The underlying assumption when using an obligor-weighted average of annual DRs is that all obligors of a certain credit rating are associated with a unique TTC PD (independent of their time of default; this contrasts with the PIT PD model discussed in the next subsection). This can be justified with mathematical equations.

Assume that we have $T$ years of data, and for each year there are $N_t$, $t=1,\mathrm{\dots },T$, obligors. Whether a specific obligor defaulted or not can be modeled by independent and identically distributed (iid) Bernoulli random variables $x_{k,t}$, $k=1,\mathrm{\dots },N_t$. The annual DR would be expressed as ${\mathrm{DR}}_t=\sum x_{k,t}/N_t$. The obligor-weighted long-run average PD (${\mathrm{LRPD}}_{\mathrm{TTC}}$) would be

which implies that $x_{k,t}\sim \mathrm{Bern}(p)$ for all $k,t$, and $p$ can be estimated with (2.1). In other words, ${\mathrm{LRPD}}_{\mathrm{TTC}}{\sum }_t{N}_t$ follows a binomial distribution, $B({\sum }_t{N}_t,p)$. The variance of ${\mathrm{LRPD}}_{\mathrm{TTC}}$ can be computed by

Usually, ${\sum }_t{N}_t$ is large, so ${\mathrm{LRPD}}_{\mathrm{TTC}}$ can be estimated with high precision. The confidence intervals evaluated using all types of methodologies (eg, Clopper–Pearson, Wald, etc) would be narrow and close to one another. However, these small confidence intervals offer false security. The simple derivation of the mean and variance of LRPD is based on the over-simplified assumption of all historical obligors following the same ${\mathrm{LRPD}}_{\mathrm{TTC}}$ at the cost of losing the annual DR information. Note how ${\mathrm{DR}}_t$ disappeared in (2.1).

In recognition of the fact that the realized annual DRs of a certain credit rating (from a TTC rating system) do vary from time to time, a more reasonable assumption is that all obligors of a certain credit rating are associated with a PIT PD (a time-varying PD dependent on the time of default). This PIT PD is a function of time and, in practice, is assumed to be constant over a year. Therefore, in mathematical terms, the obligors are represented by iid Bernoulli random variables $x_{k,t}$, $k=1,\mathrm{\dots },N_t$, and $x_{k,t}\sim \mathrm{Bern}(p_t)$, $t=1,\mathrm{\dots },T$, where $p_t$ are iid random variables following an arbitrary distribution; this contrasts with the TTC PD model in the previous subsection, where $p$ was assumed to be constant throughout time. The optimal estimate of the long-run PD is therefore the expectation of $p_t$ (not a function of $t$), which can be calculated as the simple average of the estimate of $p_t(t=1,\mathrm{\dots },T)$, ${\widehat{p}}_t={\mathrm{DR}}_t$, as the realized annual DR:

To evaluate the deviation of this estimate of the LRPD, note that ${\widehat{p}}_t\sim B(N_t,p_t)$. By the law of total variance, the variance of ${\mathrm{LRPD}}_{\mathrm{PIT}}$ can be broken down into two parts:

The first part captures the variance generated by the binomial process:

The second part captures the variance of the mean of $p_t$ itself:

The total variance of ${\mathrm{LRPD}}_{\mathrm{PIT}}$ is the combination of the two:

Since the distribution of $p_t$ is unknown, without loss of generality, $E[p_t]$ and $\mathrm{Var}[p_t]$ could be replaced with their approximations $\overline{p}=(1/T){\sum }_t{\widehat{p}}_t$ and $(1/(T-1)){\sum }_t{({\widehat{p}}_t-\overline{p})}^2$. Except for $E[p_t]$ in the first component, all other terms are second-order values. So, $\mathrm{Var}({\mathrm{LRPD}}_{\mathrm{PIT}})$ is expected to be dominated by the first component (the binomial process) when $N_t\ll 1/\overline{p}$ and by the second component (the time series) when $N_t\gg 1/\overline{p}$. In the next section, we will use the S&P data to evaluate exactly how the two components affect the deviation of the estimate.

A special case assuming the iid normal distribution of $p_t$ was assessed by Cantor and Falkenstein (2001). Our derivation below shows that a minor error exists in their final result. Based on this assumption, $p_t\sim N(p,{\sigma }^2)$. We could replace the terms in (2.3) using $E[p_t]=p$ and $\mathrm{Var}[p_t]={\sigma }^2$ and get

For each year ($T=1$), the observed ${\widehat{p}}_t$ (or ${\mathrm{DR}}_t$) follows the distribution

Note the missing term in Cantor and Falkenstein (2001, Equation (3)).

The two mathematical models are built on the same assumption; namely, for each credit rating there is a fundamental implied long-term PD. For the TTC PD estimation model, the annual PDs are assumed to be constant and equal to this long-term PD. For the PIT PD estimation model, the annual PDs are assumed to be realizations of a random variable whose expectation is the long-term PD. As the number of years $T$ gets larger, ${\mathrm{LRPD}}_{\mathrm{TTC}}$ and ${\mathrm{LRPD}}_{\mathrm{PIT}}$ will converge.

Despite having the same assumption on the existence of long-term PDs, the two mathematical models have different focuses. The TTC PD model ignores the fluctuation of the implied PD of credit ratings over time and is inappropriate for the PD estimation of a practical credit system whose ratings have a TTC component.²² 2 Having a TTC component in the rating means that the rating is not pure PIT; a constant PD assumption is only appropriate for a PIT rating system where the economic conditions are 100% (ideally) absorbed in the rating so that the implied PD can stay constant. However, it is more suitable (or easier to use) for some processes in PD validation, eg, the calibration of the rating system and the evaluation of rating consistency between different portfolios. It may best be used to answer the following question: in the long run, does the rating system yield reliable/consistent rankings of obligors’ credit? In other words, when the monotonicity of implied PDs and the separability of their confidence intervals are evaluated (Hanson and Schuermann 2005), the TTC PD model is the simplest and best option. On the other hand, the PIT PD model focuses on the annual realized PD and provides reliable answers to the following questions: what is the best PD estimate for the next $T$ years, and what is its confidence level?

The results for the TTC PD evaluation are summarized in Table 2.

The most notable issue in Table 2 is that the monotonicity of the implied PD is broken for ratings above BB$+$. This result is consistent with that in Hanson and Schuermann (2005), which indicates that, due to the limited default data, fine rating grades in high credit rankings are difficult to calibrate and deemed unreliable. In our subsequent analyses, we will therefore combine the ratings BB$-$ and above by ignoring their modifiers.³³ 3 Although we are tempted to combine modifiers in investment grades only, Table 11 in the online appendix shows that rating BB$+$ is not distinguishable from BBB. Table 2 becomes Table 3. The monotonicity of implied PDs is conserved and the TTC PD estimates of ratings below AA are separated by at least 2.44 standard deviations (98.5% confidence).

The validation results of the above monotonicity and confidence interval analyses indicate that banks should not try to keep fine rating notches with high credit ratings. In the PD estimation process, rather than applying techniques such as line-of-best-fit to get monotonic PD numbers, fine rating notches could be combined into coarse rating grades over which PD estimation may be done. The advantage of this alternative approach is two-fold. First, it would help improve product pricing. Second, it would not distort the true underlying PDs of the rating system.

Back to the discussion on PD parameter estimation, under the TTC PD methodology with the common assumption by most practitioners that the annual DR follows an iid normal distribution with mean $p$, we can derive the 95% confidence interval of the estimate of $p$ for each rating grade. The upper limit is considered an estimate with a sufficient margin of conservatism. However, as we mentioned in the introduction of this paper, if we conduct a backtest, excessive breaches (highlighted in bold in Table 4) may be observed in all rating grades with meaningful historical defaults, even though only one breach is expected over the twenty-one-year history. This inconsistency indicates that the TTC PD assumption is not appropriate for a TTC rating system, whose rating stays stable over the economic cycle while its implied PD fluctuates. This problem could be fixed with PIT PD modeling, which we will show in the next subsection.

When we tried to evaluate the annual realized DRs for S&P, we discovered that, for certain rating grades, data is not available for all of the years from 1995 to 2015. Namely, for some years, there are no obligors in the S&P data set rated at certain rating grades. These are regarded as missing data points, and the total number of years $T$ is adjusted accordingly. Table 5 summarizes the key statistics from applying the PIT PD model to the S&P large corporate data set (1995–2015).

As expected, the S&P data set is large enough that the LRPDs computed under the two methodologies are quite close to each other. However, some deviations are obvious at ratings “B” and “B$-$” in Figure 2. These are where neither the number of obligors nor the PD is large enough. Figure 2 also shows that the LRPDs from both models can be well fitted into lines, indicating that the S&P rating system is nicely calibrated. The two lines are close to each other too. Although we do not suggest using the PDs from fitted lines for ratings with default data (“A” and below), the extrapolated PDs for ratings “AAA” and “AA” are good candidates. The readings are 0.003% for “AA” and 0.001% for “AAA”, which will be used as the PD estimations for these two ratings.

The last two columns in Table 5 show the contributions of variance from binomial processes and time series to the total variance. The variances from time series are always larger than those from binomial processes, especially in the middle credit rating ranges, where $N_t$ are relatively large compared to $1/p_t$. However, at both ends of the spectrum, they are comparable. This observation suggests that the deviation from the TTC PD model is not appropriate to derive the confidence interval when LRPD is used as the prediction of PD for the coming year; this is evident from the backtest result shown earlier. It misses the variance coming from the annual realization of $p_t$. Actually, if we take the square root of the first term of variance in Table 5, the numbers are very close to the deviation column in Table 3.

Conducting the same backtest using the deviation under the PIT PD model, we see (in Table 6) that the numbers of breaches are as expected.

One major application of a PD estimate is for IRB capital calculation. In this case, the PD estimate is used as a prediction of the coming year’s DR. Apparently, the PIT PD estimation model would be more appropriate for this purpose because the assumption of the coming year’s $\mathrm{DR}\sim B(N_t,p_t)$, with its parameter $p_t$ following the same distribution as previous years, and with an expectation of $E[p_t]$ and a variance of $\mathrm{Var}[p_t]$, is much more realistic than the assumption of every year’s DR being equal to $p$. So, we have the expected DR in the coming year as $E[p_t]$, estimated as the average of the historical annual realized DRs (${\widehat{p}}_t$), $\overline{p}=(1/T){\sum }_t{\widehat{p}}_t$, also known as ${\mathrm{LRPD}}_{\mathrm{PIT}}$. The expected deviation of DR is

by replacing $T$ with 1 in (2.3). Here, like $E[p_t]$, $\mathrm{Var}[p_t]$ can be approximated as the standard deviation (STD) of ${\widehat{p}}_t$ squared, $(1/(T-1)){\sum }_t{({\widehat{p}}_t-\overline{p})}^2$. Some might be tempted to use the STD $\sqrt{(1/(T-1)){\sum }_t{({\widehat{p}}_t-\overline{p})}^2}$ directly to compute the confidence interval for the coming year’s PD. However, as we have shown in the previous subsection, the deviation of the PD prediction would be underestimated, especially for credit ratings at the two extremes (good and bad), where the first term of $(E[p_t]-E^2[p_t]-\mathrm{Var}[p_t])/N_t$ would play an important role because $N_t$ is not much larger than $1/p_t$.

Back to the S&P large corporate data set (1995–2015), if we assume that the numbers of obligors do not change between 2015 and 2016, Table 7 shows our PD estimate/prediction for 2016.

The right two columns in Table 7 provide the 80% and 90% one-sided confidence intervals. These can be interpreted as an expectation that there will be no more than two and one breaches in ten years, respectively. A simple check of the S&P historic data found that two years for “CCC$-$” and three years for “CC” observed 100% default rates; this confirms that the 100% PD estimate for these low credit ratings is not overly conservative, considering their high expected PD and low number of obligors. However, the evaluation of the confidence interval is done on each individual rating grade independently. Since the $p_t$ for all ratings are not perfectly correlated (see Table 8 for the realized correlation coefficients), the 80% one-sided confidence intervals might be good enough for the overall portfolio level.

Recently, a lot of research has been conducted to predict the PIT PD term structure in response to the IFRS 9 requirement (International Financial Reporting Standards 2014, Section 5.5.10): “At each reporting date, an entity shall measure the loss allowance for a financial instrument at an amount equal to the lifetime expected credit losses if the credit risk on that financial instrument has increased significantly since initial recognition.”

By lifetime expected credit losses (ECLs), we are to understand the “worst annual” ECL in a lifetime. Since ECL is computed based on the parameters PD, loss given default (LGD) and exposure at default (EAD), banks try to compute lifetime (usually five years) PIT PDs. There are several approaches to this, and each has different drawbacks.

1. (1)

   Derive the five-year cumulative PD term based on the historical data. This is a direct extension of the one-year PD estimate based on the Cohort method. However, the insufficiency of data becomes more serious as more years are considered (eg, very few obligors have rating data for five years). Therefore, more nonmonotonicity and missing data points for realized PD mean that techniques like curve-fitting and extrapolation are overused. This will result in unreliable PD term estimation and oftentimes unreasonable values ( or $\mathrm{PD}>1$).

2. (2)

   Project the multiperiod PD based on rating transition models. The two main issues with this approach are as follows. First, it assumes the rating migration follows a stationary Markov chain process. This assumption has been proven to be questionable because of the momentum in credit rating. Second, again because of limited data availability, it is extremely difficult to get a reliable estimate of the transition matrix.

3. (3)

   Predict the forward PD by some macroeconomic variables. As shown in the beginning of this section, the variation of realized PD in a TTC rating system could be explained in part by macroeconomic conditions. For a PIT rating system, however, the changes of the macroeconomic condition have been embedded into the rating assigned, and the observed PD therefore varies less and is uncorrelated with macroeconomic parameters. Even though for a TTC rating system the macroeconomic variables (current or one- or two-years latent) might provide good explanatory power for a one- or two-year PD, we should not expect them to be predictable for long-term PD. The reasoning behind this is as follows. Assume that some macroeconomic variables are able to reliably predict the five-year PD; at the same time, the PD is highly correlated with the recent macroeconomic parameters. This would imply that these macroeconomic parameters could be reliably predicted in a five-year range too, which is contrary to the basic principles of economics.

If we cannot hold much faith in these long-term PD predictions, are there any other options? If our understanding of lifetime ECL as worst annual ECL in a lifetime is correct, its estimate can be based on the worst annual PD in a lifetime. Here, the PD should be the one-year PIT PD: after all, IFRS 9 is an accounting standard for annual reports. According to common practice, five years can be used as a proxy for “lifetime”. The problem is thus reduced to find the worst one-year PD in five years. Assuming the underlying annual PDs follow a normal distribution $p_t\sim N(p,{\sigma }^2)$, and based on our result in (2.5) in Section 2.2, the realized PD prediction ${\widehat{p}}_t\sim N(p,((p(1-p)-{\sigma }^2)/N_t)+{\sigma }^2)$. The worst one-year PD in five years would be the expected value of order statistics of five iid $N(p,((p(1-p)-{\sigma }^2)/N_t)+{\sigma }^2)$ random variables. Using the results of Teichroew (1956) directly (partially quoted in Table 9), we get the worst one-year PD in five years:

where $p$ and $\sigma$ are the mean and STD of the historical observed PD, respectively. The numerical results applied to the S&P data set are shown in Table 10.

Another area involving the PIT versus TTC discussion is the rating conversion from PIT to TTC. One typical objective of such a conversion is to get stable regulatory capital that does not vary greatly from year to year in response to changes in the economic cycle. The optimal way of developing another TTC rating system based on borrowers’ long-term credit characteristics involves too much effort. Moody’s (Hamilton et al 2011) proposed a method to convert its expected default frequency (EDF), a PIT credit measure based on market data (distance to default, DD), directly to TTC EDF by transforming DD. This simple shortcut, as recognized by Moody’s, comes at the cost of reduced efficiency and default prediction accuracy. Given the current input DD, the original mapped EDF is the best estimate at the moment. Any change to the mapping will result in a distorted/suboptimal EDF estimate. Keeping this in mind, we evaluate Moody’s PIT to TTC transformation. The fundamental transform is done using the simple linear equation

where . This keeps the zero-frequency component and attenuates all other frequency components equally. ${\mathrm{DD}}_t^{\mathrm{TTC}}$ is thus a suppressed version of ${\mathrm{DD}}_t$ around its mean $\mu$, and the TTC EDF is a suppressed version of EDF around $\mathrm{EDF}(\mu )$. In terms of variation of EDF, the amplitude is decreased. However, the assumption of the constant $\mu$ is a big flaw in this conversion. ${\mathrm{DD}}_t$ includes not only the cyclical component, but also the idiosyncratic credit factors whose changes are not mean-reverting. The exponentially weighted moving average (EWMA) seems to be a better alternative:

In general, the idea of the direct PIT to TTC rating conversion is to apply a low pass (LP) filter to smooth out the resultant rating generated by the rating system. Whether this LP filter should be in the form of a simple moving average (SMV) or an EWMA, and whether the LP filter should be applied to the rating system input (risk drivers, ie, DD in Moody’s case) or to the final rating are choices left up to the practitioners, as long as they are theoretically correct and fulfill the practical objectives. It should always be kept in mind that the original PIT ratings are the best ratings assigned based on the best current knowledge. The converted TTC ratings are compromised ratings for capital calculation only. PIT ratings should still be kept and used for pricing purposes, at least for short-term loans, since PIT ratings incorporate more recent information and provide risk rankings that reflect obligors’ most current credit situations. Araten et al (2004) actually showed that the discriminatory power of Moody’s KMV is higher than Moody’s TTC ratings based on its rating methodology papers (71% versus 64.4% during 1997–2002 in terms of accuracy ratios).