Measuring the Performance of Bank Loans Under Basel II/III and IFRS 9/CECL

In the last two decades, both internal and external risk management of banks has undergone significant developments. Substantial investments into data collection have been made and this data is used for estimating internal credit risk models. The resulting risk parameters are required for various regulatory purposes. Banking supervision encourages banks to use a risk-based approach for computing minimum regulatory capital. Accounting rules have been tightened requiring more timely loss reserves for impaired loans. In this article, we propose a comprehensive scheme for calculating the profitability of a loan that could be used both for setting risk-based interest rates when originating a loan and for accurately determining the profitability of existing clients. The scheme utilizes the credit models developed for regulatory purposes and takes the impact of regulation on loan performance into account. We show that accounting loan loss provisions cannot be applied in a performance measurement scheme because they do not reflect true economic loss. In addition, we demonstrate that it is crucial to measure loan performance over the full life cycle of a loan. Restricting profitability measurement to a time horizon of one year as often observed in practice could be misleading.


Introduction
Since the 2000s, a number of initiatives have been undertaken by regulators and accounting boards worldwide to increase the stability of the financial system. First, the Basel Committee on Banking Supervision (BCBS) allowed banks under the Basel II accord to use internal risk parameters to compute minimum required capital to bring capital requirements more in alignment with economic risk (BCBS, 2006). Second, Basel III improved the quality of bank capital and and gave supervisors more flexibility in setting capital levels for individual institutions (BCBS, 2011). In addition, the accounting boards IASB (International Accounting Standards Board) and FASB (Financial Accounting Standards Board) have released new standards on building loan loss provisions (LLP) for impaired loans, IFRS 9 (IASB, 2014) and CECL (FASB, 2016), respectively. These reforms have a direct impact on the volume of loans a bank can originate and their profitability.
To measure the performance of a loan, we propose a scheme based on RAROC (risk-adjusted return on capital). Basically, RAROC is computed as interest income minus all costs (funding, operational, expected loss) divided by the capital that is used as a buffer against unexpected losses. RAROC is an established measure in banking practice. It can be shown that capital budgeting rules based on RAROC have an economic foundation (Stoughton & Zechner, 2007). Besides that, the measure itself shows a reasonable behavior under extreme scenarios, i.e. a loan's RAROC as a function of its interest rate has a maximum and cannot become arbitrarily large when a bank increases rates (Engelmann & Pham, 2019).
There is a large stream of literature addressing the statistical aspects of credit scoring and estimating credit risk parameters for Basel regulation and provisioning. 1 In comparison, literature on applying these parameters in banking practice beyond their regulatory purpose is comparatively scarce. An evaluation of the impact of Basel regulation on loan prices is done in Repullo and Suarez (2004) while the impact of IFRS 9 is studied in Abad and Suarez (2018). An improvement of the loan origination process by integrating credit scoring models with loan performance measures is suggested by Stein (2005). However, we are not aware of any research that combines both the credit risk modeling for Basel regulation and the new accounting standards into a multi-period loan performance measurement scheme.
The main contribution of this article is threefold. First, we will propose a comprehensive scheme for loan performance measurement that is based on the risk parameters banks have estimated for Basel and IFRS 9 / CECL purposes. The scheme enables a credit risk manager to make decisions about loan applications and the suitability of a bank's interest rate setting process taking into account all internal and external factors affecting a bank's business. Second, we will show that LLP under IFRS 9 / CECL are not suitable for direct inclusion into a loan profitability scheme. In general they are different from the true economic expected loss a bank is suffering in its lending business when all cost components are considered appropri-ately. Third, we will demonstrate that an accurate picture of a loan's performance can only be obtained if the full lifetime of the loan is analyzed including past periods. Just looking at the current year might give a misleading picture, especially for loans with collateral that changes its value over time.
The structure of this article is as follows. Section 2 briefly reviews the latest accords developed by regulators and accounting boards as far as they are relevant for loan performance measurement. In Section 3, the framework for loan performance measurement is developed and its parameterization is discussed. In Section 4 a detailed example for residential mortgages is presented using stylized but realistic credit risk models. The final section concludes.

Recent Developments in Accounting and Banking Supervision
The two major aspects of IFRS 9 / CECL and the Basel regulation having an impact on loan performance are loan loss provisioning and minimum capital requirements, respectively. Each will be discussed in a separate subsection.

Loan Loss Provisioning under IFRS 9 and CECL
IFRS 9 prescribes a three-stage algorithm for provisioning: • Stage 1: Normally performing loans, banks have to reserve one-year expected loss • Stage 2: Loan with substantially deteriorated credit quality, banks have to reserve lifetime expected loss • Stage 3: Defaulted loans, banks have to build a specific loan loss provision IFRS 9 does not prescribe criteria for determining when a loan's credit quality is deteriorated substantially. This has to be defined internally by a bank. Some suggestions are made by Chawla, Forest, and Aguais (2016b). CECL can be roughly considered as a special case of IFRS 9 where all loans are treated as Stage 2 loans (European Systemic Risk Board, 2019). For this reason, the focus of this article will be on IFRS 9.
Consider a fixed-rate n-year loan with interest rate z. Denote with p i the probability that a borrower will default until year i, l i the loss given default associated with a default in year i, and N i the outstanding balance in year i. In this context, LLP for Stage 1 loans is computed as For Stage 2 loans, LLP is computed as lifetime expected loss. It is defined as the difference between the present value of all future cash flows of a loan and the expected present value of future cash flows. The calculation of the latter includes default probabilities and loss rates and leads, therefore, to a lower result then the former. As shown in Engelmann (2018) the following expression for LLP can be derived: where p 0 = 0. A key assumption in the derivation of (2) is that l i is measured relative to outstanding balance plus one interest rate. This means, in case of a default a bank can expect to receive (1 − l i ) · N i · (1 + z).
Banks are required to reserve LLP at the beginning of a year before any interest income is generated. This means that LLP has to be funded from bank capital. The exact relation of provisions and capital will be discussed in the next subsection.
Over the year expected losses should be covered by a component of the interest rate income, the expected loss margin. When the expected loss margin is well-designed it should cover observed losses on average. Only in those years where something unexpected happens, LLP and bank capital have to be utilized to preserve depositors' funds.

Minimum Capital Requirements under Basel II/III
Minimum capital requirements have a crucial impact on loan performance. The higher the capital required to back a loan against unexpected losses, the lower is the return on capital a bank can generate. Basel II defines three different frameworks of increasing complexity for calculating regulatory capital, the standardized approach, the foundation internal ratings-based (IRB) approach, and the advanced IRB approach. Throughout this article, we assume a bank is applying the advanced IRB approach. Its cornerstone is a formula for computing required minimum regulatory capital K min : where EAD is the exposure at default, LGD the loss given default, PD the one-year default probability, Φ the cumulative standard normal distribution, and ρ the asset correlation which is prescribed by supervisors (BCBS, 2006).
How is the formula parameterized? Since its parameters are similar to p 1 and l 1 in (1), one might suppose PD = p 1 and LGD = l 1 . However, this is not true. IFRS 9 requires risk parameters to be forward-looking (IASB, 2014). Such parameters are called point-in-time (PIT) estimates, i.e. best possible predictions of their real-world realizations over the forecast horizon. Basel II, however, requires for default probabilities "long-term averages" and for loss given default an estimate corresponding to an economic downturn. This means, different sets of risk parameters have to be estimated for IFRS 9 and Basel II.
Long-term average default probabilities are also referenced through-the-cycle (TTC) default probabilities, i.e. default probabilities representing the average default rate over an economic cycle. From a statistical point of view, TTC probabilities are more complex to estimate because of the long time horizon they refer to. In addition, a direct comparison with realized default rates is not meaningful because by construction TTC probabilities should underestimate default rates during a recession and overestimate default rates during a boom. An established procedure in practice is adapting (3) to transform PIT probabilities into TTC probabilities (Aguais, Forest, King, Lennon, & Lordkipanidze, 2007;Carlehed & Petrov, 2012): where Z is a standard normally distributed systemic factor. While for capital calculation under Basel II Z was set to the 99.9% quantile, for the transformation between p 1 and PD the systemic factor Z has to be estimated and represents the state of the economy.
To determine Z, one could use a time series of default rates d i observed in a particular sector like all Vietnamese corporations or all Dutch mortgages. An observed default rate can be interpreted as realization of an average PIT PD because by definition these PDs are forecasts of default rates. Computing Φ −1 (d i ) and estimating the parameters PD and ρ should lead to reasonable values for Z i if PD is stable over time and the data history is sufficiently long. If predictions of Z i are needed, a time series model could be built that links Φ −1 (d i ) with macroeconomic factors X and predicts Z i under an assumed macroeconomic scenario. This procedure will be illustrated in Section 4.
We conclude this section describing a recent amendment to Basel II linking capital requirements with loss provisions. In (3) expected loss EL B = PD · LGD · EAD (Basel EL) is subtracted from the term involving the normal distribution. Economically, this means that supervisors have assumed that expected losses of a loan portfolio are covered by loss provisions which should be offset by a component of interest income, the expected loss margin. Only losses beyond expectations are backed by capital. This rule was changed in BCBS (2019) where the Basel Committee requires banks to provision at least EL B . If LLP is less than Basel EL on portfolio level additional capital is required while if it is higher than Basel EL capital can be released up to a cap of 0.6% of risk-weighted assets (RWA). RWA is computed as 12.5 · K min with K min from (3). 2 This means that (3) has to be adjusted to reflect this additional requirement:K A further complication in the practical implementation of this rule is that if portfolio LLP is lower than total Basel EL a bank has to build an additional capital buffer in Core Tier 1 capital. If LLP is higher than Basel EL the difference will be added to Tier 2 capital up to a cap of 0.6% of RWA which makes this rule asymmetric. For a discussion of the impact of this regulation on these bank capital components see Krüger, Rösch, and Scheule (2018).

A Framework for Loan Performance Measurement
In this section, the framework for loan performance measurement will be developed. We will consider profitability on a loan-by-loan instead of a portfolio basis allowing banks to identify their most valuable customers. The performance of a loan will be measured by RAROC (riskadjusted return on capital) which is calculated by the following scheme: To get a correct picture of a loan's performance, it is essential to take the full lifetime of a loan into account. Risk parameters might change over time and looking into one single period only might be misleading. The lifetime of a loan can be split into past periods, the current period and future periods. The key difference is that in future periods the IFRS 9 stage of a loan is unknown which makes the calculation of expected profitability more demanding.
We start with the past periods. We denote the interest rate in period i with z i . For a fixed-rate loan z i is always the fixed rate z. For a floating-rate loan, z i = λ i + s, where λ i is the realized Libor rate in this period and s a period-independent spread. The outstanding balance is N i , the funding costs f i , the operational costs c i , the coverage for expected losses ELC, and the required capitalK min,i . The expected loss coverage is designed to cover losses in loan balance, operational costs and funding costs. It is computed as where d r,i is the realized default rate of all borrowers in a borrower's rating grade r. The expected loss coverage (which is a realized quantity for past periods) is computed as the sum of realized loss in deposits N i · d r,i · (1 + f i ) and the realized loss in operational costs N i · d r,i · c i minus the expected recovery from collateral liquidation of defaulted loans The result is divided by 1 − d r,i because only surviving borrowers can make up for the loss. Remember from Section 2.1 that we use the convention that l i measures the loss rate of outstanding balance plus one interest rate. Note, that the estimated loss rate l i could be replaced with some realization if it was known. However, since this usually takes more time to observe than the default rate, we suggest sticking with the risk parameters that was used in the past if no better alternative is available.
The intuition behind using the default rate of a rating grade d r,i is that within a rating grade all loans have similar default probabilities. The surviving loans have to cover the losses of the defaulting loans on average. Roughly speaking, the rating grade is interpreted as an "insurance pool" and (6) measures the realized loss per surviving borrower the pool has to cover.
If a loan was in Stage j in period i, we find for its RAROC We relate the risk-adjusted return to regulatory capital (RegCap). Other research uses economic capital (ECap) instead (e.g. Crouhy, Turnbull, & Wakeman, 1999) which is computed from a credit portfolio model that, at least in theory, more accurately reflects the true economic risk of a loan. There are, however, conflicts of using a pure ECap model with regulation that are difficult to resolve. If ECap is lower than RegCap, a bank has to allocate by law Reg-Cap which reduces a loan's RAROC. Only if ECap is higher than RegCap it might be safely used to penalize loans with large contributions to portfolio risk. Over time ECap might be fluctuating considerably if a bank's portfolio changes which is difficult to anticipate at loan origination. For these reasons, we prefer regulatory capital in our performance measurement scheme. Note, that we implement (5) on loan level, although it is implemented on portfolio level in the Basel accord. The motivation is penalizing loans that contribute to high LLP which could eventually require a bank to build provisions so high that it will breach the 0.6% · RWA cap.
For the current period, we can use exactly the same formulas because the stage of a borrower is known. However, the realized default rates of the portfolio is still unknown. Therefore, the PIT PD p j 1 has to be used instead of d r,i in (6) leading to which has to be used in (7).
To compute expected RAROC for future periods, we have to estimate a number of quantities. First, we do not know the stage of a loan in the future. We denote by t i the probability that a loan is in Stage 2 in period i. Depending on Stage j = 1, 2, we need parameters p j i , the cumulative PIT default probabilities until period i, l i , the PIT loss rate in period i, expected balanceN i , exposure at default EAD i , TTC probabilities PD j i , downturn LGDs LGD i , expected interest rateẑ i , expected funding costsf i and operational costs c i . Note, that EAD i might be different fromN i . By the expected balanceN i the outstanding loan amount that earns interest should be modeled while EAD i refers to the outstanding exposure in case of a default. This could include additional expected drawings from credit lines. In any case EAD i ≥N i . 3 For future periods, ECL depending on stage j is computed as where p j (i|i−1) is the default probability of a borrower in period i conditional that he survived until period i − 1. Since we design our performance measurement scheme period-by-period, a calculation of RAROC in a future period is only meaningful conditional on survival of a borrower. From the term-structure of default probabilities p i the conditional probability To summarize all the period RAROCs RAROC i into a single performance measure, we use capital-weighted RAROC: The motivation for using a weighted average is giving those periods where more capital is required more weight. We abstain from introducing discount factors because they dampen potentially negative effects in the future. To ensure stability of a bank's business over time, there should be no incentives created to favor higher short-term gains over lower profit or even losses in the future. A second argument is that the use of discount factors would imply that for past periods we have to compound interest. This would be inconsistent with reality since banks mainly distribute profit to its shareholders instead of reinvesting it.
The profitability measure (11) has several applications. It can be used by a bank to evaluate the profitability of its clients or to monitor existing portfolios. In addition, it can play an important role in loan origination. For this purpose, banks define an internal profitability target RAROC target they would like to achieve. Usually, this is a management decision possibly guided by an application of the CAPM to determine the required return of a bank's shareholders. This approach has some shortcomings (Crouhy et al., 1999) and alternatives based on the Merton model (Merton, 1974) exist leading to borrower-specific hurdle rates (Miu, Ozdemir, Cubukgil, & Giesinger, 2016). However, the proposed alternatives can be applied sensibly only in markets where borrowers have listed equity which is true only for a small minority. Therefore, we stick to the assumption of a borrower-independent target profitability RAROC target .
In markets where a bank is a price taker, i.e. where it has very limited power to set the interest rate z due to strong competition, it could use (11) to define an acceptance rule for loan applications. Only those applications should be approved where RAROC ≥ RAROC target . When a bank is a price setter which could be the case in more specialized markets, it could define the hurdle rate z h by solving the optimization problem

s.t.RAROC ≥ RAROC target
A bank should offer loans only at an interest rate z h or higher to ensure it fulfills its profitability requirements.
The remainder of this section is devoted to discussing how the parameters needed for applying (11) are estimated. This will be done in two separate subsections where the first one is devoted to credit risk parameters. The second subsection deals with the internal costs for funding and operations.

Estimation of Credit Risk Parameters
The existing literature on estimating credit risk parameters is huge. Overviews with a focus on the Basel regulation are Engelmann and Rauhmeier (2011) and Ong (2007). The most complex risk parameter for IFRS 9 is the term-structure of default probabilities. It could be either estimated by techniques from survival analysis (Banasik, Crook, & Thomas, 1999;Malik & Thomas, 2010) or by translating a macroeconomic scenario into default probabilities which is more common in IFRS 9 models (Skoglund, 2017;Xu, 2016). The estimation of LGD and EAD for IFRS 9 purposes is treated in Chawla, Forest, and Aguais (2016a).
The exact model design depends on factors like portfolio segment, data availability, data quality, or predictive power of risk factors. For the purpose of this article, we assume a generic modeling framework that could be realized if data is available for more than one economic cycle and the cyclical behavior of macroeconomic factors is well reflected in the data. This assumption reflects an ideal world from a modeling point of view and may not be feasible in all financial institutions and for all portfolios. How to make adjustments if these assumptions are not fulfilled cannot be stated in general but has to be decided on a case-by-case basis.
To be more precise, we assume that a macroeconomic scenario X k,i for future periods i = 1, . . . , n exists where X k , k = 1, . . . , l are l macroeconomic factors that have a strong link to the credit risk in a particular loan segment. For instance, X could be GDP growth for a corporate portfolio, it could be the unemployment rate for a retail portfolio or it could be a house price index for a residential mortgage portfolio. In addition to the future scenarios, the past realizations of X k , X k,−m , . . . , X k,0 are known. The macroeconomic scenario could be generated by an econometric model or by expert judgment. Usually, there is some opinion over the next 1-5 years and after that the scenario converges to some neutral long-term average since predicting over time horizons of 5 years or more is too unreliable.
For the calculation of TTC default probabilities, it is essential to understand in which point of the cycle the economy is. This could be done by computing abstract systemic factors Z and applying (4). A necessary input for this procedure are realized default rates d i , i = −m, . . . , 0 that are representative for the market the bank is operating in. For instance, for a corporate portfolio d i could be country-wide default rates obtained by some government agency. From the time-series d i the parameters B and ρ of the slightly transformed equation (4) can be estimated using the techniques described in Carlehed and Petrov (2012) and the references therein. Once, these parameters are known, Z i can be computed as To obtain future predictions of Z, a link between the macroeconomic scenario X k,i and Z is needed. This could be established by estimating a time series model like Once the parameters α 0 , . . . , α l are estimated, (15) and (14) can be applied to translate the macroeconomic scenario into a scenario value for the systemic factor Z.
To compute PIT credit risk parameters, we assume that at least one of the macroeconomic variables X k is included in the model equation. In a generic setup, a bank could have a model for the PIT default probability in the next period, p 1 , the PIT loss rate l 1 , the PIT prepayment rate cpr 1 , and the PIT credit conversion factor cc f 1 . 4 All these parameters are estimated conditional on current (period 0) macroeconomic data and risk factors. The estimation of these parameters is done by the equations where X = (X 1 , . . . , X l ) is the vector of macroeconomic factors and Y p , Y l , Y cpr , and Y cc f are risk factors related to the borrower, the loan or the collateral that is backing the loan. These risk factors can differ from model to model. The estimation of p has to include the IFRS 9 stage j = 1, 2. A possible solution is presented in the next section. The functions g p , g l , g cpr , and g cc f are the link functions between the risk parameters and the risk factors. For p, cpr and cc f logistic regression is a common choice while for l a linear regression or some other form of regression function could be used.
These risk models can be used to compute risk parameters in all future periods i of a loan. This results in Note, that by p j i we denoted the cumulative default probability until period i. The risk models deliver PIT parameters in period i, i.e. in the case of default probabilities the probability of default in period i conditional on survival in period i − 1, p j (i|i − 1). Cumulative default and prepayment probabilities are then computed as (1 − cpr(k|k − 1)) .
In (20) to (23) we have used Y * i as inputs. Does that mean we need in addition to the macroeconomic scenarios also scenarios for the risk factors Y? Remember that Y is borrower-or loan-specific making this a much more demanding task. There is no clear answer to this question. In most practical cases, Y is period-independent or changes deterministically over periods. This will become clear from the detailed example provided in Section 4. However, it cannot be ruled out that in some cases it might be beneficial to define some scenario also for a particular risk factor Y .
The main purpose of all these risk models is providing the inputs for (10). In particular, we need the four quantitiesN i , ELC LGD where L is the credit limit of a loan facility in case there is some credit line included and p 2 (k|i − 1) the term structure of default probabilities of a borrower conditional on survival until period i − 1 and being in Stage 2. The function h translates PIT LGDs into regulatory downturn LGDs. This might be a function prescribed by regulators as in the US or it might be something defined internally by a bank as in other jurisdictions. An explanation of why (28) and (29) withN i instead of N i are still correct can be found in Engelmann (2018).
To complete the explanation of (10), we have to know where t i ,f i and c i comes from. The parameter t i depends on the staging rule implemented by a bank. It could either be derived from the existing models or it could be estimated by a separate model, e.g. a logistic regression where instead of the event "moving to default" the event "moving to Stage 2" will be modeled. The example in Section 4 will clarify how this could work. The remaining cost parametersf i and c i will be discussed in the next subsection.

Determination of Cost Components
A bank's operational costs have to be covered by fees or interest income. How to allocate costs to a certain product is an internal procedure that might differ from bank to bank. For the purpose of this article, we assume that operational costs are modeled as a percentage of outstanding loan balance. It might also be modeled as an absolute amount or a combination of both. There might also be cases where a borrower pays some fees in addition to interest. To include all these cases in (10) is quite straightforward.
More interesting are funding costs. This is the interest a bank has to pay to its depositors and bond investors which is in general a term-structure that is provided by the treasury department. A stylized example is given in year loan it would have to pay 12M Libor + s i on the funds to its depositors every year. There might be alternative ways a treasury quotes these rates depending on the institutional setup. We use Table 1 as an example to illustrate how to derive funding costs for every year from the data of this table. A generalization to different market conventions is not difficult.
The reason why a treasury quotes funding rates in this way is that swap rates in the interbank market are much more volatile than the spread a bank has to pay over interbank market rates for its funding. To compute future expected funding costsf i , swap rates S i for each expiry i = 1, . . . , n are collected and discount factors and forward rates are bootstrapped from them. For the purpose of this article, we assume swaps exchange once per year a fixed rate for a 12M Libor rate and the payment frequencies of swaps and loans are identical.
The calculation off i is a two-step procedure. First, discount factors for the interbank curve have to be computed from the swap rates S i observed in the market. For current market quotes the value of the floating leg equals the value of the fixed leg resulting in where δ M j is the discount factor corresponding to year j. These discount factors can be computed iteratively using the bootstrap algorithm From δ M i , we can compute forward ratesλ i that a bank expects to pay on its funding,λ i = δ M i−1 /δ M i − 1. Using the forward ratesλ i and the spread s i in Table 1 allows us to computef i using a similar bootstrap procedure.
The condition for bootstrapping the funding curve δ i is that the present value of future cash flows for depositors equals the deposited amount: which leads to the bootstrapping algorithm Future expected funding costs are then computed asf i = δ i−1 /δ i − 1. These expected funding costsf i are suitable for measuring the profitability of floating-rate loans. Therefore, we denote them byf f loat,i for clarity. For fixed-rate loans a bank commonly uses a swap to manage interest rate risk. This results in fixed ratesĝ i a bank has to pay depending on the maturity of its funds. They are computed from the relation The funding costs f f ix,i a bank has to stem in each period for a fixed-rate loan can be computed from the loan's amortization schedulê where N n+1 = 0. The intuition is to split the loan into parts with different maturities according to its amortization schedule and chargeĝ j on the part maturing in period j.

An Example for Residential Mortgages
We provide a fully worked-out example for residential mortgages for illustration. The data and models we are using are not derived from real-world data. They are a bit simpler and stylized compared to real-world models but still contain the most important features. We consider a 10-year fixed-rate annuity loan as an example which pays interest and amortization annually. We assume the loan has just been originated and we measure its expected future performance using the scheme described in Section 3.
We start with the funding curve and the calculation of future funding costsf f ix,i for the years i = 1, . . . , 10. The results are summarized in Table 2. Both swap rates S and funding spreads s are increasing with expiry leading to a rather steep curve forf f loat . Since we use a fixed-rate loan in our example, the ratesf f ix will be used for computing funding costs later using (31). Operational costs are assumed at c = 0.50% of outstanding balance.
Next, we describe the stylized macroeconomic data. We assume that three macroeconomic factors are used in modeling credit risk for residential mortgages, the unemployment rate (UR), the growth in a house price index (HPIgr) and the mortgage rate (MR). The unemployment rate should be an indicator for default rates, since growth in unemployment should result in a higher number of people unable to repay their loan. Changes in house prices are usually modeled by an index since modeling prices of individual properties is in most cases not feasible. This variable should affect both default rates and loss rates. The mortgage rate in a macroeconomic sense is not the interest rate of an individual mortgage but some average rate for currently originated mortgages. Central banks often publish such rates. If a loan market permits prepayments, this interest rate is usually a good indicator for the prepayment likelihood. For these three macroeconomic factors, we assume the scenario in Table 3. The value for Year 0 would be a realized value while for Years 1 to 9 the values reflect a scenario, in this example the gradual return from an economic boom to a neutral state.
Year UR (%) HPIgr ( 5.00 0.00 4.00 Table 3. Scenario for the macroeconomic factors UR, HPIgr, and MR. For the determination of Z i for each future year by means of (14), our stylized macroeconomic model is Furthermore, we assume that the parameters B and ρ in (14) have been estimated as B = −2.25 and ρ = 3%. Using the macro scenario in Table 3 allows the calculation of a scenario for Z which is displayed in Table 4. Note, that negative Z reflect booms while positive Z signals recession.
Year Φ −1 (d) Z To measure the profitability of a mortgage, we need additional information. We assume that the original balance N 1 = 500, 000 is equal to the house price, i.e. the loan-to-value (LTV) at origination was 100%. The loan amortizes at 2% per year and its fixed interest rate z is 3.5%. This leads to an annual annuity payment of A = 500, 000 · (2% + 3.5%) = 27, 500. For Basel II and IFRS 9, we assume the bank has developed the following risk models for borrower and collateral: logit (p(i|i − 1))) = −6.0 + 3.0 · AD i−1 + 4.0 ·UR i−1 + LTV i−1 + 2.0 · DSC i−1 , where logit(x) = log 1−x x is the logistic transformation and AD an arrears dummy, i.e. AD = 1 if the borrower's payments are in arrears for 10 days or more and AD = 0 otherwise. The risk factor DSC is the debt service coverage ratio and is computed as the ratio of all annual payments a borrower has to make on all his loan products divided by his net income. To be meaningful all loans of a borrower including those from other banks have to be included. Whether this information in easily available or not depends on the institutional setup of a country. In some countries (e.g. Malaysia) the central bank supports the collection of this data, while in other countries credit bureaus might be useful sources of information.
Model equations (33) to (35) have been expected from the discussions in Section 3. There is no model for cc f because mortgages usually come without credit lines making EAD equal to outstanding balance. The other two model equations are related to the staging rule required for IFRS 9. One option to model "deteriorated credit quality" is using "payment missing for 10 days or more" as a criterion. The probability for being in Stage 2 is in this case equal to the probability of being in arrears but not yet in default at the end of a year. This probability is modeled by the arrears rate ar(i|i − 1) which is the probability that a loan is in arrears in Year i conditional on being performing in Year i − 1. Once a loan is in arrears it could either move into default or back to Stage 1. The cure rate cr(i|i − 1) models the probability that a loan is in Stage 1 in Year i conditional on being in Stage 2 in Year i − 1. This allows the calculation of cumulative probabilities ps i ( j) of a currently performing loan for being in Stage j = 1, 2, 3 in Year i by the following algorithm: In Year i the Stage 2 loans consist of loans that moved from Stage 1 to Stage 2 or were in Stage 2 already at the end of Year i − 1 but did neither default nor cure. This is possible since a loan can be in arrears, pay back the outstanding amount during the year and move into arrears again. The higher default probability for a loan in Stage 2 is reflected by the arrears indicator AD in (33). The probability t i is the likelihood that a loan is in Stage 2 conditional on survival in Year i. From the above it can be easily computed as t i = ps i (2) 1−ps i (3) .
To make the framework of Section 4 applicable, we have to explain how to project the risk factors LTV and DSC over 10 years and how to compute downturn LGD for regulatory capital calculations. For LTV we project the house price into the future using the house price index growth scenario that was already used in the macroeconomic model (32). The outstanding loan balance is computed according to the amortization schedule of the annuity. For DSC we assume that the net annual income of the borrower is 100,000 and this value stays constant over the 10 years. We further assume that the mortgage is the only loan of the borrower. Then, DSC is computed as the ratio of the annual interest and amortization payments divided by net income. Finally, we have to define a rule for downturn LGD. In some countries, like the US, this is prescribed by regulators, in other countries this has to be defined by the bank. We assume downturn LGD is defined by the bank and it is computed using (34) under a worst case scenario of a 25% decline of the house price. The resulting risk parameters in this setup are presented in Table 5.
Finally, we have all parameters at hand that are needed to compute loan performance. We can compute ELC j , LLP 1 and LLP 2 as well as required minimum capital conditional on Stage j = 1, 2. The results are displayed in Table 6. Computing RAROC of this loan using (11) leads to 8.586%. We see that period-by-period RAROC takes values between 7.26% and 10.24%. This means that looking into the performance of the loan in a single period and ignoring past and future periods could be quite misleading. Usually credit risk parameters change with loan balance. Loans could become less risky over time as in the mortgage example over the first seven years but could also become more risky if collateral value is declining or the macroeconomic environment is expected to deteriorate.

Conclusion
In this article, we discussed the measurement of loan performance taking into account the latest Basel regulation and the recent revision of accounting standards, IFRS 9 and CECL, respectively. We have discussed the different risk parameters that are needed for each framework, through-the-cycle, point-in-time and downturn parameters, and shown how they can be obtained in principle. We have proposed a framework where the performance of a loan is measured period-by-period using RAROC as performance measure. Taking a weighted average over all periods results in a particular loan's total performance. An example illustrated the application of this framework for residential mortgage lending.
The contribution of this article has been threefold. First, we have provided a comprehensive scheme for loan performance measurement that takes into account all relevant regulatory rules and utilizes the credit risk parameters that have been estimated for regulatory and accounting purposes. It allows bank to make more informed decisions about loan applications and supports them in monitoring their existing loan portfolios. Second, we have shown that LLP computed under IFRS 9 or CECL are not suitable as a measure for expected loss and cannot be use in the numerator of a RAROC scheme. Instead we have derived an expression for the expected loss coverage (ELC) which reflects the true economic loss. Third, we have seen that for an accurate measurement of loan performance, it is insufficient to consider the current or the next year only but the full lifetime of a loan has to be taken into account.
Finally, we remark that the framework we have proposed could be used for numerous risk management applications beyond performance measurement. It could be used for balance sheet stress testing where a macroeconomic stress scenario is provided and the evolution of provisions, capital, and profit can be monitored under this scenario over time. In addition, the impact of proposed new regulation can be evaluated to see whether a bank is severely impacted or not. An example would be BCBS (2017) where a new floor on regulatory capital is introduced and our proposed framework could be very helpful in assessing its impact.

Declaration of Interest
The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper.