## Bibliography

**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.

### 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." 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.

### 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 `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.

### 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.