If there is a two security portfolio, its variance can be defined as:
where r D and r E are the rate of return for security D and security E respectively; w D and w E are weight associated with security D and E respectively; σ 2 D and σ 2 E are variance of security D and E respectively; and Cov(r D , r E ) is the covariance between r D and r E .
The problem is choosing optimal w D to minimize the portfolio variance, σ 2 p
We can solve the minimization problem by differentiating the σ 2 p with respect to w D and setting the derivative equal to 0 i.e., we want to solve
Since, WD + WE = 1 or, WE = 1 − WD therefore, the variance, σ 2 p , can be rewritten as
Now, the first order conditions of equation (E3) can be written as
Rearranging the above equation,
Finally, we have
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© 2006 Springer Science+Business Media, Inc.
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Lee, CF., Lee, A.C. (2006). Derivation of minimum-variance portfolio. In: Lee, CF., Lee, A.C. (eds) Encyclopedia of Finance. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-26336-6_81
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DOI: https://doi.org/10.1007/978-0-387-26336-6_81
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