# Default Distances Based on the KMV-CEV Model

Wen Su

###### Abstract

This paper presents a new method to assess default risk based on applying non
constant volatility to the KMV model, taking the CEV model as an instance. We
find the evidence that the classical KMV model could not distinguish ST
companies in China stock market. Aiming at improve the accuracy of the KMV
model, we assume the firm’s asset value dynamics are given by the CEV process
$\frac{dV_{A}}{V_{A}}=\mu_{A}dt+\delta V_{A}^{\beta-1}dB$ and use fixed
effects model and equivalent volatility method to estimate parameters. The
estimation results show the $\beta>1$ for non ST companies while $\beta<1$ for
ST companies and the equivalent volatility method estimate the parameters much
more precisely. Compared with the classical KMV model, our CEV-KMV model fits
the market better in forecasting the default probability. We also provide an
insight that other volatility model can be applied, too.

Keywords: KMV, Default Risk, CEV Process, Equivalent Volatility

##  1 Introduction

Due to the outbreak of the worldwide financial crisis in 2008, risk management
especially credit risk management becomes one of the most centralized topics
in finance. Bankrupt or default forecasting models, like the KMV model and
Credit Metrics model, now play a key role in assessing credit risk.

The well-known KMV model was developed by Kealhofer, McQuown, Vasicek and
their company which is also called KMV Corporation in 1997, based on the
seminal study of Merton (1974) [1]. What KMV model contributes is it offers an
insight that we can regard the equity as a call option on the firm’s asset. In
the past several decades, the power of the KMV model in assessing credit risk
has been tested. Kealhofer and Kurbat (2001) [2] focus on the listed companies
in North America and find the KMV model captures all the information of the
traditional accounting variables in financial statements when forecasting the
default risk. Compared with Altman’s (1968) [3] Z-Score and Ohlson’s (1980)
[4] O-Score, Hillegeist et al. (2004) [5] also argue that the KMV model
provides significantly more information than either of the two accounting-
based measures.

In order to improve accuracy of forecasting default probability, various
advanced theories and methods are applied by other researchers. Bharath and
Tyler (2008) [6] put up a naive predictor and find it performs slightly better
in hazard models and in out-of-sample forecasts than both the KMV model and a
reduced-form model that uses the same inputs. In order to find the best
default point, Zhang and Shi (2016) [7] use particle swarm optimization and
fuzzy clustering to modify the KMV model and the results show their FC-PSO-KMV
model provides more accurate predictions. Focusing on the default point, too,
Song et al. (2020) [8] also use particle swarm optimization to adjust the KMV
model since they find the classical KMV model could not do distinguish the ST
companies, a fact which will be also seen in this paper. However, very few pay
attention to modification on the GBM assumption in the KMV model, this paper
will point out disadvantages of the GBM assumption and try to modify it.

Initially, Black and Scholes (1973) [9] assumes the underlying stock returns
dynamics are given by a GBM, i.e., constant volatility. However, the implied
volatility smile, skew or smirk indicates that constant volatility may be
wrong. More and more volatility models are developed in order to deal with
volatility smile, like CEV model proposed by Cox (1975) [10], Heston model
(1993)[11] and Hagan’s SABR model(2002) [12].

In this paper, we take the CEV model as an instance. The constant elasticity
of variance (CEV) model is considered to be an effective model which deal with
the volatility smile. After Cox proposed the CEV model, more and more
researchers are taking further study and testing the power of the CEV model,
like Beckers (1980) [13] and Davydov et al. (2001) [14]. Though we can see
these volatility models including CEV model now play an important role in
finance, yet still few study apply the them to the KMV model. In this paper,
we will offer an insight for how to apply the non-constant volatility to the
KMV model.

The remainder of this paper proceeds as follows. Section 2 introduces some
preliminary methodology, including the KMV model, CEV process, fixed effect
model and equivalent volatility. In Section 3, we describe how we collect the
data and deduce some test approaches. Section 4 tests the classical KMV model
and find it hard to distinguish ST and non ST companies. After we analyse the
volatility problem, in section 5 we propose a CEV-KMV model and the empirical
results show the CEV process do improve the forecasting ability of the KMV
model. Finally, section 6 concludes our results and looks into the future.

##  2 Methodology

###  2.1 The KMV model

Default forecasting is one of the most enduring themes of credit risk. In the
well-known KMV model, the key point is computing default probability. Firstly,
we assume that the firm’s asset value ($V_{A}(t)$) dynamics are given by a
Geometric Brownian Motion (GBM):

$\frac{dV_{A}(t)}{V_{A}(t)}=\mu_{A}dt+\sigma_{A}dB(t),\quad t\in[0,T],$ (1)

where $B(t)$ is a standard Brownian motion, $\mu_{A}$ and $\sigma_{A}$
represents the firm’s asset value drift rate and volatility respectively. Let
$D$ represents the debt of the company at a given maturity $T$, then naturally
$V_{A}(T)<D$ means the company has no ability to afford its debt, which
finally leads to default. In the KMV model $D$ is named default point and
usually chosen to be a half of the long-term debt (LTD) plus the short-term
debt (STD) at $t=0$. The default probability can be expressed as

$P\left(V_{A}\left(T\right)<D\right)=N\left(-\frac{\ln\frac{V_{A}\left(0\right)}{D}+\left(r-\frac{\sigma_{A}^{2}}{2}\right)T}{\sigma_{A}\sqrt{T}}\right)=N\left(-d_{2}\right),$
(2)

where $N(\cdot)$ is the standard normal cumulative distribution function, $r$
is the risk free interest rate, SHIBOR may be a suitable choice in China
market.

Unfortunately, both the firm’s asset value $V_{A}(0)$ and the volatility
$\sigma_{A}$ may be not easy to obtained, which makes it hard to compute (2).
However, the volatility of the firm’s equity can be estimated by using the
historical stock data. In order to make use of this, we regard the firm’s
equity ($V_{E}$) as a call option on $V_{A}$ with strike price $D$, so $V_{E}$
can be expressed as, of course, Black-Scholes-Merton formula:

$V_{E}=V_{A}N(d_{1})-e^{-r(T-t)}DN(d_{2}),$ (3)

where $d_{2}$ is defined in (2) and $d_{1}=d_{2}+\sigma_{A}\sqrt{T}$. Then
$dV_{E}$ follows from Itô’s lemma:

$dV_{E}=\left(\frac{\partial V_{E}}{\partial t}+rV_{A}\frac{\partial
V_{E}}{\partial
V_{A}}+\frac{\sigma^{2}V_{A}^{2}}{2}\frac{\partial^{2}V_{E}}{\partial
V_{A}^{2}}\right)dt+\sigma_{A}V_{A}\frac{\partial V_{E}}{\partial
V_{A}}dB_{t},$ (4)

implying that $V_{E}$ have local volatility

$\sigma_{E}=\sigma_{A}\frac{V_{A}}{V_{E}}\frac{\partial V_{E}}{\partial
V_{A}}=\sigma_{A}\frac{V_{A}}{V_{E}}N\left(d_{1}\right).$ (5)

Since $V_{E}(0)$ and $\sigma_{E}$ can be determined or estimated, $V_{A}(0)$
and $\sigma_{A}$ can be calculated by combining (3) and (5).

Besides, it is an interesting fact that rather than just pay attention to the
theoretical default probability $N(-d_{2})$, the KMV model defines a default
distance (or distance to default) $\mathrm{DD}=d_{2}$ or

$\mathrm{DD}^{\prime}=\frac{V_{A}-D}{V_{A}\sigma_{A}}\approx\frac{\ln\frac{V_{A}}{D}+\left(r-\frac{\sigma_{A}^{2}}{2}\right)}{\sigma_{A}}=d_{2},$
(6)

and then creates a map from default distance to the historical expected
default frequency (EDF). The more details about EDF can be found in Crosbie
and Bohn (2003) [15]. However, the rare default data in China do not allow us
to do this, but it may bring some advantages in statistical inference for us
to focus on the default distances.

###  2.2 The CEV Model

The constant elasticity of variance (CEV) model was proposed by Cox (1975)
[10]. Cox assumes that the firm’s asset value dynamics are given by

$\frac{dV_{A}\left(t\right)}{V_{A}\left(t\right)}=\mu_{A}dt+\delta
V_{A}^{\beta-1}dB\left(t\right),\quad t\in\left[0,T\right],$ (7)

where $\delta>0$, $\beta>0$ are constant. 0The CEV process means $V_{A}$ has
local volatility $\sigma_{A}=\delta V_{A}^{\beta-1}$, implying

$\ln\sigma_{A}=\ln\delta+(\beta-1)\ln V_{A}.$ (8)

From (7) we find if $\beta=1$ then (7) turns back to the GBM. If $\beta>1$,
the volatility is increasing with respect to the underlying. If $\beta<1$, the
volatility is decreasing with respect to the underlying.

Similarly, if we know $\delta$ and $\beta$, it is theoretically available to
compute default probability $P(V_{A}(T)<D)$. However, since $V_{A}(T)$ is not
normal distributed, an analytical solution for default probability is hard to
get, so the numerical solution may be a better choice.

By the Feynman-Kac Theorem, $g(t,x)=E[1_{\\{V_{A}(T)<D\\}}|\mathcal{F}_{t}]$
solves the partial differential equation (PDE):

$\begin{cases}\frac{\partial u}{\partial t}+rx\frac{\partial u}{\partial
x}+\frac{\delta^{2}x^{2\beta}}{2}\frac{\partial^{2}u}{\partial
x^{2}}=0,&t\in\left[0,T\right],\\\
u\left(T,x\right)=1_{\left\\{x<D\right\\}},&\\\ \end{cases}$ (9)

it is very convenient for us to use the Finite Difference Method (FDM) to
solve (9) and then

$P(V_{A}(T)<D)=u(0,V_{A}(0)).$ (10)

###  2.3 Fixed Effect Model

An efficient and simple method to estimate the parameters in the CEV model is
that we could assume every company in a group (ST group or non ST group) has a
common $\beta$ but a unique $\delta$ in CEV process. Thus (8) implies a fixed
effects model:

$\ln\sigma_{A}(i,t)=\ln\delta_{i}+\left(\beta-1\right)\ln V_{A}(i,t),\quad
i=1,\cdots,n,\quad t=1,\cdots,T.$ (11)

Under general conditions, ordinary least squares (OLS) is a consistent
estimation procedure for these parameters. The OLS estimators using the
previous data are

$\hat{\beta}=1+\frac{\sum_{i=1}^{n}{\sum_{t=1}^{T}{\left(\ln
V_{A}\left(i,t\right)-\overline{\ln
V_{A}\left(i\right)}\right)\left(\ln\sigma_{A}\left(i,t\right)-\overline{\ln\sigma_{A}\left(i\right)}\right)}}}{\sum_{i=1}^{n}{\sum_{t=1}^{T}{\left(\ln
V_{A}\left(i,t\right)-\overline{\ln V_{A}\left(i\right)}\right)^{2}}}},$ (12)
$\ln\delta_{i}=\overline{\ln\sigma_{A}\left(i\right)}-\left(\hat{\beta}-1\right)\overline{\ln
V_{A}\left(i\right)},\quad i=1,\cdots,n,$ (13)

where
$\overline{\ln\sigma_{A}\left(i\right)}=\frac{1}{T}\sum_{t=1}^{T}{\ln\sigma_{A}\left(i,t\right)}$
and $\overline{\ln V_{A}\left(i\right)}=\frac{1}{T}\sum_{t=1}^{T}{\ln
V_{A}\left(i,t\right)}$. Despite great convenience, this method has a “model
risk”: the model we use to calculate default distance is CEV process however
the data ($V_{A}$, $\sigma_{A}$) for the fixed effects model are computed
under GBM. To overcome this drawback, the main results of Hagan and Woodward
(1999) [16] offer an insight, which we will discuss in next section.

###  2.4 Equivalent Volatility

Consider an European call option pricing problem, if the underlying asset
value dynamics $dS$ (maybe not GBM) and terminal payoff $c(T,S)=(S_{T}-K)^{+}$
are given, consequently the option price $c(0,S)$ could be obtained
theoretically. Define

$d_{1}\left(\sigma\right)=\frac{\ln\frac{S}{K}+\left(r+\frac{\sigma^{2}}{2}\right)T}{\sigma\sqrt{T}},d_{2}\left(\sigma\right)=d_{1}\left(\sigma\right)-\sigma\sqrt{T}.$
(14)

Because of the 1-1 relation between Black-Scholes-Merton formula and implied
volatility, option prices are often quoted by stating implied volatility. Thus
the equivalent volatility $\sigma_{B}$ for $dS$ is defined as the unique
solution of the following equation

$c\left(0,S\right)=SN\left(d_{1}\left(\sigma\right)\right)-e^{-rT}KN\left(d_{2}\left(\sigma\right)\right),$
(15)

where $c(0,S)$ is the option price calculated under assumption $dS$, and the
“B” in the equivalent volatility $\sigma_{B}$ means the right hand side of
(16) is Black’s model (Black-Scholes-Merton model).

Hagan and Woodward (1999) [16] use singular perturbation methods to deduce the
equivalent volatility for CEV model:

$\sigma_{B}=\frac{\delta}{f^{1-\beta}}\left\\{1+\frac{\left(1-\beta\right)\left(2+\beta\right)\left(F-K\right)^{2}}{24f^{2}}+\frac{\left(1-\beta\right)^{2}\delta^{2}T}{24f^{2-2\beta}}+\cdots\right\\},$
(16)

where $F=e^{rT}S_{0}$ is the future price and $f=\frac{1}{2}(F+K)$.

Obviously, if we assume the asset value dynamics $dV_{A}$ is given by CEV
process, the $\sigma_{A}$ computed by combining (3) and (5) is nothing but the
equivalent volatility. Go on considering every company has a common $\beta$
and a unique $\delta$, it is easy for us to estimate these parameters by
methods like calibration. Since equivalent volatility method overcome the
“model risk”, we guess if we use this method instead of the fixed effects
model as an estimation procedure, the performance of the CEV-KMV model will be
better.

##  3 Empirical Design

###  3.1 Data

In China, the listed companies under special treatment (ST) are always be
considered as the ones with higher credit risk, while the non specially
treated (non ST) listed companies usually have a lower default probability. In
order to test the performance of the KMV model, we will use the data in China
stock market to see if the default distances of the ST companies are
significantly less than those of non ST companies.

We obtain data from Choice Database established by EastMoney. For ST
companies, we only focus on those which at least have one-year continuous
transaction records, ensuring there are enough data for us to estimate
$\sigma_{E}$, the estimator of $\sigma_{E}$ is the annualized sample standard
deviation of the latest 250 daily return:

$\hat{\sigma}_{E}(i,t)=\sqrt{\frac{250}{249}\sum_{s=t-250}^{t-1}{\left(r_{is}-\bar{r}_{it}\right)^{2}}},\quad\bar{r}_{it}=\frac{1}{250}\sum_{s=t-250}^{t-1}{r_{is}}.$

We calculate the $\sigma_{E}$ quarterly (at the last day in every quarter),
from 2019Q1 to 2021Q1. The equities $V_{E}$ and default points $D$ could be
obtained directly or computed from database or financial statements, too. The
missing data have been filled with the nearest neighbor. We find 186 ST
companies are available during 2019Q1-2021Q1. Correspondingly, we randomly
choose 186 non ST companies.

Table 1 reports the descriptive statistics of our data. The average equity
amount of ST companies is 3.819 billion CNY, far less than that of non ST
companies, 19.648. Similarly, the mean default point of non ST companies is
approximately five times as much as that of ST companies. However, the
volatility of non ST companies is a little less than that of ST companies. In
both ST and non ST companies, default point is greater than equity. Since we
collect the all available ST companies and the non ST companies sampled are
chosen randomly, the summary statistics could reveal the general rule: ST
companies may be more unstable than non ST companies.

Table 1: Summary statistics of equity, default point and volatility Variable | ST/Non ST | Mean | Median | Std. | Obs.
---|---|---|---|---|---
Equity (billion CNY) | ST | 3.819 | 2.530 | 4.942 | 1674
| Non ST | 19.648 | 8.165 | 45.481 | 1674
Default point (billion CNY) | ST | 4.899 | 1.140 | 16.511 | 1674
| Non ST | 28.404 | 4.433 | 108.625 | 1674
Volatility of equity | ST | 0.480 | 0.471 | 0.116 | 1674
| Non ST | 0.411 | 0.402 | 0.102 | 1674

###  3.2 Test Approaches

It can be seen from Fig.1 that the approximate distribution of default
distances is gamma distribution, so testing whether the default distances of
ST companies are less than those of non ST companies turns to be deducing a
large sample test under gamma assumption. If the default distances of ST and
non ST companies are assumed to be $X\sim Ga(\alpha_{1},\beta_{1})$, $Y\sim
Ga(\alpha_{2},\beta_{2})$, respectively, then the Central Limit Theorem
applies and the asymptotic sampling distribution of the sample mean
$\overline{X}_{m}$ and $\overline{Y}_{n}$ is normal. Under null hypothesis
$H_{0}:\frac{\alpha_{1}}{\beta_{1}}=\frac{\alpha_{2}}{\beta_{2}}$, the test
statistics

$Z_{1}=\frac{\overline{X}_{m}-\overline{Y}_{n}}{\sqrt{\frac{\hat{\alpha}_{1}}{m\hat{\beta}_{1}^{2}}+\frac{\hat{\alpha}_{2}}{n\hat{\beta}_{2}^{2}}}}$
(17)

is asymptotically standard normal implied by the Slutsky Theorem, where
$\hat{\alpha}_{1}$, $\hat{\alpha}_{2}$, $\hat{\beta}_{1}$, $\hat{\beta}_{2}$
are the maximum likelihood estimators for these parameters. The $p$-value for
alternative hypothesis
$H_{1}:\frac{\alpha_{1}}{\beta_{1}}<\frac{\alpha_{2}}{\beta_{2}}$ can also be
derived.

(a) ST companies

(b) non ST companies

Figure 1: _Histogram and gamma density of default distances (2021Q1)_

In order to check the robustness of results based on parametric tests, the
Wilcoxon test may be helpful. By using only the information of the sample
ranks, Wilcoxon test proposed by Wilcoxon (1945) is one of the most popular
non parametric tests. Define $W_{X}$ is the sum ranks of the ST default
distance $X$ in the hybrid sample. Fixed the ratio $m/n$, when $n$ is large,
under the null hypothesis the statistic

$Z_{2}=\frac{W_{X}-\frac{m\left(m+n+1\right)}{2}}{\sqrt{\frac{mn\left(m+n+1\right)}{12}}}$
(18)

is asymptotically standard normal too. We reject the null hypothesis if
$Z_{2}<c$, where $c$ is a given constant corresponding to the significance
level. Since the Pitman’s asymptotic relative efficiency (ARE) of Wilcoxon
test on a skewed distribution is far less than $1$, $Z_{2}$ may reject more
often than $Z_{1}$.

##  4 The Performance of the KMV Model

###  4.1 Empirical Results

By using the KMV model, default distances could be computed by combining (3)
and (5). Table 2 summarizes the results. The intuitive evidence against the
effectiveness of the classical KMV model is that we could see seldom
difference between the average default distances of ST and non ST companies in
most of these quarters. Only in two quarters (2020Q2 and 2021Q1) we find the
default distances of ST companies is far less than that of non ST companies.
However, we find the standard deviations of ST companies default distances are
almost in $[2.5,2.7]$, while those of non ST companies default distances are
almost in $[1.9,2.1]$. Besides, the histograms of default distances have been
shown in Fig.1.

Table 2: Default distances based on the KMV model

Quarter |  ST/Non ST |  Mean |  Std. |  Alpha |  Beta
---|---|---|---|---|---
2019Q1 |  ST |  4.498 |  2.524 |  4.183 |  1.075
|  Non ST |  4.484 |  2.006 |  6.686 |  0.671
2019Q2 |  ST |  4.264 |  2.617 |  3.771 |  1.131
|  Non ST |  4.245 |  1.9 |  6.642 |  0.639
2019Q3 |  ST |  4.215 |  2.686 |  3.616 |  1.166
|  Non ST |  4.264 |  1.941 |  6.384 |  0.668
2019Q4 |  ST |  4.578 |  2.874 |  3.531 |  1.297
|  Non ST |  4.617 |  2.124 |  6.202 |  0.744
2020Q1 |  ST |  4.366 |  2.774 |  3.447 |  1.267
|  Non ST |  4.361 |  2.085 |  6.087 |  0.716
2020Q2 |  ST |  4.349 |  2.778 |  3.277 |  1.327
|  Non ST |  4.657 |  2.168 |  6.046 |  0.77
2020Q3 |  ST |  4.263 |  2.659 |  3.38 |  1.261
|  Non ST |  4.306 |  2.018 |  6.019 |  0.715
2020Q4 |  ST |  4.109 |  2.412 |  3.667 |  1.121
|  Non ST |  4.106 |  1.79 |  6.549 |  0.627
2021Q1 |  ST |  4.026 |  2.266 |  3.845 |  1.047
|  Non ST |  4.305 |  1.999 |  5.878 |  0.732

  * •

_Note: The default distances are computed by using $\mathrm{DD}=d_{2}$. Alpha
and beta in the table means the maximum likelihood estimators in gamma
distribution._

Both parametric and non parametric tests results are reported in the Table 3.
The parametric tests results present evidence against the KMV model. The
$p$-values for all quarters are greater than 0.1, indicating that the null
hypothesis could not be rejected. The results of 2019Q1, 2019Q2, 2020Q1 and
2020Q4 suggest most strongly that there is no evidence for ST default
distances are less than non ST default distances. A comparison of the results
for $Z_{1}$ versus $Z_{2}$ illustrates the previously discussed fact that the
non parametric test tends to reject too often. In a word, though by using
$Z_{2}$, the KMV model performs well in distinguishing ST and non ST
companies, the parametric tests results can not satisfy anyone. We believe
that the classical KMV model do have some disadvantages.

Table 3: Tests results for the KMV model

Quarter |  $Z_{1}$ |  $p$-value |  $Z_{2}$ |  $p$-value
---|---|---|---|---
2019Q1 |  0.045 |  0.518 |  -1.294 |  0.098
2019Q2 |  0.059 |  0.523 |  -1.785 |  0.037
2019Q3 |  -0.160 |  0.436 |  -2.239 |  0.013
2019Q4 |  -0.146 |  0.442 |  -2.188 |  0.014
2020Q1 |  0.020 |  0.508 |  -2.071 |  0.019
2020Q2 |  -1.213 |  0.113 |  -3.452 |  0.000
2020Q3 |  -0.159 |  0.437 |  -2.096 |  0.018
2020Q4 |  0.011 |  0.504 |  -1.597 |  0.055

  * •

_Note: $Z_{1}$ represents the test statistic in parametric test given by (17),
$Z_{2}$ represents the test statistic in non parametric test given by (18)._

###  4.2 Further Discussion

Recently, some researchers have pointed out disadvantages of the KMV model,
like Zhang and Shi (2018) [17] find that KMV model varies with firm size and
what they pay attention to is the default point. Actually, we focus on the
relation between volatility and firm size: does volatility varies with the
firm size?

(a) ST companies

(b) non ST companies

Figure 2: _The relation between $\ln V_{A}$ and $\ln\sigma_{A}$. The
horizontal axis represents $\ln V_{A}$ and the longitudinal axis represents
the $\ln\sigma_{A}$._

In theory, the volatility $\sigma_{A}$ in the classical KMV model is a
constant. In practice, however, $\sigma_{A}$ may be not a constant. Fig.2
illustrates that at least with respect to $V_{A}$, volatility $\sigma_{A}$ is
not a constant. Instead, $\sigma_{A}$ is negatively and positively correlated
to $V_{A}$ in the picture (a) and (b), respectively. Besides companies in
figures, there is a general linear correlation between $\ln\sigma_{A}$ and
$\ln V_{A}$ in all companies which matches (8) deduced from the CEV process.
What’s more? It seems in the group of ST companies $\beta$ is less than $1$
while for non ST companies $\beta$ is great than $1$. The tests results and
the facts above suggest that KMV model do vary with the firm size, applying
non-constant volatility may improve it.

##  5 The CEV-KMV model

###  5.1 Default Distance

We propose a CEV-KMV model by applying CEV process rather than GBM in the
classical KMV model. We have illustrated that the default probability under
CEV process can be calculated by FDM, but in order to in order to match the
test approaches used previously and gain advantages in statistical inference,
like we used to do in the classical KMV model, though $V_{A}(T)$ is not normal
distributed, we can redefine the default distance as

$\mathrm{DD}=N^{-1}\left(P\left(V_{A}\left(T\right)<D\right)\right),$ (19)

where $N^{-1}(\cdot)$ is the inverse function of the standard normal
cumulative distribution function. Therefore, to test the performance of CEV
process, we need only estimate $\delta$ and $\beta$.

Table 4: Default distances based on the CEV-KMV model

Quarter |  ST/Non ST |  Mean |  Std. |  Alpha |  Beta
---|---|---|---|---|---
Panel A: Fixed Effects Model
2019Q1 |  ST |  4.803 |  1.842 |  7.274 |  0.660
|  Non ST |  4.910 |  1.697 |  9.147 |  0.537
2019Q2 |  ST |  4.406 |  1.886 |  6.171 |  0.714
|  Non ST |  4.795 |  1.749 |  8.374 |  0.573
2019Q3 |  ST |  4.353 |  1.944 |  5.595 |  0.778
|  Non ST |  4.714 |  1.794 |  8.018 |  0.588
2019Q4 |  ST |  4.449 |  1.989 |  5.593 |  0.795
|  Non ST |  4.682 |  1.722 |  8.313 |  0.563
2020Q1 |  ST |  4.406 |  2.045 |  5.120 |  0.861
|  Non ST |  4.575 |  1.840 |  7.459 |  0.613
2020Q2 |  ST |  4.388 |  2.062 |  4.922 |  0.892
|  Non ST |  4.654 |  1.822 |  7.630 |  0.610
2020Q3 |  ST |  4.398 |  1.988 |  5.260 |  0.836
|  Non ST |  4.612 |  1.711 |  8.389 |  0.550
2020Q4 |  ST |  4.557 |  2.160 |  4.906 |  0.929
|  Non ST |  4.753 |  1.799 |  8.052 |  0.590
2021Q1 |  ST |  4.494 |  2.184 |  4.776 |  0.941
|  Non ST |  4.939 |  1.998 |  7.133 |  0.692
Panel B: Equivalent Volatility Method
2019Q1 |  ST |  4.725 |  1.883 |  6.686 |  0.707
|  Non ST |  5.712 |  2.133 |  7.949 |  0.719
2019Q2 |  ST |  4.365 |  1.925 |  5.757 |  0.758
|  Non ST |  5.600 |  2.393 |  6.487 |  0.863
2019Q3 |  ST |  4.412 |  1.946 |  5.581 |  0.791
|  Non ST |  5.489 |  2.489 |  5.841 |  0.940
2019Q4 |  ST |  4.464 |  2.019 |  5.273 |  0.847
|  Non ST |  5.424 |  2.410 |  6.200 |  0.875
2020Q1 |  ST |  4.340 |  2.020 |  5.027 |  0.863
|  Non ST |  5.445 |  2.489 |  5.582 |  0.975
2020Q2 |  ST |  4.352 |  1.999 |  5.071 |  0.858
|  Non ST |  5.371 |  2.467 |  5.802 |  0.926
2020Q3 |  ST |  4.505 |  1.911 |  5.825 |  0.773
|  Non ST |  5.470 |  2.311 |  6.403 |  0.854
2020Q4 |  ST |  4.424 |  1.953 |  5.366 |  0.825
|  Non ST |  5.591 |  2.366 |  6.461 |  0.865
2021Q1 |  ST |  4.344 |  1.947 |  5.280 |  0.823
|  Non ST |  5.750 |  2.458 |  6.133 |  0.938

  * •

_Note: Alpha and beta in the table means the maximum likelihood estimators in
gamma distribution._

###  5.2 Empirical Results

The estimation results under fixed effects model and equivalent volatility
model differ. The estimator of $\beta$ are $0.9957$ and $1.1850$ under fixed
effects model, for ST companies and non ST companies, respectively. While
under equivalent volatility model, the $\beta$ estimated for ST companies is
$0.9841$ and for non ST companies is $1.1413$. Besides, the estimated
$\delta$’s under two models are quite dissimilar, which may lead to the
difference between the two groups of default distances. The estimation results
imply that for ST companies, local volatility is a decreasing function with
respect to $V_{A}$, while for non ST companies local volatility is an
increasing function, illustrating the previously discussed fact.

Table 4 contains some statistics for default distances under the CEV-KMV
model. Panel A reports the mean and standard deviation of default distances by
using fixed effects model. The mean of the ST default distances is less than
that of the non ST default distances in every quarter, which reveals that the
CEV-KMV model performs better in distinguishing ST companies than the
classical KMV model. But what panel A of Table 5 summaries tells us though in
most of these quarters the null hypothesis can be rejected by using $Z_{2}$,
the test results by using $Z_{1}$ still satisfy nobody. The only good news is,
the $p$-values for the parametric tests decrease a little, by contrast to the
classical KMV model.

Focusing on the equivalent volatility method, panel B of Table 4 reports the
main statistics of the default distances. Recall we have discussed that the
performance of the CEV-KMV model will be better if we use the equivalent
volatility method instead of the fixed effects model as an estimation
procedure. The supportive evidence for that is the average default distance of
ST companies is below $4.75$ in all quarters while that of non ST companies is
larger than $5.35$ in all quarters. Besides, panel B of Table 5 reports the
tests results. The values of $Z_{1}$ and the corresponding $p$-values imply
the null hypothesis is rejected. The story is the same for the non parametric
tests, which ensures the robustness.

Figure 3: _$\overline{\mathrm{DD}}_{nST}-\overline{\mathrm{DD}}_{ST}$ under
different models. The black line represents the classical KMV model, the blue
line represents the CEV-KMV model using the fixed effects model as estimation
procedure, the purple line represents the CEV-KMV model using the equivalent
volatility method._

In order to show the improvement of applying CEV process to the KMV model
clearly, Fig.3 plots the differences between the mean non ST default distances
and mean ST default distances
$\overline{\mathrm{DD}}_{nST}-\overline{\mathrm{DD}}_{ST}$. The greater this
difference is, the better the model performs. Obviously from the figure, it
brings great advantages in distinguishing ST companies that we assume the
firm’s asset value follows CEV process rather than GBM in the KMV model.
Besides, the different methods for estimation procedure also lead to the
different results. Equivalent volatility is a more suitable method in
estimation than the fixed effects model since it overcomes the “model risk”.

Table 5: Tests results for the CEV-KMV model

Quarter |  $Z_{1}$ |  $p$-value |  $Z_{2}$ |  $p$-value
---|---|---|---|---
Panel A: Fixed Effects Model
2019Q1 |  -0.209 |  0.417 |  -0.897 |  0.185
2019Q2 |  -0.865 |  0.193 |  -2.788 |  0.003
2019Q3 |  -0.862 |  0.194 |  -2.676 |  0.004
2019Q4 |  -0.538 |  0.295 |  -2.199 |  0.014
2020Q1 |  -0.448 |  0.327 |  -1.659 |  0.049
2020Q2 |  -0.703 |  0.241 |  -2.270 |  0.012
2020Q3 |  -0.491 |  0.312 |  -1.970 |  0.024
2020Q4 |  -0.499 |  0.309 |  -1.882 |  0.030
2021Q1 |  -1.349 |  0.089 |  -2.870 |  0.002
Panel B: Equivalent Volatility Method
2019Q1 |  -2.508 |  0.006 |  -4.659 |  0.000
2019Q2 |  -3.893 |  0.000 |  -5.657 |  0.000
2019Q3 |  -3.727 |  0.000 |  -4.633 |  0.000
2019Q4 |  -3.332 |  0.000 |  -4.357 |  0.000
2020Q1 |  -4.244 |  0.000 |  -4.791 |  0.000
2020Q2 |  -3.762 |  0.000 |  -4.463 |  0.000
2020Q3 |  -3.059 |  0.001 |  -4.265 |  0.000
2020Q4 |  -3.916 |  0.000 |  -5.089 |  0.000
2021Q1 |  -4.988 |  0.000 |  -5.875 |  0.000

  * •

_Note: $Z_{1}$ represents the test statistic in parametric test given by (17),
$Z_{2}$ represents the test statistic in non parametric test given by (18)._

###  5.3 Further Discussion

The fact we find that the monotonicity of local volatility vary with whether
the company is ST is surprising. In general, for listed companies, since the
greater volume and higher transaction frequency of a larger size firm must
lead to a greater volatility (see in Weigand (1996) [18] and Lee et al. (2000)
[19]), most people may consider the larger firm size is, the greater the
volatility becomes. However, we find ST companies does not follow this rule,
the volatility of ST companies decreases when the firm size increases.

We guess one of the main reasons for ST companies disobey the general rule may
be the ST companies always have higher risk, the less firm size is, the higher
bankrupt or delisted probability will be. A higher risk in stock market often
manifests itself as a great volatility. In another aspect concerned about
internal control, McMullen et al. (1996) [20] found overall 26.5% firms offer
their internal control report but among those companies in whose financial
statements there are mistakes only 10.5% are willing to report their internal
control, which maybe also contributes to a higher credit risk. In a word, this
interesting fact we found may be caused by many other reasons and deserves
further study.

##  6 Conclusion

Yet still no consensus exists in assessing default risk even if it has
intrigued researchers for nearly 70 years. More and more tools and techniques
have been used to assess default risk. This paper presents some useful
approaches to apply CEV process to the KMV model. We focus on the default
distances and compare them under different methods.

The first thing we find is, the classical KMV model performs not well in
distinguishing ST companies in China. A contributing factor we point out is
that volatility may be not a constant, and the figures suggest applying CEV
process instead of GBM. We also notice that it is not easy to estimate the
parameters in CEV model. We present two types of estimation procedure in the
paper: fixed effects model and equivalent volatility. The estimation results
of both two methods show the local volatility is an increasing function with
respect to the non ST firm’s asset, but it is a decreasing function for the ST
companies, i.e. $\beta$ is greater than $1$ for non ST companies while $\beta$
is less than 1 for ST companies. Besides, after we applied these two methods,
the tests results show the equivalent volatility method is much more suitable
in estimating CEV parameters. Finally, we find that if the estimated
parameters are accurate, the CEV-KMV model can bring great advantages in
distinguishing ST companies.

What can be improved is that we can try some advanced parameters estimating
procedures like MCMC or other machine learning algorithm. Besides, we offer an
insight that other volatility model like Heston and SABR can be used based on
the KMV model if suitable because the 1-1 relation between equivalent
volatility and option price as well as the effective numerical methods allow
us to do it.

## References

  * [1] R. Merton, On the pricing of corporate debt: the risk structure of interest rates. The Journal of Finance, 1974.
  * [2] S. Kealhofer, M. Kurbat, The Default Prediction Power of the Merton Approach, Relative to Debt Ratings and Accounting Variables. Research Paper, Moody’s KMV, 2001.
  * [3] E.I. Altman, Financial Ratios, Discriminant Analysis and the Prediction of Corporate Bankruptcy. The Journal of Finance, 1968.
  * [4] J.A. Ohlson, Financial Ratios and the Probabilistic Prediction of Bankruptcy. Journal of Accounting Research, 1980.
  * [5] S.A. Hillegeist, E.K. Keating, D.P. Cram, K.G. Lundstedt, Assessing the Probability of Bankruptcy. Review of Accounting Studies, 2004.
  * [6] S.T. Bharath, S. Tyler, Forecasting Default with the Merton Distance to Default Model. Review of Financial Studies, 2008.
  * [7] Y.J. Zhang, B. Shi, Non-Tradable Shares Pricing and Optimal Default Point Based on Hybrid KMV Models: Evidence from China. Knowledge-Based Systems, 2016.
  * [8] Y.N. Song, F.R. Zhang, C.C. Liu, The Risk of Block Chain Financial Market Based on Particle Swarm Optimization. Journal of Computational and Applied Mathematics, 2020.
  * [9] F. Black, M. Scholes, The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 1973.
  * [10] J. Cox, Notes on Option Pricing I: Constant Elasticity of Variance Diffusion. Working Paper, Stanford University, 1975.
  * [11] S.L. Heston, A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. Review of Financial Studies, 1993.
  * [12] P.S. Hagan, D. Kumar, A.S. Lesniewski, D.E. Woodward, Managing Smile Risk. Wilmott, 2002.
  * [13] S. Beckers, The Constant Elasticity of Variance Model and Its Implications For Option Pricing. The Journal of Finance, 1980.
  * [14] D. Davydov, V. Linetsky, Pricing and Hedging Path-Dependent Options under the CEV Process. Management Science, 2001.
  * [15] P. Crosbie, J. Bohn, Modeling Default Risk. Research Paper, Moody’s KMV, 2003.
  * [16] P.S. Hagan, D.E. Woodward, Equivalent Black Volatilities. Applied Mathematical Finance, 1999.
  * [17] Y.J. Zhang, B.S. Shi, Does Default Point Vary with Firm Size?. Applied Economics Letters, 2018.
  * [18] R.A. Weigand, Trading Volume and Firm Size: A Test of the Information Spillover Hypothesis. Review of Financial Economics, 1996.
  * [19] C.F. Lee, O.M. Rui, Does Trading Volume Contain Information to Predict Stock Returns? Evidence from China Stock Markets. Review of Quantitative Finance and Accounting, 2000
  * [20] D.A. McMullen, K. Raghunandan, D.V. Rama, Internal Control Reports and Financial Reporting Problems. Accounting Horizons, 1996.


Andreas S. Kronfeld

# More on minimal renormalon subtraction

###### Abstract

The minimal renormalon subtraction (MRS) [1, 2, 3, 4] technique is summarized.
A new result is a study of the scale dependence of the pole-
mass–$\overline{\rm MS}$-mass ratio in MRS perturbation theory. As expected,
the scale dependence is much milder than in standard perturbation theory, but
it is a bit larger than other truncation effects such as omitting the N3LO
term or varying the normalization of the renormalon subtraction.

##  1 Introduction

Many quantities calculated with lattice gauge theory have to be “matched” to
continuum QCD. For quantities that run, such as $\alpha_{\text{s}}$ and quark
masses, it is conventional to quote results in the $\overline{\rm MS}$ scheme,
making perturbation theory unavoidable. The uncertainty introduced by
truncating the perturbative series is notoriously difficult to estimate.

It is therefore noteworthy that the error budget for the $\overline{\rm MS}$
quark masses of ref. [1] quotes a negligible uncertainty from perturbative
truncation. Figure 4 of ref. [1] demonstrates a small shift in the final
results when the order of perturbation theory changes from NLO to NNLO and an
imperceptible shift from NNLO to N3LO (counting the one-loop term as LO). The
total uncertainty is dominated by statistics, fitting systematics, and the
external parametric error of $\alpha_{\text{s}}$. It ranges from subpercent
for bottom, charm, and strange to 1–2% for down and up.

To tame the perturbative series, which in this case is the relation between
the pole mass and the $\overline{\rm MS}$ mass, the key ingredient was a
rearrangement and reinterpretation of the series dubbed “minimal renormalon
subtraction” (MRS) [2], which in turn relied on an expression for the
normalization of factorial growth of the series coefficients [3]. The MRS
method has been generalized and clarified in work underway during Lattice 2023
and meanwhile published [4]. This report summarizes the main points and shows
plots examining the scale dependence of the MRS quark mass. At the conference,
I showed plots of the static energy (the others were not yet ready). The two
sets of plots look very similar. References [1, 2, 3] did not study how MRS
perturbation theory depends on $s=\mu/\bar{m}$, where $\mu$ is the scale in
$\alpha_{\overline{\rm MS}}(\mu)$, and $\bar{m}=m_{\overline{\rm
MS}}(\bar{m})$. Showing the quark-mass plots here fills a gap while avoiding
rote repetition of ref. [4]. In particular, figure 3 (right) suggests that the
truncation uncertainty in the quark-mass results of ref. [1] may have been
underestimated, but arguably not so much that it would alter the quadrature
sum of all uncertainties.

##  2 Statement of the problem

References [1, 2, 3] start with a relation for the mass of a heavy-light
hadron,

$M=m_{\text{pole}}+\bar{\Lambda}+\frac{\mu_{\pi}^{2}}{2m_{\text{pole}}^{2}}+\cdots,$
(1)

where $M$ is the hadron mass, $m_{\text{pole}}$ is the quark “pole” mass,
$\bar{\Lambda}$ is the energy of gluons and light quarks, and the last term is
the heavy quark’s kinetic energy. A spin-dependent term of order
$1/m_{\text{pole}}^{2}$ can also arise; for this discussion, it can be lumped
into the last term. In perturbation theory, the pole mass is obtained from the
self-energy by finding the pole of the propagator and substituting
iteratively, because the self-energy itself depends on $m_{\text{pole}}$ [5].
The relation can be written

$m_{\text{pole}}=\bar{m}\left(1+\sum_{l=0}r_{l}(\mu/\bar{m})\alpha_{\text{s}}^{l+1}(\mu)\right),$
(2)

and the coefficients grow rapidly: $r_{l}(1)=\\{0.424,1.035,3.693,17.21\\}$
for $l=\\{0,1,2,3\\}$. (The numbers correspond to the $\overline{\rm MS}$
scheme with $n_{f}=3$ massless quarks and no massive quark sea.)

If $m_{\text{pole}}$ solves the perturbative pole condition, so does
$m_{\text{pole}}-\delta m$, where $\delta m$ is anything of order
$\Lambda\sim\mu\mathrm{e}^{-1/2\beta_{0}\alpha_{\text{s}}(\mu)}$ (the QCD
scale). In the hadron mass, $\delta m$ can be absorbed into $\bar{\Lambda}$,
which isn’t defined without specifying exactly what $m_{\text{pole}}$ means in
eq. 1. This ambiguity, often called the “renormalon” ambiguity [6, 7], is
related to the growth of the coefficients, which is factorial. Indeed, the MRS
procedure introduces some rearrangements and eventually snips off a $\delta m$
(given below, eq. 10), absorbing it into $\bar{\Lambda}$ to define
$\bar{\Lambda}_{\text{MRS}}$. Then
$M=m_{\text{MRS}}+\bar{\Lambda}_{\text{MRS}}+\cdots$.

There are many problems of the form eq. 1. Generically,

$Q^{p}\mathcal{R}(Q)=Q^{p}R(Q)+C_{p}\Lambda^{p}+\cdots,\qquad
R(Q)=\sum_{l=0}r_{l}(\mu/Q)\alpha_{\text{s}}^{l+1}(\mu),$ (3)

where $\mathcal{R}$ is some dimensionless observable with one short-distance
scale $Q$. In the following, formulas are written for arbitrary $p$ [4],
although we are mostly interested in $p=1$.

##  3 Factorial growth

It has been known for a long time (see, e.g., ref. [8]) that the coefficients
in the series of eqs. 2 and 3 grow at _large_ orders as

$r_{l}\sim\left(\frac{2\beta_{0}}{p}\right)^{l}\frac{\Gamma(l+1+pb)}{\Gamma(1+pb)}R_{0}\equiv
R_{l}$ (4a) where $b=\beta_{1}/2\beta_{0}^{2}\stackrel{{\scriptstyle
n_{f}=3}}{{=}}32/81\approx 0.4$. Komijani [3] found an expression for the
$l$-independent normalization $R_{0}$
$R_{0}=\sum_{k=0}^{{\color[rgb]{.75,0,.25}\infty}}(k+1)\frac{\Gamma(1+pb)}{\Gamma(k+2+pb)}\left(\frac{p}{2\beta_{0}}\right)^{k}f_{k},$
(4b)

where the coefficients $f_{k}$ are straightforwardly related to the $r_{l}$
(cf. eq. 6, below). Komijani observed that since $\bar{\Lambda}$ in eq. 1
(more generally, $C_{p}\Lambda^{p}$ in eq. 3) does not depend on $\bar{m}$ (or
$Q$ in eq. 3 [4]), a derivative with respect to $\bar{m}$ (or $Q$) yields a
series without the leading factorial growth; the coefficients of that series
are the $f_{k}$. In eq. 4b, the upper limit ${\color[rgb]{.75,0,.25}\infty}$
is an asymptotic consideration. If $L$ of the $r_{l}$ are known, then only $L$
of the $f_{k}$ are known. Thus, any practical application must truncate the
sum in eq. 4b to have upper limit ${\color[rgb]{.75,0,.25}L-1}$.

An unexpected—perhaps remarkable—finding [4] is a similar set of formulas that
hold at every order, not just large orders. My version reads

$r_{l}=f_{l}+\left(\frac{2\beta_{0}}{p}\right)^{l}\frac{\Gamma(l+1+pb)}{\Gamma(1+pb)}\sum_{k=0}^{{\color[rgb]{.75,0,.25}l-1}}(k+1)\frac{\Gamma(1+pb)}{\Gamma(k+2+pb)}\left(\frac{p}{2\beta_{0}}\right)^{k}f_{k}.$
(5)

Note that the relation is $=$ (equals) not $\sim$ (asymptotically goes like).
Gaze at eq. 5 and see that the growing part (in front of the sum) is exactly
like eq. 4a and that the sum itself is almost like the normalization, eq. 4b,
albeit with upper limit the depending on $l$.

Equation 5 is nothing but the inverse of the formula for the $f_{k}$ in a
carefully chosen scheme. In any scheme for $\alpha_{\text{s}}$,

$f_{k}=r_{k}-\frac{2}{p}\sum_{l=0}^{k-1}(l+1)\beta_{k-1-l}r_{l}.$ (6)

The algebra can be simplified by changing from the $\overline{\rm MS}$ scheme,
which is used in the literature to report the $r_{l}$ [9, 10, 11, 12, 13], to
a scheme with $\beta_{j}=\beta_{0}(\beta_{1}/\beta_{0})^{j}$. With these
$\beta_{j}$, the $\beta$-function can be summed as a geometric series, so this
scheme is known as the “geometric scheme”. Without loss, it is possible to
define the coupling in this scheme, $\alpha_{\text{g}}$, such that
$\Lambda_{\text{g}}=\Lambda_{\overline{\rm MS}}$. Equation 6 implies a matrix
that in the geometric scheme has a simple recursive structure. The matrix has
infinite extent but is lower triangular and can be inverted row-by-row. The
recursive nature of the matrix elements yields the $\Gamma$ functions in eq.
5.

Has anything been gained? After all, results are published for the $r_{l}$ and
the $f_{k}$ are obtained (after deciding what $p$ should be) from eq. 6. For
the first $L$ terms, eq. 5 simply returns the original information. For $l\geq
L$, however, eq. 5 suggests an approximation as follows. For the first $L$
coefficients $r_{l}$, use the published information. For the rest, namely
$r_{l}$ with $l\geq L$, truncate eq. 5, which amounts to dropping the first
term and using eq. 4 with upper limit on the sum in $R_{0}$ at $L-1$. The
truncation is systematic, once $l$ is large enough, because both $f_{l}$ and
any terms omitted from the sum in eq. 5 are factorially smaller than the terms
that are retained.

##  4 Reinterpreting the series

###  4.1 Approach of ref. [4]

In the approximation just described, the series is approximated with

$R(Q)=\sum_{l=0}^{L-1}r_{l}\alpha^{l+1}+\sum_{l=L}^{\infty}R_{l}\alpha^{l+1},$
(7)

using the $L$ terms in the literature [9, 10, 11, 12, 13] and approximating
the rest from the obvious truncation of eq. 5. If the lower limit of the
second sum is lowered to $0$, then the sum can be defined via Borel summation.
To add those $L$ terms in, they also have to be subtracted out, i.e.,

$R(Q)=\sum_{l=0}^{L-1}[r_{l}-R_{l}]\alpha^{l+1}+\sum_{l=0}^{\infty}R_{l}\alpha^{l+1}.$
(8)

Then (see ref. [4] for details)

$\sum_{l=0}^{\infty}R_{l}\alpha_{\text{g}}^{l+1}=R_{0}\frac{p}{2\beta_{0}}\mathcal{J}(pb,1/2\beta_{0}\alpha_{\text{g}})+\delta
R,$ (9)

where $\mathcal{J}$ has a convergent expansion in its second argument, i.e.,
in $1/\alpha_{\text{g}}$. The last term in eq. 9 is explicitly of order
$\Lambda_{\overline{\rm MS}}^{p}$, so it can be absorbed into the power term
in eq. 3 (or $\bar{\Lambda}$ in eq. 1). For example, for the pole mass (where
$p=1$)

$\delta m=\bar{m}\,\delta
R=-R_{0}\mathrm{e}^{\pm\mathrm{i}b\pi}\frac{\Gamma(-b)}{2^{1+b}\beta_{0}}\left[\bar{m}\frac{\mathrm{e}^{-1/[2\beta_{0}\alpha_{\text{g}}(\bar{m})]}}{[\beta_{0}\alpha_{\text{g}}(\bar{m})]^{b}}\right],$
(10)

where the sign in $\mathrm{e}^{\pm\mathrm{i}b\pi}$ exposes an ambiguity (from
a choice of contour in a Borel integration), and the quantity in brackets [ ]
is immediately recognizable as $\Lambda_{\text{g}}=\Lambda_{\overline{\rm
MS}}$.

In summary, the “minimal renormalon subtraction” (MRS) interpretation of a
perturbative series is

$R(Q)\to
R_{\text{MRS}}(Q)\equiv\sum_{l=0}^{L-1}[r_{l}-R_{l}]\alpha^{l+1}+R_{0}\frac{p}{2\beta_{0}}\mathcal{J}(pb,1/2\beta_{0}\alpha_{\text{g}}).$
(11)

For the pole mass, $r_{l}-R_{l}=\\{-0.096,-0.004,0.128,-0.127\\}$ (at
$\mu=\bar{m}$), much smaller than the $r_{l}$ given in section 2. For $l=3$,
for example, $0.127$ is much smaller than $17.2$.

###  4.2 Contrast with ref. [2]

The reasoning given in ref. [2] is a bit different. If the formulas hold only
for asymptotically large orders, it is a prescription to add and subtract them
and a practical concession to truncate:

$\sum_{l=0}^{\infty}r_{l}\alpha_{\text{g}}^{l+1}\to\sum_{l=0}^{\infty}[r_{l}-R_{l}]\alpha_{\text{g}}^{l+1}+\sum_{l=0}^{\infty}R_{l}\alpha_{\text{g}}^{l+1}\to\sum_{l=0}^{L-1}[r_{l}-R_{l}]\alpha_{\text{g}}^{l+1}+R_{0}\frac{p}{2\beta_{0}}\mathcal{J}.$
(12)

The reasoning of ref. [4] is that medium and large orders are introduced in a
well-founded approximation, as sketched in eqs. 7, 8, 9 and 11:

$\sum_{l=0}^{\infty}r_{l}\alpha_{\text{g}}^{l+1}\to\sum_{l=0}^{L-1}r_{l}\alpha_{\text{g}}^{l+1}+\sum_{l=L}^{\infty}R_{l}\alpha_{\text{g}}^{l+1}\to\sum_{l=0}^{L-1}[r_{l}-R_{l}]\alpha_{\text{g}}^{l+1}+R_{0}\frac{p}{2\beta_{0}}\mathcal{J}.$
(13)

Standard, fixed-order perturbation theory is merely

$\sum_{l=0}^{\infty}r_{l}\alpha_{\text{g}}^{l+1}\to\sum_{l=0}^{L-1}r_{l}\alpha_{\text{g}}^{l+1},$
(14)

the poor convergence of which is the _raison d’être_ for more sophisticated
approaches.

##  5 Scale-dependence of the pole-mass–$\overline{\bf MS}$-mass relation

This section shows results for MRS applied to the quark mass. As noted in the
introduction, at the conference I showed preliminary versions of plots for the
static energy, which have been finalized and published [4]. It is more
interesting to show the quark-mass plots here, in particular to fill a gap in
the information in refs. [1, 2, 3]. That work evaluated
$\alpha_{\text{s}}(\mu)$ and the corresponding coefficients at $\mu=\bar{m}$
without examining other choices. Here, $\mu=s\bar{m}$ is considered, with the
plots showing the variation for $s\in\\{{\textstyle\frac{1}{2}},1,2\\}$.

To be specific, this section shows results for a massive valence quark with
three massless sea quarks. In refs. [1, 2], the charmed sea was included
albeit decoupled, following ref. [14]. For simplicity, the charmed sea is
omitted here.

For orientation, figure 1 shows examples of a good series and an excellent
series.

Figure 1: Left: momentum-space static energy
$\tilde{R}(q)=-q^{2}\tilde{V}(q)/C_{F}$ vs. $\Lambda/q$ for $\mu=sq$; right:
coordinate-space static force $F(r)=(r^{2}/C_{F})\mathrm{d}V/\mathrm{d}r$ vs.
$r\Lambda$ for $\mu=s/r$; in both cases,
$s\in\\{{\textstyle\frac{1}{2}},1,2\\}$. From ref. [4].

The details are in ref. [4]. The important feature here is how little
variation with $s$ arises with four nontrivial terms in the series.

Figure 2: Left: fixed-order perturbation theory for $R(\bar{m})\equiv
m_{\text{pole}}/\bar{m}-1$ vs. $\Lambda/\bar{m}$ for $\mu=sq$; right: MRS
perturbation theory for $R(\bar{m})$ for $\mu=s/r$; in both cases,
$s\in\\{{\textstyle\frac{1}{2}},1,2\\}$.

Fixed-order ($L=4$) and MRS definitions of the series are shown in figure 2.
The fixed-order estimate of $R(\bar{m})\equiv m_{\text{pole}}/\bar{m}-1$ is a
disaster, but the MRS prescription exhibits a variation with the choice of $s$
that is as mild as the (renormalon-free) static force. As with the static
energy, the subtracted, truncated series and the Borel sum on the right-hand
side of eq. 11 both vary significantly with $s$, but the total does not.

Figure 3 (left) shows the convergence of MRS perturbation theory for the pole-
mass–$\overline{\rm MS}$-mass ratio.

Figure 3: Left: accumulating MRS perturbation theory order-by-order for
$R(\bar{m})\equiv m_{\text{pole}}/\bar{m}-1$ vs. $\Lambda/\bar{m}$ for
$\mu=sq$. Right: the uncertainty on $R_{0}$ propagated to $R_{\text{MRS}}$;
here $\mu^{2}=(2~{}\text{GeV})^{2}+(s\bar{m})^{2}$.

The color code is them same as in figures 1 and 2, and the order is indicated
by the length of the dashes, with dotted curves show the $\mathcal{J}$ term
only and solid showing the full N3LO result. The overlapping curves show that
the changes made for each term are small for NLO, very small for NNLO, and
extremely small for N3LO.

In refs. [1, 2, 3], the uncertainty in the normalization of the factorial
terms, $R_{0}$, is estimated to be the size of the last term, which in pole
mass as for the static energy is the fourth term. Figure 3 (right) shows how
this error estimate propagates to $R_{\text{MRS}}$. Superficially, it is 2–4
times less than the uncertainty from scale variation. That means the
truncation uncertainty in the quark-mass results [1] may be a bit larger than
thought. After refitting the data (not yet done), it seems unlikely that it
would become larger than the other, dominant uncertainties.

## References

  * [1] Fermilab Lattice, MILC, TUMQCD collaboration, _Up-, down-, strange-, charm-, and bottom-quark masses from four-flavor lattice QCD_ , _Phys. Rev. D_ 98 (2018) 054517 [1802.04248].
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