2

My course notes define

Suppose now that there are many different investments $A_1,\dots,A_n$ available. We can invest our one unit of currency by investing $t_i$ in $A_i$ for each $1 \leq i \leq n$ as long as $\sum_{i=1}^n t_i=1.$ What are all possible pairs $(\sigma, r)$ corresponding to these portfolios? This set of points in the $\sigma-r$ plane is called the feasible set.

and

An efficient portfolio is a feasible portfolio that provides the greatest expected return for a given level of risk, or equivalently, the lowest risk for a given expected return. (This is also called an optimal portfolio.) The efficient frontier is the set of all efficient portfolios.

and

We now add a risk-free asset B. Let $r_B$ be its (expected) return. [...] Consider the set $S$ consisting of all the slopes $s$ of lines $\ell_s$ in the $\sigma-r$ plane which pass through the point $(0, r_B)$ and intersect the feasible set. Let $m$ be the supremum of $S$. Consider now the line $\ell_m$ which is above all the others: The line $\ell_m$ will either be tangent to the efficient frontier or asymptotic to it. This point of tangency is called the market portfolio and we shall denote the corresponding portfolio with $M$. The new efficient frontier, $\ell_m$ is called the capital market line.

enter image description here

How can the Capital Market Line (CML) be an efficient frontier (set of all efficient portfolios) when efficient portfolios must be feasible and (with the exception of the market portfolio) the CML is outside the feasible set?

mjc
  • 2,249
  • 1
    when the risk free asset is present, the combinations of assets are more than the case in which only risky assets are present. The envelope only refers to the latter case. Also this question is offtopic here. – Snoop Nov 05 '23 at 10:23
  • Echoing @Snoop , I think you should look at "The efficient frontier is the set of all efficient portfolios" & at "The new efficient frontier is called the capital market line" : In other words , the Shaded Area was before adding B & that Area will change when adding B. I am not very sure , but Whole of CML might not be frontier , it might be that Part of CML is the frontier ! – Prem Nov 05 '23 at 10:56
  • When adding risk-free asset , feasible area will enlarge. When adding risky asset , feasible area will shrink. In Current Case , the enlargement is enough to contain the CML. – Prem Nov 05 '23 at 11:00
  • @Snoop re 'off-topic', I posted a couple of questions on the economics stack without getting much response, before discovering the quantitative finance stack, which I now think is the right place for this sort of question. – mjc Nov 15 '23 at 11:17

0 Answers0