A NEW APPROACH TO MERTON MODEL DEFAULT AND PREDICTIVE ANALYTICS WITH APPLICATIONS TO RECESSION ECONOMICS TOMMY LEWIS

Transcription

1 A NEW APPROACH TO MERTON MODEL DEFAULT AND PREDICTIVE ANALYTICS WITH APPLICATIONS TO RECESSION ECONOMICS TOMMY LEWIS

2 BACKGROUND/MOTIVATION Default risk is the uncertainty surrounding how likely it is that a given firm will be unable to service its debts and obligations- enter bankruptcy. This is important information to be able to quantify as it allows us price a company s debt in terms of credit spreads on corporate bonds that the issuer can default upon. This gives investors an incentive to purchase bonds that are inherently risky- something proven to be necessary for proper portfolio diversification.

3 BACKGROUND/MOTIVATION 1950 s Miller Modigliani (MM) Model F t,t = D t,t + E t,t (Assets = Debt + Equity) s Merton Model (M) uses (MM) to price Equity and Debt: E t,t = E[e r T t F T D + F t ] D t,t = F T E t,t s Merton KMV implementation of (M) to use public data, E and σ E to derive private data, F and σ F.

4 BASIC SET UP OF THE MERTON MODEL This is a version of the Black Scholes option pricing model. It assumes the market value of a firm s equity (market cap) is an option, with it s default point acting as a floor value for the firm s implied asset value. The idea parallels that of a call option- implied asset value falling below the current default point leaves shareholders with nothing and, lenders with whatever is left.

5 WHY USE THE MERTON MODEL The Merton Model is an attempt to quantify the default risk of a company at a given point in time. Accomplished by calculating the Default Probability of a borrower (in this case firm) which is the likelihood of that borrower failing to fully service it s debts and obligations.

6 PRELIMINARY INFO The following are results from calibrating the Merton model to data ranging from Results were gathered for the companies, AIG, Ford, and Morgan Stanley. Considerable increase both in magnitude and volatility regarding probability of default for every company in 2008 and Interesting trends in day to day variation for both stock prices and probability terms.

7 COMPUTATION AND DATA COLLECTION Python (programming language) was used for the implementation and graphical representation of results. The Wharton Research Database was used to gather data for each of the companies which was then compiled in excel in order to feed into the IDE, Spyder.

8 SYMBOL VARIABLE EXPLANATION E i Market Value of Equity Share Price * Total Shares Outstanding N x i μ i σ E i Number of trading days in a year Daily log return of stock prices Mean of log returns at time i for the previous trailing year, updated daily Volatility of Equity at time i for the previous trailing year, updated daily Assumed to always equal 252 = = ln ( S i S i 1 ) = 1 N 1 N 1 D i Default Point Total Book Liabilities N j=1 N j=1 x i j x i j μ i 2 r Riskless Rate 3-month Treasury Bill Rate T-t Time to maturity for debt obligations Taken to equal 1

9 DERIVING THE SET OF NONLINEAR EQUATIONS FOR IMPLIED FIRM VALUE AND VOLATILITY f 1 F, σ F : Relationship of equity volatility to firm asset volatility via elasticity of equity as a call option on firm assets with strike D. Fσ F N(d 1 ) = Eσ E. f 2 F, σ F : Assumption that observed and theoretical equity prices are equal. e y T t FN d 1 e rt DN d 2 = E.

10 SOLVING THE SET OF NONLINEAR EQUATIONS FOR IMPLIED Equations: FIRM VALUE AND VOLATILITY f 1 F, σ F = e y T t Fσ F N(d 1 ) E σ E = 0 f 2 F, σ F = e y T t FN d 1 e rt DN d 2 E = 0 Let f: A R n R n be a function that is differentiable on A. As it is a vector-valued function, it has component functions f i x, i = 1,2 and thus we have: f x = e y T t Fσ F N(d 1 ) E σ E e y T t FN d 1 e rt DN d 2 E, x A

11 J f (x) is the Jacobian Matrix of f x and is defined by, [ J f x ] ij = f i (x) xj DEFINING THE JACOBIAN Explicitly: J f x = e y T t FN d 1 + e y T t σ F F e.5d 1 2 2π N (σ F ) e y T t (σ F N d 1 + e.5d 1 2 e y T t F T t e.5d 1 2 2π 2π(T t) ) e y T t N d 1

12 USING NEWTON S METHOD I use Newton s Method is an iterative approach in computing an approximate solution to the system of equations, f x = 0. E + D Initial guess, x (0) = σ E Linear approximation for f x,which is near the current iterate x k : f k x = f(x k ) + J f (x k )(x x k ) = 0 Tolerance level, ε = The iteration loop will then stop if x k+1 x k < ε or the maximum amount of iterations (a specified value, 1000) has been reached. *Note that rearranging the linear approximation yields, x (k+1) = x k J f x 1 f(x k )

13 ITERATION RESULTS Once this system of equations is solved meaning both σ F and F are calculated (or very closely approximated), they are used to calculate the term for probability of default. Where probability of default, PD = N ln F D + r y 0.5σ F 2 T t σ F T t = Pr(F < D) Now we can view the dynamic relationship between S i /S i 1 and PD i /PD i 1.

14 PR[DEFAULT] VS TIME AMERICAN INSURANCE GROUP, AIG

15 PR[DEFAULT] VS TIME FORD MOTOR COMPANY, F

16 PR[DEFAULT] VS TIME MORGAN STANLEY, MS

17 RESULTS: AMERICAN INSURANCE GROUP, AIG S i /S i 1 vs PD i /PD i 1. DAILY CHANGE IN SHARE PRICE VS DAILY CHANGE IN PR(DEFAULT) Aug 6, 2008: AIG has an increase in unrealized loss from credit default swaps from $9.1 billion to $14.7 billion and $16.5 billion in collateral Nov 10, 2008: AIG discloses that it had a total of $37.3 billion in collateral on swaps and estimates total swap looses in to be $33.2 billion Mar 2, 2009: AIG reports a record fourthquarter loss of more that $60 billion and gets an additional Treasury injection of as much as $29.8 billion Aug 7, 2009: AIG posts first profit in seven quarters

18 DAILY CHANGE IN SHARE PRICE VS DAILY CHANGE IN PR(DEFAULT) MORGAN STANLEY RECESSION HIGHLIGHTS S i /S i 1 vs PD i /PD i 1 Dec 2007: The recession officially begins, unemployment rate at 5% Sept 15, 2008: Lehman Brothers files the largest bankruptcy case in US history/ Sep 16, 2008: Bailout of AIG FORD Dec 16, 2008: For the fist time in history, Fed lowers benchmark interest rate to zero/ Dec 19, 2008: Bailout of General Motors and Chrysler Mar 9, 2009: The Dow hits the low point of the recession (down 54% from Oct 2007) June 1, 2009: General Motors files for bankruptcy and says it will close 14 US plants * * * * * * AIG Oct 2, 2009: The unemployment rate peaks at 10%- hitting double digits for the first time in 26 years

19 IMPLICATIONS Clearly there is some relationship between market surges, and volatility of default probability, thus: The main implication of this study is that it motivates further examination of daily change in probability of default as a random variable and specifically if it has any precise relationship with change in daily stock price. A secondary implication is that the Merton Model has some value as a predictive analytics tool. Special thanks to Albert Cohen and Aditya Viswanathan for their assistance and advice with this project