portfolio management & diversification

8/9/2019 Portfolio Management & Diversification

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Portfolio Management &Diversication


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Diversication

Statistical Terms

Expected Return ()

Expected Return of a random variale E(R)!

is mean or t"e average of t"e sample or t"epopulation#


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Diversication

Statistical Terms

Ris$ (%)

n statistics and proailit t"eor standard

deviation (represented t"e smol sigma %) s"o*s"o* muc" variation or dispersion exists from t"eaverage (mean) or expected value#

+ lo* standard deviation indicates t"at t"e data

points tend to e ver close to t"e mean, "ig"standard deviation indicates t"at t"e data points arespread out over a large range of values#


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Significance of & %


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Diversication

Statistical Terms

-ariance (%./)

T"e variance is also e0uivalent to t"e

second movement of t"e proailit distriutionfor 1#

T"e variance is tpicall designated as -ar(1) simpl%/ (pronounced 2sigma s0uared2)#


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Diversication

Statistical Terms

3orrelation coe4cient (r or R"o)

r is a measure of t"e linear correlation (dependence)

et*een t*o variales 1 and 5 giving a value et*een67 and 87 inclusive#

67 indicates perfect positive correlation# 97 indicatesperfect negative correlation#

Purel from t"e diversication point of vie* lesser t"enumer or closer to 97 is etter#


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Diversication

Statistical Terms

3o9variance (r:%7:%/)

3ovariance is a measure of "o* muc" t*o random

variales c"ange toget"er

T"e sign of t"e covariance t"erefore s"o*s t"etendenc in t"e linear relations"ip et*een t"evariales# i#e# t"e variales tend to s"o* similare"avior t"e covariance is positive & vice versa#


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Diversication enets

Statistical Explanation

3onsider portfolio of / assets

Expected return

E(R) ! *776 *//

;"ere *7 & */ are t"e *eig"tings of t"e individual securities inportfolio

mplied Ris$

port= square root ( sum (w1^2*1^2+ w2^2*2^2 + 2*w1*w2*

Covariance(1,2))

Where covariance(1,2)= 1*2*r(1,2)

At any r (1,2) ess than 1 you wi !et re"uction in ris# $i$e$ As on! assecurities are not per%ecty positivey correate" you wi !et

re"uction in ris#$


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Diversication enets

Expected Return

mplied Ris$

mplied Ris$


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Diversication enets

Practical Explanation

t is reall unli$el t"at / securities *it"in an asset classor across t"e asset classes *ill e perfectl correlated#

&$!$ 'oar appreciation wi e positive %or most

companies ut it wi e ne!ative %or auto companies$

avin! sai" that "e!ree o% positivity ne!ativity wi e"i-erent %or "i-erent companies within the sector aswe$

.o more than i#ey that n%osys/C. , 00 ata motorswi e positivey correate" ut coecient o% correationetween them won3t e e4acty 1$

.impe e4panation ein! there wi e simiar companies

ut not e4acty same company$


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Modern Portfolio T"eor


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Modern Portfolio T"eor (MPT)

+ssumptions

7#Rational nvestors

/#+sset Returns are normall distriuted#

#?o transaction cost is involved

@#Expected returns diAerent investors are same#

B#nvestors can orro* or lend mone at ris$ free rate (Rfr)

C#?o investor is large enoug" to inuence t"e mar$et price#

n real *orld none of t"ese assumptions are true ut still MPT "as lot ofexplanation to*ards *a ris$9return trade oA "appens in real *orld#


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Modern Portfolio T"eor (MPT)

arr Mar$o*itF introduced MPT in a 7G>/

Explanation of MPT

. MPT is an investment t"eor *"ose purpose is to maximiFe aportfolioHs expected return altering and selecting t"eproportions of t"e various assets in t"e portfolio#

. MPT explains "o* to nd t"e est possile diversication#

. f investors are presented *it" t*o portfolios of e0ual value t"atoAer t"e same expected return MPT explains "o* t"e investor*ill prefer and s"ould select t"e less ris$ one#

. nvestors assume additional ris$ onl *"en faced *it" t"eprospect of additional return#

. n rief MPT explains "o* investors can reduce overall ris$ "olding a diversied portfolio of assets#


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E4cient Irontier

Jne of t"e mostimportant and *idelused concept ofModern PortfolioT"eor

Ever possilecomination of assetsplotted on grap"#

Plots return K vs Ris$K (Standard Deviation)

Jptimal portfolio lieson t"e e4cient frontiercurve (paraola)


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3apital Mar$et Line (3ML)


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3apital Mar$et Line (3ML)

19 axis is Ris$ or Standarddeviation(%)

59 axis is Expected Return orE(r)

3ML is t"e tangent linedra*n from t"e point of t"eris$9free asset tot"e feasile region for ris$assets#

T"e tangenc point Mrepresents t"e mar$etportfolio so named since

all rationalinvestors (minimum variancecriterion) s"ould "old t"eirris$ assets in t"e sameproportions as t"eir *eig"tsin t"e mar$et portfolio#


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3ML E0uation

;"ere

E(r) Expected Return from t"e stoc$N Portfolio

E(rM)-Expected Return from t"e mar$et or t"e index

rf or Rfr Ris$ free Rate

% Standard deviation of t"e stoc$NPortfolio

%M Standard deviation of t"e mar$et


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3ML and S"arpe Ratio

Mat"ematicall

+ll portfolioHs on 3ML satisf S"arpe Ratio as stated elo* ine0uation (/)

E(r) Expected Return from t"e stoc$

E(rM)9 Expected Return from t"e mar$et or t"e index

rf or Rfr Ris$ free Rate

% Standard deviation of t"e stoc$

%M Standard deviation of t"e mar$et

E0uation (/)


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Signicance of 3ML and S"arpe Ratio

+ stoc$ pic$ing rule of t"um is to u assets *"ose S"arperatio *ill e aove t"e 3ML and sell t"ose *"ose S"arpe ratio*ill e elo* 3ML#

i#e# nvest in portfolio *"ic" oAer "ig"er returns for given ris$and sell t"ose portfolioHs *"ic" oAer lesser returns for givenris$#

o*ever as per e4cient mar$et "pot"esisit follo*s t"at itOsimpossile to eat t"e mar$et and suc" underperforming oroutperforming portfolioHs s"ould not exist#

f t"ere are an suc" outperforming stoc$sNportfolioHsinvestors *ould u t"em increasing t"eir prices and reducing

t"eir future excess returns and ring t"em ac$ on 3ML#

3onversel f t"ere are an suc" underperformingstoc$sNportfolioHs investors *ould sell t"em reducing t"eirprices and increasing t"eir future excess returns and ringt"em ac$ on 3ML#


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Securit Mar$et Line (SML)


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Securit Mar$et Line (SML)

19 axis is Sstematic

Ris$ or eta(Q)

59 axis is ExpectedReturn or E(r)

SML is t"erepresentation oft"e 3apital assetpricing model#

t displas t"eexpected rate ofreturn of an individualsecurit as a functionof sstematic non9diversiale ris$


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SML E0uation

;"ere

E(Ri) Expected Return from t"e stoc$


rf or Rf Ris$ free Rate

Q non9diversifiale or sstematic ris$


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Q (eta) and itHs significance

+ measure of t"e volatilit or sstematic ris$ of a securit or a portfolio incomparison to t"e mar$et as a *"ole#

eta is used in t"e capital asset pricing model (3+PM) a model t"atcalculates t"e expected return of an asset ased on its eta and expectedmar$et returns#

Statisticall eta is calculated using regression analsis# t is t"e slope oft"e line in case of a linear regression#

+ eta of 7 indicates t"at t"e securitOs price *ill move in line *it" t"emar$et# E#g# f mar$et goes up 7K stoc$ *ill go up 7K and viceversa#

+ eta of more t"an 7 indicates t"at t"e securitOs price *ill e morevolatile t"an t"e mar$et# E#g# f mar$et goes up 7K stoc$ *ill go up more t"an 7K and vice versa# Tpicall stoc$s from sectors suc" as

metals capital goods automoiles real estate an$ing etc *ill "ave etaof more t"an 7#

+ eta of less t"an 7 indicates t"at t"e securitOs price *ill e less volatilet"an t"e mar$et# E#g# f mar$et goes up 7K stoc$ *ill go up lesst"an 7K and vice versa# Tpicall stoc$s from sectors suc" as IM3P"armac T etc *ill "ave eta of less t"an 7#


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SML and Trenor Ratio

Mat"ematicall

+ll portfolioHs on SML satisf Trenor Ratio as stated elo* in e0uation(/)

;"ere




Qi non9diversifiale or sstematic ris$

QM sstematic ris$ for mar$et is e0ual to 7#

E0uation (/)


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Signicance of SML and Trenor Ratio

+ stoc$ pic$ing rule of t"um is to u s"ares *"ose Trenorratio *ill e aove t"e SML and sell t"ose *"ose Trenorratio *ill e elo* SML#

i#e# nvest in s"ares *"ic" oAer "ig"er returns for givensstematic ris$ and sell t"ose s"ares *"ic" oAer lesserreturns for given sstematic ris$#

o*ever as per e4cient mar$et "pot"esisit follo*s t"at itOsimpossile to eat t"e mar$et and suc" underperforming oroutperforming s"ares s"ould not exist#

f t"ere are an suc" outperforming stoc$s investors *ouldu t"em increasing t"eir prices and reducing t"eir future

excess returns and ring t"em ac$ on SML#

3onversel f t"ere are an suc" underperforming stoc$sinvestors *ould sell t"em reducing t"eir prices andincreasing t"eir future excess returns and ring t"em ac$on SML#


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3apital +sset Pricing Model(3+PM)


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istor of 3+PM Model

T"e 3+PM *as introduced ac$ Trenor (7G@7

7G@/) ;illiam S"arpe (7G@=) o"n Lintner (7G@>)and an Mossin (7G@@) independentl#

T"is *as done *it" uilding on t"e earlier *or$of arr Mar$o*itF on diversication and modernportfolio t"eor# S"arpe Mar$o*itF and Merton

Miller Uointl received t"e 7GG ?oel Memorial PriFe inEconomics for t"is contriution to t"e eld of nancialeconomics#

Iisc"er lac$ (7GB/) developed anot"er version of3+PM called lac$ 3+PM or Fero9eta 3+PM t"at doesnot assume t"e existence of a ris$less asset#

T"is version *as more roust against empirical testingand *as inuential in t"e *idespread adoption of t"e3+PM#


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3+PM E0uation

;"ere




Qi non9diversifiale or sstematic ris$


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Signicance of 3+PM Model

3+PM is used to determine a t"eoreticall

appropriate re0uired rate of return of an asset fort"e given sstematic ris$#

T"e model ta$es into account t"e assetOs sensitivitto non9diversiale ris$ (also $no*n as sstematic

ris$ or mar$et ris$) often represented t"e0uantit eta (Q) in t"e financial industr as *ell ast"e expected return of t"e mar$et and t"e expectedreturn of a t"eoretical ris$9free asset#

3+PM Vsuggests t"at an investorHs cost of e0uitcapital is determined eta# ig"er t"e eta"ig"er is t"e cost of e0uit and vice versa#

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