The Loanable Funds Cycle and the Variability of the Deposit Base

Abstract

The analysis lays the foundation for Minsky’s theory, which exposes the states of the economy it goes through over evolutionary time: expansion and significant progress, then downturn either in the form of recession with negative development (or growth recession) or full-blown depression with heightened uncertainty and risk that seems uncontrollable. At some stage, the economy goes into recovery mode from Darwin’s ‘survival of fittest’ account of the intense market competition, travelling back to the expansion stage with fresh consumption and investment opportunities to explore and exploit on account of Schumpeter’s process of creative destruction. This will have significant implications for the variability of the banking sector’s deposit base, which can be modelled within the catastrophe framework to explain abrupt changes in money as loanable funds in relation to the build-up of uncertainty and default risk within the monetary economy.

Appendices

Appendix A

Assuming that the firm adopts the profit -maximising position, it produces $Q$, where ${\text{SMC}} = {\text{SMR}}$ in Fig. 7.8. If the price, $P$, is above ${\text{SATC}}_{1}$, then the firm is making an abnormal profit as shown in the area as indicated. If $P$ is between ${\text{SATC}}_{1}$ and ${\text{SAVC}}_{1}$, partly covering fixed costs as well contributing to interest payments on loans , although, overall, the firm is making a loss. If price is below ${\text{SAVC}}_{1}$, production is zero, and therefore, there is no contribution to fixed costs and interest expenditures.

Appendix B: A Portfolio Theory of Loanable Funds : Default and Risk

7.1.1 Introduction

The book has partially developed a new model based on the principle that inside money comes from the formation of real income in production, paid as bank transfers, which leads to endogenous loan creation by the banking sector. The money supply is not only outside money , exogenously controlled by the Central Bank with the monetary base via reserves . Thus, the purpose of the analysis within this Appendix is to apply a Modern Portfolio Theory (MPT) framework to the area of loans granted by the retail (or commercial) banks to borrowers in the midst of asymmetric information . The objective is to adapt the MPT as the modelling process in terms of the risk -weighted loans of default embodied in the total number of assets on the balance sheets of the banks. This discussion leads to a hybrid theory of default and risk concerning loans made by the commercial banks . Before starting the development of the model, a recap of the main features of the MPT would not go amiss by clicking on the following links, 1 (https://www.youtube.com/watch?v=lPKtI90f_sE) and 2 (https://www.youtube.com/watch?v=zVsCgU26U_8).

7.1.2 Development of the Theoretical Analysis

Suppose that a selection of loans , denoted by $W$, are perceived to be at risk of default in the midst of asymmetric information that lies within the portfolio of assets on the balance sheets. It is the speculative and Ponzi borrowers of loans that reflect the inclusion and measurement of the high degree of credit risk rate added to the mark-up , ${\text{MU}}$, of the rate of interest , $i$, outlined in the new theory. ¹⁰ This relates to the unsystematic, specific (or idiosyncratic) risk of the individual borrowers , firms or industries in which they operate.

Moreover, this implies the remaining cluster, $\left( {1 - W} \right)$, is the market group of loans within the assortment of assets, representing the pure, normal market risk , which depends on the equilibrium states of the macroeconomy and cannot be diversified away unlike specific risk . They represent the hedge borrowers within the economy. It should be noted, however, in abnormal times like the current Great Recession following the events of 2007/2008, the market rate of interest changes its form on account of the mutable market risk within the ${\text{MU}}$, the so-called systematic component is altering to reflect the growing insecurity and indecision. In fact, whilst the monetary authorities are reducing the bank rate within the market to stimulate growth and increase the level of confidence, as in the 2007/2008 downturn, the commercial banks are raising market rates on loans as well as restricting the growth of the money supply , reflecting the heightened uncertainty and higher pure market risk within the macroeconomy. This is because of the deteriorating state of the macroeconomy, leading to more likelihood of borrowers defaulting on loans .

Moreover, the combination of the two determinants, the systematic and the unsystematic risk , determines the expected, mean rate of return on a portfolio of loans , ${\text{Ei}}_{\text{p}}$, representing interest-earning assets of the banks, which is as follows:

$${\text{Ei}}_{\text{p}} = {\text{Wi}}_{\text{s}} + \left( {1 - W} \right)i_{\text{M}} ,$$

(7.3)

where $i_{\text{s}}$ is the expected mean rate of interest on a subset, $s = 1, 2, \, \ldots \,,n$, of risky loans within the portfolio of interest-earning assets. $i_{\text{m}}$ is the market, mean rate of interest , which can range from virtually no credit risk being added, nearing the risk -free rate, $i_{\text{rf}}$, on loans such as Treasury bills to low doses of credit risk costs , depending on the length of time to maturity, determining a ray of values forming the ‘pure’ market risk within $W$.

This analysis implies that a measure of risk would be the standard deviation between the two, that is

$${\sigma _{{\text{p}}} = \left( {W^{2} \sigma _{{\text{s}}}^{2}+ \left( {1 - W} \right)^{2} \sigma _{{\text{M}}}^{2} + 2W\left( {1 - W} \right){\text{Cov}}\left( {i_{{\text{s}}} ,i_{{\text{M}}} } \right)} \right)^{{\frac{1}{2}}} ,}$$

(7.4)

or

$${ = \sqrt {W^{2} \sigma _{{\text{s}}}^{2} + \left( {1 - W} \right)^{2} \sigma _{{\text{M}}}^{2} + 2W\left( {1 - W} \right){\text{Cov}}\left( {i_{{\text{s}}} ,i_{{\text{M}}} } \right)} ,}$$

where $\sigma_{\text{s}}^{2}$ and $\sigma_{\text{M}}^{2}$ are the variances of the risky and the market loans . The first part of expression (7.4), $W^{2} \sigma_{\text{s}}^{2} + \left( {1 - W} \right)^{2} \sigma_{\text{M}}^{2}$, is the weighted average of the variances of each set of loans , whereas the interesting part is the second portion in the form of $2W\left( {1 - W} \right){\text{Cov}}\left( {i_{\text{s}} , i_{\text{M}} } \right)$, which contains the covariance, ${\text{Cov}}\left( {i_{\text{s}} , i_{\text{M}} } \right)$, of the returns between the two sets of loans .

The market trade-off between the mean rate of interest on the portfolio and the factor of risk at any point is given by ${\partial }{\text{Ei}}_{\text{p}} /{\partial \sigma }_{\text{p}}$, which is essentially equal to

$$\frac{{\partial {\text{Ei}}_{\text{p}} }}{\partial W} . \frac{\partial W}{{\partial \sigma_{\text{p}} }},$$

(7.5)

this is taking into account the weighting component of risky loans that prevail in the banks’ portfolio of assets.

Differentiating ${\text{Ei}}_{\text{p}}$ of (7.3) with respect to $W$ gives

$$\frac{{{\text{Ei}}_{\text{p}} }}{\partial W} = \left( {i_{\text{s}} - i_{\text{M}} } \right).$$

(7.6)

Denoting the expression within the square root of (7.4) by $x$, the analysis uses the inverse rule of $\frac{\partial W}{{\partial \sigma_{\text{p}} }} = \frac{1}{{\frac{\partial x}{\partial W} . \frac{{\partial \sigma_{\text{p}} }}{\partial x}}}$.

In the case of $\sigma_{\text{p}} = x^{{\frac{1}{2}}}$, so that, $\frac{{\partial \sigma_{\text{p}} }}{\partial x} = \frac{1}{2}x^{{ - \frac{1}{2}}} = \frac{1}{2.\sqrt x }$, which is equal to the following format:

$$= \frac{1}{{2.\sqrt {W^{2} \sigma_{\text{s}}^{2} + \left( {1 - W} \right)^{2} \sigma_{\text{M}}^{2} + 2W\left( {1 - W} \right){\text{Cov}}\left( {i_{\text{s}} , i_{\text{M}} } \right)} }}.$$

(7.7)

Now multiplying out $x$ and simplifying before differentiating with regard to $W$ leads to the following form:

$$\frac{\partial x}{\partial W} = 2W\sigma_{\text{s}}^{2} - 2\sigma_{\text{M}}^{2} + 2W\sigma_{\text{M}}^{2} + 2{\text{Cov}}\left( {i_{\text{s}} , i_{\text{M}} } \right) - 4W{\text{Cov}}\left( {i_{\text{s}} , i_{\text{M}} } \right).$$

(7.8)

Putting all the threads together, that is (7.6), (7.7) and (7.8) to form expression (7.5) with the inverse rule, which is

$$\begin{aligned} \frac{{\partial {\text{Ei}}_{\text{p}} }}{{\partial \sigma_{\text{p}} }} & = \frac{{\partial {\text{Ei}}_{\text{p}} }}{\partial W} . \frac{\partial W}{{\partial \sigma_{\text{p}} }} = \frac{{\partial {\text{Ei}}_{\text{p}} }}{\partial W}. \frac{1}{{\frac{\partial x}{\partial W} . \frac{{\partial \sigma_{\text{p}} }}{\partial x}}} = \left( {i_{\text{s}} - i_{\text{m}} } \right) \\ & \quad. \frac{1}{{\left( {2W\sigma_{\text{s}}^{2} - 2\sigma_{\text{M}}^{2} + 2W\sigma_{\text{M}}^{2} + 2{\text{Cov}}\left( {i_{\text{s}} , i_{\text{M}} } \right) - 4W{\text{Cov}}\left( {i_{\text{s}} , i_{\text{M}} } \right)} \right).\left( {\frac{1 }{{2.\sqrt {W^{2} \sigma_{\text{s}}^{2} + \left( {1 - W} \right)^{2} \sigma_{\text{M}}^{2} + 2W\left( {1 - W} \right){\text{Cov}}\left( {i_{\text{s}} , i_{M} } \right)} }}} \right)}}. \hfill \\ \end{aligned}$$

This can be rearranged to derive the following expression:

$$= \left( {i_{\text{s}} - i_{\text{M}} } \right).\frac{{2.\sqrt {W^{2} \sigma_{\text{s}}^{2} + \left( {1 - W} \right)^{2} \sigma_{\text{M}}^{2} + 2W\left( {1 - W} \right){\text{Cov}}\left( {i_{\text{s}} , i_{\text{M}} } \right)} }}{{2W\sigma_{\text{s}}^{2} - 2\sigma_{\text{M}}^{2} + 2W\sigma_{\text{M}}^{2} + 2{\text{Cov}}\left( {i_{\text{s}} , i_{\text{M}} } \right) - 4W{\text{Cov}}\left( {i_{\text{s}} , i_{\text{M}} } \right)}}.$$

(7.9)

If the market is in equilibrium at M, as indicated in Fig. 7.9, then $W$ can be set to zero, because diversification of risky loans removes an element of unsystematic risk until only the systematic (or market) one prevails, and therefore, the portfolio contains the appropriate proportion (or combination) of $S$, and consequently no need to change the weighting. This means at point $M,\,W = 0$, which reduces (7.9) to

Fig. 7.9

The equilibrium of the portfolio possibilities and the efficient frontier

$$= \left( {i_{\text{s}} - i_{\text{M}} } \right).\left( {\frac{{\sigma_{\text{M}} }}{{{\text{Cov}}\left( {i_{\text{s}} , i_{\text{M}} } \right) - \sigma_{\text{M}}^{2} }}} \right).$$

(7.10)

If the market line (ML) is tangential to the efficient frontier, EE, at M, as shown in Fig. 7.9, where the slope of the ML curve is

$$\frac{{i_{\text{m}} - i_{\text{rf}} }}{{\sigma_{\text{M}} }},$$

(7.11)

where $i_{\text{rf}}$, is the risk -free rate of interest and $\sigma_{\text{M }}$ is the market standard deviation. Equating (7.10) with (7.11) gives $\left( {i_{\text{s}} - i_{\text{M}} } \right).\frac{{\sigma_{\text{M}} }}{{{\text{Cov}}\left( {i_{\text{s}} , i_{\text{M}} } \right) - \sigma_{\text{M}}^{2} }} = \frac{{\left( {i_{\text{M}} - i_{\text{rf}} } \right)}}{{\sigma_{\text{M}} }}$. Now simplifying for $i_{\text{s}}$ derives the following expression:

$${i_{{\text{s}}}= i_{{{\text{rf}}}}+ (i_{{\text{M}}}- i_{{{\text{rf}}}} )\left( {\frac{{{\text{Cov}}\left( {i_{{\text{s}}} ,i_{{\text{M}}} } \right)}}{{\sigma _{{\text{M}}}^{2} }}} \right),}$$

or

$${i_{{\text{s}}}= i_{{{\text{rf}}}}+ \beta _{{\text{S}}} (i_{{\text{M}}}- i_{{{\text{rf}}}} ),}$$

(7.12)

since $\beta_{\text{S}} = \left( {\frac{{{\text{Cov}}\left( {i_{\text{s}} ,i_{\text{M}} } \right)}}{{\sigma_{\text{M}}^{2} }}} \right)$, it formulates the loan asset subset’s risk relative to the dangers of the whole market portfolio. ¹¹ This is the additional rate of interest over and above $i_{\text{rf}}$ that is required on each asset, whose risk characteristic is compared with the whole market portfolio via the $\beta$. In other words, Eq. (7.12) displays the theory how to find the expected mean rate of interest on a risky set of loanable funds , $i_{\text{s}}$. Clearly, this depends on the risk -free rate of interest , $i_{\text{rf}}$, plus a weighted market premium , which depends on the loan set’s risk relative to the market one.

If $\beta_{\text{S}} = 1$, then the loan subset has the equivalent risk characteristic as the whole market portfolio and its expected mean rate of interest is the same as the market one. If $\beta_{\text{S}} \text{ < }1$, then the expected mean rate of interest is less than the whole, which is a ‘hedge ’ loan set of assets, and finally, if $\beta_{\text{S}} \text{ > }1$, then the loans are riskier than the market portfolio and attracts higher mean rate of interest to reflect the greater credit risk on account of a speculative or Ponzi loan set of assets. If $\beta \, = 0$, then the expected mean rate of return is equal to the risk -free, mean rate, $i_{\text{rf}}$.

So far, the analysis is at the micro-level with regard to its implications for theory. At the macro-level of aggregation, the risky loans granted by the commercial banks at higher interest rates than the norm represent the ‘speculative ’ and ‘Ponzi ’ borrowers . If the bubble bursts like in the case of the housing market in 2007/2008, then these borrowers can cause the banking sector to freeze up, providing the initial condition for a slump in economic activity in the form of a recession or depression . There is a Minsky Moment , a sudden catastrophic fall in asset prices that represents a snowball effect of multiplication of the initial, increasing state of risk . Briefly, the risk builds on itself. Clearly, these categories prevail within the mortgage sector as well as borrowers on low incomes to who find it difficult to reach the end of the month and require rolling overdraft facilities. As retail banks withdraw from this segment of the market, they leave the doors open for loan sharks to enter, like Wonga. ¹²

Moreover, the end result is an unstable financial environment, caused by the increase in F. Knight uncertainty because the rate of default increases on Ponzi loans , resulting an accumulation of insolvent debt carried by the banking system and the adoption of credit -rationing practices (Stiglitz and Weiss 1981). ‘Speculative ’ borrowers find they cannot refinance the principal loans even if they are able to cover the higher interest rate payments. This chain reaction even affects the ‘hedge ’ borrowers who are able to cover the interest and principal from their current investment income. If hedge financing dominates the banking sector, then it is more likely to be stable. In the opposite case, with a greater weight of speculative and Ponzi borrowers , then it is likely to be unstable.

7.1.3 Conclusion/Summary

In this Appendix, the focus has been on the determination of the rate of interest on loanable assets, whereby the required rate of interest is made up from the risk -free rate of interest plus the risk premium , which is the difference between the market portfolio of risky loans and the risk -free rate in collaboration with $\beta \, -$ coefficient. This measures the risk on a subset of loanable assets if it is included in the portfolio, taking into account the whole market together with its standard deviation. This framework develops an understanding of the financial crisis with the concepts provided by Minsky, which is centrepiece in the next chapter on catastrophe theory .