Online Appendix - Aggregate Risk and the Choice between Cash and Lines of Credit - Appendix

Q: What does Appendix A establish about the equilibrium when Ls < Ls1(θ)?

It characterizes the equilibrium cost of cash q* and shows how equilibrium outcomes depend on whether the budget constraint ever binds. It also identifies the unique equilibrium and introduces a liquidity threshold Lsmin below which q* stays at q2.

Q: How are Beta KMV and Var KMV computed in Appendix B?

The appendix maps equity to a call option on firm value and uses a set of equations to infer asset value dynamics. It estimates equity volatility from historical stock returns, iteratively solves for asset volatility, unleveres to obtain asset beta, and sets Var KMV equal to σV.

Q: How is Bank VIX forecasting constructed in Appendix C?

It computes daily forecasts of bank return volatility using financial sector index return data, then applies a Gaussian GARCH(1,1) model. Parameters are estimated using rolling sample windows up to day t, producing forecasts for day t+106 in the described setup.

Aug 28, 2026

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Online appendix material develops equilibrium characterization and empirical implementation details for a study of how firms choose between holding cash and using lines of credit under aggregate risk. Appendix A derives the unique equilibrium cost of cash under cases where the liquidity supply Ls lies below or above key thresholds, including conditions ensuring budget constraints bind or never bind. Appendix B explains the computation of KMV-based asset beta and variance using an option-pricing framework with iterative estimation. Appendix C outlines construction of bank VIX forecasts via sector return volatility series and a Gaussian GARCH(1,1) model.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages21

Reading Time53 min

AccessFree to download

Appendix A. Characterization of the equilibrium when Ls < Ls1(θ)

q2">Case 1: q1 > q2

Case 2: q1 < q2

Definition of Lsmin

Appendix B. Computing Beta KMV and Var KMV

Option-pricing setup and equity valuation

Iterative estimation and implied asset returns

Unlevering beta and variance assignments

Appendix C. Computing Bank VIX

Data-based daily volatility estimates

Gaussian GARCH(1,1) forecasting procedure

Online Appendix: Aggregate Risk and the Choice between Cash and Lines of Credit* VIRAL V. ACHARYA, HEITOR ALMEIDA, and, MURILLO CAMPELLO First draft: September 2009; This draft: June 2012 Key words: Bank lines of credit, cash holdings, liquidity management, systematic risk, loan spreads, loan maturity, asset beta. JEL classiﬁcation: G21, G31, G32, E22, E5 . *Acharya, Viral, Heitor Almeida, and Murillo Campello, 2012, Internet Appendix to “Aggregate Risk and the Choice between Cash and Lines of Credit,” Journal of Finance [vol \#],[pages], [http://www.afajof.org/IA/](http://www.afajof.org/IA/) [year].asp. Please note: Wiley-Blackwell is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing material) should be directed to the authors of the article. " Appendix A. Characterization of the equilibrium when Ls < Ls1(θ) Suppose ﬁrst that q1 > q2 , such that the ﬁrm’s budget constraint never binds in equilibrium. In this case, if Ls < Ls1 we will have that q ∗ = q2 > 1. To see why, notice that if q < q2 then systematic ﬁrms would choose xθ = 1, which is not compatible with equilibrium. If q > q2 , then xθ = 1, generating an excess supply of cash. Thus, we must have q ∗ = q2 . Since systematic ﬁrms are indiﬀerent between any xθ between 0 and 1 when q = q2 , we can sustain an equilibrium such that: θ [xθ(q2 )(ρ − ρ0 ) − wmax ] = Ls. (1) This is the unique equilibrium of the model. To see why, notice that for xθ > xθ(q2 ) , cash demand would be larger than supply, and if xθ < xθ(q2 ) , cash supply would be greater than demand and thus the cost of cash would drop to q = 1 . If q1 < q2 , then the ﬁrm’s budget constraint will bind in equilibrium, and we will have q1 < q ∗ ≤ q2 . The cost of cash q ∗ is such that the demand for cash exactly equals supply: θ [xθ(q∗ )(ρ − ρ0 ) − wmax ] = Ls. (2) Since q1 < q ∗ , then xθ(q∗ ) < 1. Since q ∗ ≤ q2 , then systematic ﬁrms would like to increase their demand for cash beyond xθ(q∗ ) , but they cannot aﬀord to do so. Thus, q ∗ is the equilibrium cost of cash in this case. Finally, notice that since the cost of cash cannot be greater than q2 , there is a level of liquidity supply (denoted by Lsmin) such that for all Ls < Lsmin, the equilibrium is q ∗ = q2 . Lsmin is such that the maximum level of xθ that satisﬁes the budget constraint when q = q2 yields a demand for cash exactly equal to Lsmin : θ [xθ(q2 )(ρ − ρ0 ) − wmax ] = Lsmin. (3) Appendix B. Computing Beta KMV and Var KMV To compute Beta KM V and Var KM V we make the following assumptions. First, suppose that the total value of a ﬁrm follows: dVV = μdt + σVdW (4) where V is the total value, μ is the expected continuously compounded return on V , σ V is the volatility of ﬁrm value, and dW is a standard Wiener process. In addition, assume that the ﬁrm issued one discount bond maturing in T periods. Under these assumptions, the equity of the ﬁrmis a call option on the underlying value of the ﬁrm with a strike price equal to the face value of the ﬁrm’s debt and a time-to-maturity of T. The value of the “call option” is: E = VN (d1 ) − e−rT FN(d2 ) (5) where E is the market value of a ﬁrm’s equity, F is the face value of the ﬁrm’s debt, r is the instantaneous risk-free rate, N ( .) is the cumulative standard normal distribution function, d1 is given by ln (V/F )+(r + 1 σ2V)T d 1 = 2 , (6) σ V √T and d2 is given by d2 = d 1 − σV √T Given the value of equity, the underlying value of the ﬁrm, or market value of asset is: V = E~~ ~~+~~ ~~e−NrF1N)(d2~~ ~~) (7) Since the value of equity is a function of the value of the ﬁrm and time, using Ito’s lemma we obtain: V ∂E V 1 σ E = σ V = σ V (8) E ∂V E N(d1 ) To implement the model, we need to simultaneously solve equations (7) and (8) . We follow Bharath and Shumway (2008), and adopt an iterative procedure as follows. First, equity volatility σE is estimated

FAQs

What does Appendix A establish about the equilibrium when Ls < Ls1(θ)?

It characterizes the equilibrium cost of cash q* and shows how equilibrium outcomes depend on whether the budget constraint ever binds. It also identifies the unique equilibrium and introduces a liquidity threshold Lsmin below which q* stays at q2.

How are Beta KMV and Var KMV computed in Appendix B?

The appendix maps equity to a call option on firm value and uses a set of equations to infer asset value dynamics. It estimates equity volatility from historical stock returns, iteratively solves for asset volatility, unleveres to obtain asset beta, and sets Var KMV equal to σV.

How is Bank VIX forecasting constructed in Appendix C?

It computes daily forecasts of bank return volatility using financial sector index return data, then applies a Gaussian GARCH(1,1) model. Parameters are estimated using rolling sample windows up to day t, producing forecasts for day t+106 in the described setup.

Aug 28, 2026

Upload by Cart