt arrows can be arranged into a single global graph of indebtedness. The main objective is to help people conceptualize the global interlocking network of debt obligations and gain some intuition of the weather patterns that affect this dynamic system. We are much better at forecasting and ameliorating the impact of bad weather than we are debt storms, so perhaps we should have supercomputers modelling global debt networks based on solid underlying data.

Finally, holding the debts of others appears to be the greatest source of wealth in the world. By its nature and the direction of the vectors, it may be an accelerating engine of inequality as well as a major driver of economic growth.