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Return and Risk

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Finance with Monte Carlo

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Abstract

This chapter is about the fundamentals of investment growth. It introduces important calculations with interest rates, returns, and discounting. These ideas will be needed in later chapters. It is also about investment risk, how it can be measured and how it can be minimized in the formation and maintenance of an investment portfolio. This is possible through one of the great financial breakthroughs, the mean-variance theory and CAPM, the capital asset and pricing model, due to Markowitz and his followers. In the 50 years since its introduction and subsequent development shortcomings of the theory have surfaced and improvements offered. Yet it is a starting point for these advanced theories and basic to a financial course of study.

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Notes

  1. 1.

    GM was declared bankrupt in 2009.

  2. 2.

    This is so for US government bonds. A financial crisis was precipitated in 1998 when the Russian government defaulted on its debt.

  3. 3.

    Some authors define the rate of return as \({\Delta P \over P}\), (CZ03; Lun98); others as we have done here, (Sch03). As in most science and engineering applications, we prefer to reserve the term rate for changes per unit time.

  4. 4.

    The exponent \(3.25186\ldots = 3 +\log (1 + 0.02/4)/\log (1 + 0.02)\) is required.

  5. 5.

    Because the log function is concave down, for positive arguments, \({1 \over 2}(\log x_{1} +\log x_{2}) <\log ({1 \over 2}(x_{1} + x_{2}))\) unless x 1 = x 2. So \({((1+x_{1})(1+x_{2}))}^{1/2}-1 <\exp ({1 \over 2}(\log (1+x_{1})+\log (1+x_{2})))-1 <\exp (\log ({1 \over 2}((1+x_{1})+(1+x_{2}))))-1 ={ 1 \over 2}(x_{1}+x_{2})\); same argument for n terms.

  6. 6.

    Search “NYSE Companies Delisted for Noncompliance” for this lengthy reference. See also www.moneycontrol.com/stocks/marketinfo/delisting/index.php

  7. 7.

    The ex-dividend day and afterward is when a stock purchase does not qualify for the current dividend. Buying prior to ex-dividend day is required.

  8. 8.

    The calculated market path is a possible realization of a geometric random walk.

  9. 9.

    By its definition, − 1 ≤ ρ XY  ≤ 1. This follows from the well-known Cauchy-Schwarz inequality as indicated by the following. If x i and y i for \(i = 1,\ldots,n\) are empirical values of X and Y, then statistically

    $$\displaystyle\begin{array}{rcl} {\text{covar}}(X,Y )& =&{ 1 \over n}\sum _{i}(x_{i} -\mu _{X})(y_{i} -\mu _{Y }) {}\\ & \leq &\sqrt{{ 1 \over n}\sum _{i}{(x_{i} -\mu _{X})}^{2}}\sqrt{{ 1 \over n}\sum _{i}{(y_{i} -\mu _{Y })}^{2}} = \sqrt{\mbox{ var} _{X}}\sqrt{\mbox{ var} _{Y}}. {}\\ \end{array}$$
  10. 10.

    It is also positive semi-definite but that is not needed for a Cholesky decomposition.

  11. 11.

    An arbitrarily constructed real symmetric matrix is not necessarily positive-semi-definite. A little algebra shows that

    $$\displaystyle{h_{33}^{2} ={ \sigma _{3}^{2} \over (1 -\rho _{12}^{2})}\left (1 -\rho _{12}^{2} -\rho _{ 13}^{2} -\rho _{ 23}^{2} +\rho _{ 12}\rho _{13}\rho _{23}\right ).}$$

    This can be negative for the choices \(\rho _{12} =\rho _{13} = -\rho _{23} ={ 3 \over 4}\). Of course if 1 and 2 are highly correlated then 1 and 3 can’t be highly correlated while 2 and 3 highly uncorrelated.

  12. 12.

    A sorting subroutine is given in Appendix E.

  13. 13.

    An extension formula for a portfolio consisting of a single GBM constituent can be derived from (2.40). But the day-to-day value of a portfolio consisting of several GBM constituents does not itself follow a GBM. Thus an accurate n-day VaR for such a portfolio requires a direct n-day simulation as above.

  14. 14.

    We use 360 days here for illustrative and comparison purposes. In actual practice 252 “trading day” years is more likely to be used by company management. Further, the international Basel regulations specify the following VaR parameters: 10 day horizon, 99 % confidence level, and at least 1 year of historical data.

  15. 15.

    Throughout this section we will measure returns in percent.

  16. 16.

    In this section and the next we are, more exactly, plotting the rate of return versus risk. Since the time period is fixed, the two only differ by a constant factor. In the next section we will add the risk-free rate to the diagram.

  17. 17.

    \(\mathbf{x}\cdot \mathbf{y} = \left (\begin{array}{cccccccccccccc} x_{1} & x_{2} & \ldots & x_{n} \end{array} \right )\cdot \left (\begin{array}{cccccccccccccc} y_{1} & y_{2} & \ldots & y_{n} \end{array} \right ) =\sum _{ 1}^{n}x_{i}y_{i} ={ \mathbf{x}}^{T}\mathbf{y}\).

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Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA

    Ronald W. Shonkwiler

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Shonkwiler, R.W. (2013). Return and Risk. In: Finance with Monte Carlo. Springer Undergraduate Texts in Mathematics and Technology. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8511-7_2

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