The set of admissible portfolios Indifference curves The set of efficient portfolios
Efficient frontier
Ludˇek Benada
Department of Finance, office - 402
e-mail: benada@econ.muni.cz
The set of admissible portfolios Indifference curves The set of efficient portfolios
Content
1 The set of admissible portfolios
2 Indifference curves
3 The set of efficient portfolios
The set of admissible portfolios Indifference curves The set of efficient portfolios
Content
1 The set of admissible portfolios
2 Indifference curves
3 The set of efficient portfolios
The set of admissible portfolios Indifference curves The set of efficient portfolios
Shape of the set of admissible portfolios
The model of H. Markowitz;
Wealth is defined
The set of admissible portfolios Indifference curves The set of efficient portfolios
Shape of the set of admissible portfolios
The model of H. Markowitz;
Wealth is defined
Portfolio holding period
The set of admissible portfolios Indifference curves The set of efficient portfolios
Shape of the set of admissible portfolios
The model of H. Markowitz;
Wealth is defined
Portfolio holding period
Problem of portfolio selection
The set of admissible portfolios Indifference curves The set of efficient portfolios
Shape of the set of admissible portfolios
The model of H. Markowitz;
Wealth is defined
Portfolio holding period
Problem of portfolio selection
Two extremes can occur when creating an investment portfolio;
The set of admissible portfolios Indifference curves The set of efficient portfolios
Shape of the set of admissible portfolios
The model of H. Markowitz;
Wealth is defined
Portfolio holding period
Problem of portfolio selection
Two extremes can occur when creating an investment portfolio;
Wealth (assets) cannot be divided
The set of admissible portfolios Indifference curves The set of efficient portfolios
Shape of the set of admissible portfolios
The model of H. Markowitz;
Wealth is defined
Portfolio holding period
Problem of portfolio selection
Two extremes can occur when creating an investment portfolio;
Wealth (assets) cannot be divided
Assets are divisible without limits
The set of admissible portfolios Indifference curves The set of efficient portfolios
Indivisible assets
The set of admissible portfolios will consist only of the finite
set
The set of admissible portfolios Indifference curves The set of efficient portfolios
Indivisible assets
The set of admissible portfolios will consist only of the finite
set
Variants of possible portfolios are determined/limited by
pseudo-short sell
The set of admissible portfolios Indifference curves The set of efficient portfolios
Two components portfolio - Basic concept
Return of the portfolio:
The set of admissible portfolios Indifference curves The set of efficient portfolios
Two components portfolio - Basic concept
Return of the portfolio:
Rp = r1 ∗ w1 + r2 ∗ w2
Risk of the portfolio:
The set of admissible portfolios Indifference curves The set of efficient portfolios
Two components portfolio - Basic concept
Return of the portfolio:
Rp = r1 ∗ w1 + r2 ∗ w2
Risk of the portfolio:
σp = w2
1 ∗ σ2
1 + w2
2 ∗ σ2
2 + 2 ∗ w1 ∗ w2 ∗ σ1,2
The set of admissible portfolios Indifference curves The set of efficient portfolios
Two components portfolio - Basic concept
Return of the portfolio:
Rp = r1 ∗ w1 + r2 ∗ w2
Risk of the portfolio:
σp = w2
1 ∗ σ2
1 + w2
2 ∗ σ2
2 + 2 ∗ w1 ∗ w2 ∗ σ1,2
where,
σ1,2 = ρ1,2 ∗ σ1 ∗ σ2
The set of admissible portfolios Indifference curves The set of efficient portfolios
Two components portfolio - Basic concept
Return of the portfolio:
Rp = r1 ∗ w1 + r2 ∗ w2
Risk of the portfolio:
σp = w2
1 ∗ σ2
1 + w2
2 ∗ σ2
2 + 2 ∗ w1 ∗ w2 ∗ σ1,2
where,
σ1,2 = ρ1,2 ∗ σ1 ∗ σ2
and,
w1 + w2 = 1
The set of admissible portfolios Indifference curves The set of efficient portfolios
Two components portfolio - Basic concept
Return of the portfolio:
Rp = r1 ∗ w1 + r2 ∗ w2
Risk of the portfolio:
σp = w2
1 ∗ σ2
1 + w2
2 ∗ σ2
2 + 2 ∗ w1 ∗ w2 ∗ σ1,2
where,
σ1,2 = ρ1,2 ∗ σ1 ∗ σ2
and,
w1 + w2 = 1 → w2 = 1 − w1
The set of admissible portfolios Indifference curves The set of efficient portfolios
Two risky assets by ρ1,2 = 1
Basic assumptions:
The set of admissible portfolios Indifference curves The set of efficient portfolios
Two risky assets by ρ1,2 = 1
Basic assumptions:
w1, w2 ≥ 0
The set of admissible portfolios Indifference curves The set of efficient portfolios
Two risky assets by ρ1,2 = 1
Basic assumptions:
w1, w2 ≥ 0
r1 < r2 ∧ σ1 < σ2
Thus,
The set of admissible portfolios Indifference curves The set of efficient portfolios
Two risky assets by ρ1,2 = 1
Basic assumptions:
w1, w2 ≥ 0
r1 < r2 ∧ σ1 < σ2
Thus,
Rp = w1 ∗ r1 + (1 − w1) ∗ r2
The set of admissible portfolios Indifference curves The set of efficient portfolios
Two risky assets by ρ1,2 = 1
Basic assumptions:
w1, w2 ≥ 0
r1 < r2 ∧ σ1 < σ2
Thus,
Rp = w1 ∗ r1 + (1 − w1) ∗ r2
σp = w1 ∗ σ1 + (1 − w1) ∗ σ2
The set of admissible portfolios Indifference curves The set of efficient portfolios
Two risky assets by ρ1,2 = 1
Basic assumptions:
w1, w2 ≥ 0
r1 < r2 ∧ σ1 < σ2
Thus,
Rp = w1 ∗ r1 + (1 − w1) ∗ r2
σp = w1 ∗ σ1 + (1 − w1) ∗ σ2
Risk minimization can be achieved:
The set of admissible portfolios Indifference curves The set of efficient portfolios
Two risky assets by ρ1,2 = 1
Basic assumptions:
w1, w2 ≥ 0
r1 < r2 ∧ σ1 < σ2
Thus,
Rp = w1 ∗ r1 + (1 − w1) ∗ r2
σp = w1 ∗ σ1 + (1 − w1) ∗ σ2
Risk minimization can be achieved:
w1 = 1
The set of admissible portfolios Indifference curves The set of efficient portfolios
Two risky assets by ρ1,2 = −1
Basic assumptions:
The set of admissible portfolios Indifference curves The set of efficient portfolios
Two risky assets by ρ1,2 = −1
Basic assumptions:
w1, w2 ≥ 0
The set of admissible portfolios Indifference curves The set of efficient portfolios
Two risky assets by ρ1,2 = −1
Basic assumptions:
w1, w2 ≥ 0
r1 < r2 ∧ σ1 < σ2
Thus,
The set of admissible portfolios Indifference curves The set of efficient portfolios
Two risky assets by ρ1,2 = −1
Basic assumptions:
w1, w2 ≥ 0
r1 < r2 ∧ σ1 < σ2
Thus,
Rp = w1 ∗ r1 + (1 − w1) ∗ r2
The set of admissible portfolios Indifference curves The set of efficient portfolios
Two risky assets by ρ1,2 = −1
Basic assumptions:
w1, w2 ≥ 0
r1 < r2 ∧ σ1 < σ2
Thus,
Rp = w1 ∗ r1 + (1 − w1) ∗ r2
σp = w1 ∗ σ1 − (1 − w1) ∗ σ2
The set of admissible portfolios Indifference curves The set of efficient portfolios
Two risky assets by ρ1,2 = −1
Basic assumptions:
w1, w2 ≥ 0
r1 < r2 ∧ σ1 < σ2
Thus,
Rp = w1 ∗ r1 + (1 − w1) ∗ r2
σp = w1 ∗ σ1 − (1 − w1) ∗ σ2
Risk minimization can be achieved:
The set of admissible portfolios Indifference curves The set of efficient portfolios
Two risky assets by ρ1,2 = −1
Basic assumptions:
w1, w2 ≥ 0
r1 < r2 ∧ σ1 < σ2
Thus,
Rp = w1 ∗ r1 + (1 − w1) ∗ r2
σp = w1 ∗ σ1 − (1 − w1) ∗ σ2
Risk minimization can be achieved:
w1 =
σ2
σ1 + σ2
The set of admissible portfolios Indifference curves The set of efficient portfolios
Two risky assets by ρ1,2 = −1
Basic assumptions:
w1, w2 ≥ 0
r1 < r2 ∧ σ1 < σ2
Thus,
Rp = w1 ∗ r1 + (1 − w1) ∗ r2
σp = w1 ∗ σ1 − (1 − w1) ∗ σ2
Risk minimization can be achieved:
w1 =
σ2
σ1 + σ2
→ σp = 0
The set of admissible portfolios Indifference curves The set of efficient portfolios
Two risky assets by ρ1,2 = 0
Basic assumptions:
The set of admissible portfolios Indifference curves The set of efficient portfolios
Two risky assets by ρ1,2 = 0
Basic assumptions:
w1, w2 ≥ 0
The set of admissible portfolios Indifference curves The set of efficient portfolios
Two risky assets by ρ1,2 = 0
Basic assumptions:
w1, w2 ≥ 0
r1 < r2 ∧ σ1 < σ2
Thus,
The set of admissible portfolios Indifference curves The set of efficient portfolios
Two risky assets by ρ1,2 = 0
Basic assumptions:
w1, w2 ≥ 0
r1 < r2 ∧ σ1 < σ2
Thus,
Rp = w1 ∗ r1 + (1 − w1) ∗ r2
The set of admissible portfolios Indifference curves The set of efficient portfolios
Two risky assets by ρ1,2 = 0
Basic assumptions:
w1, w2 ≥ 0
r1 < r2 ∧ σ1 < σ2
Thus,
Rp = w1 ∗ r1 + (1 − w1) ∗ r2
σp = w2
1 ∗ σ2
1 + (1 − w1)2 ∗ σ2
2
The set of admissible portfolios Indifference curves The set of efficient portfolios
Two risky assets by ρ1,2 = 0
Basic assumptions:
w1, w2 ≥ 0
r1 < r2 ∧ σ1 < σ2
Thus,
Rp = w1 ∗ r1 + (1 − w1) ∗ r2
σp = w2
1 ∗ σ2
1 + (1 − w1)2 ∗ σ2
2
Risk minimization can be achieved:
The set of admissible portfolios Indifference curves The set of efficient portfolios
Two risky assets by ρ1,2 = 0
Basic assumptions:
w1, w2 ≥ 0
r1 < r2 ∧ σ1 < σ2
Thus,
Rp = w1 ∗ r1 + (1 − w1) ∗ r2
σp = w2
1 ∗ σ2
1 + (1 − w1)2 ∗ σ2
2
Risk minimization can be achieved:
w1 =
σ2
2
σ2
1 + σ2
2
The set of admissible portfolios Indifference curves The set of efficient portfolios
A multi-component portfolio
Expansion to three assets . . .
The set of admissible portfolios Indifference curves The set of efficient portfolios
A multi-component portfolio
Expansion to three assets . . .
w1, w2, w3 ≥ 0
The set of admissible portfolios Indifference curves The set of efficient portfolios
A multi-component portfolio
Expansion to three assets . . .
w1, w2, w3 ≥ 0
r1 = 3, r2 = 4, r3 = 5
σ1 = 4, σ2 = 5, σ3 = 6
The set of admissible portfolios Indifference curves The set of efficient portfolios
A multi-component portfolio
Expansion to three assets . . .
w1, w2, w3 ≥ 0
r1 = 3, r2 = 4, r3 = 5
σ1 = 4, σ2 = 5, σ3 = 6
Rp =?
σp =?
The set of admissible portfolios Indifference curves The set of efficient portfolios
A multi-component portfolio
Expansion to three assets . . .
w1, w2, w3 ≥ 0
r1 = 3, r2 = 4, r3 = 5
σ1 = 4, σ2 = 5, σ3 = 6
Rp =?
σp =?
, by
ρ1,2 = −1, ρ1,3 = 0, ρ2,3 = 1
The set of admissible portfolios Indifference curves The set of efficient portfolios
3-assets Portfolio with varying proportions
...
The set of admissible portfolios Indifference curves The set of efficient portfolios
3-assets Portfolio with varying proportions
...
With the help of R . . .
The set of admissible portfolios Indifference curves The set of efficient portfolios
Combination of a risky and a risky-free assets
Assumptions:
wr , wf ≥ 0
The set of admissible portfolios Indifference curves The set of efficient portfolios
Combination of a risky and a risky-free assets
Assumptions:
wr , wf ≥ 0
Thus,
Rp = wr ∗ rr + (1 − wr ) ∗ rf
The set of admissible portfolios Indifference curves The set of efficient portfolios
Combination of a risky and a risky-free assets
Assumptions:
wr , wf ≥ 0
Thus,
Rp = wr ∗ rr + (1 − wr ) ∗ rf
and,
σp = wr ∗ σr
The set of admissible portfolios Indifference curves The set of efficient portfolios
Content
1 The set of admissible portfolios
2 Indifference curves
3 The set of efficient portfolios
The set of admissible portfolios Indifference curves The set of efficient portfolios
The investor’s Indifference curves
Map of indifference curves
The set of admissible portfolios Indifference curves The set of efficient portfolios
The investor’s Indifference curves
Map of indifference curves
An indifference curve IC represents all desirable combinations
of portfolios for a particular investor
The set of admissible portfolios Indifference curves The set of efficient portfolios
The investor’s Indifference curves
Map of indifference curves
An indifference curve IC represents all desirable combinations
of portfolios for a particular investor
Properties of ICs:
All portfolios that lie on a specific IC are equally desirable for a
particular investor
The set of admissible portfolios Indifference curves The set of efficient portfolios
The investor’s Indifference curves
Map of indifference curves
An indifference curve IC represents all desirable combinations
of portfolios for a particular investor
Properties of ICs:
All portfolios that lie on a specific IC are equally desirable for a
particular investor
A rational investor prefers portfolios from higher IC
The set of admissible portfolios Indifference curves The set of efficient portfolios
The shape of an IC
IC has a convex shape, which is given by the following
axioms:
The set of admissible portfolios Indifference curves The set of efficient portfolios
The shape of an IC
IC has a convex shape, which is given by the following
axioms:
Axiom of IN-SATURATION
The set of admissible portfolios Indifference curves The set of efficient portfolios
The shape of an IC
IC has a convex shape, which is given by the following
axioms:
Axiom of IN-SATURATION
Axiom of RISK AVERSION
The set of admissible portfolios Indifference curves The set of efficient portfolios
The shape of an IC
IC has a convex shape, which is given by the following
axioms:
Axiom of IN-SATURATION
Axiom of RISK AVERSION
All investors are risk averse, but the level of aversion is
individual
The set of admissible portfolios Indifference curves The set of efficient portfolios
Content
1 The set of admissible portfolios
2 Indifference curves
3 The set of efficient portfolios
The set of admissible portfolios Indifference curves The set of efficient portfolios
The set of efficient portfolios
Asset dominance principle:
The set of admissible portfolios Indifference curves The set of efficient portfolios
The set of efficient portfolios
Asset dominance principle:
Let A & B are assets with ri and σi
The set of admissible portfolios Indifference curves The set of efficient portfolios
The set of efficient portfolios
Asset dominance principle:
Let A & B are assets with ri and σi
Thus,
The set of admissible portfolios Indifference curves The set of efficient portfolios
The set of efficient portfolios
Asset dominance principle:
Let A & B are assets with ri and σi
Thus,
A dominates B ⇐⇒ ra ≥ rb ∧ σb ≥ σa
Equality does not occur in both cases
Definition of the effective set:
The set of admissible portfolios Indifference curves The set of efficient portfolios
The set of efficient portfolios
Asset dominance principle:
Let A & B are assets with ri and σi
Thus,
A dominates B ⇐⇒ ra ≥ rb ∧ σb ≥ σa
Equality does not occur in both cases
Definition of the effective set:
There is no portfolio in the set of permissible portfolios that
has less risk given the same or higher level of return
The set of admissible portfolios Indifference curves The set of efficient portfolios
The set of efficient portfolios
Asset dominance principle:
Let A & B are assets with ri and σi
Thus,
A dominates B ⇐⇒ ra ≥ rb ∧ σb ≥ σa
Equality does not occur in both cases
Definition of the effective set:
There is no portfolio in the set of permissible portfolios that
has less risk given the same or higher level of return
There is no portfolio in the set of permissible portfolios that
has a higher return for the same or lower risk
The set of admissible portfolios Indifference curves The set of efficient portfolios
Optimal portfolio
The optimal portfolio should lie at the intersection of the efficient
set and the indifference curve
The set of admissible portfolios Indifference curves The set of efficient portfolios
Optimal portfolio
The optimal portfolio should lie at the intersection of the efficient
set and the indifference curve
Efficient set according to Sharpe:
The set of admissible portfolios Indifference curves The set of efficient portfolios
Optimal portfolio
The optimal portfolio should lie at the intersection of the efficient
set and the indifference curve
Efficient set according to Sharpe:
For chosen rp → min σp
The set of admissible portfolios Indifference curves The set of efficient portfolios
Optimal portfolio
The optimal portfolio should lie at the intersection of the efficient
set and the indifference curve
Efficient set according to Sharpe:
For chosen rp → min σp
Efficient set according to Markowitz:
The set of admissible portfolios Indifference curves The set of efficient portfolios
Optimal portfolio
The optimal portfolio should lie at the intersection of the efficient
set and the indifference curve
Efficient set according to Sharpe:
For chosen rp → min σp
Efficient set according to Markowitz:
For chosen σp → max rp