Main Content


Bond Pricing and Yields
Term Structure of Interest Rates
Derivatives Pricing and Yields
Portfolio Analysis
Investment Performance Metrics
Financial Statistics
Standard References
Credit Risk Analysis
Portfolio Optimization
Stochastic Differential Equations
Life Tables


For the well-known algorithms and formulas used in Financial Toolbox™ software (such as how to compute a loan payment given principal, interest rate, and length of the loan), no references are given here. The references here pertain to less common formulas.

Bond Pricing and Yields

The pricing and yield formulas for fixed-income securities come from:

[1] Golub, B.W. and L.M. Tilman. Risk Management: Approaches for Fixed Income Markets. Wiley, 2000.

[2] Martellini, L., P. Priaulet, and S. Priaulet. Fixed Income Securities. Wiley, 2003.

[3] Mayle, Jan. Standard Securities Calculation Methods. New York: Securities Industry Association, Inc. Vol. 1, 3rd ed., 1993, ISBN 1-882936-01-9. Vol. 2, 1994, ISBN 1-882936-02-7.

[4] Tuckman, B. Fixed Income Securities: Tools for Today's Markets. Wiley, 2002.

In many cases these formulas compute the price of a security given yield, dates, rates, and other data. These formulas are nonlinear, however; so when solving for an independent variable within a formula, Financial Toolbox software uses Newton's method. See any elementary numerical methods textbook for the mathematics underlying Newton's method.

Term Structure of Interest Rates

The formulas and methodology for term structure functions come from:

[5] Fabozzi, Frank J. “The Structure of Interest Rates.” Ch. 6 in Fabozzi, Frank J. and T. Dessa Fabozzi, eds. The Handbook of Fixed Income Securities. 4th ed. New York, Irwin Professional Publishing, 1995, ISBN 0-7863-0001-9.

[6] McEnally, Richard W. and James V. Jordan. “The Term Structure of Interest Rates.” Ch. 37 in Fabozzi and Fabozzi, ibid.

[7] Das, Satyajit. “Calculating Zero Coupon Rates.” Swap and Derivative Financing. Appendix to Ch. 8, pp. 219–225, New York, Irwin Professional Publishing., 1994, ISBN 1-55738-542-4.

Derivatives Pricing and Yields

The pricing and yield formulas for derivative securities come from:

[8] Chriss, Neil A. Black-Scholes and Beyond: Option Pricing Models. Chicago, Irwin Professional Publishing, 1997, ISBN 0-7863-1025-1.

[9] Cox, J., S. Ross, and M. Rubenstein. “Option Pricing: A Simplified Approach.” Journal of Financial Economics. Vol. 7, Sept. 1979, pp. 229–263.

[10] Hull, John C. Options, Futures, and Other Derivatives. 5th edition, Prentice Hall, 2003, ISBN 0-13-009056-5.

Portfolio Analysis

The Markowitz model is used for portfolio analysis computations. For a discussion of this model see Chapter 7 of:

[11] Bodie, Zvi, Alex Kane, and Alan J. Marcus. Investments. 2nd. Edition. Burr Ridge, IL, Irwin Professional Publishing, 1993, ISBN 0-256-08342-8.

Investment Performance Metrics

The risk and ratio formulas for investment performance metrics come from:

[12] Daniel Bernoulli. "Exposition of a New Theory on the Measurement of Risk." Econometrica. Vol. 22, No 1, January 1954, pp. 23–36 (English translation of "Specimen Theoriae Novae de Mensura Sortis." Commentarii Academiae Scientiarum Imperialis Petropolitanae. Tomus V, 1738, pp. 175–192).

[13] Martin Eling and Frank Schuhmacher. Does the Choice of Performance Measure Influence the Evaluation of Hedge Funds? Working Paper, November 2005.

[14] John Lintner. "The Valuation of Risk Assets and the Selection of Risky Investments in Stocks Portfolios and Capital Budgets." Review of Economics and Statistics. Vol. 47, No. 1, February 1965, pp. 13–37.

[15] Malik Magdon-Ismail, Amir F. Atiya, Amrit Pratap, and Yaser S. Abu-Mostafa. "On the Maximum Drawdown of a Brownian Motion." Journal of Applied Probability. Volume 41, Number 1, March 2004, pp. 147–161.

[16] Malik Magdon-Ismail and Amir Atiya. "Maximum Drawdown.", October 2004.

[17] Harry Markowitz. "Portfolio Selection." Journal of Finance. Vol. 7, No. 1, March 1952, pp. 77–91.

[18] Harry Markowitz. Portfolio Selection: Efficient Diversification of Investments. John Wiley & Sons, 1959.

[19] Jan Mossin. "Equilibrium in a Capital Asset Market." Econometrica. Vol. 34, No. 4, October 1966, pp. 768–783.

[20] Christian S. Pedersen and Ted Rudholm-Alfvin. "Selecting a Risk-Adjusted Shareholder Performance Measure." Journal of Asset Management. Vol. 4, No. 3, 2003, pp. 152–172.

[21] William F. Sharpe. "Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk." Journal of Finance. Vol. 19, No. 3, September 1964, pp. 425–442.

[22] Katerina Simons. "Risk-Adjusted Performance of Mutual Funds." New England Economic Review. September/October 1998, pp. 34–48.

Financial Statistics

The discussion of computing statistical values for portfolios containing missing data elements derives from the following references:

[23] Little, Roderick J.A. and Donald B. Rubin. Statistical Analysis with Missing Data. 2nd Edition. John Wiley & Sons, Inc., 2002.

[24] Meng, Xiao-Li, and Donald B. Rubin. “Maximum Likelihood Estimation via the ECM Algorithm.” Biometrika. Vol. 80, No. 2, 1993, pp. 267–278.

[25] Sexton, Joe and Anders Rygh Swensen. “ECM Algorithms That Converge at the Rate of EM.” Biometrika. Vol. 87, No. 3, 2000, pp. 651–662.

[26] Dempster, A.P., N.M. Laird, and Donald B. Rubin. “Maximum Likelihood from Incomplete Data via the EM Algorithm.” Journal of the Royal Statistical Society. Series B, Vol. 39, No. 1, 1977, pp. 1–37.

Standard References

Standard references include:

[27] Addendum to Securities Industry Association, Standard Securities Calculation Methods: Fixed Income Securities Formulas for Analytic Measures. Vol. 2, Spring 1995. This addendum explains and clarifies the end-of-month rule.

[28] Brealey, Richard A. and Stewart C. Myers. Principles of Corporate Finance. New York, McGraw-Hill. 4th ed., 1991, ISBN 0-07-007405-4.

[29] Daigler, Robert T. Advanced Options Trading. Chicago, Probus Publishing Co., 1994, ISBN 1-55738-552-1.

[30] A Dictionary of Finance. Oxford, Oxford University Press., 1993, ISBN 0-19-285279-5.

[31] Fabozzi, Frank J. and T. Dessa Fabozzi, eds. The Handbook of Fixed-Income Securities. 4th Edition. Burr Ridge, IL, Irwin, 1995, ISBN 0-7863-0001-9.

[32] Fitch, Thomas P. Dictionary of Banking Terms. 2nd Edition. Hauppauge, NY, Barron's. 1993, ISBN 0-8120-1530-4.

[33] Hill, Richard O., Jr. Elementary Linear Algebra. Orlando, FL, Academic Press. 1986, ISBN 0-12-348460-X.

[34] Luenberger, David G. Investment Science. Oxford University Press, 1998. ISBN 0195108094.

[35] Marshall, John F. and Vipul K. Bansal. Financial Engineering: A Complete Guide to Financial Innovation. New York, New York Institute of Finance. 1992, ISBN 0-13-312588-2.

[36] Sharpe, William F. Macro-Investment Analysis. An “electronic work-in-progress” published on the World Wide Web, 1995, at

[37] Sharpe, William F. and Gordon J. Alexander. Investments. Englewood Cliffs, NJ: Prentice-Hall. 4th ed., 1990, ISBN 0-13-504382-4.

[38] Stigum, Marcia, with Franklin Robinson. Money Market and Bond Calculations. Richard D. Irwin., 1996, ISBN 1-55623-476-7.

Credit Risk Analysis

The credit rating and estimation transition probabilities come from:

[39] Altman, E. "Financial Ratios, Discriminant Analysis and the Prediction of Corporate Bankruptcy." Journal of Finance. Vol. 23, No. 4, (Sept., 1968), pp. 589–609.

[40] Basel Committee on Banking Supervision, International Convergence of Capital Measurement and Capital Standards: A Revised Framework, Bank for International Settlements (BIS). comprehensive version, June 2006.

[41] Hanson, S. and T. Schuermann. "Confidence Intervals for Probabilities of Default.” Journal of Banking & Finance. Vol. 30(8), Elsevier, August 2006, pp. 2281–2301.

[42] Jafry, Y. and T. Schuermann. "Measurement, Estimation and Comparison of Credit Migration Matrices." Journal of Banking & Finance. Vol. 28(11), Elsevier, November 2004, pp. 2603–2639.

[43] Löffler, G. and P. N. Posch. Credit Risk Modeling Using Excel and VBA. West Sussex, England: Wiley Finance, 2007.

[44] Schuermann, T. "Credit Migration Matrices." in E. Melnick and B. Everitt (eds.), Encyclopedia of Quantitative Risk Analysis and Assessment. Wiley, 2008.

Credit Derivatives

Beumee, J., D. Brigo, D. Schiemert, and G. Stoyle. “Charting a Course Through the CDS Big Bang.” Fitch Solutions, Quantitative Research. Global Special Report. April 7, 2009.

Hull, J., and A. White. “Valuing Credit Default Swaps I: No Counterparty Default Risk.” Journal of Derivatives. Vol. 8, pp. 29–40.

O'Kane, D. and S. Turnbull. “Valuation of Credit Default Swaps.” Lehman Brothers, Fixed Income Quantitative Credit Research. April, 2003.

O'Kane, D. Modelling Single-name and Multi-name Credit Derivatives. Wiley Finance, 2008, pp. 156–169.

Portfolio Optimization

The Markowitz model is used for portfolio optimization computations.

[45] Kelley, J. E. "The Cutting-Plane Method for Solving Convex Programs." Journal of the Society for Industrial and Applied Mathematics. Vol. 8, No. 4, December 1960, pp. 703–712.

[46] Markowitz, H. "Portfolio Selection." Journal of Finance. Vol. 7, No. 1, March 1952, pp. 77–91.

[47] Markowitz, H. M. Portfolio Selection: Efficient Diversification of Investments. John Wiley & Sons, Inc., 1959.

[48] Rockafellar, R. T. and S. Uryasev. "Optimization of Conditional Value-at-Risk." Journal of Risk. Vol. 2, No. 3, Spring 2000, pp. 21–41.

[49] Rockafellar, R. T. and S. Uryasev. "Conditional Value-at-Risk for General Loss Distributions." Journal of Banking and Finance. Vol. 26, 2002, pp. 1443–1471.

[50] Konno, H. and H. Yamazaki. "Mean-Absolute Deviation Portfolio Optimization Model and Its Application to Tokyo Stock Market." Management Science. Vol. 37, No. 5, May 1991, pp. 519–531.

[51] Cornuejols, A. and R. Tütüncü. Optimization Methods in Finance. Cambridge University Press, 2007.

Stochastic Differential Equations

The SDE formulas come from:

[52] Ait-Sahalia, Y. “Testing Continuous-Time Models of the Spot Interest Rate.” The Review of Financial Studies. Spring 1996, Vol. 9, No. 2, pp. 385–426.

[53] Ait-Sahalia, Y. “Transition Densities for Interest Rate and Other Nonlinear Diffusions.” The Journal of Finance. Vol. 54, No. 4, August 1999.

[54] Glasserman, P. Monte Carlo Methods in Financial Engineering. Springer-Verlag, New York, 2004.

[55] Hull, J. C. Options, Futures, and Other Derivatives. 5th edition, Englewood Cliffs, NJ: Prentice Hall, 2002.

[56] Johnson, N. L., S. Kotz, and N. Balakrishnan. Continuous Univariate Distributions. Vol. 2, 2nd ed. New York: John Wiley & Sons, 1995.

[57] Shreve, S. E. Stochastic Calculus for Finance II: Continuous-Time Models. Springer-Verlag, New York, 2004.

Life Tables

The Life Table formulas come from:

[58] Arias, E. “United States Life Tables.” National Vital Statistics Reports, U.S. Department of Health and Human Services. Vol. 62, No. 7, 2009.

[59] Carriere, F. “Parametric Models for Life Tables.” Transactions of the Society of Actuaries. Vol. 44, 1992, pp. 77-99.

[60] Gompertz, B. “On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies.” Philosophical Transactions of the Royal Society. Vol. 115, 1825, pp. 513–582.

[61] Heligman, L. M. A., and J. H. Pollard. “The Age Pattern of Mortality.” Journal of the Institute of Actuaries. Vol. 107, Pt. 1, 1980, pp. 49–80.

[62] Makeham, W. M. “On the Law of Mortality and the Construction of Annuity Tables.” Journal of the Institute of Actuaries. Vol. 8, 1860. pp. 301–310.

[63] Siler, W. “A Competing-Risk Model for Animal Mortality.” Ecology. Vol. 60, pp. 750–757, 1979.

[64] Siler, W. “Parameters of Mortality in Human Populations with Widely Varying Life Spans.” Statistics in Medicine. Vol. 2, 1983, pp. 373–380.