Financial Economics 4: Portfolio Theory

Financial Economics 4: Portfolio Theory Stefano Lovo HEC, Paris

What is a portfolio? Definition A portfolio is an amount of money invested in a number of financial assets. Example Portfolio A is worth Eu 100, 000 and consists of Eu 5, 000 invested in Alcatel shares. Eu 0, 000 invested in Michelin shares. Eu 18, 000 invested in JP Morgan shares. Eu, 000 Invested in gold. Eu 15, 000 invested in German Treasury bills. Vector representation: X A = {x Al, x M, x Jp, x g, x Tb } = {0.5, 0., 0.18, 0., 0.15} Stefano Lovo, HEC Paris Portfolio Theory / 40

Notation Let S = {s 1, s,..., s N } be the set of financial assets available in the economy. Let {v 1, v,..., v N } be the amount of money invested in each one of the N assets to obtain portfolio A. Today the value of portfolio A is V A = N v i = v 1 + v + + v n. i=1 The composition of the portfolio A is X A = {x 1, x,..., x N } where and x i = v i V A. N x i = x 1 + x + + x n = 1 i=1 Stefano Lovo, HEC Paris Portfolio Theory 3 / 40

Some portfolios S = {s 1, s, s 3 } = {Alcatel, BMW, Treasury bill} 1 Only long positions V A = 5, 000 +, 000 + 10, 000 = 17, 000 X A = { 5,000 17,000,,000 17,000, 10,000 17,000 } {0.9, 0.1, 0.59} Short position V B = 15, 000 + 6, 000 4, 000 = 17, 000 X B = { 15,000 17,000, 6,000 17,000, 4,000 17,000 } {0.88, 0.353, 0.35} Stefano Lovo, HEC Paris Portfolio Theory 4 / 40

Short selling an asset Example Let X B = {0.88, 0.353, 0.35}, then portfolio B has a short position in asset s 3. Stefano Lovo, HEC Paris Portfolio Theory 5 / 40

Short selling an asset Example Let X B = {0.88, 0.353, 0.35}, then portfolio B has a short position in asset s 3. How to short sell an asset i At time 0: 1 You borrow the asset i from your broker. You sell the asset in the stock market at price P i (0). At time 1: 1 You pay your broker whatever dividend the asset has paid at time 1. You buy the asset in the stock market at price P i (1). 3 You return the asset to your broker. The broker charges you a fee for lending you the asset and asks for a collateral. Stefano Lovo, HEC Paris Portfolio Theory 5 / 40

Portfolio return rate example Asset p i (0) p i (1) D i r i Alcatel 50 49 4 6% BMW 0 0 10% T.B. 100 100 % You invest Eu 100 into portfolio X C = {x Al, x BM, x TB } = {0.8, 0., 0}. What is the rate of return of your portfolio? At t = 1 portfolio C is worth 100 0.8(1+r Al )+100 0.(1+r BM ) = 80(1.06)+0(1.1) = 106.8 hence r C = 106.8 100 100 = 6.8% = 80(1.06)+0(1.1) (80+0) 100 = = 80 0.06+0 0.1 100 = 0.8 6% + 0. 10% = x Al r Al + x BM r BM Stefano Lovo, HEC Paris Portfolio Theory 6 / 40

Return rate of a portfolio Return rate on a long position (buy) on portfolio A: 1 At time 0 you buy the portfolio at X a = {x 1, x,..., x n }. At time 1 you receive dividends from each stock 3 At time 1 you resell the portfolio. The return rate r A on your investment in Portfolio A is r A = n r i x i = r 1 x 1 + r x + + r n x n. i=1 Stefano Lovo, HEC Paris Portfolio Theory 7 / 40

Uncertainty Theorem Let S = {s 1, s,..., s N } be the set of available assets. Let r 1, r,..., r N be N random variables representing the return rates on assets s 1, s,..., s N, respectively. Then, the rate of return of a portfolio A with composition X A = {x 1, x,..., x N } is a random variable r A := N x i r i i=1 Stefano Lovo, HEC Paris Portfolio Theory 8 / 40

Uncertainty Example The rate of return of a portfolio composed of risky asset is uncertain: S = {Alcatel, BMW } Portfolio A: X A = {0.8, 0.} Event r Al r BM r A The economy is booming 1% 0% 0.8 1%+0. 0%=13.6% The economy is in recession -3% -15% -0.8 3%-0. 15% =-5.4% Stefano Lovo, HEC Paris Portfolio Theory 9 / 40

Expected return on a portfolio Theorem The expected return rate on a portfolio with composition X = {x 1, x,..., x n } is: E [ r X = x1 E [ r 1 + x E [ r + + xn E [ r n = n x i E [ r i. i=1 Example S={Al,BM} X A ={0.8,0.} E [ r Al =1.5% E [ r BM =-4.5% E [ r A Event Prob r Al r BM r A Boom 0.3 1% 0% 13.6% Recession 0.7-3% -15% -5.4% = 0.3 13.6% 0.7 5.4% = 0.3% = 0.8 1.5% 0. 4.5% = 0.3% Stefano Lovo, HEC Paris Portfolio Theory 10 / 40

Expected return on a portfolio: QCQ Assets s 1 s s 3 s 4 E [ r i 30% 5% 8% % What are the expected returns of the following portfolios? x 1 x x 3 x 4 X A 0 1 0 0 X B 0. 0.1 0.3 0.4 X C 0.3 0.8 0.6 0.7 X D 0.5 0. 0.8 0.1 Answers: E [r A = 5%, E [r B = 11.7%, E [r C = 3.4%, E [r D = 16.%. Stefano Lovo, HEC Paris Portfolio Theory 1 / 40

Variance of the return rate of a portfolio Theorem The variance of the return rate on a portfolio with composition X = {x 1, x,..., x n } is: σ X := Var [ r X = n n x i x j σ ij i=1 j=1 where σ ii = Var [ r i and σij := Cov [ r i, r j. for n = : σ X = x 1 σ 1 + x σ + x 1x Cov [ r 1, r Since x 1 + x = 1 and Cov [ r i, r j = ρ1, σ 1 σ, σ X = x 1 σ 1 + (1 x 1) σ + x 1(1 x 1 )ρ 1, σ 1 σ Stefano Lovo, HEC Paris Portfolio Theory 13 / 40

Variance of the return rate of a portfolio: example Example S = {s 1, s } s 1 s σ i 30% 1% ρ 1 0.56 If the composition of portfolio A is X A = {0.8, 0.}, then σ A = x 1 σ 1 + (1 x 1) σ + x 1(1 x 1 )ρ 1 σ 1 σ = = 0.8 0.3 + 0. 0.1 + 0.8 0. 0.56 0.3 0.1 = 0.05 σ A = σ A = 0.05 =.7% Stefano Lovo, HEC Paris Portfolio Theory 14 / 40

Summing-up about portfolios 1 Ingredients: Set of available assets S = {s 1, s,..., s n } Recipe: Portfolio composition X = {x 1, x,..., x n } satisfying n x i = 1. i=1 3 Gain: Expected return of a portfolio E [ r n X = x i E [ r i. i=1 4 Risk: variance of the portfolio s return Var [ r X = n n x i x j σ ij i=1 j=1 Stefano Lovo, HEC Paris Portfolio Theory 15 / 40

Risk aversion Definition A Mean-Variance investor is someone who likes high expected return portfolios and exhibits risk aversion. More precisely: Stefano Lovo, HEC Paris Portfolio Theory 16 / 40

Risk aversion Definition A Mean-Variance investor is someone who likes high expected return portfolios and exhibits risk aversion. More precisely: 1 The investor only cares about E [ r X and σ X. For a given level of risk σx, he/she prefers the portfolio with the largest expected return E [ r X. 3 For a given level of expected return E [ r X, he/she prefers the portfolio with the lowest risk σ X. Stefano Lovo, HEC Paris Portfolio Theory 16 / 40

Risk aversion: Graphically Portfolio X is at least as good as portfolio X iff U(X) U(X ). E [ r X D B A C B C D A? D B? D A B C σ X Stefano Lovo, HEC Paris Portfolio Theory 17 / 40

Diversification Rule 0: Diversification of your investment allows to reduce the risk of your portfolio. Example E [ r 1 Cov [ r 1, r Event Prob. r 1 r ω 1 0.5 0% 5% ω 3 0.5 0% 15% ω 0.5 0% 5% ω 4 0.5 0% 15% = E [ r = 10%, σ1 = 10%, σ = 5% = ρ 1, = 0 If X = {0., 0.8}, then E [ r X = 10% σ X = 0. 0.1 + 0.8 0.05 = 4.47% < 5% Stefano Lovo, HEC Paris Portfolio Theory 18 / 40

Diversifying Example E [ r 1 Event Prob. r 1 r ω 1 0.5 0% 15% ω 3 0.5 0% 15% ω 0.5 0% 5% ω 4 0.5 0% 5% = E [ r = 10%, σ1 = 10%, σ = 5% ρ 1, = 1 If X = { 1 3, 3 }, then E [ r X = 10% (1 σ X = 3 0.1 ) ( + 3 0.05 ) 4 0.05 0.1 = 0 9 Stefano Lovo, HEC Paris Portfolio Theory 19 / 40

Minimum Variance Portfolio S = {s 1, s }. What is the composition of the portfolio (x 1, x ) that has the minimum variance? min x x 1,x 1 σ 1 + x σ + x 1x ρ 1 σ 1 σ s.t. x 1 + x = 1 x min 1 = σ (σ ρ 1 σ 1 ) σ 1 +σ ρ 1σ 1 σ, x min = 1 x min 1 σ min = (1 ρ 1 )σ 1 σ σ 1 +σ ρ 1σ 1 σ Remarks: If ρ 1 1, then x min 1 > 0 and x min > 0 If σ 1 > σ and ρ 1 is close enough to 1 then short sell s 1. If ρ 1 = ±1, then σ min = 0. Stefano Lovo, HEC Paris Portfolio Theory 0 / 40

Limit of risk reduction through diversification Definition The idiosyncratic risk on stock i is the uncertainty on r i linked to risk factors that are specific to firm i. Examples: changes in management, production process innovation, local strikes, etc. The systematic risk on stock i is the uncertainty on r i linked to risk factors that affect the whole economy and are common to other stocks. Examples: Wars, political change, general economic downturn, pandemics, etc. Theorem Diversification intended as buying a large number of different assets can eliminate the idiosyncratic risk but not the systematic risk. Stefano Lovo, HEC Paris Portfolio Theory 1 / 40

Limit of diversification σ X Idiosyncratic risk Systematic risk Number of securities Stefano Lovo, HEC Paris Portfolio Theory / 40

Portfolio choice Assumptions 1 S = {s 1, s,..., s n } Any assets i < n is a risky asset with return r i. 3 Asset n is a risk-free asset with r n = r f. (Ex. Treasury bill) 4 Investors are mean-variance investors. Implication: Choose thex = {x 1,..., x f } that maximizes U (X) := E [ r X a σ X subject to 1 = x 1 + x + + x f, E [ r X = n x i E [ r i, Var [ r X = i=1 n i=1 j=1 n x i x j σ ij. Stefano Lovo, HEC Paris Portfolio Theory 3 / 40

Efficient Portfolio Definition A portfolio A is said to be efficient if there exists no other portfolio B satisfying: σ B σ A and E [ r B E [ r A with at least one strict inequality. Portfolio choice methodology 1 Determine the set of couple risk-return (E [ r X, σ X ) reachable by combining the available assets S. Identify the set of portfolios that are efficient. 3 Among the set of efficient portfolios choose the one that best fits the investor s risk aversion. Stefano Lovo, HEC Paris Portfolio Theory 4 / 40

Return-risk region. Case 0 Just one risky asset: S = {s 1 } with E [ r 1 > 0 and σ 1 > 0. E [ ~ r X E [ ~ r1 σ 1 σ X Stefano Lovo, HEC Paris Portfolio Theory 5 / 40

Return-risk region. Case 1 One risky asset and one risk-free asset: S = {s 1, s f } with E [ r 1 > 0, σ 1 > 0, r f = r f < E [ r 1 and σ f = 0. E [ r X = x 1 E [ r 1 + (1 x1 )r f σx = x1 σ 1 E [ r X = r f ± E [ r 1 rf σ X σ 1 Example E [ r 1 = 0%, σ 1 = 0.01, r f = % and σ f = 0 What is the composition of a portfolio with E [ r X = 11%? What is the composition of a portfolio with risk σ X = %? The expected return on portfolio A is E [ r A = 40%. What is its risk σ A? Stefano Lovo, HEC Paris Portfolio Theory 6 / 40

Return-risk region. Case 1 One risky asset and one risk-free asset: S = {s 1, s f } { 0,1} E [ ~ r X E [ ~ r1 x > 0 1 x f > 0 { 1,0} x > 1 1 x f < 0 Short-sell s f r f Buy s 1 and s f x < 0 1 x f > 1 Short-sell s 1 σ 1 σ X Stefano Lovo, HEC Paris Portfolio Theory 7 / 40

Capital Allocation Line Theorem In Case 1 a risk-return combination corresponds to an efficient portfolio if and only if E [ r X = rf + λσ X where λ := E[ r 1 r f σ 1 E [ ~ r X E [ ~ r1 CAL r f σ 1 Stefano Lovo, HEC Paris Portfolio Theory 8 / 40 σ X

Return-risk region. Case Two risky assets : S = {s 1, s } with E [ r 1 > 0, σ 1 > 0, E [ r > 0, σ > 0. E [ r X = x 1 E [ r 1 + (1 x1 )E [ r σx = x1 σ 1 + (1 x 1) σ + x 1(1 x 1 )ρ 1 σ 1 σ. ( ) ( ) σx E[ r = X E[ r σ E[ r 1 E[ r 1 + E[ r 1 E[ r X σ E[ r 1 E[ r + + (E[ r X E[ r )(E[ r 1 E[ r X) (E[ r 1 E[ r ) ρ 1 σ 1 σ. Stefano Lovo, HEC Paris Portfolio Theory 9 / 40

Return-risk region. Case Two risky assets : S = {s 1, s } E [ ~ r X E[ ~r 1 x 1 > 0 x > 0 { 1,0} x 1 >1 x < 0 Short-sell s E[ r { 0,1} Buy s 1 and s Short-sell s 1 x 1 < 0 x >1 σ min σ σ 1 σ X Stefano Lovo, HEC Paris Portfolio Theory 30 / 40

Case with perfect positive correlation Show that if S = {s 1, s } and ρ = 1, then 1 the return risk region is as depicted below. find the composition, the expected return and the risk of the minimum Var portfolio 3 find the equation of the two lines. E [ ~ r X [ ~ { 1,0} E r1 E [ ~ r { 0,1} x > 0 1 x > 0 x < 0 1 x > 1 ρ 1 = 1 x > 1 1 x < 0 Short-sell s Buy s 1 and s Short-sell s 1 σ σ 1 σ X Stefano Lovo, HEC Paris Portfolio Theory 31 / 40

Return-risk region. Case with perfect negative correlation Show that if S = {s 1, s } and ρ = 1, then 1 the return risk region is as depicted below. find the composition, the expected return and the risk of the minimum Var portfolio. 3 find the equation of the two lines. E [ ~ r X [ ~ { 1,0} E r1 x1 > 0 x > 0 ρ 1 = -1 x > 1 1 x < 0 Short-sell s E [ ~ r { 0,1} x < 0 1 x > 1 Buy s 1 and s Short-sell s 1 σ σ 1 σ X Stefano Lovo, HEC Paris Portfolio Theory 3 / 40

Return-risk region. Case 3 Two risky assets and one risk-free asset: S = {s 1, s, s f } with E [ r 1 > 0, σ 1 > 0, E [ r > 0, σ > 0, r f = r f, σ f = 0. E [ ~ r X Tangency portfolio: { x T,1 x T,0} {αx T, α(1-x T ), 1- α } E[ ~r 1 { 1,0,0} r f [ E r { 0,1,0} {x 1,1-x 1, 0 } σ min σ σ 1 σ X Stefano Lovo, HEC Paris Portfolio Theory 33 / 40

Case 3: efficient portfolios Theorem In Case 3 a risk-return combination corresponds to an efficient portfolio if and only if: 1 where E [ r X = rf + λσ X λ := E [ r T rf σ T and E [ r T and σt are the expected return and risk of the tangency portfolio. It is obtained combining the tangency portfolio with the risk-free asset. Stefano Lovo, HEC Paris Portfolio Theory 34 / 40

Tangency Portfolio composition for Case 3: S = {s 1, s, s f } What is the composition of the tangency portfolio? E[ r T r max λ = max f x1 T x1 T σ T subject to E [ r T = x1 T E [ r 1 + (1 x T 1 )E [ r σ T = ( (x1 T σ 1) + ((1 x1 T )σ ) + x1 T (1 x 1 T )ρ 1,σ 1 σ ) 1 This gives x T 1 = (E [ r rf )ρ 1, σ 1 σ (E [ r 1 rf )σ (E [ r 1 + E [ r rf )ρ 1, σ 1 σ (E [ r 1 rf )σ (E [ r rf )σ 1 Stefano Lovo, HEC Paris Portfolio Theory 35 / 40

General case n 1 risky assets and one risk-free asset: S = {s 1,..., s n 1, s f } E [ ~ r X Tangency portfolio: Capital Allocation Line Efficient portfolio frontier E [ ~ r T r f σ min σ T σ X Stefano Lovo, HEC Paris Portfolio Theory 36 / 40

Security Market Line Relation Theorem For any asset or portfolio s i, E [ r i rf = β i ( E [ r T rf ) where β i := Cov[ r i, r T σ T E [ ~ r i E [ ~ r T r f β i Stefano Lovo, HEC Paris Portfolio Theory 37 / 40

Optimal Portfolio Theorem An investor with a mean-variance utility function and a risk aversion A will choose an efficient portfolio such that: x T = E[ r T r f Aσ T, x f = 1 x T E [ ~ r X E [ ~ r T Optimal portfolio Indifference courve CAL rf Tangency portfolio σ min σ T σ X Stefano Lovo, HEC Paris Portfolio Theory 38 / 40

Exercise Suppose that S = {s 1, s, s 3, s f } {E [ r 1, E [ r, E [ r 3, rf } = {9%, 3%, 1%, 1%} The tangency portfolio is X T = {1.48, 0.3, 0.78, 0} and σ T = 18%. Questions: 1 What is E [ r T? (Ans.: 1%) What are β 1, β, β 3? (Ans.: 0.4, 1.1, 0.1) 3 What is the composition of the optimal portfolio for an investor with risk aversion A = 4? (Ans.: 1.543 in the tangency portfolio and 0.543 in the risk free asset, that is X P = {.84, 0.463, 1.05, 0.543}) Stefano Lovo, HEC Paris Portfolio Theory 39 / 40

Summary Definition of efficient portfolio. Mean-Variance investors choose efficient portfolios. An efficient portfolio is composed of the tangency portfolio and the risk free asset. if X is efficient, then E [ r X = rf + E[ r T r f σ T The tangency portfolio satisfies σ x E [ r i rf = β i ( E [ r T rf ) where β i = Cov[ r T, r i σ T Stefano Lovo, HEC Paris Portfolio Theory 40 / 40