Tangent portfolio Enlargement of portfolio deversication Portfolio Theory Lecture 5 Lud¥k Benada Department of Finance - 402, benada.esf@gmail.com March 21, 2016 Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Structure 1 Tangent portfolio 2 Enlargement of portfolio deversication Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Risky and Risk free Asset rf ... treasury bills, deposits at a bank Model example (ri ,rf ,rp) Return and risk Graphycal representation Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Risky and Risk free Asset rf ... treasury bills, deposits at a bank Model example (ri ,rf ,rp) Return and risk Graphycal representation Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Risky and Risk free Asset rf ... treasury bills, deposits at a bank Model example (ri ,rf ,rp) Return and risk Graphycal representation Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Risky and Risk free Asset rf ... treasury bills, deposits at a bank Model example (ri ,rf ,rp) Return and risk Graphycal representation Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication The Shape of EPF All combination of risky and risk free asset ⇒a line The mean of the security could be determined rp = rf + ¯rA−rf σA σp The impact on the permissible and eective set (EPF) Which line to choose? Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication The Shape of EPF All combination of risky and risk free asset ⇒a line The mean of the security could be determined rp = rf + ¯rA−rf σA σp The impact on the permissible and eective set (EPF) Which line to choose? Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication The Shape of EPF All combination of risky and risk free asset ⇒a line The mean of the security could be determined rp = rf + ¯rA−rf σA σp The impact on the permissible and eective set (EPF) Which line to choose? Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication The Shape of EPF All combination of risky and risk free asset ⇒a line The mean of the security could be determined rp = rf + ¯rA−rf σA σp The impact on the permissible and eective set (EPF) Which line to choose? Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication The Shape of EPF All combination of risky and risk free asset ⇒a line The mean of the security could be determined rp = rf + ¯rA−rf σA σp The impact on the permissible and eective set (EPF) Which line to choose? Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Lending and Borrowing A part of our fund to rf ⇒borrow sources to someone We can invest more than we own ...we must borrow The boundary between lending and borrowing represents the situation when all funds are invested in ri !In accordance with the condition: ∑N i=1 pijk = 1! The hypothetical shape of EPF and the real shape of EPF Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Lending and Borrowing A part of our fund to rf ⇒borrow sources to someone We can invest more than we own ...we must borrow The boundary between lending and borrowing represents the situation when all funds are invested in ri !In accordance with the condition: ∑N i=1 pijk = 1! The hypothetical shape of EPF and the real shape of EPF Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Lending and Borrowing A part of our fund to rf ⇒borrow sources to someone We can invest more than we own ...we must borrow The boundary between lending and borrowing represents the situation when all funds are invested in ri !In accordance with the condition: ∑N i=1 pijk = 1! The hypothetical shape of EPF and the real shape of EPF Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Lending and Borrowing A part of our fund to rf ⇒borrow sources to someone We can invest more than we own ...we must borrow The boundary between lending and borrowing represents the situation when all funds are invested in ri !In accordance with the condition: ∑N i=1 pijk = 1! The hypothetical shape of EPF and the real shape of EPF Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Lending and Borrowing A part of our fund to rf ⇒borrow sources to someone We can invest more than we own ...we must borrow The boundary between lending and borrowing represents the situation when all funds are invested in ri !In accordance with the condition: ∑N i=1 pijk = 1! The hypothetical shape of EPF and the real shape of EPF Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Finding a portfolio with respect to rf Four possible scenarious: The Short Sell is allowed and there is rf The Short Sell is allowed, but there is not rf The Short Sell is not allowed and there is rf There is neither SS allowed, nor rf exists Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Finding a portfolio with respect to rf Four possible scenarious: The Short Sell is allowed and there is rf The Short Sell is allowed, but there is not rf The Short Sell is not allowed and there is rf There is neither SS allowed, nor rf exists Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Finding a portfolio with respect to rf Four possible scenarious: The Short Sell is allowed and there is rf The Short Sell is allowed, but there is not rf The Short Sell is not allowed and there is rf There is neither SS allowed, nor rf exists Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Finding a portfolio with respect to rf Four possible scenarious: The Short Sell is allowed and there is rf The Short Sell is allowed, but there is not rf The Short Sell is not allowed and there is rf There is neither SS allowed, nor rf exists Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Finding a portfolio with respect to rf Four possible scenarious: The Short Sell is allowed and there is rf The Short Sell is allowed, but there is not rf The Short Sell is not allowed and there is rf There is neither SS allowed, nor rf exists Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Short Sell allowed with existance of rf Maximalization of an objective function with restrictions The objective function is tg of the angle (rf ,T) The restrictions are weights ... f −−→ (X) = ¯rp−rf σp Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Short Sell allowed with existance of rf Maximalization of an objective function with restrictions The objective function is tg of the angle (rf ,T) The restrictions are weights ... f −−→ (X) = ¯rp−rf σp Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Short Sell allowed with existance of rf Maximalization of an objective function with restrictions The objective function is tg of the angle (rf ,T) The restrictions are weights ... f −−→ (X) = ¯rp−rf σp Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Short Sell allowed with existance of rf Maximalization of an objective function with restrictions The objective function is tg of the angle (rf ,T) The restrictions are weights ... f −−→ (X) = ¯rp−rf σp Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Diversication of assets Meaning of diversication The Central Limit Theorem ... A portfolio of N assets created with same weights: rp = 1 N ∑N i=1 wi ∗ri Variance of portfolio on conditions (N(µ,σ2, σi,j = 0, N → ∞) ⇒0 If the distribution deviates from gaussian, then the mean-variance approach exhibits shortcomings Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Diversication of assets Meaning of diversication The Central Limit Theorem ... A portfolio of N assets created with same weights: rp = 1 N ∑N i=1 wi ∗ri Variance of portfolio on conditions (N(µ,σ2, σi,j = 0, N → ∞) ⇒0 If the distribution deviates from gaussian, then the mean-variance approach exhibits shortcomings Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Diversication of assets Meaning of diversication The Central Limit Theorem ... A portfolio of N assets created with same weights: rp = 1 N ∑N i=1 wi ∗ri Variance of portfolio on conditions (N(µ,σ2, σi,j = 0, N → ∞) ⇒0 If the distribution deviates from gaussian, then the mean-variance approach exhibits shortcomings Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Diversication of assets Meaning of diversication The Central Limit Theorem ... A portfolio of N assets created with same weights: rp = 1 N ∑N i=1 wi ∗ri Variance of portfolio on conditions (N(µ,σ2, σi,j = 0, N → ∞) ⇒0 If the distribution deviates from gaussian, then the mean-variance approach exhibits shortcomings Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Diversication of assets Meaning of diversication The Central Limit Theorem ... A portfolio of N assets created with same weights: rp = 1 N ∑N i=1 wi ∗ri Variance of portfolio on conditions (N(µ,σ2, σi,j = 0, N → ∞) ⇒0 If the distribution deviates from gaussian, then the mean-variance approach exhibits shortcomings Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Diversication of assets Meaning of diversication The Central Limit Theorem ... A portfolio of N assets created with same weights: rp = 1 N ∑N i=1 wi ∗ri Variance of portfolio on conditions (N(µ,σ2, σi,j = 0, N → ∞) ⇒0 If the distribution deviates from gaussian, then the mean-variance approach exhibits shortcomings Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Reciprocal correlation of assets If any assets are correlated the ability to minimize the risk is limited ... ... Elimination of risk involves only the nonsystematic risk! Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Reciprocal correlation of assets If any assets are correlated the ability to minimize the risk is limited ... ... Elimination of risk involves only the nonsystematic risk! Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Reciprocal correlation of assets If any assets are correlated the ability to minimize the risk is limited ... ... Elimination of risk involves only the nonsystematic risk! Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Reciprocal correlation of assets If any assets are correlated the ability to minimize the risk is limited ... ... Elimination of risk involves only the nonsystematic risk! Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Alternative approaches to risk Variation rate Rate of negative risk (Downside risk) Value at Risk Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Alternative approaches to risk Variation rate Rate of negative risk (Downside risk) Value at Risk Lud¥k Benada MPF_APOT Tangent portfolio Enlargement of portfolio deversication Alternative approaches to risk Variation rate Rate of negative risk (Downside risk) Value at Risk Lud¥k Benada MPF_APOT