VALUING DEBT & ANALYSING CREDIT RISK: A QUANTITATIVE MODEL

KIRAN KUMAR K V

In the last two decades a number of financial crises have led to the growth of Credit Risk Analysis as a specialized proficiency. Events like the below have made the credit risk analysis models highly demanded by the financial world:


  • The “Collapse of Japanese Bubble”: Due to a continued overinvestment that led for overvaluation of Japanese real estate and equity prices during the 80s, Japanese economic growth slowed down and the slowing down lasted for more than 2 decades by now.
  • The “Tequila Crisis” of Mexico: Due to devaluation of Mexican Peso in Dec-1994, the Mexican Government Bonds caused severe losses to the investors.
  • The “Asian Contagion”: The crash of Thai Baht in July 1997 spread its wings to the equity markets of other related Asian economies and also subsequently the global markets.
  • The “Russian Debt Crisis”: In August, 1998, Russia defaulted on its debt and thus its securities caused significant losses and later this spread to affect the global equities.
  • The “Global Financial Crisis”: The famous ‘Housing Bubble Burst’ in US, resulted in significant losses on securitized instruments based on home mortgages. This was one event that shook the whole world’s security markets equally. A related crisis is the “Greek Sovereign Debt Crisis”, this led to crisis of confidence, indicated by a widening bond-yield spreads and rising cost of risk insurance on credit default swaps.
Mostly caused by an unexpected shift of a macroeconomic factor, these events have caused the default risk increase during these times. Traditionally, CREDIT RATINGS were the only measures of credit risk analysis, which were only partly effective in capturing the changes in these correlated default risks. As a result, additional tools to quantify and manage risks have emerged. These tools include methods for estimating correlated default probabilities and recovery rates based on macroeconomic factors. Such new models incorporated the systemic risk similar to those occurred during the above crises.

Before we understand the new tools, it may be useful to have an idea of the basic credit risk measures that are generally estimated:
  • Probability of Default – The probability that the bond will default before maturity.
  • Loss given Default – The amount of the remaining coupon and principal payments lost in the event of default (expressed as a percentage of the position of exposure).
  • Recovery Rate – The percentage of position received or recovered in default. Therefore, percentage of loss given default plus recovery rate would be 100%
  • Expected Loss – The probability of default multiplied by the loss given default.
  • Present Value of the Expected Loss – Conceptually the largest price once would be willing to pay on a bond to  a third party (for example, an insurer) to entirely remove the credit risk of purchasing and holding the bond.
Traditional models for credit risk analysis have been primarily – (1) Credit Scorings order credit riskiness of the borrowers in an ordinal ranking (applicable for small owner-operated businesses and individuals) and (2) Credit Ratings provide a simple statistics that summarized a complex credit analysis of a potential borrower (used for companies, sovereigns, sub-sovereigns and asset backed securities)

As these traditional models fail to capture quantitatively the basic credit risk measures discussed above, but, only give an approximation ide0a of the risk involved, a number of analytical tools have been developed. Here we discuss one such quantitative model called STRUCTURED MODEL.


STRUCTURED MODEL EXPLAINED

This model attempts to understand the economics of company’s liabilities and build on the insights of option pricing theory. This can be called structural model because this is based on the structure of a company’s balance sheet. Let’s say we assume the structure of an issuing company’s balance sheet that has
  • Assets of At,
  • Debt of D(t,T), which is a zero-coupon bond with maturity and face value K and
  • Equity of St
Then the accounting equation of the company’s balance sheet can be given as: At= D(t,T) + St. The company’s shareholders have limited liability, that is to the maximum of At. In other words, if the equity holders default on the debt payment at time T, the debt holder’s only recourse is to take over the company and assume the ownership of At. Thus, we say equity shareholders’ liability in this case is limited, which becomes the basis of analogy between company’s equity and a call option. A call option’s payoff at time T is max[ST – K, 0]. In our example, if we consider company’s asset as the underlying of the call option and the asset price at time T, (AT) as the spot price of the underlying at maturity, the payoff of the said option would be max[AT – K, 0].(Note that K is the principal repayment of the zero coupon bond). Debt holders in this case, become the writer of the call option, and their payoffs would be min[K, AT]. What is means is that, debt holders would get K, if the At>K, as the equity holders find it beneficial to exercise the option (i.e., repay K to debt holders) and debt holders would get AT, if the At<K, as the equity shareholders find it unviable to exercise the option and they would default, and thus debt holders get to own the assets at price AT.
Thus, it implies that:
  • Probability of default at time T is probability that the AT<K, and
  • Loss given default  is K-AT
The assumptions we make implicitly, in the above construct are:
  1. The company’s assets trade in frictionless markets that are arbitrage free
  2. The riskless rate of interest, r, is constant over time and
  3. The time value of the company’s assets has a lognormal distribution with µT and variance σ2T
The above three assumptions are identical to those for stock price behaviour in the original Black-Scholes option pricing model. Hence, we can apply the BS model to find out the equity’s time t value because it is a European call option on the company’s assets. Thus, the valuation model becomes:

The first term in this expression At N(-d1 )  refers to the present value of the payoff on a company’s debt if default occurs. The second term Ke^(-r(T-t) ) N(d2) refers to present value of the payoff on the company’s debt if default does not occur. Thus, the sum of these two terms, therefore, gives the present value of the company’s debt.

A Case Study
Let us take the case of the below bond currently traded at NSE WDM:

NSE & BSE – Corporate Bonds Trade Repository for 08-Sep-2017
xchange
ISIN
Security Description
Coupon Rate
Maturity Date
Last Traded Price (in Rs.)
Price (Average weighted price)
Yield (Weighted Avg. Yield ) %
Turnover (in Lacs)
Credit Rating
NSE
INE155A08183
TATA MOTORS LIMITED 10.3 NCD 30NV18 FVRS10LAC LOA UPTO 11MR14
10.30
30-Nov-2018
103.7962
103.7962
6.9000
10500.00


The information required to conduct a credit analysis have been obtained from the financial statements of the company as below:
  • Time t asset value= = 2737.54 crores (Non-current assets of Tata Motors as on 31-03-2017)
  • Expected return on assets = µT = 4.97% per year (Average return on assets of Tata Motors between FY 2006 to FY 2017)
  • Risk free rate = r = 6.01% per year (Yield on 10-year GOI from www.rbi.org)
  • Face Value of debt =  = Rs. 2050.608 (Non-current interest bearing liabilities of Tata Motors as on 31-03-2017)
  • Time to maturity =  = 445 days or 1.22 years (number of days between 11-Sep-2017 to 30-Nov-2018)
  • Asset return volatility = σT = 1.52% (standard deviation of return on assets of Tata Motors between FY 2006 to FY 2017)
Using the above model various credit analysis outputs are obtained as below:

Input
Value (in crores)
At
145019.6
u
          0.83500
r
0.0501
K
60629
T-t
1.22
sd
0.4764
d1
2.03660
d2
1.510399705
N(-d1)
0.0208
N(-d2)
0.0655
e1
3.85640
e2
3.3301948
N(-e1)
0.0000575
N(-e2)
0.0004339
Probability of Default
0.0004339
Expected Loss
3.200
Present Value of Expected Loss
711.133


The above present value of expected loss can be reduced from the value of the debt, based on a certain market discount rate to arrive at the value of total debt of this compan
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