1. Introduction
Default can happen if the value of the issuer’s total assets is less than the value of the debt obligations should the issuer be unable to make the required payments. There are many different approaches to evaluate default probabilities. The structural model and reducedform model are the main measures of default probabilities. In the structural model, or the Merton distance to default (DD) model, which is inspired by Merton’s [
1] bond pricing model, a defaulttriggering event is explicitly defined as a firm’s failure to pay debt obligations by means of modeling the equity value of the firm as a call option on the firm’s value, with the firm’s face value of the debt as a strike price. On the other hand, for the reducedform model, a defaulttriggering event is defined as an unexpected event that is governed by an exogenous defaultintensity process.
The Merton model is widely used to measure the distance to default by Moody’s KMV (Kealhofer, McQuown and Vasicek). This was used previously to provide quantitative credit analysis tools to creditors and investors until the acquisition in 2002 by Moody’s Analytics and is now part of Moody’s Analytics Enterprise Risk Solutions. However, the original formula is complicated. Campbell, Hilscher and Szilagyi [
2] concluded that the strong predictive power of the Merton DD model came from the functional form imposed by the Merton model. Bharath and Shumway [
3] also provide evidence to show that it is the functional form of the Merton DD model that makes it useful and important for predicting defaults. Rather than simultaneously solving two nonlinear equations or implementing a complicated iterative procedure, as suggested by Crosbie and Bohn [
4] and Vassalou and Xing [
5] to obtain default probabilities, they proposed a naive default forecasting model, which retains the structure of the Merton DD model, but with a simpler and easier calculation procedure.
It is recognized that simplicity does not mean simplistic. Recently, significant works in the literature have focused on the issue of ambiguity aversion. However, the discussion of ambiguity aversion is not a novel issue, but has a long history. We use the following example, which was adapted from Ellsberg’s [
6] famous experiment, to illustrate the meaning of ambiguity aversion. The experiment subjects are provided with two urns, each of which contains several red and white balls. There is no difference in the total number of balls in each urn. The balls in one urn are half red and half white. However, the ratio in the other urn is unobservable to the subjects. The rule is that $1 will be awarded when a red ball is drawn. Although the two urns promise the same expected return, an ambiguityaverse subject will be reluctant to draw a ball from the latter, because of a tendency to think pessimistically, leading to an expectation that the other urn contains lower odds.
Ambiguity aversion, or uncertainty aversion, is a wholly different concept from risk aversion. Risk aversion is an attitude that penalizes the expected return of a risky investment. Consider the following experiment. The experiment subjects are provided with two potential investments. One investment pays off either $1 or $0 with equal probability. Another guarantees $0.5 for sure. Although the two investments offer the same expected return, a riskaverse investor will resist investing in the former. Ambiguity aversion, however, represents a lack of confidence in probability estimates. According to Bewley ([
7], p. 1134), “… an increase in uncertainty aversion increases the size of the set of subjective distributions. An uncertaintyneutral decision maker has only one subjective distribution and so is Bayesian.”
Generally speaking, the approaches of addressing ambiguity aversion may be divided roughly into two categories. One approach is called the recursive multipriors utility approach, which addresses ambiguity aversion through the multiprior expected utility (for example, [
8,
9,
10,
11]). Gilboa and Schmeidler [
12] contribute to this field by initially establishing a welldefined maxmin expected utility with multiple priors. They are famous for developing a useful axiomatic theory in this field. Following the approach of Gilboa and Schmeidler [
12], Easley and O’Hara [
13] discussed microstructure and ambiguity and showed that ambiguity could reduce participation by both investors and issuers. The incomplete preferences of Bewley [
14], the smooth ambiguity model of Klibanoff, Marinacci, and Mukerji [
15] and the variational preferences of Maccheroni, Marinacci and Rustinchini [
16] are interesting extensions to this approach.
This paper uses the other approach, called the robust control approach, to depict investors’ ambiguity aversion. This describes ambiguity through a set of priors and introduces the penalty function to a general utility function in order to capture investors’ ambiguity aversion (for example, [
17,
18,
19,
20,
21]). Theoretically, economic agents are assumed to possess perfect knowledge of the data generating process and to know the underlying probability law exactly. However, we cannot ignore the fact that investors are not certain of the correctness of the model. Investors with higher ambiguity aversion worry about a worstcase scenario. Therefore, the investor will choose alternative models that are further from the reference model. Hence, the robust control approach assigns a smaller penalty to more distant perturbations. However, if ambiguity aversion is low, the investor will choose alternative models that are similar to the reference model. As a result, the robust model assigns a greater penalty to more distant perturbations. That is, the penalty is inversely related to the investor’s degree of ambiguity aversion.
There have been numerous studies investigating ambiguity issues in asset prices and equity premia (for example, [
22,
23,
24] for a literature review on ambiguity in asset pricing and portfolio choice). By using the robust control technique, So [
25] derives an adjusted Black–Scholes pricing formula, which maintains the functional form of the original Black–Scholes formula, but adds an additional parameter depicting uncertainty aversion to compute the riskfree interest rate and dividend yield. We find that managers’ uncertainty aversion about the project value raises his subjective evaluation of the real options. If we compare the firm value and the equity value to the project value and the real options in that paper, we can infer that investors’ ambiguity about the firm’s value will change the valuation of the equity. In addition, since the riskfree interest rate has been adjusted, it is possible that the default rate of the fixed income debt will also be modified. This paper could be treated as an extension of [
25] from the theoretical real options analysis to the applied risk management field. The purpose of this paper is to develop a structural credit pricing model with ambiguity aversion as additional information and to derive a riskneutral default probability under uncertainty about the firm’s value. In addition, we use the new model to examine the validation of Merton’s DD model under ambiguity. To the best of our knowledge, ambiguity aversion has not been applied to examine the default prediction issue addressed here.
As in risk aversion, investors’ ambiguity aversion is a subjective parameter, so one cannot easily observe its aggregate value or assign it a reasonable value. The Consumer Confidence Index (CCI), however, is an observed indicator, which is defined as the degree of optimism on the state of the economy that consumers are expressing through their activities surrounding saving and spending. In the USA, the CCI is issued monthly by the Conference Board, which was started in 1967 and is benchmarked to 1985. AitSahalia and Brandt [
26] also mentioned that the CCI and other economic indexes are suitable to capture market uncertainty. Buraschi and Jiltsov [
27] believe that the CCI has an important link with market conditions, so they used the CCI to control uncertainty and added it to construct the model and investigate option markets.
Following the literature, we use the CCI in our empirical study as an inverse proxy to the level of ambiguity aversion. We modify Bharath and Shumway’s naive model with the CCI and focus on how accurate the default predictability of our new model is compared with the original model.
The organization of the remainder of the paper is as follows. The models are presented in
Section 2. The data are described in
Section 3. The empirical results are analyzed in
Section 4, and the conclusions are given in the final section.
3. Data
The sample is from 2000 to 2011, giving 144 monthly observations, and December, 2007, to 2011 in terms of the CDS spread regression to match the CDS contract. The sample of observed firms was based on an article in CFO Magazine that was published in November, 2003, “Ranking America’s top debt issuers by Moody’s KMV Expected Default Frequency.” We choose 52 firms with complete data as the nondefault sample and added 5 firms entering bankruptcy between 2000 and 2011.
Table 1.
Summary statistics.
Panel A.
Variables in the Merton distance to default (DD) model and hazard models.
Panel A.
Variables in the Merton distance to default (DD) model and hazard models.
Variable  Quantiles 

Mean  SD  Min  0.25  Median  0.75  Max 

E  59,213.18  74,361.40  0.45  13,575.92  28,167.40  74,009.42  519,044.42 
F  41,562.61  57,808.95  684.97  17,509.28  23,839.00  38,574.00  761,203.00 
r (%)  2.29  2.00  −0.02  0.21  1.72  4.04  6.39 
CCI  86.94  28.22  25.30  61.40  92.95  106.13  144.70 
r spread (%)  0.77  10.45  −69.01  −4.58  0.80  6.48  89.40 
1/σ_{E}  1.91  0.44  0.52  1.58  1.92  2.24  3.13 
naive σ_{v} (%)  37.00  10.03  17.92  29.84  35.38  42.43  76.44 
π^{naive} (%)  6.43  9.53  0.00  0.23  2.58  8.20  54.21 
π^{CCI} (%)  6.59  9.36  0.00  0.33  2.81  8.59  53.56 
Panel B.
Correlation.
Corr(π^{naive}, π^{CCI}) = 0.994

We obtained much of the data from the Bloomberg Database, including the 3month riskfree rate, daily stock prices and shares outstanding. We then calculated the value of firms, the volatility of equity, and stock returns of firms in the USA. Following Bharath and Shumway [
3], we assume that the debts in firms have a oneyear horizon. The value of the firms’ debts is obtained from the data of the book value of firms’ total liabilities in Bloomberg. The information of CCI is also available in Bloomberg, which is monthly data, for 7,906 firmmonths’ of data, and the data of the firm’s CDS spread were derived from the Datastream database. We adopt a 5year CDS contract with a base date starting from December, 2007, containing 2,107 firmmonths’ of data.
Table 1 shows the descriptive statistics of the variables used in the paper. The information about the variables used in the Merton DD, naive DD and hazard models can be found in Panel A. A simple comparison of π
^{naive} and π
^{CCI} is also given in Panel A. Broadly speaking, π
^{CCI} has a higher mean and lower standard deviation than does π
^{naive}. From Panel B, we can see that the correlation of π
^{naive} and π
^{CCI} is extremely high. Therefore, we can infer that π
^{naive} and π
^{CCI} have a high gearing effect.
5. Conclusions
Campbell, Hilscher and Szilagyi [
2] and Bharath and Shumway [
3] provided evidence that the default predictability of the Merton DD model can be attributed to its specific functional form. Bharath and Shumway [
3] constructed a naive default forecasting model. The feature of their naive model is that it retains the functional form of the Merton DD model and computes default probability in a naive way, rather than simultaneously solving two nonlinear equations or implementing a complicated iterative procedure.
Recently, the issue of ambiguity aversion has drawn a lot of attention in the academic field. Although there have been many studies dealing with ambiguity issues in asset prices and equity premia, little is available regarding the influence of ambiguity aversion on forecasting default probability. The purpose of this paper was to construct a default prediction model under ambiguity and to examine the validation of the Merton DD model under ambiguity. To the best of our knowledge, ambiguity aversion has not previously been analyzed in terms of the default prediction issue considered here.
By using CCI in our empirical analysis as an inverse proxy for the level of ambiguity aversion, we constructed a new default forecasting model. Applying the Cox proportional hazard model, this paper showed that the new model was superior to Bharath and Shumway’s naive model. In addition, although the statistical significance of Bharath and Shumway’s naive default probability is retained, its sign is changed due to the effect of our model in the CDS spread regressions. Therefore, our model could serve as a useful reference for modifying the Merton DD model when ambiguity aversion is accommodated.