Investments and Financial Markets

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1 Arisoy, Hamo ad Raposo Ivestmets ad Fiacial Markets Retur ad risk: Portfolio maagemet ad fiacial theory UE 06 Master SOM Retur ad Risk Toolbox Idices ad ETFs (Chapters 4 & 5) Rate of retur (Chapter 7) Volatility of a sigle asset (Chapter 8) Volatility of a portfolio (Chapter 0) Sesitivity ad diversifiable risk (Chapter ) Portfolio maagemet ad fiacial theory 6. Attitude towards risk ad the risk premium (Chapter ) 7. Naïve diversificatio (Chapter 3) 8. Optimal diversificatio/efficiet frotier (Chapter 4) 9. Capital asset pricig model (Chapter 5) Exercises Uiversité Paris-Dauphie

Arisoy, Hamo ad Raposo Attitude towards risk ad the risk premium. Sait Petersburg's paradox. Icreasig ad cocave utility fuctio 3. The risk premium ad its determiats 4.

2 Arisoy, Hamo ad Raposo Attitude towards risk ad the risk premium. Sait Petersburg's paradox. Icreasig ad cocave utility fuctio 3. The risk premium ad its determiats 4. Utility fuctio: examples Chapter 3 St. Petersburg paradox The game starts at ad the pot is doubled every time the toss is a head. The gambler is paid euros if tail appears for the first time at the th draw (Beroulli, 738) What would be the fair price to pay the casio to eter ito this game? 4 Uiversité Paris-Dauphie

Arisoy, Hamo ad Raposo Cocave utility fuctio Let's cosider a lottery i which a player ca either ear C for sure or a risky amout whose expected value is equal to C The player's behavior ca be

3 Arisoy, Hamo ad Raposo Cocave utility fuctio Let's cosider a lottery i which a player ca either ear C for sure or a risky amout whose expected value is equal to C The player's behavior ca be described by a utility fuctio (U) that allows to covert moetary gais / losses ito satisfatio U( C) ³ U ( C - h ) + U( C + h ) Utility U( C) - U( C - h) ³ U( C + h) - U( C) u(c+h) u(c) U(C - h) + U(C + h) u(c-h) c-h c c+h Future cosumptio Which implies that the utility fuctio is cocave Beroulli (738) Vo Neuma ad Morgester (947) 5 Idifferece curve (iso-utility) A idifferece curve is a combiatio of portfolios that yields the same level of utility to its holder, regardless of their retur ad risk characteristics E(R) s(r) See Sectio. 6.3 Ituitively, a icrease i the level of risk should be accompaied by a icrease i expected returs i such a way that the expected utility remais uchaged. The margial rate of substitutio (MRS) betwee expected retur ad risk is positive, resultig i covex idifferece curves. The higher the curve is the higher the utility gaied. 6 Uiversité Paris-Dauphie 3

4 Arisoy, Hamo ad Raposo Ordial utility fuctio Here, the utility fuctio is used to take a decisio about the rakig of various competig projects. What matters is the rakig rather tha the utility level. Ay liear traformatio of a utility fuctio (e.g. A + B*U(W), where A ad B are costats), will yield differet values for the utility, but preserves rakig A quadratic utility fuctio W-(/)W² yields the same rakig as aother quadratic utility fuctio computed as 3+W-W² (A=3 ad B=). 7 The risk premium From the defiitio of risk aversio Let p deote the value of the risk premium that oe has to pay to the gambler for him to accept participatig i the game ~ W ( ) ~ E W p : the certaity equivalet of What is the value of the risk premium? Assume h is small compared to the level of cosumptio C. It implies that the risk premium p is small too. Equivaletly, if the portfolio risk is small, W(ω) is close to E(W) for all states of ature ω We are lookig for p such that: ~ ~ [ ( ) ] = ( ) [ ] U E W p E U W % % ( ) ( ) U E W E U W th e ag e t is b etter o ff w ith a risk free b et co m p ared w ith a risk y b et w ith sam e ex p ected v alu e 8 Uiversité Paris-Dauphie 4

5 Arisoy, Hamo ad Raposo Prefers a secure amout U[ E ( W ~ ) ] What is the amout we shall substract for the alteratives to become equivalet? U rather tha a lottery with the same expected gai [ ( )] E U W ~ ~ ~ [ E( W ) p] = E[ U ( W )] 9 Risk premium ad its determiats Approximatio of Pratt(964) % ( ( ω )) U W Taylor expasio aroud E(W) ~ ( ( ) ~ ~ U W ω ) U( E[ W] ) ( ) ~ ~ + ( W ω E[ W] ) U ( E[ W] ) ~ ~ U + ( W( ω ) E[ W] ) Takig the expectatio ( [ ]) E[ U( W ~ ( ))] U( E[ W ~ ]) ( W ~ ) U E W ~? ω + σ Approximatio of U E W ~ p U E W ~ p U E W ~ [ ( ) ] ( ) p must be such that : U [ ( ~ ) ] [ ( ~ = )] Hece ~ U ( E[ W] ) ~ p = ( W ) ~ σ U ( E[ W] ) [ ] [ ( )]. p >0 if U is icreasig ad cocave. p depeds + o the volatility of W 3. p depeds + o the curvature of U ~ ( E[ W] ) 0 Uiversité Paris-Dauphie 5

Arisoy, Hamo ad Raposo Absolute risk aversio (ARA) «Absolute» i the sese that the aversio is computed for a give level of wealth Absolute risk aversio decreases (or stays costat) whe the aget's

If her relative risk aversio is costat this percetage will be left uchaged whereas if her relative risk aversio is decreasig, the percetage will icrease.

6 Arisoy, Hamo ad Raposo Absolute risk aversio (ARA) «Absolute» i the sese that the aversio is computed for a give level of wealth Absolute risk aversio decreases (or stays costat) whe the aget's wealth icreases Relative risk aversio (RRA) A ivestor ows ad ivests half of his wealth i stocks. How will this percetage evolve whe his wealth is equal to ? If her relative risk aversio is costat this percetage will be left uchaged whereas if her relative risk aversio is decreasig, the percetage will icrease. A CRRA utility fuctio iduces a periodic rebalacig of ivested fuds. Risk aversio U ''(E[W]) ARA = U '(E[W]) RRA = W ARA Quadratic utility fuctio U b ( W ) = W W ( b > 0) U(W) udesirable /b W However, absolute risk aversio icreases with wealth (!) Uiversité Paris-Dauphie 6

Arisoy, Hamo ad Raposo Quadratic utility fuctio () However it has ice properties such as: The expeted utility ca be expressed usig oly the first two momets of the probability desity fuctio of the

7 Arisoy, Hamo ad Raposo Quadratic utility fuctio () However it has ice properties such as: The expeted utility ca be expressed usig oly the first two momets of the probability desity fuctio of the wealth (irrespective of its shape), See 6.4 Ivestors utilitity: Icreases with expected wealth/cosumptio Decreases with risk (measured by the variace of expected wealth/cosumptio) 3 Quadratic utility fuctio ad idifferece curves (curves of iso-satisfactio) Right pael, 0.5 ad Cost = [.75,.875] Oly the covex part ( is preseted. The satisfactio icreases goig to the orth west of the risk-retur payoff diagram..875 Cost=.75 4 Uiversité Paris-Dauphie 7

8 Arisoy, Hamo ad Raposo Expoetial utility bw U( W) = e ( b 0) bw U ( W) = b e bw U ( W) = b e AAR = b (CARA) R ( ) A W = 0 W Nice property If wealth is ormally distributed, the expected utility ca be expressed usig oly the first two momets. (See 6.) 5 Naïve diversificatio. Diversificatio ad average covariace. Simulatig aïve diversificatio 3. Actual diversificatio. Iteratioal diversificatio. Compositio of household portfolios 4. Efficiecy of aïve diversificatio. Diversificatio ad volatility 5. The determiats of firm-specific risk Chapter 3 6 Uiversité Paris-Dauphie 8

9 Arisoy, Hamo ad Raposo Naïve diversificatio ad average covariace What is the expressio of the variace of a portfolio whose umber of assets goes to ifiity? i= j= σp = xixjσi, j = xixiσi, i + xixjσi, j i= j= i= i j 4 34 j i α α V (x i = ad V = Max σ i, i ) i Limα = β 0 ( ) σ σ β = = σ { Lim= Lim= σ Whe the umber of stocks is large, the variace of the portfolio ca be approximated by the average covariace of the costituet stocks. 7 Diversificatio (with radom selectio of stocks) Sample: 04 Eurozoe stocks from Ja 004 to Jue 007 Aualized volatility 75% of obs 95% of obs media No. of stocks i the portfolio 8 Uiversité Paris-Dauphie 9

10 Arisoy, Hamo ad Raposo The effect of diversificatio σ Diversifiable risk Systematic risk Number of stocks 5 stocks Volatility i % of idividual risk The average volatility of a -stock portfolio is ormalized to 00% Aualized volatility Volatility (i %) 004-0/ Aualized volatility /0-0 Number of stocks i the portfolio Asymptot is movig Variatios i the risk of the idex Variace-covariace matrix chages too Uiversité Paris-Dauphie 0

11 Arisoy, Hamo ad Raposo Iteratioal diversificatio Iteratioal diversificatio Several stocks i the CAC 40 idex operate busiesses outside the home market Currecy risk Home bias The home bias idicator (last colum) is the lowest for the UK ad the US Coutry Worldwide Foreig stocks weight (%) (%) Home bias US Japa UK Frace Deutschlad Switzerlad Spai Lau, Ng & Zhag (00) Based o the compositio of ivestmet fuds over The home bias colum reports the value of l[(00 Foreig stocks) / worldwide weight] Correlatio betwee idices ad volatility of Cac40 idex Volatility i Paris 35% 33% 3% 9% 7% 90% 80% 70% 60% Correlatio Cac40-DJIA 5% 3% 50% % 9% 7% Volatility CAC40 Correlatio CAC40-DJIA 40% 30% 5% % M J Jt A S O N D J F M A M J Jt A S O N D J F M Uiversité Paris-Dauphie

12 Arisoy, Hamo ad Raposo Optimal diversificatio ad the efficiet frotier. Locus of feasible portfolios. Determiig the compositio of a frotier portfolio 3. The risk-free asset 4. Markowitz + risk-free asset Chapter 4 3 Efficiet frotier (U.S, 96-0) Bods MVP The locus of efficiet portfolios correspods to the frotier above the MVP With assets, the possible portfolios lie dow o a parabola Whe the o. of assets icrease, a surface (i gray) icludig all possible portfolios appears The curve (Markowitz frotier ) o the top (with bald ad dashed lies) covers all possible portfolios; Oe characteristic poit o the frotier is the MVP (miimum variace portfolio) The efficiet frotier is the part of the curve that lies above the MVP. 4 Uiversité Paris-Dauphie

13 Arisoy, Hamo ad Raposo Compositio of a portfolio Remark : Istead of ivestig i EON, a ratioal ivestor would prefer ivestig i a portfolio with same risk but twice expected retur as EON For Markowitz, if a ivestor is risk-averse (with icreasig ad cocave utility fuctio ad covex idifferece curves) The ivestor will choose the portfolio which lies o the efficiet fotier ad which is taget to her idifferece curve. This is the portfolio that maximizes ivestor s utility MVP 5 Compositio of the optimal portfolio Objective fuctio Max å x i E( R i ) {xi } i = subject to åå x x s i = j = åx i = i i = j i, j =s The objective: The ivestor has to determie the compositio of ivestmet weights (vector x) that maximizes the expected retur while achievig the variace s² she has selected The solutio: Costraied optimizatio Þ Lagragia We form Lagragia L with m ad q/ as the correspodig Lagrage multipliers ü ì ì ü L = å x i E( R i ) + q ís - å å x i x j s i, j ý + m í - å x i ý î i = i = j = i = î þ þ 6 Uiversité Paris-Dauphie 3

14 Arisoy, Hamo ad Raposo Optimal portfolio compositio Solutio L x i ( ) = E R θ x σ µ = 0 ( i, i =, L, ) i j i, j j= L = σ x ix jσ i, j = 0 θ i= j= L = x i = 0 µ i= st order coditios j= Partially differetiatig L with respect to each compoet of vector (x) θ ad µ yields (+) equatios. Theorem: a ecessary coditio for portfolio efficiecy is that there exist umbers µ et θ such that: E( R i ) = µ + θ x jσ i, j = µ + θ x jcov( R i ; R j) = µ + θcov R i ; x jr j j= j= ( i ) = µ + θ ( i ; p ) E R Cov R R A ecessary coditio for portfolio efficiecy is that there exists a liear relatioship betwee the expected retur of each costituet asset ad its covariace with portfolio p 7 Necessary coditio for portfolio efficiecy E(R i ) ( i ) = µ + θ ( i ; p ) E R Cov R R θ Cov(R i,r p ) measures the margial risk of asset i. The cotributio ca be either positive or egative µ is the expected retur of a portfolio whose cotributio to the risk of portfolio p is zero µ Cov(R i ;R p ) θ >0 is the icremetal retur ivestors require to hold a asset that icreases the risk of portfolio p by uit (relative risk aversio) Dumas ad Allaz (995, p 67) 8 Uiversité Paris-Dauphie 4

15 Arisoy, Hamo ad Raposo Markowitz's efficiet frotier E(R p ) MVP Efficiet portfolio for a give level of risk The slope of the tagecy lie is equal to 0,5θ σ²(r p ) Solvig the problem (equatios to 3), allows to establish the equatio of the curve for the locus of efficiet portfolios (bold curve) It correspods to a parabola (if the horizotal axis is expressed i terms of variace) or a hyperbola (if the horizotal axis is expressed i terms of stadard deviatio) 9 Portfolio risk ad retur i the presece of a risk-free asset R f R a R p risk-free asset portfolio with risky assets oly resultig portfolio Let portfolio p comprise x% i risky asset (a) ad (-x)% i risk-free asset. Expected retur of p : Variace of p : E σ ( R p ) = ( x ) R f + x E ( R a ) ( R ) = x σ ( R ) [ σ ( R ) = 0, Cov ( R, R ) = 0] p a f f a Tobi (958) 30 Uiversité Paris-Dauphie 5

portfolio i this regio Optimal M The higher the slope the higher the

16 Arisoy, Hamo ad Raposo Risk-free borrowig X=0 Risk-free ledig R a X> R f R p X= X=0,5 3 Markowitz + risk-free asset () E(R) Impossible: o ivestable portfolio i this regio Optimal M The higher the slope the higher the ivestor's satisfactio A R f MVP σ(r) Possible but ot optimal 3 Uiversité Paris-Dauphie 6

17 Arisoy, Hamo ad Raposo Markowitz + risk-free asset () E(R) The locus of optimal portfolios is ow a straight lie goig through R f ad M At poit M : 00% ivested i risky assets At Rf: 00% risk-free asset Betwee Rf ad M: ledig Beyod M: borrowig A M Ivestor's satisfactio moves from A to A (higher idifferece curve) R f PVM σ(r) 33 The Capital Asset Pricig Model (CAPM). -fud separatio theorem. CAPM ad the Security Market Lie (SML) 3. Provisioal SML ad portfolio maagemet. Market timig ad stock pickig. Computig the SML 3. Time-varyig SML ad market risk premium Chapter5 Sharpe (964) Liter (965) 34 Uiversité Paris-Dauphie 7

18 Arisoy, Hamo ad Raposo -fud separatio theorem What is the compositio of portfolio M? As aticipatios are homogeeous, the efficiet frotier as well as M are the same for all ivestors Market equilibrium implies that M icludes all risky assets: M correspods to the market portfolio. Stocks are selected idepetly of agets' risk aversio separatio portfolios must be perfectly diversified icrease i risk aversio does ot imply lower betas for selected stocks selected portfolios differ i their weights ot i their compositio. If a aget is highly risk averse, he will icrease the proportio of the risk-free asset ad decrease the proportio of the market portfolio CAPM ad Security Market Lie ( ) = + θ ( ; ) E R R Cov R R i f i M E(R i ) E(R m ) µ=r f θ Market At poit M : E R θ = ( m ) σ ( R ) m R f [ ] ( ) β ( ) E R = R + E R R i f i m f Security Market Lie (SML) Cov(R m ;R m ) Cov(R i ;R m ) Dumas ad Allaz (995, p 84-85) 36 Uiversité Paris-Dauphie 8

19 Arisoy, Hamo ad Raposo The SML ad the price of risk [ ] ( ) β ( ) E R = R + E R R i f i m f E(R) Rm: retur o market portfolio Oly systematic risk is priced Beta correspods to stock risk per uit of market risk The slope of the SML correspods to the price of risk E(R m )-R f R f β 37 SML ad portfolio maagemet SML slope ad market timig Ivestors do ot like risk If takig risk ivolves o retur (the slope of the SML is small) the ivestor should ivest i the riskfree asset If the market price of risk is high (the slope of the SML is large), the ivestor should ivest i risky assets Deviatios from the SML ad stock pickig 38 Uiversité Paris-Dauphie 9

20 Arisoy, Hamo ad Raposo SML ad stock pickig P D t 0 = t t= ( + IRR) IRR, which is ukow, correspods to market expectatio regardig future returs D: provisioal stream of cash flows (divideds) P 0 : curret tradig price Negative relatioship betwee IRR ad P If a stock lies above the SML: For this stock to revert to a retur that is cosistet with the CAPM, its IRR must decrease (P must icrease) Ivestors will demad (sell) stocks which lie above (below) the SML because they are uder(over)valued. 39 Provisioal SML ad portfolio maagemet Retur 8% August 996 6% Slope= 3.97 R²=33,5% 4% % 0% 8% ERR=3.9% Mouliex IRR=.07% Risk=.7 6% 0,0 0,5,0,5,0 Risk,5 40 Uiversité Paris-Dauphie 0

21 Arisoy, Hamo ad Raposo Spread relative to Germa Bud (April 03) Relative spread + σ Spread relative to Germa Bud STOXX 50 Idex Eurozoe Relattive spread to Bud (%) STOXX 50 Idex Eurozoe Relative spread - σ 4 Uiversité Paris-Dauphie

Models of Asset Pricing

Models of Asset Pricing APPENDIX 1 TO CHAPTER 4 Models of Asset Pricig I this appedix, we first examie why diversificatio, the holdig of may risky assets i a portfolio, reduces the overall risk a ivestor faces. The we will see