Figure 7. The recursive model illustrated in this figure applies to a typical corporate bond portfolio and is discussed below. The figure plots the change in PRINCIPAL given reinvestment of the "return-on-investment" (ROI) accrued. The line occurs at the 7% "interest rate paid" (IRP) level -- the "unsustainable" level beyond which the PRINCIPAL steadily decline with time.
The purpose of this section is to address the “7% Paradox” and the nature of the recursive behavior national debt imposes on the solvency of countries. I would like to point out that the recursive model used is very simple. I will state below the nature of the recursion and the reader can judge for herself.
A general "sustainability of interest rate" model. This model simply looks at an asset base (PRINCIPAL, PT, that we start with), and assumes a constant return-on-investment (ROI). The owner then pays out an interest rate RT for owning the PRINCIPAL in year T -- much as a sovereign would pay for a loan of PRINCIPAL -- and tracks the change in PRINCIPAL over future years. We always start with PRINCIPAL = 100%. The basic recursion becomes:
PT = PT-1 [1 + (ROI - RT)] , (1)
Where PT is the current PRINCIPAL (in year T) and PT-1 is the PRINCIPAL one year earlier (last year = T-1 ).
It follows from Figure 7, that for interest rates paid less than 5% (the return on the investment) the PRINCIPAL actually increases with time. However, for interest rates paid greater than the 5% ROI, the PRINCIPAL decreases, the decrease is more rapid with the more interest paid out. It’s pretty clear where the 7% rule applies in this case (remember we did assume a return of 5%, so the rule needs to be generalized).
Discussion
We have assumed:
- The ROI is 5%
- The PRINCIPAL reopresents a corporate fixed-income asset, that receives the indicated ROI and pays out to investors an "interest rate paid" (IRP) at the yield indicated by the left-hand ordinate.
- The “interest rate paid” is assumed to represent the "cost" to the corporation for the bond at current market yields.
- The IRP is for the entire period indicated by the abscissa, “Future Years (Y).”
- It is assumed that the bond PRINCIPAL is 100% at the inception of the period in question (Future Year = 0).
- The IRP represents a weighted rate, across the range of bonds in question, all with ROI = 5%, in the portfolio.
- All ROI is reinvested in the asset account to increase the "remaining PRINCIPAL" as akin to the "PRICE" of the asset.
Generally speaking, one would expect that the “interest rate paid” (IRP) would not exceed the ROI for the bond; however, once the bond is issued it is subject to further modification as the risk profile for the bond changes in the secondary market…the IRP is presumed to be “dynamic.”
Clearly, for IRP < ROI, over time the “remaining PRINCIPAL” increases. When IRP > ROI this PRINCIPAL decreases.
Equation (1)_ represents a “recursion.” As the IRP increases, the history of the “remaining PRINCIPAL” declines (the “PRICE” of the asset drops). Granted this “PRICE” decline may not be quite as drastic as evidenced in Equation (1), but absent psychological impact (from say an external risk assessment) the trend is clear.
It is clear that as IRP goes above the 7% level (indicated by the line), the “remaining PRINCIPAL” drops off fairly rapidly. This has been used to represent the “unsustainable” rate suggested to surround the “sovereign debt” issue currently of interest. Of course, the actual paradoxical level at which the debt becomes “unsustainable” depends on the overall ROI for the asset. Hence, this “7% Paradox” is somewhat loosely defined.
For the case of the corporate bond portfolio represented ideally in Figure 7, this level is suggested to be appropriate. Clearly, in discussions regarding “sovereign debt” this level suggests a typical ROI for a country in the 5% range if the so-called “7%” yield on a country’s “sovereign bond” does represent an “unsustainable” figure, as is frequently indicated in the news.
