EXAMINING THE IMPACT OF DEBT MATURITY TIME, EXPECTED RETURN AND VOLATILITY ON PROBABILITY OF DEFAULT IN CREDIT RISK MODELLING: THE CASE OF MERTON AND MKMV MODELS
George Jumbe ^{1}, Ravi Gor ^{2}
^{1}^{ }Research Scholar, Department of Mathematics, Gujarat University, India
^{2}^{ }Department of Mathematics, Gujarat
University, India

ABSTRACT 

In order to
model default risk, this article examines the impact of debt maturity time,
volatility, and expected asset return on probability of default (PD). The
study compares the probability of default produced by the Merton and Moody's
KMV (MKMV) methodologies and add modifying time, volatility, and expected
return on assets to see how they affect the probabilities of default produced.
It utilizes the balance sheet from Apple Inc. (AAPL) recorded from 2019
September 29 to 2022 September 29 for the current and total liabilities and
asset values in order to calculate the Probability of Default. The process
begins by determining the distances to default (DD) for Merton and MKMV using
the balance sheet, and then use the DDs to determine the likelihood of
default (PD). Results indicates that, the MKMV approach compares favorably to
the Merton approach. 

Received 04 December 2022 Accepted 03 January 2023 Published 18 January 2023 Corresponding Author George Jumbe, georgejumbe@yahoo.com DOI 10.29121/IJOEST.v7.i1.2023.442 Funding: This research
received no specific grant from any funding agency in the public, commercial,
or notforprofit sectors. Copyright: © 2023 The
Author(s). This work is licensed under a Creative Commons
Attribution 4.0 International License. With the
license CCBY, authors retain the copyright, allowing anyone to download,
reuse, reprint, modify, distribute, and/or copy their contribution. The work
must be properly attributed to its author. 

Keywords: Default Risk,
Debt Maturity Time, Volatility, Merton Model, MKMV Model, Expected Return,
Distance to Default, Probability of Default 
1. INTRODUCTION
Default risk, also known as default probability, is the likelihood that a borrower won't pay back the principal and interest owed on a debt security in whole and on time. One of the two elements of credit risk is default risk, along with loss severity. The chance of a default over a specific time horizon is described by the financial risk management term probability of default (PD). It offers a prediction of the probability that a borrower won't be able to pay back its debt. The PD is normally determined by performing a migration study of loans with similar ratings over a certain time period and calculating the proportion of defaulted loans. The PD is then assigned to the risk level and each risk level will only have one PD percentage Joseph (2021).
The PD can be used in a wide range of different credit analysis or risk management scenarios. It is influenced by the borrower's qualities as well as the overall state of the economy. In order to make up for the increased default risk, creditors (lenders) often demand a higher interest rate. Various credit evaluations and risk management frameworks make use of the PD especially in the determination of the economic or regulatory capital for a banking organization.
The PD is closely related to the expected loss, which is calculated as the sum of the Product Derivative (PD), Loss Given Default (LGD), and Exposure at Default (EAD). This study explore the effects of volatility, expected return, and debt maturity time on probability of default in credit risk using Merton and Moody's KMV model.
2. LITERATURE
Peresetsky et al. (2011) conducted an econometric analysis of Russian bank
failures from 1997 to 2003, concentrating on the degree to which information
from quarterly bank balance sheets that is made publicly available can be used
to anticipate defaults in the future. Their probability of default model was
built using binary choice models. The model produced a fair predictive power on
calculating the chance of default that Russian banks can utilize, despite the
low quality of the balance sheet data from Russia.
To calculate the implied likelihood of default from stock and option
market prices, Câmara et al. (2012) modified Merton (1976) ruin option pricing model. By examining all
international financial institutions having traded options in the US and
concentrating on the subprime mortgage crisis era, they tested their model. The
performance of the implied default probability produced by their methodology
was compared to the anticipated default frequencies based on the Moody's KMV
model. Their model's outcomes surpassed credit scores and agreed with those of
the KMV model.
A method for expressing credit risk was created by Valášková and Klieštik (2014) and uses either the probability of debtor default (businesses) or the difference between the asset value of the company and the default barrier expressed as a number of standard deviations. They demonstrate that the bond maturity is the key point in the Merton model and that defaults at this time occur when the market value of the asset is less than the bond's nominal value. In this situation, the company's financial resources won't be sufficient to pay off all of its debtors.
Based on Moody's KMV model, Voloshyn (2015) created a straightforward method for explicitly predicting a credit limit for a company. Their method allowed for the consideration of loan term to maturity, asset quality, balance sheet structure, and required degree of default probability. The proposed technique explained wellknown intuitive phenomena like the lower the credit limit, the higher the level of confidence, and the lower the credit limit, as well as the lower the credit limit, the higher the volatility of return on assets. Their method allowed for the consideration of the possibility that a company might invest fresh debt in assets with a different quality than existing assets.
Sariev and Germano
(2020) developed an enhanced Bayesian regularization
method to train artificial neutral networks (ANNs) and contrasted it to the
traditional regularizations method used to train
feedforward networks, which uses the backpropagation technique. On three
separate data sets, they examined the categorization accuracy of various
network designs. Profitability, leverage, and liquidity were discovered to be
significant financial default driver groups.
Joseph (2021) calculated and calibrated the credit rating default probability using Bayesian statistics and MonteCarlo simulations. They used their approach on banks and other financial institutions to address the problem of nonmonotonicity that arises when default rates are calculated empirically. By assuming that the default rate parameter is a nonrandom variable that follows the Beta distribution, they were able to implement their strategy. They first calculated a posterior density of the default rate parameter using historical data, and then they used simulations to calculate an estimate of the actual default rate parameter. Their outcomes were found to be equivalent to those of other wellliked calibration techniques frequently employed in the literature. They take a long time, though, with their method. However, their approach consumes large amount of time to execute.
3. THE MERTON MODEL
Merton (1974) is based on the work of Black and Scholes (1973) on option pricing and offers a framework for valuing debt issued by a firm. The model assesses the structural credit risk of a company by modeling the company's equity as a call option on its assets. The model calculates the theoretical pricing of European put and call options without considering dividends paid out during the life of the option. The original Merton model is based on some simplifying assumptions about the structure of the typical firm’s finances. The event of default is determined by the market value of the firm’s assets in conjunction with the liability structure of the firm. When the value of the assets falls below a certain threshold (the default point), the firm is considered to be in default. A critical assumption is that the event of default can only take place at the maturity of the debt when the repayment is due. Other assumptions include as described in Tudela and Young (2005), include;
1) All options are European and are exercised only at the time of expiration.
2) No dividends are paid out.
3) Market movements are unpredictable (efficient markets).
4) No commissions are included.
5) Underlying stocks' volatility and riskfree rates are constant.
6) Returns on underlying stocks are normally distributed.
The firm issues two classes of securities: equity and
debt. The equity receives no dividends. The debt is a pure discount bond where
a payment of D is promised at date T. If at date T the firm’s asset value exceeds the promised
payment, D, the debtholders are paid the promised amount and the shareholders
receive the residual asset value, (). If the asset value is insufficient to meet the
debtholders’ claims the firm defaults, the debtholders receive a payment equal
to the asset value, and the shareholders get nothing. The equation connecting
between asset, equity and debt values is given by Majumder (2006):
Equation 1
where is the total asset
value of the firm, is the equity value of
the firm and is the total amount of the firm’s debt.
All debts are mapped into a zerocoupon bond by selecting a
debt maturity . When shareholders' stock still has value and the company's debt
holders are paid in full. If , the business fails
on its debt. In this scenario, shareholders would receive nothing, and debt
holders would have priority claim to the remaining asset. The equity value at
time can be expressed as follows:
Equation 2
This is the payout of a European call option with a maturity of and a strike price of written on an underlying asset. The returns on the firm’s assets are assumed to be normally distributed and their behavior can be described with the following Brownian motion Zieliński (2013):
Equation 3
where is a standard Brownian motion, is the expected return
on assets which can be equal to the
riskfree interest rate and is the volatility of the firm’s assets (the standard
deviation of annualized rate of return).
The firm’s asset value is assumed to obey lognormal diffusion
process with a constant volatility given by:
Equation 4
where is initial value of
the assets specified at and is the value of the
asset at time . The expected value of the assets at the time is given by:
Equation 5
Equation 6
Using the
BlackScholes assumptions, we get:
Equation 7
for the call
option value, and
Equation 8
for the put option.
where is the standard normal
cumulative distribution probability function, and
Equation 9
Equation
10
The value of the debt is determined by . The probability of the company's debt default under
riskneutral conditions is . Here, the event that shareholders' call
option matures outofmoney is what triggers a credit default at time , with the following riskneutral probability:
Equation 11
Since equity is an option on form value,
the volatility of equity, denoted as , is also a function of and
. Using another geometric Brownian motion for equity we can obtain and and use Ito's
Lemma to demonstrate that instantaneous volatilities satisfy:
Equation 12
using BlackScholes equation, it can be
shown that , then (12) we becomes:
Equation 13
Equation 14
where is essentially the
delta of equity with respect to firm value. The price of an equity and the volatility of its return are
observed in the equity market. Finally, (7) and (14), can be solved
simultaneously for and .
3.1. Distances to
Default (DD) by Merton Approach
Distance to default (DD) is the difference
in standard deviations between the debt threshold and the anticipated asset
value at maturity. It serves as the basis for assessing credit risk. It is a
standard index that evaluates a company's creditworthiness and enables
comparisons between different companies and over time. There are fewer
likelihood of defaults with the higher values of DDs because the corporation is
more likely to repay debts on schedule. The DD measures how far a company's
assets are from the obligations whose value would cause a default Chen et al. (2010). The
distance to default (DD) is computed using the following formula:
Equation 15
3.2. Probability of Default (PD) by Merton Approach
The probability that the asset value will fall below the debt threshold at the end of the time horizon is known as the likelihood of default (PD) and is determined by:
Equation 16
4. MOODY
KMV (MKMV) MODEL
Oldrich Vasicek, John
McQuown, and Stephen Kealhofer
founded and debuted KMV in 1989. In 2002, Moody's Corporation bought the model
and gave the name of MKMV model. The model has been continuously updated and
improved since the acquisition in 2002 by Moody's Corporation. The KMV
structural model, now known as Moody's KMV (MKMV), is similar
to the Merton model but makes more reasonable assumptions by including
novel ideas such the default point, multiclass liabilities, distance to
default, and expected default frequency Voloshyn (2015).
4.1. Default Point
(DPT)
The Merton model assumes that all
liabilities are mapped to zero coupon bonds; instead, different classes of
liabilities, such as shortterm or current liabilities (CL) and longterm or
total liabilities (LTL), are used in the MKMV model. The MKMV model allows for
prematurity default, which is activated anytime the asset value falls below a
predetermined level known as the default point (DPT). Typically, MKMV suggests that, a company's default point falls between its shortterm
debt and half of its longterm debt Jumbe and Gor (2020).
Equation 17
The key takeaway from this is that the
company will always have to put shortterm obligations (debt)(CL) ahead of
longterm responsibilities (LTL). Additionally, default doesn't always happen
when a company's asset worth equals the book value of all of its liabilities. A
firm will, however, go into default when there is no longer any difference
between its assets and its default point.
4.2. Distance to Default (DD)
In
the MKMV model, the default point is a crucial component for calculating the
distance to default (DD). The DD, which is determined using the supplied
relation below, is the number of standard deviations the asset value is from
the default point.
Equation 18
From
the Merton model, the 1year probability of default is given as
where
Equation 19
where is the expected return
on assets which can be equal to the riskfree interest rate. According to
empirical research, the quantity is very close to zero hence negligible. So, we
approximate (19) to the expression below:
Equation 20
Replacing the liability D
with the default point DPT in (20) we
obtain the DD in MKMV.
Equation 21
Equation
22
4.3. Expected Default Frequency (EDF)
The MKMV model uses the Expected Default Frequency (EDF), a
fundamental quantity, to determine the likelihood that a specific firm will go
out of business within a year. In contrast to the EDF calculated from empirical
data in the MKMV, the chance of default in the Merton model is estimated from a
normal distribution, and this does not adhere to the genuine probability. According
to Voloshyn (2015), a company's EDF is as follows:
Equation 23
In the MKMV model, a declining empirical function takes the place of the normal distribution
function used by the Merton model. The function converts a company's distance to default into
the percentage of companies with similar DD values in an extensive historical
database that have defaulted in the past. As a result, it follows that two
distinct businesses with the same DD will also have the same EDF. To make a
comparison, in order to obtain the EDF for more than a year, the liability D is
substituted with the default point DPT in (15);
Equation
24
where
Equation 25
5. DATA ANALYSIS AND DISCUSSION
The information
on Apple Inc.'s (AAPL) asset values (A), current liabilities (CL), and
longterm liabilities (LTL) documented from 2019 September 29 to 2022 September
29 is shown in Table 1.
Table 1
Table 1 Current Liabilities, Long Term Liabilities, Total Asset Values and Default Points 

Time (T) 
9/29/2019 
9/29/2020 
9/29/2021 
9/29/2022 
Total Asset (A) 
338,516,000 
323,888,000 
351,002,000 
352,755,000 
Total debt (D) 
108,047,000 
112,436,000 
124,719,000 
120,069,000 
CL 
105,718,000 
105,392,000 
125,481,000 
153,982,000 
LTL 
142,310,000 
153,157,000 
162,431,000 
148,101,000 
DPT 
176,873,000 
181,970,500 
206,696,500 
228,032,500 
Source (Apple Inc.(AAPL) , https://finance.yahoo.com/quote/AAPL/balancesheet?p=AAPL). 
Table 2 shows the
calculated default probabilities (PDs and EDF) and distances to defaults (DDs)
based on information from Table 1. Combining
current liabilities (CL) with longterm liabilities (17) yields default points
(DPT) (LTL). In order to calculate the distances to default (DDs), (15) and
(25) are used, respectively. In order to determine the probability of default
(PDs), (16) and (24) are used, respectively. The table demonstrates that as the
number of years until debt maturity rises, the distances to default (DDs)
shorten. The table also demonstrates that as debt maturity durations in years
rise, so do the probabilities of default (PDs).
Table 2
Table 2 Effect of Time on Distance to Default and Probability of Default from Table 1 (, ) 

Time (T) 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

2.3243 
1.7637 
1.5382 
1.4171 
1.3435 
1.2959 
1.2640 
1.2425 
1.2281 
1.2188 

0.0101 
0.0389 
0.0620 
0.0782 
0.0895 
0.0975 
0.1031 
0.1070 
0.1097 
0.1114 

2.2184 
1.6888 
1.4771 
1.3642 
1.2962 
1.2527 
1.2240 
1.2050 
1.1928 
1.1853 

0.0133 
0.0456 
0.0698 
0.0863 
0.0975 
0.1052 
0.1105 
0.1141 
0.1165 
0.1179 

1.7958 
1.3900 
1.2331 
1.1529 
1.1072 
1.0801 
1.0643 
1.0556 
1.0519 
1.0517 

0.0363 
0.0823 
0.1088 
0.1245 
0.1341 
0.1400 
0.1436 
0.1455 
0.1464 
0.1464 

1.2564 
1.0086 
0.9220 
0.8832 
0.8660 
0.8599 
0.8604 
0.8649 
0.8721 
0.8811 

0.1045 
0.1566 
0.1783 
0.1886 
0.1932 
0.1949 
0.1948 
0.1935 
0.1916 
0.1891 
Figure 1 displays the
distances to default (DDs) plotted against the number of years until debt
maturity (T). The figure indicates that as loan maturity time lengthens, DDs
decrease. However, DDs generated using the firm's total debt (TD) are higher
than DDs generated using CL, LTL, and DPT. According to this result, businesses
that think about using current or shortterm liabilities are more likely to
default. However, as investment duration increases, the firm's stability to
default declines; as a result, this study suggests that firms should think
about adopting shortterm investments for their stability. The image likewise
depicts the DDS and PDs as having an inversely proportional connection. The PDs
rise when the DDs fall, and vice versa. This suggests that businesses with
larger DDs will experience a decreased likelihood of default. The odds of
default are displayed against the dates of the debt maturities in Figure 2 The graph
demonstrates that the PDs produced by DPT are more than those produced by TD,
CL, and LTL. This result demonstrates that companies utilising DPT in their
investment schemes are much more likely to default than companies using TD, CL,
or LTL. The image also depicts the exponential rise in default probability
before they begin to fall at some point in the future.
Figure 1
Figure 1 Distances to Default vs Time 
Figure 2
Figure 2 Probability of Default vs Time 
The impact of altering volatility () on the DDs and PDs is seen in Table 3. The table
demonstrates a decline in DDs when volatility rises. This research suggests
that because volatility lowers a firm's stability to default, higher volatility
firms will default more frequently. The table also demonstrates an increase in
PDs as volatility rises, suggesting that businesses with high volatility have a
higher chance of defaulting.
Table 3
Table 3 Effect of Volatility on Distance to Default and Probability of Default from Table 1 (, ) 


0.1 
0.2 
0.3 
0.4 
0.5 
0.6 
0.7 
0.8 
0.9 
1.0 

12.8214 
6.3357 
4.1405 
3.0178 
2.3243 
1.8452 
1.4888 
1.2089 
0.9801 
0.7871 

12.2919 
6.0709 
3.9640 
2.8855 
2.2184 
1.7570 
1.4131 
1.1427 
0.9213 
0.7342 

10.1788 
5.0144 
3.2596 
2.3572 
1.7958 
1.4048 
1.1113 
0.8786 
0.6865 
0.5229 

7.4821 
3.6661 
2.3607 
1.6830 
1.2564 
0.9553 
0.7260 
0.5415 
0.3869 
0.2532 

0.0 
1.1813e10 
1.7330e05 
1.2729e03 
1.0055e02 
3.2502e02 
6.8274e02 
1.1335e01 
1.6350e01 
2.1560e01 

0.0 
6.3586e10 
3.6859e05 
1.9542e03 
1.3265e02 
3.9461e02 
7.8810e02 
1.2657e01 
1.7844e01 
2.3142e01 

0.0 
2.6599e07 
5.5784e04 
9.2067e03 
3.6266e02 
8.0040e02 
1.3323e01 
1.8981e01 
2.4619e01 
3.0053e01 

3.6526e14 
1.2315e04 
9.1198e03 
4.6184e02 
1.0448e01 
1.6970e01 
2.3391e01 
2.9407e01 
3.4941e01 
4.0005e01 
The decline of DDs versus volatilities is depicted in Figure 3 When
volatility rises, DDs also fall, and vice versa. The DDs decrease with
increasing volatilities. The exponential increase of the PDs versus volatilities
is depicted in Figure 4 With rising
volatilities, PDs grow exponentially, raising the risk of default.
Figure 3
Figure 3 Distances to Default vs Volatility 
Figure 4
Figure 4 Probability of Default vs Volatility 
The impact of different interest rates (r) on the DDs
and PDs is shown in Figure 4. The table
displays the DDs and rates as a linear
relationship. With an increase in interest rates, the DDs rise. This indicates
that high interest rates lead to an increase in DDs, making it more likely that
the company will eventually default. Lower PDs are seen with higher DDs. The
table also demonstrates the decline in PDs in comparison to the rate increase.
High interest rates thereby lessen the likelihood of default (PD).
Table 4
Table 4 Effect of Rates on Distance to Default and Probability of Default from Table 1 (, ) 

Rate (r) 
0.1 
0.2 
0.3 
0.4 
0.5 
0.6 
0.7 
0.8 
0.9 
1.0 

15.7214 
7.7857 
5.1071 
3.7428 
2.9043 
2.3286 
1.9031 
1.5714 
1.3024 
1.0771 

15.1919 
7.5209 
4.9306 
3.6105 
2.7984 
2.2403 
1.8274 
1.5052 
1.2435 
1.0242 

13.0788 
6.4644 
4.2263 
3.0822 
2.3758 
1.8881 
1.5255 
1.2411 
1.0088 
0.8129 

10.3821 
5.1161 
3.3274 
2.4080 
1.8364 
1.4387 
1.1403 
0.9040 
0.7091 
0.5432 

1.7677e02 
1.060e02 
6.1350e03 
3.4223e03 
1.8405e03 
9.5371e04 
4.7610e04 
2.2892e04 
1.060e04 
4.7253e05 

2.2838e02 
1.3961e02 
8.2341e03 
4.6833e03 
2.5680e03 
1.3571e03 
6.9103e04 
3.3894e04 
1.6011e04 
7.2825e05 

0.0575 
0.0379 
0.0241 
0.0148 
0.0088 
0.0050 
0.0027 
0.0015 
0.0007 
0.0004 

0.1500 
0.1081 
0.0754 
0.0509 
0.0331 
0.0209 
0.0127 
0.0074 
0.0042 
0.0023 
The linear relationship between DDs and rates is
depicted in Figure 5. As interest
rates rise, the DDs rise as well. Since asset values will be well outside of
the default threshold, an increase in DDs implies stability of the firm from
default. The inversely proportional link between the PDs and rates is shown in Figure 4 With rising
interest rates and vice versa, the PDs decline. This suggests that a high
interest rate lowers the risk of a default by the company.
Figure 5
Figure 5 Distances to Default vs Rates 
Figure 6
Figure 6 Probability of Default vs Rates 
6. CONCLUSION AND SUGGESTION FOR FUTURE RESEARCH
In this study, the effect of changes in interest rates, volatility, and debt maturity times on the likelihood of default was examined. The study examined the results of the Merton and MKMV strategies for determining the distances to default (DDs) and probability of defaults (PDs). The results show that DDs and PDs produced by the MKMV technique (sDPT) are significant when compared to those produced by the Merton approach in each scenario (sTD, sSTL and sLTL). The DDs appear to be contracting for longer maturities in Figure 1. This shows that as maturities get longer, the company's financial situation gets worse. The development of PDs for longer maturities is seen in Figure 2 This shows that businesses are very vulnerable to defaulting on loans with longer maturities. The decline of DDs versus volatilities is depicted in Figure 3 According to this, asset values converge to the default threshold value as volatilities rise, increasing the chance of default for greater volatilities. The development of the PDs against volatilities is depicted in Figure 4 The likelihood of default rises as volatilities rises. The linear relationship between DDs and rates is depicted in Figure 5. As interest rates rise, the DDs rise as well. Since asset values will be well outside of the default threshold, an increase in DDs implies stability of the firm from default. The inversely proportional link between the PDs and rates is shown in Figure 6. With rising interest rates and vice versa, the PDs decline. This suggests that a high interest rate lowers the risk of a default by the company. Future research will examine the impact of changes in interest rates, volatility, and debt maturity dates on credit ratings and credit quality.
CONFLICT OF INTERESTS
None.
ACKNOWLEDGMENTS
None.
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