The set of admissible portfolios Indifference curve The set of efficient portfolios Portfolio Theory Lecture 3 Luděk Benada Department of Finance - 402, benada.esf@gmail.com March 7, 2016 Luděk Benada MPF APOT O The set of admissible portfolios Q Indifference curve Q The set of efficient portfolios Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios Forms of admissible portfolios • The model of Markowitz Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios Forms of admissible portfolios • The model of Markowitz • The wealth is defined i me dcr*i^)d Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios Forms of admissible portfolios • The model of Markowitz • The wealth is defined • Time period Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios Forms of admissible portfolios • The model of Markowitz • The wealth is defined • Time period • Problem of portfolio selection Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios Forms of admissible portfolios • The model of Markowitz • The wealth is defined • Time period • Problem of portfolio selection • There are two extrems for a portfolio construction: Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios Forms of admissible portfolios The model of Markowitz • The wealth is defined • Time period • Problem of portfolio selection There are two extrems for a portfolio construction Wealth (assets) can not be devided Luděk Benada MPF APOT □ s1 The set of admissible portfolios Indifference curve The set of efficient portfolios Forms of admissible portfolios The model of Markowitz • The wealth is defined • Time period • Problem of portfolio selection There are two extrems for a portfolio construction Wealth (assets) can not be devided • Wealth (assets) can be arbitrarily devided Luděk Benada MPF APOT □ s1 The set of admissible portfolios Indifference curve The set of efficient portfolios Indivisible assets • The set of admissible portfolios will consist of only a finite set Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios Indivisible assets • The set of admissible portfolios will consist of only a finite set • The options by portfolio creation are limited to "pseudo-short sell" Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios Basic concept - two-components portfolio 9 Return of portfolio: rp = wi * r\ Vy o \a/ö i ernte in r^or*tToliO" \A/t i i/i/<-\ — p = Wl * rl + P\/l 1 v 1' 2 1 v ly 12 Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios Basic concept - two-components portfolio 9 Return of portfolio: rp = wi * r\ 9 The weights in portfolio: w\ + 1/1/2 = 1 Expected return of portfolio: rD — w\ * T\ P\/l ~^-t 2 i v ly 12 Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios Basic concept - two-components portfolio • Return of portfolio: rp = Y^'lLi wi * H • The weights in portfolio: w\ + 1/1/2 = 1 • Expected return of portfolio: rp = 1/1/1 * T\ + 1/1/2 * h Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios Basic concept - two-components portfolio 9 Return of portfolio: rp = £/=i wi * r\ 9 The weights in portfolio: w\ + 1/1/2 = 1 o Expected return of portfolio: rp = w\ * T\ + 1/1/2 * ?2 9 Covariance of two assets: g\2 = g\ * 02 * P12 0: Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios Basic concept - two-components portfolio 9 Return of portfolio: rp = £/=i wi * r\ 9 The weights in portfolio: w\ + 1/1/2 = 1 o Expected return of portfolio: rp = w\ * T\ + 1/1/2 * ?2 • Covariance of two assets: g\2 — g\ * 02 * P12 • Risk of portfolio: dp = a / 1/1/^ * g\ + (1 — 1/1/1 )2 * O"! + 2 * 1/1/1 * (1 — 1/1/1) * CJ12 Luděk Benada MPF APOT Assumptions: w\, 1/1/2 > 0, r\ < r2 A g\ < 02 Luděk Benada F APOT The set of admissible portfolios Indifference curve The set of efficient portfolios Two risky assets - Ö12 = 1 • Assumptions: wi, 1/1/2 > 0, r\ < ri A C\ < 02 • Return of portfolio... Luděk Benada MPF APOT □ The set of admissible portfolios Indifference curve The set of efficient portfolios Two risky assets - Ö12 = 1 • Assumptions: 1/1/1, 1/1/2 > 0, r\ < r2 A Oi < 02 • Return of portfolio... • Risk of expected return: op — w\ * Gi + (1 — w\) * 02 Luděk Benada MPF APOT □ e The set of admissible portfolios Indifference curve The set of efficient portfolios Two risky assets - O12 1 • Assumptions: wi, 1/1/2 > 0, r\ < r2 A di < 02 • Return of portfolio... • Risk of expected return: op — w\ * di + (1 — w\) * 02 9 The minimalization of the risk coulde be reached: Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios Two risky assets - Ö12 = 1 • Assumptions: 1/1/1, 1/1/2 > 0, r\ < r2 A Oi < 02 • Return of portfolio... • Risk of expected return: op — w\ * Gi + (1 — w\) * 02 • The minimalization of the risk coulde be reached: • 1/1/1 = 1! Luděk Benada MPF APOT □ e The set of admissible portfolios Indifference curve The set of efficient portfolios — fwo ris ky assets - Ö12 ~~ — 1 • Assumptions: 1/1/1,1/1/2 > 0,ri < r2 A 0\ < 02 Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios Two risky assets - G12 —1 • Assumptions: 1/1/1, 1/1/2 > 0, r\ < A CJi < 02 • Return of portfolio... Luděk Benada MPF APOT □ ► 4 flP The set of admissible portfolios Indifference curve The set of efficient portfolios Two risky assets - 012 = — 1 • Assumptions: 1/1/1,1/1/2 > 0, r\ < A di < 02 • Return of portfolio... • Risk of expected return: cp — w\ * Gi — (1 — w\) * 02 s1 Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios Two risky assets - O12 —1 • Assumptions: 1/1/1, 1/1/2 > 0, r\ < r2 A di < 02 • Return of portfolio... • Risk of expected return: ap = w\ * di — (1 — w\) * 02 • The minimalization of the risk coulde be reached: Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios Two risky assets - Ö12 = — 1 • Assumptions: 1/1/1, 1/1/2 > 0, r\ < A di < 02 • Return of portfolio... • Risk of expected return: cp — w\ * Gi — (1 — w\) * 02 • The minimalization of the risk coulde be reached: — °2 i ct1+ct2 ' s1 Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios wo risky assets - Ö12 = 0 • Assumptions: 1/1/1,1/1/2 > 0,ri < r2 A CJi < Ö2 a Rpturn nf nnrtfnlin i-~ ~ Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios Two risky assets - 012 = • Assumptions: 1/1/1, 1/1/2 > 0, r\ < ri A g\ < 02 • Return of portfolio... Pigk ^)f e^cpected return * — \ 11/1/^ ^ ^ I v Luděk Benada MPF APOT □ The set of admissible portfolios Indifference curve The set of efficient portfolios Two risky assets - 012 = • Assumptions: 1/1/1, 1/1/2 > 0, r\ < ri A g\ < 02 • Return of portfolio... • Risk of expected return: op — y^vf * c\ + (1 — 1/1/1 )2 * cj| —r-1 I" j." X j-L i U L. L J Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios Two risky assets - G12 = 0 • Assumptions: 1/1/1,1/1/2 > 0, r\ < r2 A Gi < 02 • Return of portfolio... • Risk of expected return: op = * (7-j2 + (1 — w\)2 * a% • The minimalization of the risk coulde be reached: o Wi 2_i s1 Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios wo risky assets - Ö12 = 0 • Assumptions: 1/1/1,1/1/2 > 0,ri < r2 A di < 02 • Return of portfolio... Risk of expected return: ap = ywf * c\ + (1 — 1/1/1)2 * ex? The minimalization of the risk coulde be reached: 1/1/1 = _ ^2 s1 Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios More components portfolio • Assumptions: w\, 1/1/2,1/1/3 > 0 Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios More components portfolio • Assumptions: w\, 1/1/2,1/1/3 > 0 Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios More components portfolio • Assumptions: w\, 1/1/2,1/1/3 > 0 «... • ... Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios Combination risky asset and riskfree asset o Assumptions: 1/1/1,1/1/2 >0 Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios Combination risky asset and riskfree asset o Assumptions: 1/1/1,1/1/2 >0 • Return of portfolio... Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios Combination risky asset and riskfree asset o Assumptions: 1/1/1,1/1/2 >0 • Return of portfolio... • Risk of expected return: ov> = 9 9 Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios Combination risky asset and riskfree asset o Assumptions: 1/1/1,1/1/2 >0 • Return of portfolio... o Risk of expected return: ap — y 1/1/j2 * (7-j2 • The minimalization of the risk coulde be reached Luděk Benada MPF APOT □ s1 The set of admissible portfolios Indifference curve The set of efficient portfolios Combination risky asset and riskfree asset o Assumptions: 1/1/1,1/1/2 >0 • Return of portfolio... o Risk of expected return: ap — y 1/1/j2 * (7-j2 • The minimalization of the risk coulde be reached Luděk Benada MPF APOT □ s1 The set of admissible portfolios Indifference curve The set of efficient portfolios Indifference curves of investor Map of investor's ICs Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios Indifference curves of investor • Map of investor's ICs • An IC represents all desirable combinations of portfolio for an investor Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios Indifference curves of investor • Map of investor's ICs 9 An IC represents all desirable combinations of portfolio for an investor • Properties of ICs: • All portfolios on an IC are equally desirable • A rational investor prefers portfolios on higher ICs Luděk Benada MPF APOT □ The set of admissible portfolios Indifference curve The set of efficient portfolios The form of ICs The ICs are convex: 0? 1 1 ^0 0,0 Luděk Benada F APOT The set of admissible portfolios Indifference curve The set of efficient portfolios The form of ICs • The ICs are convex: • Unsaturation (return) Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios The form of ICs • The ICs are convex: • Unsaturation (return) • Risk aversion Ul fCblbldllCC LU tne risK, di J L LI IC □ Luděk Benada MPFAPOT The set of admissible portfolios Indifference curve The set of efficient portfolios The form of ICs • The ICs are convex: • Unsaturation (return) • Risk aversion • All investors have some kind of resistance to the risk, but the individual aversion is unique Luděk Benada MPFAPOT The set of admissible portfolios Indifference curve The set of efficient portfolios Princip of asset dominance • There are such assets A, B, where: ra > A <7a < Tknr , „ ,s\ r- w j. L _ j. A J ^ " _ j. D n i ic \a/Cj c j\/ i n jt la nnm i n jtoc rs Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios Princip of asset dominance • There are such assets A, B, where: ra > A <7a < • Thus we say that A dominates B Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios Definition of effective set • There is no set in between the admissible portfolios, that can has with the same or higher return less risk —i— ■ i I i ■ ■ i i . r i • .1 Luděk Benada MPF APOT □ s1 The set of admissible portfolios Indifference curve The set of efficient portfolios Definition of effective set • There is no set in between the admissible portfolios, that can has with the same or higher return less risk • There is no set in between the admissible portfolios, that can has with the same or lower risk higher return Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios The form of efficient set Sharpe's set of efficient portfolios Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios The form of efficient set 9 Sharpe's set of efficient portfolios » For chosen rp=>min op ka r»rLr/-ma/i-i--7 'c cot r>f pffiripnt nortfo Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios The form of efficient set • Sharpens set of efficient portfolios • For chosen rp=^min ap • Markowitz's set of efficient portfolios Luděk Benada MPF APOT The set of admissible portfolios Indifference curve The set of efficient portfolios The form of efficient set Sharpe's set of efficient portfolios • For chosen rp^>min op Markowitz's set of efficient portfolios • For chosen <7p^>max rp Luděk Benada MPF APOT < n ► igi ► 4 = k 4 The set of admissible portfolios difference curve The set of efficient portfolios The form of efficient set 9 Sharpe's set of efficient portfolios • For chosen rp^>min op 9 Markowitz's set of efficient portfolios • For chosen <7p^>max rp 9 The solution for an optimal portfolio is ESPs AlCs Luděk Benada MPF APOT