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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.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/online-appendix-aggregate-risk-and-the-choice-between-cash-and-lines-of-credit-appendix/156461/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/online-appendix-aggregate-risk-and-the-choice-between-cash-and-lines-of-credit-appendix/156461.png","ImageObject",300,407,{"name":92,"@type":93},"Cart","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-09-23","2026-08-28",true,{"@type":102,"interactionType":103,"userInteractionCount":24},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What does Appendix A establish about the equilibrium when Ls \u003C Ls1(θ)?","Question",{"text":112,"@type":113},"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.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How are Beta KMV and Var KMV computed in Appendix B?",{"text":117,"@type":113},"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.",{"name":119,"@type":110,"acceptedAnswer":120},"How is Bank VIX forecasting constructed in Appendix C?",{"text":121,"@type":113},"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.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},156461,1787959337,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":24,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":129,"read_time":144},18829141979164,"https://eur-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","Online Appendix:  \nAggregate Risk and the Choice between Cash and Lines of Credit*  \nVIRAL V. ACHARYA, HEITOR ALMEIDA, and, MURILLO CAMPELLO  \nFirst draft: September 2009; This draft: June 2012  \nKey words: Bank lines of credit, cash holdings, liquidity management, systematic risk, loan spreads, loan maturity, asset beta.  \nJEL classification: G21, G31, G32, E22, E5 .  \n*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. \"  \nAppendix A. Characterization of the equilibrium when Ls \u003C Ls1(θ)  \nSuppose first that q1 > q2 , such that the firm’s budget constraint never binds in equilibrium. In this case, if Ls \u003C Ls1 we will have that q ∗ = q2 > 1. To see why, notice that if q \u003C 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:  \nθ [xθ(q2 )(ρ − ρ0 ) − wmax ] = Ls. (1)  \nThis 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 \u003C xθ(q2 ) , cash supply would be greater than demand and thus the cost of cash would drop to q = 1 .  \nIf q1 \u003C q2 , then the firm’s budget constraint will bind in equilibrium, and we will have q1 \u003C q ∗ ≤ q2 . The cost of cash q ∗ is such that the demand for cash exactly equals supply:  \nθ [xθ(q∗ )(ρ − ρ0 ) − wmax ] = Ls. (2)  \nSince q1 \u003C q ∗ , then xθ(q∗ ) \u003C 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.  \nFinally, 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 \u003C 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 :  \nθ [xθ(q2 )(ρ − ρ0 ) − wmax ] = Lsmin. (3)  \nAppendix B. Computing Beta KMV and Var KMV  \nTo compute Beta KM V and Var KM V we make the following assumptions. First, suppose that the total value of a firm follows:  \ndVV = μdt + σVdW (4)  \nwhere 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:  \nE = VN (d1 ) − e−rT FN(d2 ) (5)  \nwhere 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  \nln (V/F )+(r + 1 σ2V)T  \nd 1 =   2 , (6)  \nσ V √T  \nand d2 is given by    \nd2 = d 1 − σV √T  \nGiven the value of equity, the underlying value of the firm, or market value of asset is:  \nV = E~~ ~~+~~ ~~e−NrF1N)(d2~~ ~~) (7)  \nSince the value of equity is a function of the value of the firm and time, using Ito’s lemma we obtain:  \nV ∂E   V 1   \nσ E = σ V = σ V (8)  \nE ∂V E N(d1 )  \nTo 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","cbCaigYNDJYbc0wi","https://ap.wps.com/l/cbCaigYNDJYbc0wi","pdf",360779,21,"English","# Appendix A. Characterization of the equilibrium when Ls \u003C Ls1(θ)\n## Case 1: q1 > q2\n## Case 2: q1 \u003C q2\n## Definition of Lsmin\n# Appendix B. Computing Beta KMV and Var KMV\n## Option-pricing setup and equity valuation\n## Iterative estimation and implied asset returns\n## Unlevering beta and variance assignments\n# Appendix C. Computing Bank VIX\n## Data-based daily volatility estimates\n## Gaussian GARCH(1,1) forecasting procedure","[{\"question\":\"What does Appendix A establish about the equilibrium when Ls \\u003c Ls1(θ)?\",\"answer\":\"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.\"},{\"question\":\"How are Beta KMV and Var KMV computed in Appendix B?\",\"answer\":\"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.\"},{\"question\":\"How is Bank VIX forecasting constructed in Appendix C?\",\"answer\":\"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.\"}]","Online Appendix - Aggregate Risk and the Choice between Cash and Lines of Credit - Appendix | PDF",53]