Abstract
Portfolio management is about maximizing the return of an asset portfolio by minimizing (or keeping control of) risk at the same time.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
From now on, we will suppress the argument S(t) to simplify the notation. It is understood that all quantities involving instrument values (and their derivatives) are all functions of S(t) and all quantities are also functions of t.
- 2.
In the following, we make frequent use of vector notation, i.e.,
$$\displaystyle \begin{aligned} \mathbf{w}=\left( \begin{array}[c]{c} w_{1}\\ \vdots\\ w_{M} \end{array} \right) ,\,\mathbf{R}=\left( \begin{array}[c]{c} R_{1}\\ \vdots\\ R_{M} \end{array} \right) ,\,\mathbf{1}=\left( \begin{array}[c]{c} 1\\ \vdots\\ 1 \end{array} \right) , \text{etc.} \end{aligned}$$ - 3.
Recall that all returns are in linear compounding (c.f. comment after Eq. 26.1).
- 4.
The last step follows directly from Eq. 26.6 which in terms of logarithmic (or relative) changes reads
$$\displaystyle \begin{aligned} \delta\ln V_{k}=r_{k}\delta t\approx\sum_{i}^{n}\Omega_{i}^{k}r(S_{i})\delta t=\sum_{i}^{n}\Omega_{i}^{k}\delta\ln S_{i}\;, \end{aligned}$$thus
$$\displaystyle \begin{aligned} \operatorname*{cov}\left[ \delta\ln V_{k},\delta\ln V_{l}\right] & =\operatorname*{cov}\left[ \sum_{i}^{n}\Omega_{i}^{k}\delta\ln S_{i},\sum _{j}^{n}\Omega_{j}^{l}\delta\ln S_{j}\right] \\ & =\sum_{i,j}^{n}\Omega_{i}^{k}\Omega_{j}^{l}\underset{\delta\Sigma_{ij} }{\underbrace{\operatorname*{cov}\left[ \delta\ln S_{i},\delta\ln S_{j}\right]} }=\delta C_{kl}\;. \end{aligned} $$ - 5.
Here δ ki again denotes the Kronecker delta.
- 6.
This approximation is derived in Eq. 30.9, for instance.
- 7.
A introduction to the method of Lagrange multipliers for solving optimization problems subject to constraints can be found in [34], for instance.
- 8.
Since the Lagrange function differs from the function which we actually wish to optimize by only a zero, the extremum of the Lagrange function coincides with the desired extremum. However, this is only the case if the difference between the two functions is indeed zero, i.e., only if the constraint is satisfied. This is the essential idea of the method of Lagrange multipliers.
- 9.
If C were negative definite the extremum of the Lagrange Function would correspond to a maximum risk portfolio.
- 10.
A short and simple overview of linear algebra can be found in [78], for example.
- 11.
Which it does, if C is positive definite.
- 12.
Therefore, we also have
$$\displaystyle \begin{aligned} {\mathbf{1}}^{T}{\mathbf{C}}^{-1}\mathbf{R}=\sum_{r,s=1}^{M}R_{s} \left(C^{-1}\right)_{rs}=\sum_{r,s=1}^{M}R_{s}\left(C^-1\right)_{sr} ={\mathbf{R}}^{T}{\mathbf{C}}^{-1}\mathbf{1}\;. \end{aligned}$$ - 13.
The set of available (risky) assets is sometimes called investment universe.
- 14.
Quantities belonging to the optimal portfolio carry a subscript m in the following.
- 15.
Remember that we already have committed ourselves to the plus sign in Eq. 26.40, i.e. to the upper branch of the efficient frontier.
- 16.
The algebra is:
$$\displaystyle \begin{aligned} \mathbf{w} & =\sigma_{\min}^{2}{\mathbf{C}}^{-1}\mathbf{1+}\frac{\sigma _{V}^{2}-\sigma_{\min}^{2}}{R-R_{\min}}{\mathbf{C}}^{-1}\left(\mathbf{R-1} R_{\min}\right) \\ & =\sigma_{\min}^{2}{\mathbf{C}}^{-1}\mathbf{1}+\frac{\sigma_{V}^{2} -\sigma_{\min}^{2}}{\sigma_{\text{min}}^{2}\frac{{\mathbf{R}}^{T}\mathbf{C} ^{-1}\mathbf{R}-R_{\text{min}}^{2}\mathbf{/}\sigma_{\text{min}}^{2}}{R_{\min} -r_{f}}}{\mathbf{C}}^{-1}\left(\mathbf{R-1}R_{\min}\right) \\ & =\sigma_{\min}^{2}{\mathbf{C}}^{-1}\mathbf{1}+\frac{\sigma_{\text{min}} ^{4}~\frac{{\mathbf{R}}^{T}{\mathbf{C}}^{-1}\mathbf{R-}R_{\text{min}} ^{2}\mathbf{/}\sigma_{\text{min}}^{2}}{\left(R_{\text{min}}-r_{f}\right) ^{2}}}{\sigma_{\text{min}}^{2}\frac{{\mathbf{R}}^{T}{\mathbf{C}}^{-1} \mathbf{R}-R_{\text{min}}^{2}\mathbf{/}\sigma_{\text{min}}^{2}}{R_{\min} -r_{f}}}{\mathbf{C}}^{-1}\left(\mathbf{R-1}R_{\min}\right) \\ & =\sigma_{\min}^{2}{\mathbf{C}}^{-1}\mathbf{1}+\frac{\sigma_{\min}^{2} }{R_{\min}-r_{f}}{\mathbf{C}}^{-1}\left(\mathbf{R-1}R_{\min}\right)\;. \end{aligned} $$ - 17.
We will continue to use also the expression “portfolio” and mean by it the total investment consisting of the risky part and the money market account.
References
M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1972)
C. Alexander (ed.), The Handbook of Risk Management and Analysis (Wiley, Chichester, 1996)
L.B.G. Andersen, R. Brotherton-Ratcliffe, The equity option volatility smile: an implicit finite-difference approach. J. Comput. Finance 1(2), 5–37 (1998)
L.B.G. Andersen, V.V. Piterbarg, Interest Rate Modeling (Atlantic Financial Press, New York, London, 2010)
N. Anderson, F. Breedon, M. Deacon, et al., Estimating and Interpreting the Yield Curve (Wiley, Chichester, 1996)
D.F. Babbel, C.B. Merrill, Valuation of Interest-Sensitive Instruments (Society of Actuaries, Schaumburg, IL, 1996)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2019 The Author(s)
About this chapter
Cite this chapter
Deutsch, HP., Beinker, M.W. (2019). Classical Portfolio Management. In: Derivatives and Internal Models. Finance and Capital Markets Series. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-030-22899-6_26
Download citation
DOI: https://doi.org/10.1007/978-3-030-22899-6_26
Published:
Publisher Name: Palgrave Macmillan, Cham
Print ISBN: 978-3-030-22898-9
Online ISBN: 978-3-030-22899-6
eBook Packages: Economics and FinanceEconomics and Finance (R0)