Time-Varying Volatility of Bank Betas

MATT BRIGIDA

Associate Professor of Finance (SUNY Polytechnic Institute)

Time-Varying Volatility of Bank Betas (pdf paper)

Previous Research: Drechsler et al (2021)

Banks hedge interest rate risk by matching the sensitivities of interest income and expense to the short rate
Net interest margins are insensitive to interest rate changes
Key is the deposit franchise which acts like long-term debt rather than the short rate
Maturity transformation does not cause interest rate risk

Estimate static interest income and expense beta coefficients over their entire sample (1984 to 2017)
We should expect substantial variation in interest income betas given the sensitivity of duration to the coupon rate and yield
Are able to match interest income and expense betas at each quarter?
Measure a bank's uncertainty about whether they will be matched
This uncertainty is a yet unmeasured source of bank risk

Hypothesis and Goals

Banks forecast future interest income and expense betas and then adjust their balance sheet to attempt to lessen any difference
Banks reforecast betas and adjust their balance sheet in a continual dynamic matching strategy
A natural model of this process is the Kalman filter
Models a rational Bayesian market participant which updates forecasts of estimated coefficients as new information arrives

Test whether expense and income betas are matched each quarter
Measure the uncertainty of beta estimates
Test if the market prices this uncertainty

Data

Built from a database of FDIC Call reports and ranges from October 1992 to June 2024
Total interest income to assets (FDIC BankFind code: INTINCY), expense to assets (EINTEXP), and assets for each bank over each quarter
Aggregate into deciles to control for bank size

Empirical Methods

Constant Coefficient Regressions

Interest Income Regression

$\Delta IntInc_{dt} = \alpha^{Inc}_d + \beta^{Inc}_{d,0} \Delta FedFunds_{t}$

$+ \beta^{Inc}_{d,1} \Delta FedFunds_{t-1} + \epsilon_{dt}$

and we report:

$$\beta^{Inc}_d = \beta^{Inc}_{d,0} + \beta^{Inc}_{d,1}$$

for each decile d.

Interest Income Regression

$\Delta IntExp_{dt} = \alpha^{Exp}_d + \beta^{Exp}_{d,0} \Delta FedFunds_{t}$

$+ \beta^{Exp}_{d,1} \Delta FedFunds_{t-1} + \epsilon_{dt}$

and we report:

$\beta^{Exp}_d = \beta^{Exp}_{d,0} + \beta^{Exp}_{d,1}$

for each decile d.

NIM Betas

$$\beta^{NIM}_d = \beta^{Inc}_d - \beta^{Exp}_d$$

Test for Non-Constant Coefficients

Brown, Durbin, Evans (1975)
Alternative is coefficients which follow a random walk.

State-Space Model

Kim and Nelson (1999)

"One nice thing about the Kalman Filter is that it gives us insight into how a rational economic agent would revise his estimates of the coefficients in a Bayesian fashion when new information is available in a world of uncertainty, especially under a changing policy regime"

$$\Delta IntInc_{dt} = \alpha_{d,t} + \beta^{Inc}_{d,0, t} \Delta FedFunds_{t}$$

$$+ \beta^{Inc}_{d,1, t} \Delta FedFunds_{t-1} + \epsilon_{dt}$$

where coefficients take the form of a random walk:

$$\alpha^{Inc}_{d,t} = \mu_1 + \gamma_1 \alpha^{Inc}_{d, t-1} + \nu_{1,t}$$ $$\beta^{Inc}_{d,0, t} = \mu_2 + \gamma_2 \beta^{Inc}_{d,0, t-1} + \nu_{2,t}$$ $$\beta^{Inc}_{d,1, t} = \mu_3 + \gamma_3 \beta^{Inc}_{d,1, t-1} + \nu_{3,t}$$

$$\epsilon_t \sim i.i.d. N(0, R)$$ $$\nu_t \sim i.i.d. N(0, Q)$$ $$E(\epsilon_t, \nu'_t) = 0$$

$\Delta IntInc_{dt}$ is the quarterly change interest income for decile $d$ at time $t$
$\Delta FedFunds_t$ is the quarterly change in the Federal Funds rate at time $t$
$\beta^{Inc}_{d,0,t}$ is the time-varying coefficient on the contemporaneous Federal Funds rate change for decile $d$ at time $t$
$\beta^{Inc}_{d,1,t}$ is the coefficient on the Federal Funds rate change lagged one quarter.

Conditional Variance

$H_{t|t-1} = x_{t-1}P_{t|t-1}x'_{t-1} + \sigma^2_{\epsilon}$

where:

$x_{t-1}$ is the vector is the Fed Funds rate and its lag at time t-1
$P_{t|t-1}$ covariance matrix of $\beta_t$ conditional on information up to $t-1$. This is the degree of uncertainty associated with the inference on $\beta_t$ at time $t-1$.

Results

Time-Varying Beta Coefficients

Interest Expense Betas

Interest Income Betas

Net Interest Margin Betas

Granger Causation

Conditional Volatility

Interest Expense Volatility

Interest Income Volatility

Market Pricing of Beta Volatility

Estimate an index model including time-varying volatility estimates
Dependent variable is $XLF$
Quarterly data from Q1 1999 to Q1 2024

Pricing Equation

$$\begin{equation} r_{XLF, t} = \gamma_0 + \gamma_1 \Delta CV_{Exp, t} + \gamma_2 \Delta CV_{Inc, t} + \gamma_3 r_{M, t} + \xi_t \end{equation}$$

Implications

One standard deviation increase in uncertainty $\Rightarrow$ 2.34% decline in stock values
This lowers the largest bank stocks by about \$47 billion

Conclusion

Time-Varying Betas

We find evidence that interest income and expense betas vary substantially over time
Weak evidence that expense betas Granger cause income betas

Beta Forecast Uncertainty

Beta uncertainty rose prior to the 2008 and 2023 financial crises
Expense uncertainty rose more for large banks, and income uncertainty for small banks
Uncertainty was very low from 2009 to 2019---a benefit of ZIRP
Interest expense uncertainty Granger causes income uncertainty

Pricing Beta Forecast Uncertainty

Interest expense uncertainty is negatively affects bank stock prices
One standard deviation increase in uncertainty $\Rightarrow$ 2.34% in stock values
This relationship is new to the literature and cannot be hedged

Future Research

Beta and Volatility by Bank Operational Focus

SPECGRP: title: Asset Concentration Hierarchy description: An indicator of an institution's primary specialization in terms of asset concentration if (ctx.SPECGRP == 0) { ctx.SPECGRPN = 'No Specialization Group'; } else if (ctx.SPECGRP == 1) { ctx.SPECGRPN = 'International Specialization'; } else if (ctx.SPECGRP == 2) { ctx.SPECGRPN = 'Agricultural Specialization'; } else if (ctx.SPECGRP == 3) { ctx.SPECGRPN = 'Credit-card Specialization'; } else if (ctx.SPECGRP == 4) { ctx.SPECGRPN = 'Commercial Lending Specialization'; } else if (ctx.SPECGRP == 5) { ctx.SPECGRPN = 'Mortgage Lending Specialization'; } else if (ctx.SPECGRP == 6) { ctx.SPECGRPN = 'Consumer Lending Specialization'; } else if (ctx.SPECGRP == 7) { ctx.SPECGRPN = 'Other Specialized Under 1 Billion'; } else if (ctx.SPECGRP == 8) { ctx.SPECGRPN = 'All Other Under 1 Billion'; } else if (ctx.SPECGRP == 9) { ctx.SPECGRPN = 'All Other Over 1 Billion'; } else { ctx.SPECGRPN = 'Error in Specialization Group'; }