Currently, I´m studying portfolio management and portfolio selection. The founder of the MPT is Harry Markowitz, of course. But reading his famous article from 1952 and his book from 1959 (actually, I have the 2nd edition from 1991 at hand, but that shouldn´t make a difference), I realized that Markowitz never really derived an equation to calculate the efficient frontier.

He does describe what efficient portfolios are and introduces some algorithms to attain efficient sets. But if I´m not mistaken, the first who derived an equation to calculate the efficient frontier was Robert Merton in his paper "An Analytic Derivation of the Efficient Portfolio Frontier" from 1972. Starting with the expected return $\bar{E}$ of the minimum variance portfolio, Merton derives the following equation that yields the expected return of an efficient portfolio as a function of its variance:

$$E=\bar{E}+\frac{1}{C} * \sqrt{DC(\sigma^2 - \bar{\sigma}^2)}$$

My question is: Why didn´t Markowitz derive such an equation? I guess he could have done it, he is a genius in his field of research and the founder of this theory. Moreover, it seems much easier to calculate the efficient frontier using an equation rather than algorithmic approaches, so I suppose there must have been an early interest in finding such an equation like Merton did.

It would be very nice if someone can clarify this. Maybe I´m missing out on a pivotal element.

Thanks a lot in advance.

  • $\begingroup$ I've taken the liberty to add a link to Merton's paper. $\endgroup$
    – olaker
    Dec 10, 2016 at 12:10
  • $\begingroup$ Sure, should have done this myself. :D Thank you. $\endgroup$ Dec 10, 2016 at 12:26
  • $\begingroup$ How does Merton's closed-form solution for the unconstrained efficient frontier above correspond to Niedermayer and Niedermayer's closed-form algorithm for solving the short-sale constrained efficient frontier that exactly solves for turning points (corner portfolios)? $\endgroup$
    – develarist
    Jun 28, 2019 at 14:47

1 Answer 1


It is surprising. What I think is: Markowitz became interested in the general problem when there are constraints (including inequality constraints) on the portfolio weights (in addition to the standard $\sum w_i = 1$ constraint). Once he devised a computer algorithm [the Critical Line Method] for solving this problem (he was a math programming whiz) he seems to have stopped there. Perhaps he did not realize the importance of an analytical formula in the simple case with only a fully invested constraint. Merton had a better math background I believe and saw this. Merton has also said [personal communication] that he does not like geometric arguments (like Markowitz used throughout his book) and preferred an analytic or algebraic method of deriving results from starting assumptions. He started out trying to derive Portfolio Theory in this manner, and it took him farther than Markowitz.

  • 2
    $\begingroup$ It was interesting to see that Markowitz did not derive such an equation and that it took 13 years for someone else to address this issue. But your explanation is reasonable. Harry Markowitz and Robert Merton might have different perspectives on the same problem, depending on their previous research interests and approaches. It is really exciting to see how different approaches to the same problem can solve it. Thanks a lot for your answer. $\endgroup$ Dec 10, 2016 at 18:13

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