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Financial modeling

{{Short description|Modeling financial systems}}

{{Use American English|date = January 2019}}

Financial modeling is the task of building an abstract representation (a model) of a real world financial situation.{{Cite web |author = Investopedia Staff| url=http://www.investopedia.com/terms/f/financialmodeling.asp | title=Financial Modeling|date = 2020}} This is a mathematical model designed to represent (a simplified version of) the performance of a financial asset or portfolio of a business, project, or any other investment.

Typically, then, financial modeling is understood to mean an exercise in either asset pricing or corporate finance, of a quantitative nature. It is about translating a set of hypotheses about the behavior of markets or agents into numerical predictions.{{cite journal|last1=Low|first1=R.K.Y.|last2=Tan|first2=E. |title=The Role of Analysts' Forecasts in the Momentum Effect|journal=International Review of Financial Analysis|volume=48|pages=67–84|date=2016|doi=10.1016/j.irfa.2016.09.007|url=https://espace.library.uq.edu.au/view/UQ:406166/UQ406166_OA.pdf}} At the same time, "financial modeling" is a general term that means different things to different users; the reference usually relates either to accounting and corporate finance applications or to quantitative finance applications.

Accounting

File:Cash Flow Projection.png-based Cash Flow Projection (click to view at full size)]]

In corporate finance and the accounting profession, financial modeling typically entails financial statement forecasting; usually the preparation of detailed company-specific models used for decision making purposes, valuation and financial analysis.

Applications include:

To generalize {{Citation needed|date=November 2011}} as to the nature of these models:

firstly, as they are built around financial statements, calculations and outputs are monthly, quarterly or annual;

secondly, the inputs take the form of "assumptions", where the analyst specifies the values that will apply in each period for external / global variables (exchange rates, tax percentage, etc....; may be thought of as the model parameters), and for internal / company specific variables (wages, unit costs, etc....). Correspondingly, both characteristics are reflected (at least implicitly) in the mathematical form of these models:

firstly, the models are in discrete time;

secondly, they are deterministic.

For discussion of the issues that may arise, see below; for discussion as to more sophisticated approaches sometimes employed, see {{section link|Corporate finance|Quantifying uncertainty}} and {{section link|Financial economics|Corporate finance theory}}.

Modelers are often designated "financial analyst" (and are sometimes referred to, tongue in cheek, as "number crunchers"). Typically, the modeler will have completed an MBA or MSF with (optional) coursework in "financial modeling".Example course: [https://study.unisa.edu.au/courses/013613/2022 Financial Modelling], University of South Australia Accounting qualifications and finance certifications such as the CIIA and CFA generally do not provide direct or explicit training in modeling.[http://www.ft.com/cms/s/2/db7a4838-1352-11e5-ad26-00144feabdc0.html#ft-article-comments The MiF can offer an edge over the CFA] Financial Times, June 21, 2015. At the same time, numerous commercial training courses are offered, both through universities and privately.

For the components and steps of business modeling here, see {{slink|Outline of finance|Financial modeling}}; see also {{section link|Valuation using discounted cash flows|Determine cash flow for each forecast period}} for further discussion and considerations.

Although purpose-built business software does exist, the vast proportion of the market is spreadsheet-based; this is largely since the models are almost always company-specific. Also, analysts will each have their own criteria and methods for financial modeling.See for example, [https://ssrn.com/abstract=256987 Valuing Companies by Cash Flow Discounting: Ten Methods and Nine Theories], Pablo Fernandez: University of Navarra - IESE Business School Microsoft Excel now has by far the dominant position, having overtaken Lotus 1-2-3 in the 1990s. Spreadsheet-based modelling can have its own problems,Danielle Stein Fairhurst (2009). [http://www.fimodo.com/2009/11/six-reasons-your-spreadsheet-is-not-a-financial-model/ Six reasons your spreadsheet is NOT a financial model] {{webarchive|url=https://web.archive.org/web/20100407003951/http://www.fimodo.com/2009/11/six-reasons-your-spreadsheet-is-not-a-financial-model/ |date=2010-04-07 }}, fimodo.com and several standardizations and "best practices" have been proposed.[http://www.eusprig.org/best-practice.htm Best Practice] {{Webarchive|url=https://web.archive.org/web/20180329184059/http://www.eusprig.org/best-practice.htm |date=2018-03-29 }}, European Spreadsheet Risks Interest Group "Spreadsheet risk" is increasingly studied and managed; see model audit.

One critique here, is that model outputs, i.e. line items, often inhere "unrealistic implicit assumptions" and "internal inconsistencies".{{cite book|author1=Krishna G. Palepu|author2=Paul M. Healy|author3=Erik Peek|author4=Victor Lewis Bernard|title=Business analysis and valuation: text and cases|url=https://books.google.com/books?id=DPK43Sku2PsC&pg=261|access-date=12 November 2011|year=2007|publisher=Cengage Learning EMEA|isbn=978-1-84480-492-4|pages=261–}} (For example, a forecast for growth in revenue but without corresponding increases in working capital, fixed assets and the associated financing, may imbed unrealistic assumptions about asset turnover, debt level and/or equity financing. See {{section link|Sustainable growth rate|From a financial perspective}}.) What is required, but often lacking, is that all key elements are explicitly and consistently forecasted.

Related to this, is that modellers often additionally "fail to identify crucial assumptions" relating to inputs, "and to explore what can go wrong".{{cite book|author1=Richard A. Brealey|author2=Stewart C. Myers|author3=Brattle Group|title=Capital investment and valuation|url=https://books.google.com/books?id=eKF8IBCwfy4C&pg=PA223|access-date=12 November 2011|year=2003|publisher=McGraw-Hill Professional|isbn=978-0-07-138377-6|pages=223–}} Here, in general, modellers "use point values and simple arithmetic instead of probability distributions and statistical measures"Peter Coffee (2004). [http://www.eweek.com/c/a/Database/Spreadsheets-25-Years-in-a-Cell/ Spreadsheets: 25 Years in a Cell], eWeek.

— i.e., as mentioned, the problems are treated as deterministic in nature — and thus calculate a single value for the asset or project, but without providing information on the range, variance and sensitivity of outcomes;Prof. Aswath Damodaran. [http://pages.stern.nyu.edu/~adamodar/pdfiles/papers/probabilistic.pdf Probabilistic Approaches: Scenario Analysis, Decision Trees and Simulations], NYU Stern Working Paper

[http://www.analycorp.com/uncertainty/flawarticle.htm The Flaw of Averages] {{webarchive|url=https://web.archive.org/web/20111207025740/http://www.analycorp.com/uncertainty/flawarticle.htm |date=2011-12-07 }}, Prof. Sam Savage, Stanford University.

see {{section link|Valuation using discounted cash flows |Determine equity value}}.

A further, more general critique relates to the lack of basic computer programming concepts amongst modelers,

Blayney, P. (2009). [https://www.learntechlib.org/p/31495/ Knowledge Gap? Accounting Practitioners Lacking Computer Programming Concepts as Essential Knowledge]. In G. Siemens & C. Fulford (Eds.), Proceedings of World Conference on Educational Multimedia, Hypermedia and Telecommunications 2009 (pp. 151-159). Chesapeake, VA: AACE. with the result that their models are often poorly structured, and difficult to maintain. Serious criticism is also directed at the nature of budgeting, and its impact on the organization.Loren Gary (2003). [http://hbswk.hbs.edu/item/3623.html Why Budgeting Kills Your Company], Harvard Management Update, May 2003.Michael Jensen (2001).

[https://ssrn.com/abstract=321520 Corporate Budgeting Is Broken, Let's Fix It], Harvard Business Review, pp. 94-101, November 2001.

Quantitative finance

File:OAS valuation tree (es).png - usually returned by commercial derivatives software]]

In quantitative finance, financial modeling entails the development of a sophisticated mathematical model.See discussion here: {{cite web |url=https://www.siam.org/Portals/0/Student%20Programs/Thinking%20of%20a%20Career/brochure.pdf |archive-url=https://web.archive.org/web/20190305095047/https://www.siam.org/Portals/0/Student%20Programs/Thinking%20of%20a%20Career/brochure.pdf |archive-date=2019-03-05 |url-status=live |title=Careers in Applied Mathematics|publisher=Society for Industrial and Applied Mathematics}} Models here deal with asset prices, market movements, portfolio returns and the like.

Relatedly, applications include:

These problems are generally stochastic and continuous in nature, and models here thus require complex algorithms, entailing computer simulation, advanced numerical methods (such as numerical differential equations, numerical linear algebra, dynamic programming) and/or the development of optimization models. The general nature of these problems is discussed under {{section link|Mathematical finance|History: Q versus P}}, while specific techniques are listed under {{section link|Outline of finance|Mathematical tools}}.

For further discussion here see also: Brownian model of financial markets; Martingale pricing; Financial models with long-tailed distributions and volatility clustering; Extreme value theory; Historical simulation (finance).

Modellers are generally referred to as "quants", i.e. quantitative analysts (or "rocket scientists") and typically have advanced (Ph.D. level) backgrounds in quantitative disciplines such as statistics, physics, engineering, computer science, mathematics or operations research.

Alternatively, or in addition to their quantitative background, they complete a finance masters with a quantitative orientation,Mark S. Joshi, [http://www.markjoshi.com/downloads/advice.pdf On Becoming a Quant] {{Webarchive|url=https://web.archive.org/web/20120114045604/http://www.markjoshi.com/downloads/advice.pdf |date=2012-01-14 }}. such as the Master of Quantitative Finance, or the more specialized Master of Computational Finance or Master of Financial Engineering; the CQF certificate is increasingly common.

Although spreadsheets are widely used here also (almost always requiring extensive VBA);

custom C++, Fortran or Python, or numerical-analysis software such as MATLAB, are often preferred, particularly where stability or speed is a concern.

MATLAB is often used at the research or prototyping stage [https://www.mathworks.com/solutions/finance-and-risk-management.html MATLAB for Quantitative Finance and Risk Management], MathWorks because of its intuitive programming, graphical and debugging tools, but C++/Fortran are preferred for conceptually simple but high computational-cost applications where MATLAB is too slow;

Python is increasingly used due to its simplicity, and large standard library / available applications, including QuantLib.

Additionally, for many (of the standard) derivative and portfolio applications, commercial software is available, and the choice as to whether the model is to be developed in-house, or whether existing products are to be deployed, will depend on the problem in question.

See {{slink|Quantitative analysis (finance)|Library quantitative analysis}}.

The complexity of these models may result in incorrect pricing or hedging or both. This Model risk is the subject of ongoing research by finance academics, and is a topic of great, and growing, interest in the risk management arena.Riccardo Rebonato (N.D.). [http://www.quarchome.org/ModelRisk.pdf Theory and Practice of Model Risk Management].

Criticism of the discipline (often preceding the 2008 financial crisis by several years) emphasizes the differences between finance and the mathematical / physical sciences, and stresses the resultant caution to be applied by modelers, and by traders and risk managers using their models. Notable here are Emanuel Derman and Paul Wilmott, authors of the Financial Modelers' Manifesto. Some go further and question whether the mathematical- and statistical modeling techniques usually applied to finance are at all appropriate (see the assumptions made for options and for portfolios).

In fact, these may go so far as to question the "empirical and scientific validity... of modern financial theory".Nassim Taleb (2009).[http://www.fooledbyrandomness.com/Triana-fwd.pdf "History Written By The Losers"], Foreword to Pablo Triana's Lecturing Birds How to Fly {{ISBN|978-0470406755}}

Notable here are Nassim Taleb and Benoit Mandelbrot.{{cite web |url=http://www.fooledbyrandomness.com/fortune.pdf |title=How the Finance Gurus Get Risk All Wrong |access-date=2010-06-15 |url-status=dead |archive-url=https://web.archive.org/web/20101207045925/http://www.fooledbyrandomness.com/fortune.pdf |archive-date=2010-12-07|author = Nassim Taleb and Benoit Mandelbrot}}

See also {{section link|Mathematical finance|Criticism}}, {{section link|Financial economics|Challenges and criticism}} and {{slink|Financial engineering|Criticisms}}.

== Competitive modeling ==

Several financial modeling competitions exist, emphasizing speed and accuracy in modeling. The Microsoft-sponsored ModelOff Financial Modeling World Championships were held annually from 2012 to 2019, with competitions throughout the year and a finals championship in New York or London. After its end in 2020, several other modeling championships have been started, including the Financial Modeling World Cup and Microsoft Excel Collegiate Challenge, also sponsored by Microsoft.{{Cite book|last=Fairhurst|first=Danielle Stein|url=https://books.google.com/books?id=X_9REAAAQBAJ|title=Financial Modeling in Excel for Dummies|publisher=John Wiley & Sons|year=2022 |isbn=978-1-119-84451-8|location=|oclc=1264716849}}

== Philosophy of financial modeling ==

Philosophy of financial modeling is a branch of philosophy concerned with the foundations, methods, and implications of modeling science.

In the philosophy of financial modeling, scholars have more recently begun to question the generally-held assumption that financial modelers seek to represent any "real-world" or actually ongoing investment situation. Instead, it has been suggested that the task of the financial modeler resides in demonstrating the possibility of a transaction in a prospective investment scenario, from a limited base of possibility conditions initially assumed in the model.{{cite journal |last= Mebius |first= A. |year= 2023 |title= On the epistemic contribution of financial models |journal= Journal of Economic Methodology |volume = 30 | issue = 1 |pages= 49–62 |doi= 10.1080/1350178X.2023.2172447|s2cid= 256438018 |doi-access= free }}

See also

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References

{{Reflist}}

Bibliography

{{refbegin|30em}}

General

  • {{cite book | author=Avon, Jack | title=The Financial Modellers VBA Compendium | publisher=Begawans Veranda | location=London | year=2017| isbn=978-0-9956-7254-3}}
  • {{cite book | author=Benninga, Simon | title=Financial Modeling | url=https://archive.org/details/financialmodelin00benn | url-access=registration | publisher=MIT Press | location=Cambridge, MA | year=1997| isbn=0-585-13223-2}}
  • {{cite book | author=Benninga, Simon | title=Principles of Finance with Excel | publisher=Oxford University Press | location=New York | year=2006 | isbn=0-19-530150-1}}
  • {{cite book | last=Fabozzi | author-link=Frank Fabozzi | first=Frank J. | title=Encyclopedia of Financial Models| publisher=Wiley | location=Hoboken, NJ | year=2012| isbn=978-1-118-00673-3}}
  • {{cite book | last=Ho | author-link=Thomas Ho (finance) | first=Thomas|author2=Sang Bin Lee | title=The Oxford Guide to Financial Modeling| publisher=Oxford University Press | location=New York| year=2004 | isbn=978-0-19-516962-1 | author2-link=Sang Bin Lee}}
  • {{cite book | author=Sengupta, Chandan| title=Financial Analysis and Modeling Using Excel and VBA, 2nd Edition | publisher=John Wiley & Sons | location=Hoboken, NJ | year=2009| isbn=9780470275603}}
  • {{cite book | author=Winston, Wayne | title=Microsoft Excel 2013 Data Analysis and Business Modeling | publisher=Microsoft Press | year=2014| isbn=978-0735669130}}
  • {{Cite book

| last = Yip

| first = Henry

| title = Spreadsheet Applications to securities valuation and investment theories

| publisher = John Wiley and Sons Australia Ltd.

| isbn = 0470807962

| date = 2005

}}

Corporate finance

  • {{cite book | author=Avon, Jack. | title=The Handbook of Financial Modeling |edition= 2nd | url=https://link.springer.com/book/10.1007/978-1-4842-6540-6| publisher=Springer | location=New York | year=2021 | doi=10.1007/978-1-4842-6540-6 | isbn=978-1-4842-6540-6 | s2cid=227164870 }}
  • {{cite book | last=Bastick | first=Liam | title=Introduction to Financial Modeling| publisher=Wiley | year=2020| isbn=978-1615470662}}
  • {{cite book | author=Beech, G. and Thayser, D. | title=Valuations, Mergers and Acquisitions | url=https://global.oup.com/academic/product/valuations-mergers-and-acquisitions-9780199052776| publisher=Oxford University Press | location=Oxford | year=2015| isbn=978-0-585-13223-5}}
  • {{cite book | author=Day, Alastair | title=Mastering Financial Modelling in Microsoft Excel | publisher=Pearson Education | location=London | year=2007 | isbn=978-0-273-70806-3}}
  • {{Cite book|author=Fairhurst, Danielle|url=https://books.google.com/books?id=X_9REAAAQBAJ&pg=PA120|title=Financial Modeling in Excel for Dummies|publisher=John Wiley & Sons|year=2022 |isbn=978-1-119-84451-8|location=|page=120|oclc=1264716849}}
  • {{cite book | author=Lynch, Penelope | title=Financial Modelling for Project Finance, 2nd Edition| publisher=Euromoney Trading | year=1997| isbn=9781843745488}}
  • {{cite book | author1= Mayes, Timothy R. |author2=Shank, Todd M. | title=Financial Analysis with Microsoft Excel| publisher=Cengage Learning | location=Boston| year=2014 | isbn= 978-1-285-43227-4|edition=7th}}
  • {{cite book | author=Peter K Nevitt |author2=Frank J. Fabozzi | title=Project Financing| publisher=Euromoney Institutional Investor PLC | year=2000 | isbn=978-1-85564-791-6 }}
  • {{cite book | author=Ongkrutaraksa, Worapot | title=Financial Modeling and Analysis: A Spreadsheet Technique for Financial, Investment, and Risk Management, 2nd Edition | publisher=Pearson Education Australia | location=Frenchs Forest | year=2006 | isbn=0-7339-8474-6}}
  • {{cite book | author= Palepu, Krishna G. |author2=Paul M. Healy | title= Business Analysis and Valuation Using Financial Statements, 5th Edition | publisher=South-Western College Publishing | location=Boston| year=2012| isbn= 978-1111972288}}
  • {{cite book | author=Pignataro, Paul | title=Financial Modeling and Valuation: A Practical Guide to Investment Banking and Private Equity | publisher=Wiley | location=Hoboken, NJ| year=2003 | isbn=978-1118558768}}
  • {{cite book | author=Proctor, Scott | title=Building Financial Models with Microsoft Excel: A Guide for Business Professionals, 2nd Edition | publisher=Wiley | location=Hoboken, NJ | year=2009 | isbn=978-0-470-48174-5}}
  • {{cite book | author= Rees, Michael | title= Financial Modelling in Practice: A Concise Guide for Intermediate and Advanced Level| publisher=Wiley | location=Hoboken, NJ | year=2008| isbn=978-0-470-99744-4}}
  • {{cite book | author= Rees, Michael | title= The Essentials of Financial Modeling in Excel: A Concise Guide to Concepts and Methods| publisher=Wiley | location=Hoboken, NJ | year=2023| isbn=978-1394157785}}
  • {{cite book | author=Soubeiga, Eric | title=Mastering Financial Modeling: A Professional's Guide to Building Financial Models in Excel | publisher=McGraw-Hill | location=New York | year=2013| isbn=978-0071808507}}
  • {{cite book | author=Swan, Jonathan | title=Financial Modelling Special Report | publisher=Institute of Chartered Accountants in England & Wales | location=London | year=2007 }}
  • {{cite book | author=Swan, Jonathan | title=Practical Financial Modelling, 2nd Edition | publisher=CIMA Publishing | location=London | year=2008 | isbn=978-0-7506-8647-1}}
  • {{cite book | author= Tham, Joseph|author2=Ignacio Velez-Pareja| title= Principles of Cash Flow Valuation: An Integrated Market-Based Approach | publisher=Elsevier | location=Amsterdam | year=2004 | isbn= 0-12-686040-8 }}
  • {{cite book | author=Tjia, John | title=Building Financial Models | publisher=McGraw-Hill | location=New York | year=2003 | isbn=0-07-140210-1}}

Quantitative finance

  • {{cite book | author= Hirsa, Ali| title=Computational Methods in Finance| publisher=CRC Press | location= Boca Raton| year=2013| isbn=9781439829578}}
  • {{cite book | author=Brooks, Robert | title=Building Financial Derivatives Applications with C++| publisher=Praeger | location=Westport | year=2000 | isbn=978-1567202878 }}
  • {{cite book | last=Brigo | author-link=Damiano Brigo | first=Damiano |author2=Fabio Mercurio | title=Interest Rate Models - Theory and Practice with Smile, Inflation and Credit| edition= 2nd | publisher= Springer Finance | location=London | year=2006| isbn=978-3-540-22149-4| author2-link=Fabio Mercurio }}
  • {{cite book | last=Clewlow| first=Les|author2=Chris Strickland | title=Implementing Derivative Models| publisher=Wiley | location=New Jersey| year=1998| isbn=0-471-96651-7}}
  • {{cite book | author=Duffy, Daniel | title=Financial Instrument Pricing Using C++ | publisher=Wiley | location=New Jersey| year=2004 | isbn=978-0470855096}}
  • {{cite book | last=Fabozzi | author-link=Frank Fabozzi | first=Frank J. | title=Valuation of fixed income securities and derivatives, 3rd Edition | publisher=Wiley | location=Hoboken, NJ | year=1998| isbn=978-1-883249-25-0}}
  • {{cite book | last=Fabozzi | first=Frank J. |author2=Sergio M. Focardi |author3=Petter N. Kolm | title=Financial Modeling of the Equity Market: From CAPM to Cointegration | publisher=Wiley | location=Hoboken, NJ | year=2004 | isbn=0-471-69900-4}}
  • {{Cite book

| author = Shayne Fletcher

| author2 = Christopher Gardner

| title = Financial Modelling in Python

| publisher = John Wiley and Sons

| isbn = 978-0-470-74789-6

| date = 2010

}}

  • {{cite book | last=Fusai| first=Gianluca |author2=Andrea Roncoroni | title=Implementing Models in Quantitative Finance: Methods and Cases| publisher=Springer Finance | location=London | year=2008| isbn=978-3-540-22348-1}}
  • {{cite book | author= Haug, Espen Gaarder| title=The Complete Guide to Option Pricing Formulas, 2nd edition | publisher=McGraw-Hill | year=2007 | isbn=978-0071389976}}
  • {{cite book | title = Interest Rate Modelling in the Multi-Curve Framework | author =M. Henrard | year = 2014 | isbn =978-1137374653 | publisher = Springer}}
  • {{cite book | author=Hilpisch, Yves| title=Derivatives Analytics with Python: Data Analysis, Models, Simulation, Calibration and Hedging| publisher=Wiley | location=New Jersey | year=2015 | isbn=978-1-119-03799-6 }}
  • {{cite book | last=Jackson | first=Mary |author2=Mike Staunton| title=Advanced modelling in finance using Excel and VBA| publisher=Wiley | location=New Jersey| year=2001 | isbn=0-471-49922-6}}
  • {{cite book | last=Jondeau | first=Eric |author2=Ser-Huang Poon |author3=Michael Rockinger | title=Financial Modeling Under Non-Gaussian Distributions | publisher=Springer | location=London | year=2007 | isbn=978-1849965996 }}
  • {{cite book | author1=Joerg Kienitz|author2=Daniel Wetterau| title= Financial Modelling: Theory, Implementation and Practice with MATLAB Source| publisher=Wiley | location=Hoboken, NJ | year=2012 | isbn=978-0470744895}}
  • {{cite book | last= Kwok | first= Yue-Kuen | title= Mathematical Models of Financial Derivatives, 2nd edition | publisher= Springer Finance | location= London | year=2008| isbn= 978-3540422884}}
  • {{cite book | author=Levy, George | title=Computational Finance: Numerical Methods for Pricing Financial Instruments| publisher=Butterworth-Heinemann | year=2004 | isbn=978-0750657228 }}
  • {{cite book | author=London, Justin | title=Modeling Derivatives in C++ | publisher=Wiley | location=New Jersey| year=2004 | isbn=978-0471654643}}
  • {{cite book |author1=Löeffler, G |author2=Posch, P. | title= Credit Risk Modeling using Excel and VBA| publisher=Wiley | location= Hoboken, NJ | year=2011 | isbn= 978-0470660928 }}
  • {{cite book | last= Rouah | first= Fabrice Douglas |author2=Gregory Vainberg | title= Option Pricing Models and Volatility Using Excel-VBA | publisher=Wiley | location=New Jersey| year=2007| isbn= 978-0471794646 }}
  • {{cite book | author= Antoine Savine and Jesper Andreasen | title=Modern Computational Finance: Scripting for Derivatives and xVA| publisher=Wiley| year=2018| isbn=978-1119540786}}
  • {{cite book | author= Alexander Sokol | title=Long-Term Portfolio Simulation - For XVA, Limits, Liquidity and Regulatory Capital| publisher=Risk Books| year=2014| isbn=978-1782720959}}
  • {{Cite book |last=Charles Tapiero |title=Risk and Financial Management: Mathematical and Computational Methods |publisher=John Wiley & Son |year=2004 |isbn=0-470-84908-8}}
  • {{cite book | author1= Humphrey Tung |author2= Donny Lai| author3=Michael Wong|author4 = Stephen Ng | title= Professional Financial Computing Using Excel and VBA | publisher= John Wiley & Sons| year=2010| isbn= 9780470824399 }}

{{refend}}

{{Corporate finance and investment banking}}

Category:Financial models

Category:Actuarial science

Category:Mathematical finance

Category:Corporate finance

Category:Computational fields of study