Financial Modeling


Feasible Portfolio—includes the entire range of reasonably attainable portfolios within an envelope data set—e.g., all efficient and inefficient points along and beneath envelope curve. A feasible portfolio includes the following characteristics:

A feasible portfolio includes the following characteristics:

  • Any portfolio whose proportions sum to 1;
  • In other words, it encompasses the entire area “inside and to the right of envelope”; [i]
  • A feasible portfolio never transcends the curve threshold but may incorporate efficient and/or inefficient points within its range including:
  • Points along curve; and/or
  • Points off curve to the right of envelope between efficient/inefficient points (never left of curve or below inefficient points along envelope)
  • Any region above, below, and/or left of the curve constitutes an infeasible region, and not reasonably attainable for given data set.

Financial Modeling - safeEnvelope Portfolio—portfolio of risky assets (security investments) that projects the lowest variance for all portfolios with the same expected return, mathematically instantiated by a curve. The curve includes a range of feasible efficient and/or inefficient points, with risk (demarcated by x-axis), correlating to some assumed reward (y-axis).

Efficient Portfolios—points (representing the portfolio or collection of individual investments/risky assets), along the envelope curve which maximizes return (reward, y-axis) for a given risk (x-axis) assumed. The efficient portfolio assumes a point of tangency where some imaginary line may form connecting c (hypothetical y-intercept, risk-free point), with some crossing point along the efficient frontier. See Table 1-2. This point of tangency illustrated by an imaginary line crossing the efficient frontier demonstrates efficiency maximum reward for minimum risk (line intersecting risk-free, y-intercept).

  • In other words, this envelope curve of investments maximizes expected return for “all portfolios sharing the same variance.” [ii]
  • The variance measures each number’s distance from the mean in a number distribution data set. A higher variance reflects greater volatility, which generally assumes greater risk.
  • Therefore, since the efficient portfolio projects higher expected returns for all portfolios with same variance—an efficient portfolio “efficiently” correlates with a higher reward at proportionately lower risk.

TABLE 1-2 [iii]

Feasible Set v. Efficient Set of Portfolios—A feasible set represents the set of portfolio means and standard deviations generated by feasible portfolios. A feasible set of portfolios includes both efficient and inefficient portfolios within the entire data set provided. However, an efficient set of portfolios only includes those efficient points—exclusively limited to those points which maximize return for the assumed portfolio variance provided.

Therefore, an efficient set of portfolios differs from the feasible set in that the feasible set includes both efficient and inefficient portfolios within envelope range, unlike the efficient set, which limits only to those efficient portfolios—points maximizing return.

The efficient portfolio, AKA efficient frontier, includes only those efficient points along the envelope curve. Additionally, the feasible set, unlike an efficient set, may include not only inefficient portfolios but points not necessarily along the frontier envelope curve. See Tables 1-2 & 1-3.

TABLE 1-2 [iv]                                                                        

TABLE 1-3 [v]                                

(b) Market portfolio—portfolio comprising all risky assets in the entire economy, with each asset proportioned to assigned weighted values. [vi]

Mean-variance efficientportfolio that maximizes expected return for a given level of risk based on mean portfolio construction—as graphically represented by the “efficient portfolio frontier.” The name “mean-variance efficient” aptly characterizes an efficient portfolio because it optimizes efficiency, presumably producing highest rewards at proportionately lower risk, thereby offsetting market volatility. In other words, the mean variance efficient correlates to a point of highest efficiency—achieving greatest returns per risk assumed, consequently mitigating investment price fluctuations.

[i] See Simon Benninga, “Financial Modeling, 4th Ed.” MIT Press, 2014, p. 222.

[ii] See Benninga Id. at 210.

[iii] Harvey, Campbell, Goetzmann, William N., “Introduction to Investment Theory,” Yale School of Management, pp.7,

[iv] See Id. at 4.

[v] See Id. at 6.

[vi] See Benninga Id. at 226.

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