Equity Market and Credit Cycle

Transcription

1 Washington University in St.Louis September 12, 2016

2 Motivation It is an established fact that performance of Stock Market Leads our real Economics. Real Economy should also be related with Credit market. Then it seems natural to think Stock Market and Credit market should be positively linked. Empirical fact: Credit Market lead Real Economics(i.e Azariadis et.al(2014)). For us, Stock Market leads Credit Market Our question: Why Stock Market leads Credit Market 1? Or to be more precise, How? 1 I only consider Commercial and Industry loans here, In another words, Mortgage and consumer loan are beyond our scope

3 Fact: Leading Role of Stock Market Figure: Absolute level of Commercial loans and S&P500, Monthly

4 Fact: Leading Role of Stock Market, Figure: Annual Growth rate of Commercial loans and S&P500, Monthly

5 Fact: Leading Role of Stock Market, Figure: Annual Growth rate of Commercial loans and S&P500, Monthly

6 Fact: Leading Role of Stock Market, Figure: Annual Growth rate of Commercial loans and S&P500, Monthly

7 Exception 1 There are exceptions where the relationship between Equity Market and Credit Market appears to be weak. Namely and The Period during is called Credit Crunch (Berger et.al(1995)), which coincide with the period of merge and acquisition of banks. To explain Credit Crunch goes beyond the scope of this paper But we can derive some implication on the effect of this Bank Industry deregulation.

8 Exception 2 Period During this period Our Inflation rate(in the sense of growth of CPI) attain the peak. To be more Precise, May 1980 is the month when peak of inflation is attained. After this period, I documented in another paper, QTM breaks down at around It is natural to think there may exist some transition period between QTM-period and non-qtm-period. I will bet Again, this is not within our scope OPEC: Qatar (1961), Indonesia (1962), Libya (1962), the United Arab Emirates (1967), Algeria (1969), Nigeria (1971), Ecuador (1973), Gabon (1975) and Angola (2007) Iranian Revolution; End of inflation? 1981 April: Oil Price $115, 1983 July, $71

9 Proposed Possibility It seems reasonable to think equity is traded between more informed individual, who may know more than bank Boom on Equity market reflects improvement of our economics, and Vice versa. Then this boom can lead to a demand for loan. Or information of improvement of our economics will be obtained by bank, which will implied less default risk so more loans. We design a model try to identify the quantitative importance of this part(supply Channel). So we adopted a model have implication of Default risk from Corporate Finance Literature and embedded bank s decision process in to get supply curve of loan.

10 Benchmark Model, Settup Representative Firm and Representative Bank Firm will choose from a sequence of pair of loan contract offered by bank. A typical Loan contract is a pair of interest rate and Amount of loan to lend. {(r i, X i )} i Θ Firm owns Asset, whose value will evolve as a geometric Brownian Motion: dv a V a = µ a dt + σ a dw t F t, is the sigma-algebra generated by {W t }, P(ω) would be the accordingly induced probability measure. So {W t } will be a Brownian Motion with respect to Probability space (Ω, F t, P). µ a and σ a are not known to Bank.

11 Benchmark Model, Settup We denote the book value of debt to be X and the maturity date would be T, which means it borrowed Xe rτ at the very first place. Say t is the beginning date of this loan, τ = T t, And X will not change during time t and T. Market value of this firm s equity is VE s, at time s [t, T ] with interest rate of loan r and dividends yield δ. Unsecured loan is similar to a collateral loan with the whole asset as collateral. In another word, when firm defaults, Bank will get the asset which was held by firm 3. Firm default if and only if V T a X 3 I implicitly assume recovery rate is 1 here.

12 Benchmark Model, Settup Then equity of the firm has the same payoff as a Call option, with strike price X If I assume complete market, then a risk-neutral Probability measure exist, which is using K-N theorem with a pricing kernel implied by this complete market. Where V s e = V s A e δτ Φ(d 1 ) Xe rτ Φ(d 2 ), s [t, T ] (1) d 1 = ln( V s A X ) + (r δ σ2 A )τ σ A τ, d 2 = d 1 σ A τ (2)

13 Benchmark Model, Bank s Problem I assume that our bank industry is perfectly Competitive, in another words, Bank will earn zero Profit. Xe (r f r)τ = (1 P)X + P E[V T A Default, V t A ] (3) where r f is the effective federal fund rate, the opportunity cost of Funding. Since limitation of data source, We use 3 Month T-rate to approximate federal fund rate before 1954 June. E[VA T Default] = VAe t (µ A 1 2 σ2 A )τ+ σ 2 A τ 2 ( ) ln(x ) [ln(v t A ) + (µ A 1 1 ( ) 2 Φ σ2 A)τ] ln(x ) [ln(v t A ) + (µ A Φ σ2 A)τ] σ A τ σ A τ Appendix 25

14 Benchmark Model, Rational Expectation Equilibrium: Illustration Given the contract firm has chosen, say (r, X ). If bank know (µ a, σ a ) Equation 1 can back out V a if V e was given Hence within the maturity of a loan, say 48 months, given (r, X ), the performance of {Ve t } 48 t=1 will give us a sequence of {Va t } 48 t=1, And the behavior of this sequence should be consistent with (µ a, σ a ). Further more, the potential contracts set, say Θ, should be Consistent with (µ a, σ a ), by which I mean, equation 3 holds

15 Benchmark Model, Rational Expectation Equilibrium: Calculation Algorithm Given a sequence of {V E } WL t=1 For each risk free rate, r. Guess X = X 0 σ A = σ E = var( ln(v E ) ln(v E ) ) Using Equation (3) to calculate a series of {VA t}wl t=1 Back to Step 2, Replace σ A = var( ln(v A) ln(v A ) ), till σ A converges, name it to be σ A Use σ A and equation (3) to calculate {V A }, and estimate µ A = E( ln(v A) ln(v A ) ) Use formula (3) to compute VA t, with µ A, σ A. According to equation (4), we can obtain an Level of Loan, we name it to be X new But this step does not take into consider that Bank makes decision according to recent history instead of daily price of Stock. Hence we replace this step to be.

16 Benchmark Model, Rational Expectation Equilibrium: Calculation Algorithm Given a sequence of {V E } WL t=1 For each risk free rate, r. Use formula (3) to compute {V t A }30 t daily =1, with µ A, σ A. According to equation (4), we can obtain a sequence of Level of Loan {X t }, we name the mean of this sequence to be X new. 4 Go Back to Step 1, but replace the guess to be X = X new, continue this progress until X converges. 4 We also tried use the mean of {V t A} WL t=1 to solve out a level of loan. This algorithm is much faster, with pretty similar result. But it has potential problems from our nonlinear setting.

17 Benchmark Model(48 Months Loan), Implication Figure: log of 48 and 72-months Loan, Estimate

18 Benchmark Model(48 Months Loan), Implication Figure: log of 48 and randomly 72-months Loan, Estimate

19 Benchmark Model(48 Months Loan), Implication with data Figure: log of Estimated stock of Loan with data

20 Benchmark Model(48 Months Loan), Implication with data: HP filter Figure: log of Estimated stock of Loan with data: cyclical Part

21 Benchmark Model(48 Months Loan), Implication with data: HP filter Figure: log of Estimated stock of Loan with data: cyclical Part, Coe:

22 Benchmark Model(48 Months Loan), Implication with data: HP filter Figure: log of Loans: cyclical Part; With 24 months shift

23 Extended Model Jump Process Allow µ a and σ a depends on X Bayesian Learning process similar to learning. Kalman-Burcy filter.

24 Conclusion Equity Market plays a role in Credit Cycle We propose a default risk channel of Equity market s effect.

25 Appendix E[e x e x θ] = E[e x x ln(θ)] = Φ( ln(θ) µ σ ) 1 ˆ ln(θ) = Φ( ln(θ) µ ) 1 µ+ σ2 e 2 σ e x 1 (x µ)2 dx e σ 2πσ 2 ˆ ln(θ) 1 2πσ 2 e µ+ σ2 = e 2 Φ( ln(θ) µ ) 1 ln(θ) µ σ2 Φ( ) σ σ (x µ σ 2 ) 2 2σ 2 dx P = Pr(Default) = Pr(ln(V A ) ln(x )) = Φ( ln(x ) [(ln(v t A ) + (µ A 1 2 σ2 A )τ] σ A τ ) µ+ σ2 E[V A Default] = e 2 Φ( ln(θ) µ σ where we know µ = ln(v t A ) + (µ A 1 2 σ2 A )τ, σ2 = σ 2 A τ, θ = X. ) 1 Φ( θ µ σ2 ) σ