Assessing Credit Risk for a Ghanaian Bank Using the Black- Scholes Model VK Dedu 1, FT Oduro 2 1,2 Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana. Abstract This paper examines the application of option pricing methodology to the evaluation of corporate credit risk in a banking credit portfolio context. Risk is here measured by means of the default probability of corporate entities with particular reference to a banking institution in the Ghanaian financial market using option pricing methodology. The resulting probability of default of clients was modeled as a stochastic process and the default probability of corporate entities was evaluated using the Black-Scholes default probability. Keywords Option pricing methodology, Black-Scholes default probability, asset volatility. I. INTRODUCTION Banks seek to address the issue of maximazing shareholders expectation while increasing their profit margins by issuing out more loans amidst other financial packages. For most banks, however, loans are the largest and most obvious source of credit risk [1]. Credit risk in fact accounts for about 60% of a bank s total risk, while operational risk accounts for 30%, market risk, 5% and other risks take 5% (Ibid). In recent times, increasing academic and professional interest has been directed at the issue of credit risk with the aim of being able to adequately assess, quantify or foresee possible default events. Thus, several approaches and models have been propounded to anticipate the financial crises of a firm or an individual. Nevertheless, yet over the years, several banks and individuals have filed for bankruptcy or simply folded up due to their inability to honor external debt obligations. The objective of this paper is to evaluate or assess the default probability of firms and individuals and consequently the associated credit risk by the use of the Black-Scholes-Merton option pricing methodology based on a risk neutral valuation. This methodology has been used traditionally to assess risk in the area of commodity and derivatives trading but not normally in the area of credit risk. A. Credit Risk Credit risk is the probability that a counter party will fail to honor the payment of both the principal and the interest on a given loan. In other words it is potential that a bank borrower or counterparty will fail to meet their obligations in accordance with agreed terms [1]. Several sources of credit risk arise in banking and thee include loans, acceptance, inter-bank transactions, trade financing, foreign exchange transactions, financial futures, swaps, bonds, equities, options, the extension of commitment and guarantee, and the settlement of transactions. Loans are of course the largest source of credit risk. Risk often emerges in process of investing and in the allocation of capital. Risks need to be assessed in order to achieve a sound investment decision. These decisions are made by balancing the risks against returns. Giving loans is a risky affair for banks sometimes, even though certain risks may also arise when banks offer securities and other forms of investment. The risk of losses that result in the default of payment of debtors is a kind of risk that must be anticipated. Minimizing the probability of default and hence the credit risk enables a bank to keep substantial amount of capital to protect itself from insolvency and to maintain its economic stability. The greater the bank is exposed to risks, the greater the amount of capital that must be put aside in terms its reserves, so as to maintain its solvency and stability. B. Options An option is a financial instrument that gives the holder the right but not the obligation to buy or sell an asset by a certain date at a predetermined price. The price the buyer of the option pays in order to gain the stated right is the option premium. Options come in two forms [2]: A call option gives the holder of the option the right but not the obligation to buy an asset by a certain date for a certain price. A put option gives the holder of the option the right but not the obligation to sell an asset by a certain date for a certain price. 13
An option can either be American or European. American options can be exercised at any time up to the expiration date, whereas European can be exercised only on the expiration date itself. II. METHODOLOGY A. Option Pricing Models The main idea underlying option pricing models is that a credit default will occur when the economic value of the borrower s assets falls below the economic value of its debt. A loan taken out by the company or individual can be associated with the purchase of an option which would allow the equity investors or the individual to satisfy the claims of the debt lenders by handing over the company or the individual s assets instead of repaying the debt in the case of default [3]. The price the company pays for this option corresponds to the risk premium included in the interest on the loan. The probability that the option will be exercised becomes the probability of default. B. Black Scholes Model The Black-Scholes Merton formula [2] for prices of a European call option and put option on a non-dividend paying stock is K is the strike price of the option r is the continuously compounded risk-free rate is the stock price volatility T is the time to maturity of the option. C. Default Probability (Firm s) We link the Black-Scholes Model [2] to the to the debt obligations of a firm in the form of zero coupon bonds at time T years: is the asset value, D is the debt obligation, T is the time to maturity, is the asset volatility, is the riskfree interest rate are respectively the probabilities relative to the exercise of the option and debt repayment From the basic accounting equation Implying that the value of the debt today becomes: And the value of the company s debt at time T is The Black-Scholes Merton (BSM) default probability is simply the probability that the market value of firm s assets is less than the face value of the firm s liabilities, D, at time T (i.e. ) [4]. With risk neutral valuation framework or risk neutral world, the expected value (E) of a call and put respectively at option at maturity is From the BSM model equity can be viewed as a call option on the value of the firm s assets. The expected value of the firm s equity at time T is N(x) is the cumulative probability distribution function for a standardized normal distribution is the stock price at time And the value of the equity today is (9) 14
And the asset volatility, can be calculated from is the standard normal distribution. Assume that a loan defaults if the value of the borrower s assets at the loan maturity T falls below the contractual amount B payable. Let s denote to be the value of the i-th borrower s assets and assume that it follows a stochastic process described by the equation the volatility of equity of listed companies can be calculated form historical prices of stock Equity holders (shareholders) of a company have a call option on the assets value. That is equity holders have the right but not the obligation to pay off the debt holders (liabilities of the firm). Debt holders on the other hand have a put option on the firm s assets, which is implied in the limited liability of the company. The put option in this case is the right but not the obligation to sell the firm s assets in case of default. D. Default Probability (Clients ) Default probabilities of clients of banks can be modeled to follow the BSM default probability. We consider a random variable X which follows a normal distribution with mean µ and variance where X is the loss incurred due to customers default on loans then more concisely we can deduce that losses follow a normal distribution where and are the instantaneous mean and variance, respectively and is an increment of the Wiener process. B which is the total contractual amount payable at maturity T, is assumed to be prepaid at time t >0 The logarithm of the total asset value at time normally distributed with the mean And variance is given by The assets value at Time T is is a standard normal random variable. The probability of default denoted p of the i-th loan is then is With its probability distribution function given by Then it follows that A loan will be in default if the value of the borrower s assets at the time of maturity of the loan at time T is less than the contractual amount payable It follows that the probability of default is given by We denote p the probability of default by It follows that 15
Therefore N is the cumulative normal distribution function (. III. DATA ANALYSIS AND RESULTS The analysis focused on a bank which is assumed to be a listed member of the Ghana Stock Exchange. The source of data used was from the bank s financial statement. The following assumptions were made to evaluate the default probability. 1. The bank stock is not a dividend paying stock 2. Risk neutral valuation 3. There is no arbitrage opportunity 4. There are no transaction costs. 5. A perfect market is assumed 6. The debt of the firm is considered to be in the form of zero coupon bonds. Assuming from the bank s financial statement, that the current market value of its total asset ( is GHȼ516,632,000 and that it has an impending debt obligation ( ) of GHȼ465,546,000 in five years time, the BSM default model was used to evaluate the default probability of the bank. The bank s asset volatility was evaluated by means of equation (10).The equity volatility of the bank was estimated from the assumed 15 days trading information of the company s stock on the stock exchange. The analysis focused on a bank which is assumed to be a listed member of the Ghana Stock Exchange. The source of data used was from the bank s financial statement. The following assumptions were made to evaluate the default probability. 1. The bank stock is not a dividend paying stock 2. Risk neutral valuation 3. There is no arbitrage opportunity 4. There are no transaction costs. 16 5. A perfect market is assumed 6. The debt of the firm is considered to be in the form of zero coupon bonds. Assuming from the bank s financial statement, the current market value of its total asset ( is GHȼ516,632,000 and has an impending debt obligation ( ) of GHȼ465,546,000 in five years time. The BSM default model was used to evaluate the default probability of the bank [5]. The bank s asset volatility was evaluated by means of equation (10).The equity volatility of the bank was estimated from the assumed 15 days trading information of the company s stock on the stock exchange which is displayed in Table 1. Day Thus, TABLE I DAILY SHARE PRICES Share price(ghȼ) 1 0.28 2 0.28 3 0.28 4 0.28 5 0.28 6 0.28 7 0.27 8 0.28 9 0.28 10 0.28 11 0.27 12 0.27 13 0.27 14 0.28 15 0.26
The study assumed the bank to have a total of 302,039,348 shares and therefore with a current market value of the equity to be GHȼ78,530,230.48. The bank s volatility of equity was evaluated using equation (11) yielding Expected share price of the company The BSM default probability of the bank is thus given by This implies that the bank has a virtually zero probability of defaulting on its current long term debt obligations. From equation (5) The Bank s equity five years from now is given by And IV. DISCUSSION The discussion focused on a simple situation. If the bank wants to borrow from external sources in the form of debt financing, the lender s purchases a claim on the firm's assets, and thereby becomes a partial owner of the company. The value of bank s assets increases by the amount received. The new total value of the firm's assets is equal to the value of the stock (equity) and the value of the debt. With time, the market value of the bank s assets will change as the market perception of the future earning power of the company changes. These changes obviously involve considerable uncertainty. What concerns the lender is the market value of bank s assets when the term of the debt elapses. Three situations are possible. 1. The market value of the bank s asset is greater than the face value of the debt 2. The market value of the bank s asset is at least that the debt obligation payable 3. The market value of the bank s asset value is less than face value of the debt In the first situation, the equity holders of the bank would have a call option on the asset value. That is the Equity holders would have the right but not the obligation to pay off the company s debt holders. Clearly the (GHȼ ). With an expected equity payoff The expected payoff of the company s equity at time T (5 years) = Dividing the expected payoff at time T=5 years by the total shareholder base of the company yields the expected share price of the company within five years. In such a situation the stockholders will pay off the debt since an equity value of GHȼ386,236,167.5 is an indication of the bank s stock performance. The shareholders can honor the debt obligation since the company s asset value is enough for them to do so.if the Bank however does not have enough cash to settle the debt obligation, the stockholders can raise it by selling a part of the assets at their market value. It will be in the interest of the Stockholders of the bank to pay its lenders, otherwise the lenders would force the bank to bankruptcy. The Lenders on the other hand have a put option on the assets value which is implied in the limited liability of the bank. 17
The put option in this sense becomes the right but not the obligation to sell off Bank s assets which causes the stockholders to lose control of the firm. In the situation that the market value of the bank s assets falls below the amount due the lender s, then the bank cannot repay its lenders since the bank would have an expected equity payoff The bank cannot raise additional cash be it by equity finance or debt finance since no other lender would refinance the loan, because that would mean taking over the loss from the original lender. It is also not possible to raise additional equity, since the banks stock would be nonperforming and an expected equity payoff of is an indication of the stock s non-performance. The company has to declare bankruptcy. Because of the limited liability of the shareholder the expected payoff of the shareholders.the stockholders of bank get nothing, while the lenders take over the assets. The lenders will thus realize a loss equal to the difference between the face value of the debt and the market value of assets. V. CONCLUSION In addition to the many useful applications of the Black- Scholes Merton model for evaluating or quantifying credit risk, one of its strongest attributes is its ability to resonate with two key stakeholder groups; shareholders and debt holders. The model predicted effectively the company s probability of default without the use of enormous data which makes it a mathematically elegant model for banks to work with. It is a high time Ghanaian indigenous banks considered the use of option pricing methodologies in its day to day credit risk management. The result above has shown strong and robust means of evaluating the bank s probability of default and hence the corresponding credit risk. Acknowledgment The authors are very grateful to Messieurs Bismark Owusu Tawiah, Collins Knight Arthur and Miss Victoria Naaki Armah for research assistance. REFERENCES [1] Basel Committee on Banking Supervision (1999) Principles for the Management of Credit Risk Published by the Risk Management Group of the Basel Committee on Banking Supervision [2] Hull, J.C (2009) Options, Futures and Other Derivatives, Seventh Edition, Pearson Prentice Hall, New Jersey [3] Kirmße, S. (1996): Die Bepreisung und Steuerung von Ausfallrisiken im Firmenkundengeschäft der Kreditinstitute, Frankfurt am Main 1996 [4] www.scribd.com/doc/52647586/26/option-pricing-models [5] Glasserman, P (2000). Probability Models of Credit Risk, Columbia Business School, pp 2 43 [6] World Bank (2005) Credit Risk and Emergence of Credit Derivatives [7] Alexander J. McNeil, Rüdiger Frey, & Paul Embrechts (2006) Quantitative Risk Management: Concepts, Techniques, and Tools, Princeton Series in Finance Darrell Duffie and Stephen Schaefer, Series Editors 18