**Note**

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.

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.

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.

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.

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.

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." https://www.risk.net/risk-magazine, 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.

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 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 `https://www.stanford.edu/~wfsharpe/mia/mia.htm`

.

[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.

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.

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.

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.

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.

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.