# RDP 2022-03: Macrofinancial Stress Testing on Australian Banks Appendix A: Model Specifications

## A.1 Overview

This appendix presents a set of technical model documentation in equations. It can be read alongside the documentation provided in the R language found in the supplementary information. Throughout the following description of the model, variables that are part of the assumed economic scenario are denoted with a hat accent, that is, $\stackrel{^}{variable}$. These comprise macroeconomic variables and time-varying exogenous shocks. Variables that are model parameters that can be changed are denoted with a bar, that is, $\overline{variable}$. Variables without accents are observed in the data, endogenous, or are pre-calibrated sensitivities to the macroeconomic environment.

## A.2 Notation and implicit assumptions

The model generally works by taking an observed starting value (e.g. observed mortgage PDs in December 2021) and evolving them through the simulated periods, by applying model mechanisms and other inputs. The observed starting values are denoted with subscript 0. Negative subscripts denote observed data from earlier quarters. Sometimes multiple steps are required to update a variable from t–1 to t, and in these cases, the intermediate updates are denoted with prime (e.g. for two steps f1 and f2, the notation is ${x}^{\prime }={f}_{1}\left({x}_{t-1}\right)$ then ${x}_{t}={f}_{2}\left({x}^{\prime }\right)$). Where these prime are used, the superscript L simply denotes the last iteration, that is, the iteration with the largest number of prime. Where an interim variable is used but its value does not need to be stored for use in later periods, the t subscript is sometimes dropped.

The variable subscripts are: i represents a bank; t represents a period; and j represents an ‘asset class’. The sum of all the asset classes always equals the total value of the balance sheet. For example, within the asset class ‘credit cards’, there may be various subtypes of credit cards debt, but none of those subtypes can be defined as asset classes, or there would be double counting when all the asset classes are summed. The balance of an asset class can be referred to as Balanceijt or by name, such as AC.Cashit, in which case the asset class will always be preceded by ‘AC’ (like the ‘AC_’ in the code).

Implicit model assumptions:

• Throughout the model, liabilities are implicit. All explicit changes are to assets and equity/capital, and liabilities implicitly move to balance the balance sheet.

Unless otherwise specified, the variable denominations follow the same conventions as the code. That is:

• All variables referring to Australian dollar amounts are expressed in whole dollars (e.g. not millions etc).
• All variables expressing proportions or percentages are expressed in raw form, that is, 0.01 is 1 per cent, not 1 basis point.
• All figures, rates, proportions etc are expressed on a quarterly basis, except PDs which are annualised (or more specifically 4x the quarterly value).

## A.3 Selected variables

Table A1: Selected Variables in Stress Testing Model
Variable Definition (specification) Source
RetEarn Profit retained ($) Derived NPAT Net profit after tax ($) APRA (ARF_330.0)
Dividend Dividends ($) Derived NetIncome Net income ($) Derived
NetIntInc Net interest income ($) APRA (ARF_330.0) FireSaleLoss Losses due to asset fire sales ($) Derived
Tax Taxes ($) Derived OtherInc Other income ($) APRA (ARF_330.0)
OpExp Operating expenses ($) APRA (ARF_330.0) BDD Bad and doubtful debt charge Derived CET1Capital CET1 capital ($) APRA (ARF_110.0)
TotAssets Total assets ($) APRA (ARF_322.0) Reinvestment Cash reinvested ($) Derived
TotLiabilities Total liabilities ($) Derived AvgRW Average risk weight (per cent) Derived AvgLnRW Average loan risk weight (per cent) Derived RWA RWA ($) APRA (ARF_110.0)
BankType Indicator, IRB bank = 1 APRA
CET1Ratio CET1 capital ratio (per cent) Derived
IntInc Interest income ($) APRA (ARF_330.1) IntExp Interest expense ($) APRA (ARF_330.1)
CapEffectOnFunding Effect of capital on funding (per cent) Derived
FundContagion Capital contagion effects on funding (per cent) Derived
AvgLendingRate Average lending rate (per cent) Derived
AvgFundingRate Average funding rate (per cent) Derived
TotEquity Total equity ($) APRA (ARF_322.0) OpExp Operating expense ($) APRA (ARF_330.0)
WholesaleLiabPrpn Proportion of wholesale liabilities (per cent) Derived
LMI Housing loans with LMI ($) APRA (ARF_223.0) OutstandingMortgagesLVR80 Mortgages with LVRs ≥ 80 APRA (ARF_223.0) AvailableCreditLimits Business credit lines available ($) APRA (ARF_741.0)
CLD Business credit line drawdown ($) Derived Write-off Write-offs ($) Derived
Provisions Provisions Derived
FundingRun Size of liquidity run ($) Derived RunGap Liquidity run size after cash Derived TaxRate Corporate tax rate Australian Government SpareCET1 Spare CET1 ratio available for dividends (per cent) Derived MaxDiv Maximum dividend payable Derived PayoutRatio Dividend payout ratio Derived Tier1Capital Tier 1 capital ($) APRA (ARF_110.0)
Tier2Capital Tier 2 capital ($) APRA (ARF_110.0) Replenish Replenishment of liquid assets ($) Derived
NetCashIncome Cash flow (\$) Derived

## A.4 Macroeconomic variable inputs

The model uses four macroeconomic variables. These are the GDP quarterly growth rate $\left(\stackrel{^}{GD{P}_{t}}\right)$, the unemployment rate $\left(\stackrel{^}{U{R}_{t}}\right)$, housing price growth rate $\left(\stackrel{^}{H{P}_{t}}\right)$, and commercial real estate price growth rate $\left(\stackrel{^}{CR{E}_{t}}\right)$.[38]

The evolution of these macroeconomic variables over time are usually supplied (assumed from) by a fully specified macroeconomic model such as the one documented in Ballantyne et al (2019). Alternatively they can be determined endogenously by the macroeconomic equations in the model given the history of these variables and specifying any other shocks to be included. When following the latter approach, the path of quarterly GDP growth and the unemployment rate are modelled in terms of deviations from their equilibrium rates:

(A1) $GD P t ^ − GDP* ¯ = α 1 ( GD P t−1 ^ − GDP* ¯ )+ α 2 ( GD P t−2 ^ − GDP* ¯ )+ α 3 ( U R t−1 ^ − UR* ¯ )$
(A2) $U R t ^ − UR* ¯ = β 1 ( U R t−1 ^ − UR* ¯ )+ β 2 ( U R t−2 ^ − UR* ¯ )+ β 3 ( GD P t ^ − GDP* ¯ ) + β 4 ( GD P t−1 ^ − GDP* ¯ )+ β 5 ( GD P t−2 ^ − GDP* ¯ )$

where $\overline{GDP*}$ represents the model-assumed equilibrium GDP growth, $\overline{UR*}$ represents the model-assumed equilibrium unemployment rate, and where

$α 1 =−0.177 α 2 =0.036 α 3 =0.141 β 1 =1.236 β 2 =−0.234 β 3 =−0.036 β 4 =−0.078 β 5 =−0.097$

In a similar manner, forecasts of house price growth depend on lagged house price growth and current GDP growth, while commercial real estate price growth depends on its own lag, GDP growth and current house price growth.

(A3) $H P t ^ = θ 1 + θ 2 H P t−1 ^ + θ 3 GD P t ^$
(A4) $CR E t ^ = γ 1 + γ 2 CR E t−1 ^ + γ 3 GD P t ^ + γ 4 H P t ^$

where

$θ 1 =0.322 θ 2 =0.632 θ 3 =0.212 γ 1 =−0.050 γ 2 =0.769 γ 3 =0.155 γ 4 =0.203$

The coefficients for GDP growth and the unemployment rate equations and commercial price growth are estimated using quarterly data for the period from 1995 to 2019. For house price growth we use a quarterly sample from 1980 to 1992, which has the benefit of being a period of larger property price falls along with significant negative GDP growth outcomes, which provides better estimates for the dynamics of these variables in a stress testing context.[39]

## A.5 Model overview

The model simulates profit and loss (P&L), balance sheet and capital dynamics. The main aggregate variables modelled are listed in the following equations. Beneath these aggregates, the model simulates their subcomponents in more detail.

First the P&L effects are modelled.

(A5) $RetEar n it =NPA T it −Dividen d it$
(A6) $NPA T it =NeIncom e it −BD D it −FireSaleLos s it −Ta x it$
(A7) $NeIncom e it =NeIntIn c it +OtherIn c it −OpEx p it$

Once retained earnings is formulated, capital is updated.

(A8) $CET1Capita l it =CET1Capita l it−1 +RetEar n it$

Asset balances are updated to take into account losses and reinvestments.[40]

(A9) $TotAsset s it =TotAsset s it−1 +RetEar n it +R einvestmen t it$
(A10) $Reinvestmen t it =f( BD D it ,RetEar n it ,CET1Capita l it ,AvgLnR W it )$
(A11) $TotLiabilitie s it =TotLiabilitie s it−1 +( TotAsset s it −TotAsset s it−1 )+RetEar n it$

Risk-weighted assets (RWAs) and capital ratios are updated.

(A12) $AvgLnR W it =f( BankTyp e i , GD P t ^ ,P D i0 ,P D it−1 ,P D it ,LG D i0 ,LG D it−1 ,LG D it )$
(A13) $RW A it =( ∑ k∈C Balanc e ikt L ×AvgLnR W it )*( 1+ RWAShoc k t ^ )$
(A14) $CET1Rati o it = CET1Capita l it RW A it$

## A.6 Retained earnings – net income

The components of net income are each projected based on recently observed values. The first component, net interest income ( NetIntIncit ), is further separated into interest income ( IntIncit ) and interest expenses ( IntExpit ).

(A15) $NetIntIn c it =IntIn c it −IntEx p it$

Endogenous funding costs (optional)

There are three endogenous funding cost options in the model: funding costs can depend on banks' own CET1 ratios; funding costs can experience contagion from other banks' CET1 ratios; and funding costs can depend on GDP growth. First, each bank's endogenous funding effect is formulated.

(A16) $CapEffectOnFundin g it =max{ 0, CET1FundCostThresh ¯ −CET1Rati o it−1 } × CapCoefOnFundRate ¯$

If the option for endogenous effects of a bank's own capital is switched on, then the option for funding cost contagion can also be switched on. This additional funding cost component takes the difference between the largest capital effect on funding across all banks and that bank's own capital effect, and then multiplies it by a pre-specified contagion weight parameter. If this weight is set to one, all bank's experience the same funding cost effect, equal to what the worst bank would experience on its own.

(A17) $FundContagio n it ( max j { CapEffectOnFundin g jt }−CapEffectOnFundin g it ) × ContagionWeight ¯$

Finally, if the option for endogenous effects of the macroeconomic environment (GDP growth) is switched on, then the effect of GDP growth on funding costs are formulated.

(A18) $GDPEffectOnFundin g it =max{ 0, GDPCoefOnFundRate ¯ ×( GD P t ¯ − GD P 0 ^ ) }$

When these options are switched on – that is, these components are not set to zero – the value of their sum is multiplied by the bank's proportion of funding that is wholesale (i.e that is not sourced from retail deposits), before they feed into funding costs.

Funding costs

Interest income and expenses are projected by assuming that the average lending interest rate earnt across all assets, and the average funding interest rate paid across all liabilities, remain constant over time, aside from any assumed shocks (specified in percentage points), and any endogenous effects of capital on funding costs (if this option is switched on). Aside from shocks and endogenous effects (which are often set to zero), these rates are assumed constant at their starting values (annualised).

(A19) $AvgLendingRat e it =4× IntIn c i0 TotAsset s i0 + LendingShoc k t ^$
(A20) $AvgFundingRat e it 4× IntEx p i0 TotAsset s i0 −TotEquit y i0 + FundingShoc k t ^ +WholesaleLiabPrp n i0 ×( CapEffectOnFundin g it +FundContagio n it +GDPEffectOnFundin g it )$

Income

To formulate the quarterly income components, these average rates are applied to the lagged asset and liability values.

(A21) $IntIn c it = AvgLendingRat e i 4 ×TotAsset s it−1 IntEx p it = AvgFundingRat e i 4 ×( TotAsset s it−1 −TotEquit y it−1 )$

Other income and operating expenses are assumed to grow at the same rate as the balance sheet (with a one quarter lag), plus exogenous shocks. Their starting values are the average over the four quarters up to the starting period. This averages out one-off items, without requiring discretion to remove them one by one.

(A22) $OtherIn c it =OtherIn c it−1 × TotAsset s it−1 TotAsset s it−2 ×( 1+ OtherIncShoc k t ^ )$
(A23) $OtherIn c i0 = 1 4 ∑ t=−3 0 OtherIn c it$
(A24) $OpEx p it =OpEx p it−1 × TotAsset s it−1 TotAsset s it−2 ×( 1+ OpExpShoc k t ^ )$
(A25) $OpEx p i0 = 1 4 ∑ t=−3 0 OpEx p it$
(A26) $NetIncom e it =NetIntIn c it +OtherIn c it −OpEx p it$

## A.7 Retained earnings – bad and doubtful debts charge

The bad and doubtful debt charge has two components: write-offs and net new provisions. Write-offs are losses realised in the contemporaneous quarter (for simplicity it is assumed that defaulted loans are written off in the same quarter). Net new provisions are the change in the provisions balance from one quarter to the next, capturing the change in the outlook for forward-looking losses. This captures all types of loan-loss provisions, although the model could also be interpreted as net new provisions covering collective provisions, with specific provisions assumed to crystallise into write-offs within the quarter and be captured by write-offs.

The approach for projecting net new provisions is a modification of the approach for projecting write-offs, so write-offs will be described first.

The model separates banks' balance sheets into 22 asset classes, each categorised as loans, securities or other. The bad and doubtful debts charge applies to the 10 asset classes in the ‘loans’ category. Three of these loan asset classes – domestic mortgages, overseas mortgages and domestic business loans – together comprise around 80 per cent of banks' total lending, so these asset classes are modelled in more detail.

Mortgage loss rates

The basic idea of modelling mortgage losses is to update banks' mortgage LVRs, and hence LGDs, based on the housing price scenario, and to update the default rates based on the unemployment rate.

The mortgage-loss projections utilise detailed inputs on banks' mortgage LVRs. In the following, vectors are denoted in bold script, and single elements within those vectors are denoted with a k subscript and non-bold script. The mortgage-loss inputs comprise:

• starting-period data on each bank's mortgage portfolio LVR distribution, that is, the proportion of its mortgages that are in each of 250 LVR buckets $\underset{250×1}{\underbrace{\left(L{D}_{i0}\right)}}$;
• the same LVR distribution but for only newly issued mortgages in the starting-period quarter $\underset{250×1}{\underbrace{\left(L{N}_{i0}\right)}}$; and
• a vector of multiplier parameters for capturing the correlation between mortgage LVRs and probabilities of default $\underset{250×1}{\underbrace{\left(\overline{LM}\right)}}$.

A bank's LVR distribution is updated based on the housing price scenario outcome $\left(\stackrel{^}{H{P}_{t}}\right)$ and the mortgage flows parameter $\left(\overline{MortFlows}\right)$, which here captures amortisation.

(A27) $L D ′ it = f LD ( 1 1+ H P t ^ ×( 1− MortFlows ¯ ),L D it−1 )$

The function fLD shifts each proportion in $L{{D}^{\prime }}_{it}$ across buckets in line with the housing price growth and the amortisation rate.[41]

Then the LVR distribution is updated based on the inflow of new credit, which is assumed to happen at the same rate as amortisation, plus any assumed credit growth floor (note that the parameters are each scalars). The outflow of credit captured by $\overline{MortFlows}$ in Equation (A28) does not perfectly offset the inflow captured by $\overline{MortFlows}$ in this step, because the flow in Equation (A27) shifts the LGD distribution without changing its shape, and the flow in this step acknowledges that new mortgages are likely to put more weight in the distribution at LGDs representing mortgages at origination.

(A28) $L D ′ it = L D ′ it +( MortFlows ¯ + CreditGrFloor ¯ )×L N i0 1+ MortFlows ¯ + CreditGrFloor ¯$

First the LMI coverage for loans with LVRs of greater than or equal to 80 per cent are calculated using starting balances.

(A29) $LMICoverag e it = LM I i0 OutstandingMortgagesLVR≥ 80 i0$

To convert the LVRs to LGDs, the LVRs are first inverted and subtracted from 1, to capture the proportion of the loan not covered by the collateral value (for loans with LVRs 80 or above these proportions are adjusted by the estimated LMI coverage (LMICoveragei0) and the pre-specified recovery rate $\left(\overline{LMIRecoveryRate}\right)$. Then the assumed foreclosure cost $\left(\overline{ForeclosureCost}\right)$ is added for all LVRs above a minimum threshold $\left(\overline{MinForeclosureLVR}\right)$. Denote by $L{{D}^{″}}_{ikt}$ an individual element of $L{{D}^{″}}_{it}$, and use $𝕀$ as an indicator function (also assume that if LVR bucket k is not evaluated in the max function then the function evaluates to zero).

(A30) $LG D ikt =max{ 0,1− 1 L D ″ ik∈[ 1,80 ]t } +( 1− LMICoverag e it ) *max{ 0,1− 1 L D ″ ik∈[ 81,250 ]t } +max{ 0,1− 1 L D ″ ik∈[ 81,250 ]t }*( 1− LMIR ecoveryRate ¯ )*LMICoverag e it +𝕀( L D ″ ikt > MinForeclosureLVR ¯ )× ForeclosureCost ¯$

The next step is to formulate PDs to align with each bucket-level LGD (denote them PDikt ). Bucket-level PDs are their starting value, plus the unemployment rate times the unemployment rate coefficient $\left(\overline{{\beta }_{Mort,UR}}\right)$, multiplied by a bucket-level multiplier $\left(\overline{L{M}_{k}}\right)$ that represents the correlation between LVR and PD. These multipliers are from work that became Bergmann (2020). Bucket-level PDs are also assumed to not fall below their starting value times the multiplier or a base level PD which accounts for the natural level of defaults that take place even in a strong economy $\left(\overline{PD}\right)$.

(A31) $P D ikt =max{ ( P D i0 4 + β Mort,UR ¯ × U R t ^ ) × L M k ¯ , PD ¯ }$

Finally, the aggregate mortgage loss rate is the weighted sum of loss rates across the distribution.

(A32) $L R it Mort = ∑ k P D ikt ×LG D ikt ×L D ″ ikt$

To produce the LDit input for the next quarter, subtract the defaulted loans using the bucket-level PDs, and then rescale the distribution to sum to one.

(A33) $L D ikt = L D ″ ikt ×( 1−P D ikt ) ∑ k L D ″ ikt ×( 1−P D ikt )$

Business losses and commercial real estate losses

Business loss rates (commercial real estate loans adopt the same loss rates as business loans) are decomposed into default rates and LGDs. The model updates default rates based on changes in year-ended GDP growth $\left(\stackrel{^}{GD{P}_{YE,t}}\right)$ which are constructed from quarterly GDP growth. LGDs for business loans are determined by the housing and commercial real estate prices scenario.

The business loss projections utilise information on the size (corporate, SME corporate and SME retail), industry and collateral composition (fully secured, partially secured and unsecured) of banks' business lending. In the following, vectors are denoted in bold script, and single elements within those vectors are denoted with a k subscript and non-bold script. The business loss inputs to estimate default rates comprise of:

• starting-period data on each bank's business portfolio detailing the amount of business lending for each size-industry pair reported $\underset{75×1}{\underbrace{\left(B{D}_{i0}\right)}}$;[42]
• size-industry multipliers that scale default rates for each industry (all initially set to one) $\underset{75×1}{\underbrace{\left(B{M}_{i0}\right)}}$.

For each bank and for each size-industry pairing, the proportion of a bank's overall business loan book to that size-industry pairing is calculated.

(A34) $B P i0 = B D i0 ∑ k B D ik0$

Next, PDs for the banks' business lending books are formulated by calculating PDs for each size– industry category of lending. This involves taking the starting PD, adjusting it by a preset sensitivity to the macroeconomic environment $\left({\beta }_{PD,k}^{Bus}\right)$ and including any industry-specific multipliers. Then to calculate PDs for the overall business portfolio these PDs are weighted appropriately by size–industry compositions of each bank's business lending portfolio.

(A35) $P D it =max{ P D i0 ,( P D i0 + ∑ k ( B P ik0 × β PD,k Bus GD P YE,t Δ ) )× ∑ k ( B P ik0 B M ik0 ) }$

The business loss inputs to estimate LGDs comprise of:

• starting-period data on each bank's business portfolio detailing amount of business lending for each size-collateral category $\underset{9×1}{\underbrace{\left(B{C}_{i0}\right)}}$;
• simulated starting LVR distribution for fully secured corporate lending $\underset{100,000×1}{\underbrace{\left(L{C}_{i0}\right)}}$;
• simulated starting LVR distribution for fully secured SME retail lending $\underset{100,000×1}{\underbrace{\left(L{R}_{i0}\right)}}$.

For each bank and for each size-collateral pairing, the proportion of the bank's overall business loan book to that size-collateral pairing is calculated.

(A36) $B L i0 = B C i0 ∑ k B C ik0$

The LVR distribution for fully secured corporate lending is updated based on the commercial real estate scenario $\left(\stackrel{^}{CR{E}_{t}}\right)$.

(A37) $L C it = L C it−1 ( 1+ CR E t ^ )$

The LVR distribution for fully secured SME retail lending is updated based on changes in the house price scenario $\left(\stackrel{^}{H{P}_{t}}\right)$.

(A38) $L R it = L R it−1 ( 1+ H P t ^ )$

The LGDs for partially secured and unsecured lending for each business size category is a constant set by parameter $\left(\overline{LGDNonFullySecured}\right)$.

First, LGDs for fully secured corporate lending are calculated. To convert the LVRs for fully secured lending to corporates into LGDs, the LVRs are first inverted and subtracted from 1, to capture the proportion of the loan not covered by the collateral value. Then the assumed foreclosure cost for commercial real estate $\left(\overline{ForeclosureCostCRE}\right)$ is added for all LVRs above a minimum threshold $\left(\overline{MinForeclosureLVR}\right)$. Denote by LCikt an individual element of LCit, and use $𝕀$ as an indicator function. Then the average LGD is taken as the LGD experience for the portfolio of these loans.

(A39) $FS C LG D it = ∑ k ( max{ 0,1− 1 L C ikt }+𝕀( L C ikt > MinForeclosureLVR ¯ × ForeclosureCostCRE ¯ ) ) k$

The same calculation is performed for fully secured SME retail lending using the appropriate LVR distribution (LRit ) and the appropriate foreclosure costs for house prices $\left(\overline{ForeclosureCostHP}\right)$.[43]

(A40) $FS C LG D it = ∑ k ( max{ 0,1− 1 L C ikt }+𝕀( L C ikt > MinForeclosureLVR ¯ × ForeclosureCostHP ¯ ) ) k$

LGDs for the business portfolio overall are then calculated as weighted average of the different size–collateral components.

(A41) $LG D it = f Bus,LGD ( FS C LG D it ,FS R LG D it , LGDNonFullySecured ¯ ,B L i0 )$

The function fBus,LGD weights each LGD by that LGD's proportion in BLi0 .

Finally, the aggregate business loss rate is the product of PDs and LGDs calculated earlier.

(A42) $L it Bus =P D it ×LG D it$

Credit lines

In determining business loss rates, banks' are also potentially exposed to drawdowns on their credit lines. The dollar amount of credit line drawdowns (CLD ) for each bank is determined by each bank's available credit limits for business lending and a parameter determining the share of these limits drawn down $\left(\overline{DrawdownRate}\right)$.

(A43) $CLD=AvailableCreditLimits× DrawdownRate ¯$

Credit line drawdowns feed into the calculation of PDs by increasing the share of banks' overall portfolio that is lent to corporates. LGDs for drawdown credit lines are determined by a set parameter $\left(\overline{LGDCreditLines}\right)$ and are appropriately weighted, based on the business loan LGD calculation based on the size of the drawdown relative to overall lending to business loans already on the balance sheet.

Other loss rates

The loss rates for non-mortgage and non-business asset classes are each formulated as the product of a PD and an LGD. PDs and LGDs are individually modelled as their starting value plus the inner product of a vector of pre-calibrated coefficients and the change in the macro scenario since the starting period, subject to a pre-specified minimum value.

Denote the balance (net of provisions) of asset class j as AssetBalijt . For all non-mortgage loan classes except ‘overseas and other loans’ (OOL) we have

(A44) $L R ijt =P D ijt ×LG D ijt$
(A45) $P D ijt =max{ P D ij0 _ ,P D ij0 + β PD GDP GD P t Δ + β PD UR U R t Δ + β PD CRE CR E t Δ }$
(A46) $LG D ijt =max{ LG D ij0 _ ,LG D ij0 + β LGD GDP GD P t Δ + β LGD UR U R t Δ + β LGD CRE CR E t Δ }$

The superscript $\text{Δ}$ denotes the cumulative change in the macroeconomic variable since the starting period. Specifically,

(A47) $GD P t Δ ≡ GD P t ^ − GD P 0 ^$
(A48) $U R t Δ ≡ U R t ^ − U R 0 ^$
(A49) $CR E t Δ ≡100× CR E t ^ − CR E 0 ^ CR E 0 ^$

The pre-calibrated PD coefficients used in these equations are shown in Table A2 and LGDs do not depend on the macro environment.

Table A2: Pre-calibrated PD Coefficients
Asset class ${\beta }_{PD}^{GDP}$ ${\beta }_{PD}^{UR}$ ${\beta }_{PD}^{CRE}$
AC.CreditCards 0 0.4 0
AC.DomOthPersonal 0 0.4 0
AC.DomSov 0 0 0
AC.DomFinCorp 0 0 0

For the OOL loan class, the PD and LGD are taken as the average across the other nine loan classes.

(A50) $L R it OOL =− 1 9 ∑ j≠OOL P D ijt × 1 9 ∑ j≠OOL LG D ijt$

Turning loss rates into write-offs

The dollar value of write-offs for each asset class is

(A51) $Write−of f ijt =L R ijt ×AssetBa l ijt−1$
(A52) $Write−of f ijt = ∑ j Write−of f ijt$

Provisions and bad and doubtful debt charges

The model approximately assumes that banks can accurately forecast the subsequent $\stackrel{^}{{Q}_{p}}$ quarters of the macroeconomic scenario, and that they hold sufficient provisions to cover the losses over that interval, without provision balances dropping below a pre-specified minimum. The first step in doing this is to calculate a required provision balance for each asset class (ReqProvisionsijt). To formulate this, the process for calculating write-offs is repeated, but the macro scenario is replaced with an artificial scenario that represents the cumulative effect of the subsequent four quarters.

Specifically, ReqProvisionsijt is set equal to the value obtained for Write-offijt when the processes described previously in Section A.7 are repeated, but with the actual macroeconomic scenario replaced with an artificial macroeconomic scenario. The artificial macroeconomic scenario, denoted by P superscripts, is defined by as follows.

(A53) $GD P t PΔ ≡ ∑ q=t+1 t+ Q p ^ ( GD P q ^ − GD P 0 ^ )$
(A54) $GD P YE,t PΔ ≡ ∑ q=t+1 t+ Q p ^ ( GD P YE,q ^ − GD P YE,0 ^ )$
(A55) $U R t PΔ ≡ ∑ q=t+1 t+ Q p ^ ( U R q ^ − U R 0 ^ )$
(A56) $CR E t PΔ ≡100× ∑ q=t+1 t+ Q p ^ CR E q ^ − CR E 0 ^ CR E 0 ^$
(A57) $H P t P ^ ≡min{ H P t+1 ^ ,..., H P t+ Q p ^ ^ }$
(A58) $CREBu s t P ^ ≡min{ CR E t+1 , ^ ..., CR E t+ Q p ^ ^ }$

For GDP, UR and CRE, these manipulations are simply the values from Equations (A53) to (A56) summed across the quarters that provisions are assumed to cover. Feeding these summed variables into the non-mortgage write-off formulation process should roughly approximate write-offs over the subsequent $\stackrel{^}{{Q}_{p}}$ quarters, because the macroeconomic variables feed linearly into Equations (A45) and (A46). However, the same approach of summing future observations would not work for mortgage provisions and business provisions. This is because HP feeds into mortgage write-offs nonlinearly, and so summing future HP outcomes would significantly overstate future losses. Likewise, for business loans HP and CRE feed into business write-offs nonlinearly, as can be seen from Equations (A37) and (A38). Instead these variables are manipulated by taking the worst single outcome over the quarters that provisions are assumed to cover.[44]

Banks' actual provision balances for each asset class are ReqProvisionsijt plus an assumed minimum, set at the lowest observed value in past data, which is intended to cover losses further out than the provisioned quarters.

(A59) $Provision s ijt =ReqProvision s ijt + min t<0 { Provision s ijt }$
(A60) $Provision s ijt = ∑ j Provision s ijt$

The bad debt charge is then calculated as contemporaneous write-offs plus the increase in provisions balance (which can be negative, for example, when the subsequent quarters in the scenario are improving).

(A61) $BD D it =Write−of f it +Provision s it −Provision s it−1$

## A.8 Retained earnings – securities fire sales

These fire sale dynamics are developed from the theory presented in Garvin (2019). See Section 2.2.1 of that paper for a brief relevant discussion that is applicable to the modelling here.

The funding run

After the first period of the model, each bank potentially experiences a funding run. These runs can either be deterministic in nature (i.e. the run happens every time the bank's opening CET1 ratio is below the pre-determined threshold) or probabilistic (i.e. the difference between the bank's opening CET1 ratio and the pre-determined threshold determines the probability of the run). When the model is set to be probabilistic, the chance of a run is specified according to the relationships in Table A3.

Table A3: Probability of a Funding Run
Per cent per quarter
CET1 capital ratio Probability
≥ 8 per cent 0
5–8 per cent 5
< 5 per cent 12

The size of a bank's funding run is determined by: the bank's opening CET1 ratio relative to a predetermined ‘run threshold’ $\left(\overline{CET1RunThresh}\right)$; the bank's previous-period total value of liabilities (TotLiabilitiesit–1); and a parameter that maps these two into the dollar value of the funding run $\left(\overline{RunPerCET1Gap}\right)$.[45]

(A62) $FundingRu n it max( 0, CET1RunThresh ¯ −CET1Rati o it−1 )*TotLiabilitie s it−1 * RunPerCET1Gap ¯$

Banks use cash (AC.Cashit–1) and securities to meet the funding run. Securities are either AGS (AC.AGSit–1), semis (AC.Semisit–1), or other (OthSecit–1), which is the combination of all the remaining securities asset classes. If a bank cannot meet the run after liquidating all of its securities, it fails and the model stops running.

Banks first use cash:

(A63) $RunGa p it =max( 0,FundingRu n it −AC.Cas h it−1 )$

Then, if their run gap is still positive, they sell AGS.

AGS sale and equilibrium price depression

The equilibrium AGS market price $\left({p}_{AGS}^{*}\right)$ is determined by banks' total selling and a pre-specified price depression function. Prior to any price depression (e.g. if banks did not sell any AGS), the AGS price is assumed to be 1, as a numeraire. This assumption is made because the liquid asset balances are expressed in dollar values, but to solve the equilibrium price, it is useful to separate the concepts of quantity and price (where value is quantity multiplied by price).

Denote banks' total equilibrium AGS sales as ${Q}_{AGS}^{*}$. There are two unknowns: ${p}_{AGS}^{*}$ and ${Q}_{AGS}^{*}$. So we need two equations to solve them, which are: 1) the price depression function; and 2) the formulation of how much banks sell for a given equilibrium price.

The price depression function is

(A64) $p AGS * =1− ( DepressCoef ¯ × Q AGS * TotMarket ) 2$

where the two parameters are set to achieve an appropriate price depression curve. They are chosen such that the price will never go to zero for feasible sales quantities.

The total quantity of sales for a given equilibrium price is as follows. Each bank's quantity of AGS securities sold $\left(qSol{d}_{i}^{AGS}\right)$ equals all its AGS if it does not hold enough, or, if it does hold enough, just enough to meet the run.

(A65) $qSol d i AGS ={ A C AG S it−1 if RunGa p it > p AGS * ×AC.AG S it−1 RunGa p it _ p AGs * if RunGa p it ≤ p AGS * ×AC.AG S it−1$

And the total quantity of sales is

(A66) $Q AGS * = ∑ I qSol d i AGS$

The above three equations pin down the equilibrium price and quantities.

Semis sales and equilibrium price depression

After AGS sales, a bank's remaining run gap is

(A67) $RunGa p ′ it =RunGa p it − p AGS * × q i AGS$

If their run gap is still positive, they sell semis. The semis price equilibrium is the same as the AGS equilibrium. That is, the same expressions all hold after replacing AGS subscripts and superscripts with semis subscripts and superscripts, with one exception.

The exception results from the assumption that the semis price is always depressed at least as much as the AGS price. This reflects an implicit assumption that AGS are more liquid than semis, but does it in a way that does not require simultaneously solving the equilibrium prices for every type of security. It also ensures that banks are acting ‘rationally’ when they liquidate all their AGS before turning to other securities sales.

This assumption that ${p}_{AGS}^{*}\ge {p}_{Semis}^{*}$ is embedded by adding one term to the total quantity of sales equation:

(A68) $Q Semis * = ∑ I qSol d i Semis + Q AGS *$

Other securities sales

After semis sales, a bank satisfies its remaining run gap by selling other securities. The process of selling semis then turning to other securities is identical to the process of selling AGS then selling semis. The same assumption is made about price depression ranking, that is, that ${p}_{OthSec}^{*}\ge {p}_{Semis}^{*}$.

The outcome

As mentioned above, if a bank still has a positive run gap after liquidating all its securities, it is considered to be failed and the model stops running. But this tends to only occur under quite extreme settings.

This fire sale component of the model has a P&L consequence and a balance sheet consequence.

The P&L consequence is the loss incurred from selling securities at depressed prices (i.e. losses from the removal of securities on the balance sheet) as well as the revaluation losses (i.e. the losses from securities on the balance sheet being revalued downwards reflecting the fall in prices). The value of this loss is determined by the price depression experienced by asset classes.

(A69) $FireSaleLos s it =AC.AG S it−1 ( 1− p AGS * )+AC.Semi s it−1 ( 1− p Semis * ) +AC.OthSe c it−1 ( 1− p OthSec * )$

The balance sheet consequence is that the cash used and securities liquidated are removed from the balance sheet and the remaining securities are revalued downwards. However, banks subsequently issue new liabilities over time, at a pre-specified rate, to replenish the availability of their liquid assets. This will be described in a later section. The funding run is also a reduction of liabilities, which gets implicitly incorporated through the effect of the fire sale loss on capital, given the implicit nature of the liabilities in the modelling.

## A.9 Retained earnings – tax and dividends

Tax

Banks are assumed to pay a tax rate of 30 per cent of net income minus credit losses, after adjusting for deferred tax assets. Banks cannot treat new provisions for future credit losses as a deduction from taxable income. Instead, they accrue a deferred tax asset, which is used as a tax deduction when the provisioned loss is written off, if that happens in a profitable quarter.

Taxes are therefore calculated as follows

(A70) $Ta x it = TaxRate ^ ×max{ 0,NetIncom e it −Write−of f it }$

This formulation means that losses from forward-looking provisions don't get factored into profits for tax purposes.

Dividends

Dividends follow a pre-specified decision rule. First an upper bound ( MaxDivit ) on the dividend ratio is calculated based on the bank's capital ratio, applying requirements from Prudential Standard APS 110 Capital Adequacy (APRA 2016, Attachment B). The specific calculation is:

(A71) $AT1Shortfal l it =max{ 0,( MinTier1Ratio ¯ − MinCET1Ratio ¯ )−( Tier1Rati o it−1 −CET1Rati o it−1 ) }$
(A72) $T2Shortfal l it =max{ 0,( MinTotCapRatio ¯ − MinTier1Ratio ¯ ) −( TotCapRati o it−1 −Tier1Rati o it−1 ) }$
(A73) $SpareCET 1 it =CET1Rati o it−1 −AT1Shortfal l it −T2Shortfal l it − MinCET1Ratio ¯$
(A74) $MaxDi v it ={ 0 if NPA T it ≤0 0 if NPA T it >0 and SpareCET 1 it ≤0.25× CCB ¯ 0.2 if NPA T it >0 and SpareCET1>0.25× CCB ¯ and SpareECT 1 it ≤0.5× CCB ¯ 0.4 if NPA T it >0 and SpareCET1>0.5× CCB ¯ and SpareECT 1 it ≤0.75× CCB ¯ 0.6 if NPA T it >0 and SpareCET1>0.75× CCB ¯ and SpareECT 1 it ≤ CCB ¯ ∞ if NPA T it >0 and SpareCET 1 it > CCB ¯$

Subject to the above maximum, the dividend payout ratio is a function of the CET1 ratio. First two thresholds are formulated:

(A75) $HghDivThresh= TargetCET1Ratio ¯ + DivCapBuffer ¯$
(A76) $LowDivThresh= MinCET1Ratio ¯ +0.25* CCB ¯$
(A77) $BtwnThresh=( DivRatioNormal ¯ − DivRatioRecovery ¯ )× CET1Rati o it−1 −LowDivThresh HghDivThresh−LowDivThresh$
(A78) $PayoutRati o it ={ DivRatioNormal ¯ if CET1Rati o it−1 >HghDivThresh DivRatioRecovery+BtwnThresh ¯ if CET1Rati o it−1 ≤HghDivThresh$
(A79) $Dividen d it =min{ PayoutRati o it ,MaxDi v it }×NPA T it$

## A.10 Capital levels

The model focuses on CET1 capital, and just updates other types of capital assuming the same growth as CET1 capital. This has implicit effects on the liability structure based on the implicit liabilities assumption. That is, the rise in Tier 1 or Tier 2 capital implies an offsetting decline in other types of liabilities.

(A80) $CET1Capita l ′ it =CET1Capita l it−1 +RetEar n it$
(A81) $TotEquit y ′ it =TotEquit y it−1 +RetEar n it$
(A82) $Tier1Capita l it =Tier1Capita l it−1 × CET1Capita l ′ it CET1Capita l ′ it−1$
(A83) $Tier2Capita l it =Tier2Capita l it−1 × CET1Capita l ′ it CET1Capita l ′ it−1$

## A.11 Updating the asset side of the balance sheet

Removing impaired assets from balance sheet

Asset balances are net of provisions. First, the asset losses from the bad and doubtful debt stage are subtracted from the balance sheet. Indexing credit asset classes individually as c (rather than j ) and as a set C :[46]

(A84) $Balanc e ′ ict =Balanc e ict−1 −Write−of f ict −( Provision s ict −Provision s ict−1 )$

Removing and revaluing fire sale assets with some replenishment

Here any cash and securities liquidated are subtracted from the balance sheet. However, banks are assumed to replenish their HQLA components (cash, AGS, semis) at a pre-specified rate relative to their starting balance as long as HQLA are below their starting balances.

First work out the (pre-constrained) replenishment values. This is the replenishment rate multiplied by the period 0 balance. Specifically, for X in AC.Cash , AC.AGS and AC.Semis:

(A85) $Replenis h i X = ReplenishHQLA ¯ × X i0$

Then the asset balances are updated but replenishment cannot take assets above their starting period values.

(A86) $AC.Cas h ′ it =min{ AC.Cas h i0 ,max{ 0,AC.Cas h it−1 −FundingRu n it }+Replenis h i AC.Cash }$
(A87) $AC.AG S ′ it =min{ AC.AG S i0 ,AC.AG S it−1 −qSol d i AGS +Replenis h i AC.AGS }$
(A88) $AC.Semi s ′ it =min{ AC.Semi s i0 ,AC.Semi s it−1 −qSol d i Semis +Replenis h i AC.Semis }$

Replenish credit losses by spending net income

If net cash income – defined as retained earnings plus credit losses – is positive, which it will be almost always, it is spent on replenishing credit losses.[47]

(A89) $NetCashIncom e it =RetEar n it +BD D it$

Spending is proportional to the period 0 balances of these assets. This is regardless of the distribution of credit losses on asset classes, which embeds the assumption that a bank will not tilt its reinvestment towards the asset classes making the largest losses. The initial proportions are:

(A90) $InitialPrp n ic = Balanc e ic0 ∑ k∈C Balanc e ik0$

To condition on net cash flow being positive, define the indicator I (NCFi > 0) as one if NetCashIncomeit > 0 and zero otherwise (and vice versa for I (NCIi < 0)).

(A91) $Balanc e ″ ict =Balanc e ′ ict +I( NC I i >0 )×min{ BD D it ,NetCashIncom e it }×InitialPrp n ic$

If net cash flow is negative it is subtracted from cash.

(A92) $AC.Cas h ″ it =AC.Cas h ′ it +I( NC I i <0 )×NetCashIncom e it$

At this point the total asset balance is updated to reflect changes so far:

(A93) $TotAsset s ′ it = ∑ j Balanc e ijt L$

Updating risk densities

For IRB banks, to update total risk-weighted assets, we make a simplifying assumption that total risk-weighted assets are all attributable to banks' loan credit asset classes as specified by the model (i.e. domestic mortgage, domestic business, domestic commercial property, etc).[48] This allows us to calculate the average risk weight on loan assets by dividing total risk-weighted assets by total loan credit assets ( AvgLnRWit ). The model then adjusts the average risk weight on loan assets as the macroeconomic scenario changes based upon whether a bank is classified a standardised or IRB bank.

The growth in average loan risk weights are formulated using the model's internally generated PDs and LGDs. For each bank, a single PD and LGD are calculated across their balance sheet for starting period for all loan asset classes (c ).

(A94) $PDag g i0 = ∑ k∈C ( Balanc e ik0 *P D ik0 ) ∑ k∈C Balanc e ik0$
(A95) $LGDag g i0 = ∑ k∈C ( Balanc e ik0 *LG D ik0 ) ∑ k∈C Balanc e ik0$

Then for each period, aggregate PDs and LGDs are calculated for the start of the period (i.e. that obtained at the end of the last period) and the current period by weighting the appropriate model-generated PDs and LGDs with the initial PD and LGD calculated.[49] This mimics the through-the-cycle modelling that banks will engage in that smooth changes in risk weights over time.

(A96) $PDag g it−1 =( StartingRWAWeight ¯ )*PDag g i0 +( 1− StartingRWAWeight ¯ ) * ∑ k∈C ( Balanc e ikt−t *P D ikt−1 ) ∑ k∈C Balanc e ikt−1$
(A97) $LGDag g it−1 =( StartingRWAWeight ¯ )*LGDag g i0 +( 1− StartingRWAWeight ¯ ) * ∑ k∈C ( Balanc e ikt−t *P D ikt−1 ) ∑ k∈C Balanc e ikt−1$
(A98) $PDag g it =( StartingRWAWeight ¯ )*PDag g i0 +( 1− StartingRWAWeight ¯ ) * ∑ k∈C ( Balanc e ikt *P D ikt ) ∑ k∈C Balanc e ikt$
(A99) $LGDag g it =( StartingRWAWeight ¯ )*LGDag g i0 +( 1− StartingRWAWeight ¯ ) * ∑ k∈C ( Balanc e ikt *LG D ikt ) ∑ k∈C Balanc e ikt$

Then for the balance sheet, opening and closing average loan risk weights are calculated.

(A100) $RWOpe n it =12.5*[ LGD×N( G( PDag g it−1 )+ R ×G( 0.999 ) 1−R )−PDag g it−1 ×LGDag g it−1 ]$
(A101) $RWClos e it =12.5*[ LGD×N( G( PDag g it )+ R ×G( 0.999 ) 1−R )−PDag g it ×LGDag g it ]$

Average loan risk weights are then determined by the growth rates between opening and closing average loan risk weights, subject to a maximum limit on risk-weight growth.

(A102) $AvgLnR W it =min{ AvgLnR W i0 ( 1+ MAXRWAGrowth ¯ ),AvgLnR W it−1 ( 1+( RWClos e it RWOpe n it −1 ) ) }$

The model also assumes that average loan risk weights cannot fall below their starting levels.

(A103) $AvgLnR W it =max{ AvgLnR W it ,AvgLnR W i0 }$

For standardised banks average loan risk weights are held constant at their starting levels.

Leverage any profits into credit reinvestment at a conditional leverage rate

If the bank has positive retained profits (i.e. earnings), they are spent on new credit assets. If the bank's capital ratio is high enough, the profits are assumed to be leveraged up at the same ratio as the bank's period 0 CET1 ratio adjusted for changes in risk weights. If the bank's capital ratio is not, the earnings are spent without any releveraging.

The previous step involved banks spending the minimum of BDDs and net cash income on new assets. By definition, any net cash income left over is equal to retained earnings.

To condition on the bank's CET1 ratio being above the threshold for releveraging, define

(A104) $I( Rele v i )=1⇔ CET1Capita l ′ it AvgLnR W it × ∑ k∈C Balanc e ikt L ≥ MinCET1Ratio ¯ + CCB ¯ + AssetPurchBuffer ¯$

and I (Relevi) = 0 otherwise.

(A105) $Balanc e ‴ ict =Balanc e ″ ict +I( Rele v i )×{ 0,RetEar n it }×InitialPrp n ic ×( 1 AvgLnR W it ) ×( RW A i0 CET1Capita l i0 )$

If banks are below the releveraging CET1 threshold but above a repurchasing threshold, they just spend the retained earnings without any releveraging. This will lift their capital ratio. Define an indicator for the bank's CET1 ratio being in this region:

(A106) $I( Repurc h i )=1⇔ CET1Capita l ′ it AvgLnR W it × ∑ k∈C Balanc e ikt L ≥ MinCET1Ratio ¯ AND I( Rele v i )=0$

and I (Repurchi) = 0 otherwise.

(A107) $Balanc e ‴ ′ ict =Balanc e ‴ ict +I( Repurc h i )×max{ 0,RetEar n it }×InitialPrp n ic$

If the bank's capital is below both thresholds, then the retained earnings are not spent, which implies they are used to pay down liabilities. This does not require another equation because in this case we already have that $Balanc{{{e}^{‴}}^{\prime }}_{ict}=Balanc{{e}^{‴}}_{ict}.$

At this point the total balance sheet size is updated again

(A108) $TotAsset s ″ it = ∑ j Balanc e ijt L$

Impose any asset growth floor

The model has a parameter $\left(\overline{AssetGrthFloor}\right)$ that reflects an assumption that banks must maintain credit supply by growing their balance sheet at the specified rate. Its default value is negative infinity. Any imposed asset growth is assumed to be spread evenly across the whole balance sheet.

(A109) $ImposedGrowt h i = 1+ AssetGrthFloor ¯ TotAsset s ″ it /TotAsset s it−1$
(A110) $Balanc e ijt =Balanc e ijt L ×max{ 1,ImposedGrowt h i }$

And total assets is updated for the final time.

(A111) $TotAsset s it = ∑ j Balanc e ijt L$

## A.12 Capital ratios

Now total RWAs are updated based on the updated balance sheet value, using the average loan risk weight modelled earlier with the option to impose additional growth on total RWAs.

(A112) $RW A it =( ∑ k∈C Balanc e ikt L ×AvgLnR W it )*( 1+ RWAShoc k t ^ )$

The next step is to incorporate the triggering of any AT1 capital.

(A113) $Tier1Converte d i ={ Tier1Capita l it −CET1Capita l it if CET1Capita l it

Then the capital ratios are updated.

(A114) $CET1Capita l it =CET1Capita l ′ it +Tier1Converte d i$
(A115) $Tier1Capita l it ={ CET1Capita l it if CET1Capita l it
(A116) $TotEquit y it =TotEquit y ′ it +Tier1Converte d i$
(A117) $CET1Rati o it = CET1Capita l it RW A it$
(A118) $Tier1Rati o it = Tier1Capita l it RW A it$
(A119) $TotCapitalRati o it = Tier1Capita l it +Tier2Capita l it RW A it$

## Footnotes

Technically, house prices and commercial real estate prices are supplied as indices but are then converted by the model into growth rates. [38]

We shrink the constant on the commercial real estate equation to keep the equilibrium growth rate below estimates of equilibrium nominal GDP growth. [39]

Reinvestment here refers to both reinvestment in new assets and replenishments of assets that have been written-off. [40]

If any loans have their LVR pushed above 2.5, they are kept in the 2.49–2.5 bucket. Loans with an LVR this high are very rare. [41]

This would technically also include lending to financial institutions, but these exposures are assigned a zero dollar value because exposures to financial institutions are dealt with in another loss rate category. [42]

LGDs for SME corporate lending are calculated as the average LGD of corporate lending and SME retail lending. [43]

Another approach for calculating required provisions would be to repeat the write-offs process for each of the subsequent quarters of the scenario that provisions are assumed to cover, and sum these. This approach is not taken because it would slow down the model's run time. [44]

The predetermined run threshold is set at a different level based upon whether banks will experience a deterministic or probabilistic liquidity run. [45]

If any balance goes negative, the model is currently set up to just reset that balance to zero and to return a warning message. [46]

Credit losses enter cash flow positively because retained earnings is pulled downward by credit losses, but cash flow is not (not directly anyway). [47]

In practice this will be a reasonable assumption as a large majority of banks' total risk-weighted assets are composed of credit risk-weighted assets which will relate primarily to loan asset classes. [48]

Mortgage LGDs are constrained to be at least as high as their starting values due to regulatory imposed minima on mortgage LGD modelling. [49]