Appendix A: Pass-through Lower Bound in BA-MARTIN | RDP 2022-08: The Consequences of Low Interest Rates for the Australian Banking Sector

RDP 2022-08: The Consequences of Low Interest Rates for the Australian Banking Sector Appendix A: Pass-through Lower Bound in BA-MARTIN

Anthony Brassil

December 2022

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A.1 The immediate effect

By substituting Brassil et al (2022)'s Equations (8), (11) and (12) into their Equation (14), the immediate effect of a change in the cash rate (r_C_,t ) on lending rates is:

\frac{d r_{M, t}}{d r_{C, t}} = \frac{d r_{D, t}}{d r_{C, t}} + \underset{\begin{matrix} Endogenous response \\ to capital ratio \end{matrix}}{\underset{︸}{[\frac{- 400 λ w_{t}}{τ - 400 w_{t} e_{t} β_{M, t}}]}} \times [\underset{\begin{matrix} effect of NIM \\ on capital ratio \end{matrix}}{\underset{︸}{\frac{τ w_{t - 1} e_{t}}{400 w_{t}}}} + \underset{\begin{matrix} Effect of \\ credit growth \\ on capital ratio \end{matrix}}{\underset{︸}{- e_{t} β_{M, t}}}] \times \frac{d r_{D, t}}{d r_{C, t}}

The reversal rate exists if $\frac{d r_{M, t}}{d r_{C, t}} < 0.$ Given that $\frac{d r_{D, t}}{d r_{C, t}} > 0,$ simplification of the equation above shows that an immediate reversal rate occurs iff:

\frac{λ e_{t} (τ w_{t - 1} - 400 w_{t} β_{M, t})}{τ - 400 w_{t} e_{t} β_{M, t}} > 1

With credit demand contemporaneously responding negatively to interest rates ( $β_{M, t} < 0,$ as discussed in Brassil et al (2022)), even if the behavioural response parameter $λ$ takes its highest possible value of 1, the above condition would require w_t–1e_t >1, which can never hold as w_t–1 is the lagged aggregate risk weight (w_t–1 < 1) and e_t is the equity share of assets (e_t < 1). Therefore, in the baseline version of BA-MARTIN, cash rate cuts will always have an initial expansionary effect via the banking sector.

A.2 The low-for-long effect

While there is no immediate reversal rate, holding rates low for an extended period will amplify the credit demand response. There is also an additional amplifying mechanism when rates are held low for an extended period. The nonlinearity of the banking sector in BA-MARTIN means the amount banks want to increase their capital ratios in each period depends on the amount of capital they have at the beginning of the period. This means that smaller capital ratio increases in previous periods (resulting from a cash rate reduction) will affect pass-through in those periods, and in all future periods in which capital ratios remain below their target.

To see if these additional low-for-long mechanisms can lead to a reversal rate, I continuously substitute the expressions for past capital shortfalls into Brassil et al (2022)'s Equation (14). This provides an expression for the effect an extended period of lower cash rates $(r_{C, s \to t})$ has on current lending rates (r_M,t):^[20]

(A1)

\frac{d r_{M, t}}{d r_{C, s \to t}} \approx \underset{A}{\underset{︸}{\frac{d r_{D, t}}{d r_{C, t}}}} + \underset{B}{\underset{︸}{(\frac{- 400 λ \bar{w}}{τ - 400 \bar{w e} β_{M}}) \frac{d y_{t}}{d r_{C, s \to t}}}} + [\underset{C}{\underset{︸}{- ψ}} + \frac{- 400 λ \bar{w} (1 - λ)}{\underset{D}{\underset{︸}{τ - 400 \bar{w e β_{M}}}}}] \underset{E}{\underset{︸}{\sum_{j = 0}^{t - 1 - s} {(1 - λ)}^{j} \frac{d y_{t - 1 - j}}{d r_{C, s \to t}}}}

where

\frac{d y_{t - j}}{d r_{C, s \to t}} \approx \underset{\begin{matrix} Effect of NIM \\ on capital ratio \end{matrix}}{\underset{︸}{(\frac{τ \bar{e}}{400}) \frac{d r_{D, t - j}}{d r_{C, s \to t}}}} + - \bar{e} \underset{\begin{matrix} Effect of \\ credit growth \\ on capital ratio \end{matrix}}{\underset{︸}{\frac{d Δ a_{t + 1 - j}}{d r_{C, s \to t}}}}

Explaining the components of Equation (A1):

The effect of the current lower cash rate on current debt funding costs.
The endogenous response to the effect of the persistently lower cash rate on the current capital ratio.
The effect persistently lower capital ratios have on debt funding costs by increasing bank default risk.
The endogenous response to persistently lower capital ratios.
The effect persistently lower cash rates have had on past capital ratios.

Equation (A1) is an approximation of the true response because in places where the risk weights and equity shares act as coefficients, I have replaced their contemporaneous values with their steady-state values. The magnitude of the effects would be little different without this approximation, but the expression would be far more complex and less intuitive.

Equation (A1) does not solve to a simple inequality as with the ‘immediate effect’ above. Instead, by replacing each NIM and credit growth effect by the largest possible values these effects take over the extended period, I can construct a lower bound for pass-through:

(A2)

\frac{d r_{M, t}}{d r_{C, s \to t}} > \frac{d r_{D, t}}{d r_{C, t}} - {\frac{400 \bar{w}}{τ - 400 \bar{w e} β_{M}} [1 - {(1 - λ)}^{t - s + 1}] + \frac{ψ}{λ} [1 - {(1 - λ)}^{t - s}]} Θ_{t}

where

Θ_{t} = (\frac{τ \bar{e}}{400}) \frac{d r_{D, t}}{d r_{C, t}} + \max {- \bar{e} \frac{d Δ a_{t + 1 - j}}{d r_{C, s \to t}} \forall t \geq s}

With the baseline calibration of BA-MARTIN, Equation (A2) is bounded by:

\frac{d r_{M, t}}{d r_{C, s \to t}} > \frac{d r_{D, t}}{d r_{C, t}} - [1 - {0.85}^{t - s + 1}] [0.0525 \times \frac{d r_{D, t}}{d r_{C, t}} + 30 \max {- \frac{d Δ a_{t + 1 - j}}{d r_{C, s \to t}} \forall t \geq s}]

Following a 100 basis point cash rate reduction, the maximum increase in quarterly credit growth is less than 1 percentage point, even after holding the cash rate at this lower level for a decade. Substituting in this maximum credit growth response, the lower bound simplifies to the following:

\frac{d r_{M, t}}{d r_{C, s \to t}} > 0.94 \times \frac{d r_{D, t}}{d r_{C, t}} - 0.3

Even at the lowest debt funding cost pass-through that results from the retail deposit ELB, this lower bound remains above zero. Therefore, even with an incredibly conservative methodology – that would also require banks to be below their target capital ratio and be excluded from external equity markets for an extended period – the lending rate pass-through lower bound with the baseline BA-MARTIN calibration remains above zero in a low-for-long scenario.

A.3 Zero pass-through is sufficient to prevent capital ratio deterioration following a cash rate cut

From Equation (9) in Brassil et al (2022), changes in banks' capital ratios can be decomposed into changes in their equity, credit growth and changes to their risk weights. Once provisioning for losses has returned to normal, risk weights do not respond to changes in the cash rate. And with credit demand only directly responding to cash rate changes via the resulting change in lending rates, zero pass-through would also mute the effect of cash rate changes on credit growth. Therefore, in the stressed state, it is only through changes in banks' return on assets that cash rate changes would affect banks' capital ratios when pass-through is zero.

From Equation (8) in Brassil et al (2022), cash rate reductions increase banks' return on assets when pass-through is zero:

\frac{d R O A_{t}}{d r_{C, t}} = \frac{- τ}{400} (1 - w_{t - 1} e_{t}) \frac{d r_{D, t}}{d r_{C, t}} < 0

This means that if banks were happy with the speed at which they were replenishing their capital ratios before the cash rate reduction, they would be no worse off after the cash rate reduction if pass-through was zero. Therefore, zero pass-through is always more than sufficient to maintain banks' desired speed of capital ratio replenishment, which means a Brunnermeier and Koby (2018) reversal rate cannot exist.

This can also be shown by considering how banks would need to change their desired speed of adjustment (the $λ$ parameter in BA-MARTIN) if lending rates remaining constant following the cash rate reduction was the optimal response:

\frac{d λ}{d r_{C, t}} \approx - [\frac{(τ - 400 \bar{w e} β_{M}) + λ τ (1 - \bar{w e})}{400 \bar{w} z_{t}}] \frac{d r_{D, t}}{d r_{C, t}} < 0

This inequality shows that if the cash rate is reduced, it is optimal for banks to keep their lending rates constant only if they desire an increased speed of adjustment. The implication being that if banks keep their desired speed of adjustment constant (as is assumed in BA-MARTIN because this is considered a behavioural parameter), then zero pass-through would be excessively responsive to the cash rate change.

A.4 The credit risk channel with less responsive banks

If banks became sufficiently unresponsive to capital shortfalls (i.e. $λ \to 0$ ), Equation (A1) would simplify to the following:

\frac{d r_{M, t}}{d r_{C, s \to t}} \approx \frac{d r_{D, t}}{d r_{C, t}} - ψ \sum_{j = 0}^{t - 1 - s} \frac{d y_{t - 1 - j}}{d r_{C, s \to t}}

With $\frac{d y_{t - j}}{d r_{C, s \to t}} \geq 0$ , the longer the cash rate is held lower, the more likely this sum will become sufficiently large to generate a reversal rate. However, the longer it takes for this sum to become large, the more likely it is that banks' capital ratios will have returned to target, they will have regained access to equity markets, or some policy response is implemented to alleviate the problem. Therefore, this equation is not proof that a reversal rate would exist in a low-for-long scenario even with sufficiently unresponsive banks.

Footnote

r_{C, s \to t} = x

denotes the cash rate being held constant at

x

between periods s and t.

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