# Journal of Credit Risk

**ISSN:**

1744-6619 (print)

1755-9723 (online)

**Editor-in-chief:** Linda Allen and Jens Hilscher

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Need to know

- The adoption of IFRS9 accounting model by financial institutions mandates the estimation of forward-looking losses for credit portfolios
- However, the incorporation of forward-looking losses into the existing Basel IRB framework, is not optimal, leading to possible capital overcharges
- After highlighting the mechanics of the problem, we propose slight alterations to the Basel functions in order to combine expected credit losses with the existing Capital Adequacy framework

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Abstract

It is evident that the definition of expected credit losses (ECL) diverges between International Financial Reporting Standard 9 (IFRS 9) (the accounting model recently adopted by European banks) and the probability of default/loss given default methodology used in the Basel internal ratings-based approach to capital adequacy estimation. The ongoing discussion on the incorporation of lifetime ECL into the Basel framework – through the adoption of lifetime expected losses with the greatest possible consensus – will eventually lead to modifications, but for the time being it is not optimal. We establish that the combination of lifetime ECL and the Basel Capital Adequacy Framework, which relies on a one-year horizon, results in capital overestimation. Alongside this finding, and in order to alleviate the problem, we propose two alterations to the risk weight functions that constitute the core of the Basel advanced methodologies.

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Introduction

## 1 Framework evolution

The Basel II framework, as introduced in Basel Committee on Banking Supervision (2004), presented the concept of internal ratings-based (IRB) approaches for capital measurement purposes. Banks were encouraged to provide internal risk parameter estimates (probability of default (PD), loss given default (LGD) or both) as an input to the suggested risk weight functions, in order to derive estimates for unexpected credit losses. Expected credit losses (ECL) were also defined, and they constitute an integral part of the aforementioned functions.

Even at that initial stage, it was implicitly recognized that a discrepancy between required ECL (as defined in IRB approaches) and actual provisions held could arise, leading to an overstatement of unexpected credit losses and consequently a reduction in the Common Equity Tier 1 (CET1) ratio. The cause of this inconsistency was the implementation of an accounting model for the calculation of provisions such as IAS 39 (IFRS Foundation 2008). The recommended “cure” for this potential problem was the inclusion of part of the excess provisions (no more than 60% of the risk-weighted assets), with the agreement of the various National Supervisory Authorities, in Tier 2 supplementary capital.

However, the transition to International Financial Reporting Standard 9 (IFRS 9)^{1}^{1} 1 Or the US GAAP–ASU 2016-13 rules. (IFRS Foundation 2014), which encompasses the notion of lifetime expected losses, along with the importance assigned to the CET1 ratio by Basel III rules (Basel Committee on Banking Supervision 2017) aggravate the aforementioned deficiency.

## 2 Basel formula decomposition

Ignoring maturity, haircut and specific portfolio adjustments, the “general” formula for the estimation of unexpected losses is^{2}^{2} 2 We assume familiarity with the Basel risk weight functions.

$$\mathrm{UL}=\left[\mathrm{\Phi}\left(\left(\frac{1}{\sqrt{1-\rho}}\right){\mathrm{\Phi}}^{-1}(\mathrm{PD})+\left(\frac{\sqrt{\rho}}{\sqrt{1-\rho}}\right){\mathrm{\Phi}}^{-1}(0.999)\right)-\mathrm{PD}\right]\mathrm{LGD}\cdot \mathrm{EAD},$$ | (2.1) |

where $\mathrm{UL}$ denotes the unexpected losses, $\mathrm{PD}$ denotes the one-year expected probability of default throughout the economic cycle, $\mathrm{LGD}$ denotes the LGD observed in the worst period of the economic cycle (${\mathrm{LGD}}_{\mathrm{downturn}}$), and

$$\mathrm{\Phi}\left(\left(\frac{1}{\sqrt{1-\rho}}\right){\mathrm{\Phi}}^{-1}(\mathrm{PD})+\left(\frac{\sqrt{\rho}}{\sqrt{1-\rho}}\right){\mathrm{\Phi}}^{-1}(0.999)\right)$$ |

is the stressed one-year PD, where the systematic risk factor degrades unexpectedly.

Setting

$${\mathrm{PD}}^{*}=\mathrm{\Phi}\left(\left(\frac{1}{\sqrt{1-\rho}}\right){\mathrm{\Phi}}^{-1}(\mathrm{PD})+\left(\frac{\sqrt{\rho}}{\sqrt{1-\rho}}\right){\mathrm{\Phi}}^{-1}(0.999)\right),$$ |

the unexpected loss equation, through the cycle, for a one-year horizon is rewritten as

$$\mathrm{UL}=\underset{\text{totallossoneyear}}{\underset{\u23df}{{\mathrm{PD}}^{*}\cdot \mathrm{LGD}\cdot \mathrm{EAD}}}-\underset{\text{expectedlossoneyear}}{\underset{\u23df}{\mathrm{PD}\cdot \mathrm{LGD}\cdot \mathrm{EAD}}}.$$ | (2.2) |

Thus, ECL under Basel IRB are defined as

$${\mathrm{ECL}}_{\mathrm{Basel}}=\mathrm{PD}\cdot \mathrm{LGD}\cdot \mathrm{EAD}.$$ | (2.3) |

## 3 The definition of lifetime expected credit losses

Let us define ECL for a fixed-term loan under IFRS 9, for stage 2 and stage 3 nondefaulted loans, or under a stricter framework like the US GAAP that demands lifetime losses for all loans, as follows:

$${\mathrm{ECL}}_{\mathrm{life}}=\sum _{t=1}^{T}\frac{{p}_{t}\cdot {\mathrm{lgd}}_{t}\cdot {\mathrm{EAD}}_{t}}{{(1+i)}^{t}},$$ | (3.1) |

where $T$ is the number of annual periods^{3}^{3} 3 The periods are assumed to be annual for the sake of simplifying the analysis. Monthly periods could be used instead with an adjustment to equivalent annual payments, leading to the same conclusions. until maturity, ${p}_{t}$ is the one-year expected PD for the interval $(t,t+1]$, ${\mathrm{lgd}}_{t}$ is the loss given that the loan defaults in year $t$, ${\mathrm{EAD}}_{t}$ is the loan amount balance in year $t$, and $i$ is the annual opportunity cost of capital for the financial institution.

## 4 Comparing credit losses

We apply the following assumptions so that PD and LGD can be compared. Without them, no comparison would be possible, since the definitions of PD and LGD in the Basel framework and in the IFRS 9 (US GAAP) framework diverge.

### 4.1 Annual PD

Assume that the cumulative probability function^{4}^{4} 4 Any other function that is monotonous and bounded to the $[0,1]$ interval may be used. until time $t$ takes the form

$${P}_{\mathrm{cum},t}=1-{\mathrm{e}}^{-at}.$$ | (4.1) |

For year 1,

$${P}_{\mathrm{cum},t}={P}_{1\mathrm{y}\mathrm{r}}=1-{\mathrm{e}}^{-a}\mathit{\hspace{1em}}\Rightarrow \mathit{\hspace{1em}}a=-\mathrm{ln}(1-{P}_{1\mathrm{y}\mathrm{r}}).$$ | (4.2) |

Further, if the one-year forward probability is close to the economic-cycle one-year PD,^{5}^{5} 5 In practice, average one-year default frequencies are calculated with the use of the longer available sample in order to approximate the cycle one-year horizon PD. then the following equality holds:

$${P}_{1\mathrm{y}\mathrm{r}}=\mathrm{PD}=p.$$ | (4.3) |

Substituting (4.2) and (4.3) into (4.1),

$${P}_{\mathrm{cum},t}=1-{(1-p)}^{t}.$$ | (4.4) |

The one-year expected PD as defined in (3.1) becomes

$${p}_{t}={P}_{\mathrm{cum},t+1}-{P}_{\mathrm{cum},t}=p{(1-p)}^{t}.$$ | (4.5) |

### 4.2 LGD

Assume that ${\mathrm{lgd}}_{t}={\mathrm{LGD}}_{\mathrm{downturn}}=\mathrm{lgd}$.

### 4.3 EAD

We definite $L$ to be the initial loan amount and $B$ to be the fixed annual installment paid at each point in time $t$, where

$$B=\frac{Lr}{1-{(1/(1+r))}^{T}}.$$ | (4.6) |

If we set $L$ to equal €1, $r$ to be the annual loan rate and ${\mathrm{ead}}_{t}$ to be the percentage of remaining loan capital at each point in time $t$, then

$${\mathrm{ead}}_{t}={(1+r)}^{t-1}-B\frac{{(1+r)}^{t-1}-1}{r}.$$ | (4.7) |

With the substitution of (4.6) into (4.7), we obtain

$${\mathrm{ead}}_{t}=\frac{{(1+r)}^{T}-{(1+r)}^{t}}{{(1+r)}^{T}-1}.$$ | (4.8) |

We employ relationship (3.1) with the use of (4.5) and (4.8) to calculate the ECL for every euro of loan amount (actually the percentage of ECL, defined as ${\mathrm{ecl}}_{\mathrm{life}}$):

$${\mathrm{ecl}}_{\mathrm{life}}=\sum _{t=1}^{T}\frac{{p}_{t}\cdot \mathrm{lgd}\cdot {\mathrm{ead}}_{t}}{{(1+i)}^{t}}=\underset{{\mathrm{ecl}}_{\mathrm{Basel}}}{\underset{\u23df}{p\cdot \mathrm{lgd}}}\underset{f}{\underset{\u23df}{\sum _{t=1}^{T}{(1-p)}^{t}\frac{{(1+r)}^{T}-{(1+r)}^{t}}{{(1+i)}^{t}({(1+r)}^{T}-1)}}},$$ | (4.9) |

where ${\mathrm{ecl}}_{\mathrm{Basel}}$ is the ECL of the loan according to the Basel definition in (2.3), with $\mathrm{EAD}=\text{\u20ac1}$ and $f$ being the increasing factor of ${\mathrm{ecl}}_{\mathrm{life}}$ over ${\mathrm{ecl}}_{\mathrm{Basel}}$.

From (4.9),

$${\mathrm{ecl}}_{\mathrm{life}}=f\cdot {\mathrm{ecl}}_{\mathrm{Basel}}.$$ | (4.10) |

## 5 Impact estimation

The impact estimation is based on term loans with maturities over one year. For term loans with annual maturity, revolving loans with annual revaluation of the loan or the connected obligor, or stage 1 loans categorized by IFRS 9 rules, the Basel framework is sufficient.

The comparison therefore refers to

- •
mortgage loans,

- •
fixed-asset business loans,

- •
specific project loans and

- •
consumer loans.

Setting $\mathrm{EAD}=\text{\u20ac1}$, we redefine (2.2) in percentage terms:

$${\mathrm{ul}}_{\mathrm{Basel}}={\mathrm{tl}}_{\mathrm{Basel}}-{\mathrm{ecl}}_{\mathrm{Basel}},$$ | (5.1) |

where ${\mathrm{ul}}_{\mathrm{Basel}}$ is the unexpected loss as defined in the Basel framework for €1 of loan, for a one-year period, and ${\mathrm{tl}}_{\mathrm{Basel}}$ is the total loss as defined in the Basel framework for €1 of loan, for a one-year period.

Since ${\mathrm{ecl}}_{\mathrm{life}}\%$ of the loan is covered with the use of new IFRS 9 rules, the proper way to calculate (the percentage) unexpected loss is through

$${\mathrm{ul}}_{\mathrm{Basel}}^{\prime}={\mathrm{tl}}_{\mathrm{Basel}}-{\mathrm{ecl}}_{\mathrm{life}}.$$ | (5.2) |

The percentage impact on unexpected losses of adopting a framework that requires a lifetime horizon for the calculation of ECL would be the difference between (5.1) and (5.2):

$$\mathrm{\Delta}{\mathrm{ul}}_{\mathrm{Basel}}={\mathrm{ul}}_{\mathrm{Basel}}-{\mathrm{ul}}_{\mathrm{Basel}}^{\prime}={\mathrm{ecl}}_{\mathrm{Basel}}(f-1).$$ | (5.3) |

The increasing factor $f$ is essentially a function of the other observed variables introduced in Section 3:

$$f=f(T,r,p,i).$$ | (5.4) |

The marginal effect (derivations based on (4.9)) on the increasing factor, based on the definition of $f$ in (4.9), is given by

$\frac{\partial f}{\partial T}$ | $>0,$ | (5.5) | ||

$\frac{\partial f}{\partial r}$ | $>0,$ | (5.6) | ||

$\frac{\partial f}{\partial p}$ | $$ | (5.7) | ||

$\frac{\partial f}{\partial i}$ | $$ | (5.8) |

The theoretical effect based on derivations is verified by providing values for the parameters $T$, $r$, $p$ and $i$.

### 5.1 Numerical verification of unexpected loss sensitivity

The sensitivities depicted in (5.5)–(5.8) are verified in Figures 1–3. As an example, we use the term loan defined in Section 3 with varying parameters.

What we observe on the vertical axis, in all three figures, is the $(f-1)$ coefficient of (5.3). On the horizontal axis, denoted “PD effect”, is the borrower increase in total risk (individual plus systematic risk).

In Figure 1, which shows the effect of loan maturity ($T$) on the increase of unexpected losses, three identical term loans are considered ($r=5\%$, $i=1\%$) with corresponding maturities (in years) of $T=2$, $T=10$, $T=30$, for varying PD values. It is evident that maturity increase is accompanied by a larger $(f-1)$ factor. The unexpected loss increases in accordance with (5.3). Portfolios comprised of long-maturity loans, such as mortgage loans, are greatly affected.

In Figure 2, one of the loans used in Figure 1 ($r=5\%$, $i=1\%$, $T=10$) is contrasted with a loan with parameters $r=15\%$, $i=1\%$, $T=10$, at different PD levels. It is evident that for portfolios with higher interest rates (eg, consumer and personal loan portfolios) the costs of unexpected losses will be enhanced compared with those of lower-rate portfolios.

When the opportunity cost of funds is higher, implying increased profitability for the financial institution in other comparable projects/portfolios, the significance of extra unexpected losses tends to be reduced. This fact is shown in Figure 3.

Finally, it is easily extracted from Figures 1 to 3 that higher PDs lean toward diminishing the effect of extra unexpected losses.

What we have established so far is that, for IRB portfolios, the incorporation of lifetime loan losses will lead to higher provisioning but also to augmented capital charges in the form of unexpected losses. The effect will be

- •
- •
- •
- •

The adjustment of IRB formulas in order to avoid capital overcharging is examined in the final section.

## 6 Adapting the internal ratings-based approach formula

$?$ | lgd | Definition | |
---|---|---|---|

Loan 1 | 0.01 | 0.10 | Low-risk obligor with high-quality collateral |

Loan 2 | 0.01 | 0.75 | Low-risk obligor with low-quality collateral |

Loan 3 | 0.15 | 0.10 | High-risk obligor with high-quality collateral |

Loan 4 | 0.15 | 0.75 | High-risk obligor with low-quality collateral |

UL | ECL | ||

equation | equation | Definition | |

Method 1 | (2.1) | (2.3) | Basel approach for UL and ECL for IFRS 9 |

stage 1 loans | |||

Method 2 | (2.1) | (4.9) | Basel approach for UL and ECL for IFRS 9 |

stage 2 and stage 3${}^{*}$ loans | |||

Method 3 | (6.2) | (4.9) | Suggestion as described in Section 6.1 |

Method 4 | (6.4) | (4.9) | Suggestion as described in Section 6.2 |

Loan 1 | Method 1 | Method 2 | Method 3 | Method 4 |
---|---|---|---|---|

ECL (provisions) | €100.00 | €897.76 | €897.76 | €897.76 |

UL (capital) | €1002.65 | €1002.65 | €204.89 | €3476.13 |

Total | €1102.65 | €1900.41 | €1102.65 | €4373.89 |

Loan 2 loss | Method 1 | Method 2 | Method 3 | Method 4 |
---|---|---|---|---|

ECL (provisions) | €750.00 | €6 733.21 | €6733.21 | €6 733.21 |

UL (capital) | €7519.86 | €7 519.86 | €1536.65 | €26 070.96 |

Total | €8269.86 | €14 253.07 | €8269.86 | €32 804.17 |

Loan 3 loss | Method 1 | Method 2 | Method 3 | Method 4 |
---|---|---|---|---|

ECL (provisions) | €1500.00 | €6 894.04 | €6894.04 | €6894.04 |

UL (capital) | €4190.62 | €4 190.62 | €0.00 | €2772.80 |

Total | €5690.62 | €11 084.66 | €6894.04 | €9666.84 |

Loan 4 loss | Method 1 | Method 2 | Method 3 | Method 4 |
---|---|---|---|---|

ECL (provisions) | €11 250.00 | €51 705.31 | €51 705.31 | €51 705.31 |

UL (capital) | €31 429.67 | €31 429.67 | €0.00 | €20 796.01 |

Total | €42 679.67 | €83 134.98 | €51 705.31 | €72 501.32 |

According to the business logic applied, two slight alterations could clear the inconsistency driven by unexpected losses.

### 6.1 Keep unexpected and expected losses with a different time horizon

If we use the actual amount of provisions held under the lifetime horizon-demanding accounting framework in (2.2), then a different magnitude of unexpected losses is derived:

$${\mathrm{UL}}^{\prime}=\underset{\text{totallossoneyear}}{\underset{\u23df}{{\mathrm{PD}}^{*}\cdot \mathrm{LGD}\cdot \mathrm{EAD}}}-\underset{\text{expectedlosslife}}{\underset{\u23df}{{\mathrm{provisions}}_{\mathrm{life}}}}.$$ | (6.1) |

The idea behind such a calculation would be the preservation of capital additional to lifetime expected losses, representing adverse changes over a one-year horizon. The final adjustment of the general IRB formula (2.1) should be

${\mathrm{UL}}^{\prime}=\mathrm{max}\{\mathrm{\Phi}(\left({\displaystyle \frac{1}{\sqrt{1-\rho}}}\right){\mathrm{\Phi}}^{-1}(\mathrm{PD})$ | $+\left({\displaystyle \frac{\sqrt{\rho}}{\sqrt{1-\rho}}}\right){\mathrm{\Phi}}^{-1}(0.999))$ | |||

$\mathrm{\hspace{1em}}\times \mathrm{LGD}\cdot \mathrm{EAD}-{\mathrm{provisions}}_{\mathrm{life},0}\},$ | (6.2) |

with $\mathrm{EAD}$ the current loan amount ($t=0$).

### 6.2 Match the time horizon of unexpected and expected losses with a different PD

This solution builds on the notion of a single equivalent to lifetime probability, which stems from (4.9), thus equating the reference horizon for expected and unexpected losses. For all loans with a maturity longer than one year,

$${\mathrm{ecl}}_{\mathrm{life}}={\mathrm{PD}}_{\mathrm{life}}\cdot \mathrm{lgd}\mathit{\hspace{1em}}\Rightarrow \mathit{\hspace{1em}}{\mathrm{PD}}_{\mathrm{life}}=\frac{{\mathrm{ecl}}_{\mathrm{life}}}{\mathrm{lgd}}.$$ | (6.3) |

Adjust the general equation (2.1) to

${\mathrm{UL}}^{\prime \prime}=[\mathrm{\Phi}(\left({\displaystyle \frac{1}{\sqrt{1-\rho}}}\right){\mathrm{\Phi}}^{-1}({\mathrm{PD}}_{\mathrm{life}})$ | $+\left({\displaystyle \frac{\sqrt{\rho}}{\sqrt{1-\rho}}}\right){\mathrm{\Phi}}^{-1}(0.999))-{\mathrm{PD}}_{\mathrm{life}}]$ | |||

$\mathrm{\hspace{1em}\hspace{1em}}\mathit{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\times \mathrm{LGD}\cdot \mathrm{EAD},$ | (6.4) |

where $\mathrm{EAD}$ represents the current loan amount ($t=0$).

### 6.3 Measuring capital overcharging along with the effect of proposed changes

Using as a basis a single mortgage loan example, as defined in Section 3 and using the methodology of Section 4, we will attempt to quantify the actual effect of the aforementioned suggestions. The following assumptions are made: $L=\mathrm{\u20ac}100$, as described in Section 4.3; $r=3\%$, as described in Section 4.3; $i=3\%$, as described in Section 3; and $T=20$, as described in Section 3.

By varying the risk parameter values – $p$ as defined in (4.3) and $\mathrm{lgd}$ as defined in Section 4.2 – we create four different loans with common $L$, $r$, $i$ and $T$ values: see Table 1.

We will estimate the expected and unexpected losses for the loans in Table 1 using four different methods: see Table 2.

The following points are evident from Figures 4–7.

- (1)
The application of the methodology of lifetime expected losses, for the part of the portfolio that is applied (eg, IFRS 9 stages 2 and 3), enhances the total capital charge, as can be seen from the results of method 2.

- (2)
The correct amount of unexpected loss is given by our method 3 calculations, since part of the method 2 unexpected loss is already included in the method 3 ECL amounts.

- (3)
In our opinion, method 4 is the most accurate solution, on the assumption that lifetime expected losses are taken into account, because it reconciles the lifetime reference horizons of unexpected losses and ECL.

- (4)
If the amounts derived by method 4 are too conservative and the banking industry fails to provide the necessary credit expansion during times of economic growth, then method 3 is a suitable alternative, as it allows for an unexpected loss amount but not for a full lifetime horizon. This is a question that remains to be investigated.

The emerging unexpected losses capital overcharge (see (2) above) may be defined as

$${\mathrm{ul}}_{\mathrm{overcharge}}=\frac{{\mathrm{UL}}_{\mathrm{method}3}-{\mathrm{UL}}_{\mathrm{method}2}}{L}.$$ | (6.5) |

This overcharging effect, for each loan case, is presented in Figure 8. It is clear that the worsening of the loan risk parameters affects the predefined ${\mathrm{ul}}_{\mathrm{overcharge}}$ percentage.

## 7 Conclusion

The current adoption of forward-lifetime ECL accounting (IFRS 9) distorts the “traditional” definitions of expected and unexpected losses in the Basel framework, which are fused into the IRB risk weight functions. In this paper, the distortion is illuminated through the consideration of a term loan with a maturity of over a year, and it is quantified both theoretically and empirically. Minor modifications are proposed for the core of the risk weight functions in order to keep up with the business logic of lifetime losses.

## Declaration of interest

The author reports no conflicts of interest. The author alone is responsible for the content and writing of the paper. The proposed methodologies reflect the author’s view only and have no relation to any practices implemented by the National Bank of Greece. To the best of my knowledge, at the time of writing (March 2018), no publication describing a similar methodology exists.

## References

- Basel Committee on Banking Supervision (2004). International convergence of capital measurement and capital standards – a revised framework. Report, June, Bank for International Settlements.
- Basel Committee on Banking Supervision (2006). International convergence of capital measurement and capital standards - a revised framework (comprehensive version). Report, June, Bank for International Settlements.
- Basel Committee on Banking Supervision (2010). Basel III: a global regulatory framework for more resilient banks and banking systems. Report, Bank for International Settlements.
- Basel Committee on Banking Supervision (2015). Guidance on credit risk and accounting for expected credit losses. Report, Bank for International Settlements.
- Basel Committee on Banking Supervision (2016). Regulatory treatment of accounting provisions. Discussion Paper, Bank for International Settlements.
- Basel Committee on Banking Supervision (2017). Basel III: finalising post-crisis reforms. Report, Bank for International Settlements.
- Cohen, B. H., and Edwards, G. A., Jr. (2017). The new era of expected loss provisioning. BIS Quarterly Review March, 39–56.
- IFRS Foundation (2008). International accounting standard 39 – financial instruments: recognition and measurement (revision 12/2008). Report, International Financial Reporting Standards Foundation.
- IFRS Foundation (2014). IFRS 9 financial instruments. In Red Book 2017. International Financial Reporting Standards Foundation.

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