part ii : portfolio theorycontents.kocw.net/kocw/document/2014/hanyang/chunghyunch... ·...

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Part II : PORTFOLIO THEORY

o Risk and Return

o Efficient Diversification

o CAPM and APT

o Efficient Markets

o Behavioral Finance and Technical Analysis

BUS403 Investments 2014_Spring Prof. Chung


1

Risk and Return

2

Investment returns

• The rate of return on an investment can be calculated as follows:

(Ending price-Beginning price + Cash dividend) HPR = ________________________

Beginning price

(HPR: Holding period return)

• initial price(P0): $1,000, $1,100 after one year(P1) and $100 dividend, the rate of return for this investment is:

($1,100 - $1,000+$100) / $1,000 = 20%.

Return and Risk

Capital gain yield

Dividend yield


3

Table 5.1 : Quarterly cash flows and rates of return of a mutual fund

(unit: mil.) 1st 2nd 3rd 4th

Assets(Beg.) 1.0 1.2 2.0 .8

HPR .10 .25 (.20) .25

TA (Before Net Flows) 1.1 1.5 1.6 1.0

Net Inflows 0.1 0.5 (0.8) 0.0

Assets(End) 1.2 2.0 .8 1.0

Return and Risk

Measuring returns over multiple periods


4

Arithmetic average

ra = (r1 + r2 + r3 + ... rn) / n

ra = (.10 + .25 - .20 + .25) / 4

= .10 or 10%

Geometric (time-weighted average return) average

(1+rg)n=[(1+r1) (1+r2) .... (1+rn)]

rg = {[(1.1) (1.25) (.8) (1.25)]} 1/4 - 1

= (1.5150)1/4 -1 = .0829 = 8.29%

Measuring returns over multiple periods

Return and Risk


5

Money (dollar)-weighted average rate of return

– IRR for the portfolio,

%17.4,)1(

1

)1(

8.0

)1(

5.0

i)(1

1.010

PV(CIF)PV(COF) or 0NPV when

432

iiii

Time Outflows($mil.) Inflows($mil.) Net Cash Flows ($mil.)

0 1 (1)

1 1.1+0.1 1.1 (0.1)

2 1.5+0.5 1.5 (0.5)

3 1.6+(0.8) 1.6 0.8

4 1 1

Return and Risk


6

Case2. Money (dollar)-weighted rate of return

%39.9,)1(

10$470$

)1(

5$

i)(1

$225$200

2

i

ii

Time Outflows($) Inflows($) Return(%)

0 200

1 225+225 225 + 5(Div.)# 15[=(225-200+5)/200]

2 470(235*2) + 10(Div. 5*2) 6.67[=(470-450+10)/450]

• Arithmetic mean: 10.84% [=(15+6.67)/2]

• Geometric mean: 10.76% [=(1.15)(1.0667) – 1]

# not reinvested

Return and Risk


time Outlay

0 $200 to purchase the first share

1 $225 to purchase the second share

Proceeds

1 $5 dividend received from the first share(not reinvested)

2 $10 dividend ($5 per share * 2 shares)

2 $470 received from selling two shares at $235 per share

7

Money (dollar)-weighted rate of return (cont’d)

IRR 9.39% vs. Arithmetic mean 10.84%

More weight given to the second year when more

money was invested. ($200 vs. $450)

Drawback as a tool for money managers’

performance measure

Clients determine when and how much money is

given.

Return and Risk


8

Conventions of rate of return quotation

Nominal rate (iNOM) – also called the contracted,

quoted, stated rate or APR (annual percentage rate).

APR=(periods in year) X (rate for period)

APR = 12 * 1% = 12%

Periodic rate (iPER) – amount of interest charged

each period, e.g. monthly or quarterly.

iPER = iNOM / m, where m is the number of

compounding periods per year. m = 4 for quarterly

and m = 12 for monthly compounding.

Return and Risk


9

Effective (or equivalent) annual rate (EAR): the

annual rate of interest actually being earned.

EAR=( 1+ iPER)Periods per year - 1

=( 1 + iNOM / m )m - 1

EAR for monthly return of 1%

EAR = (1.01)12 - 1 = 12.68%

An investor would be indifferent between an

investment offering a 12.68% annual return and

one offering a 1% monthly compounding return.

Return and Risk


10

ㅇ Nominal rate of return (APR) : 8%

ㅇ Effective rate of return

EARannually : (1+0.08)-1=8.00%

EARsemiannually : (1+0.08/2)2-1=8.16%

EARquarterly : (1+0.08/4)4-1=8.24%

EARmonthly : (1+0.08/12)12-1=8.30%

EARcontinuously compounding : e0.08-1=8.33%

Why is it important to consider EAR? An investment with monthly payments is different

from one with quarterly payments. Must put each return on an EAR basis to compare rates of return.

Return and Risk


11

ㅇ effective rate (cont’d)

Ex.) Which one has a higher effective interest rate?

A 3mo. T-bill selling at $97,645 (pure discount bond,

face value : $100,000) vs.

A coupon bond selling at par and paying a 10% coupon

semiannually.

(100,000-97,645)/97,645=r3=0.0241

EART=(1+r3)4-1=(1+0.0241)4-1=0.0999

vs.

EARC=(1+r6)2-1=(1+0.1/2)2-1 =0.1025

Return and Risk


12

Nominal (Real) rate: growth rate of the money (purchasing power)

Fisher effect: Exact

(1+명목)=(1+실질)*(1+ 예상 물가상승률) (1+R)=(1+r)(1+i), r = (R - i) / (1 + i) r=(9%-6%) / (1.06) = 2.83%

Fisher effect: Approximation nominal rate ≈ real rate + expected rate of inflation

(Ex) R = 9%, i = 6% R = r + i, r = R - i r= 9% - 6%=3%

Real vs. Nominal Rates

Return and Risk


13

Probability distributions

A listing of all possible outcomes, and the

probability of each occurrence.

Expected Rate of Return

Rate of

Return (%) 100 15 0 -70

Firm X

Firm Y

Return and Risk


14

Expected Rate of Return

15.0% (0.3) (-70%)

(0.4) (15%) (0.3) (100%) k

P k k

return of rate expected k

M

^

n

1i

ii

^

^

Return and Risk


15

Stand-Alone Risk: the standard deviation

2variancedeviation Standard

2222 )ˆ( kEkEkkE

21

2

22

n

1i

i

2^

i

(0.3)15.0) - (-70.0

(0.4)15.0) - (15.0 (0.3)15.0) - (100.0

P )k (k

M

=65.84%

Return and Risk


16

Normal Distribution with Mean of 12%

and St Dev of 20%

Return and Risk Return and Risk


17

Comments on standard deviation as a measure of risk

Standard deviation (σi) measures total, or stand-alone, risk.

Larger σi is associated with a wider probability distribution of returns.

The larger σi is, the lower the probability that actual returns will be close to expected returns.

Return and Risk


18

Example of measuring Expected return and risk

States (s) 1 2 3 4 5

Prob. (ps) 0.1 0.15 0.25 0.35 0.15

Returns (RA,s) -0.1 0.0 0.1 0.2 0.3

1187.0A

n

s

A sAsRE Rp1

13.0,][

n

s

A AsAs RERp1

22

0141.0)]([ ,

222

,

2 ))(( AAAsAA RERERERE

Return and Risk


19

Portfolio return and risk : 2 risky assets

BAABBABBAA

ABBABBAAP

BBAAp

xxxx

xxxx

RExRExRE

2

2

)()()(

2222

22222

Portfolio construction

E(Rp)= 0.24, σp= 0.2588

구분 E(R) σ ρ12 x

S1 15% 15% 0.2

40%

S2 30% 40% 60%

예)

* E(aX+bY)=a*E(X) + b*E(Y),

* Var(aX+bY)=a2*Var(X) + b2*Var(Y) + 2*a*b*Cov(X,Y)


20

N

i

N

j

ijjiP

N

j

jjp

xx

RExRE

1 1

2

1

)()(

1 2 n

1 x121

2 x2x1 21 xnx1 n1

2 x1x2 12 x22 2

2 xnx2 n2

N x1xn1n x2xn 2n xn2n

2

< Variance-Covariance Matrix >

E(R) σ x

S1 15% 15% 30%

S2 30% 40% 50%

S3 25% 30% 20%

E(Rp)= 0.245; σp= 0.2625

Ρ12

S1 S2 S3

S1 1 0.45 -0.3

S2 1 0.7

S3 1

ex)

Portfolio return and risk : N risky assets



21

※ Portfolio risk

ㅇ 공분산(Covariance)

- 두 확률변수(수익률)의 공조성(co-movement)

- 수익률이 체계적인 관계없이 움직이면 AB0

기대수익률의주식

수익률의주식상황에서

확률발생할상황이

jRE

BAjjiR

niip

RERRERp

RERRERpRERRERp

RERRERE

j

ij

i

BnBAnAn

BBAABBAA

BBAAAB

:)(

),( :

),,2,1( :

))())(((

))())((())())(((

)()(((

,

,,

2,2,21,1,1



22

※ Portfolio risk

ㅇ 상관계수(Correlation Coefficient)

- 두 수익률의 표준화된 공조성

AB>0 : 양의 상관관계 (rho)

AB=+1 : 완전 양의 상관관계

AB<0 : 음의 상관관계

AB=-1 : 완전 음의 상관관계

AB=0 : (AB=0) 수익률의 움직임에 체계적인 관계가 없음

11

AB

BA

ABAB



23

※ Portfolio risk

(예) 공분산, 상관계수 산출

상태(s) 1 2 3 4 5

확률(ps) 0.1 0.15 0.25 0.35 0.15

수익률(RA,s) -0.4 -0.1 0.2 0.5 0.8

수익률(RB,s) -0.15 -0.05 0.2 0.25 0.15

78726.0

137727.0;35623.0;1375.0)(;29.0)(

038625.0))())(((

))())((())())(((

)()(((

,,

2,2,21,1,1

BA

ABAB

BABA

BnBAnAn

BBAABBAA

BBAAAB

RERE

RERRERp

RERRERpRERRERp

RERRERE



24

Investor attitude towards risk Investor’s view of risk

Risk Averse

Risk Neutral

Risk Seeking

Risk aversion – assumes investors dislike risk and require

higher rates of return to encourage them to hold riskier

securities.

Risk premium=Exp. Rate of ret. – Rf

Risk-free rate: the rate you can earn from risk-free asset such as T-bills,

MMF, etc.

Excess return=Actual rate of return – Rf

Risk premium = Expected excess return

Return and Risk


25

W1 = 150 Profit = 50

W2 = 80 Profit = -20

1-p = .4 100

Risky Inv.

Risk Free T-bills (5%) Profit = 5

Risk Premium = 17 [=0.6*50+0.4*(-20)-5]

(compensation for the risk of the investment)

Risk Premium

Return and Risk


26

Asset allocation

자산배분과 증권선택

Asset Allocation

Security Selection

Bottom-up Top-down

• (risky vs. riskless)

• (fundamental analysis)

• (technical analysis)

• (portfolio analysis)

Risky portfolio (0.6) S1: 0.25 S2: 0.4 S3: 0.35

Riskless portfolio(0.4)

B1: 0.7 CD: 0.3

+

Complete portfolio S1: 0.15; S2: 0.24; S3: 0.21; B1:0.28;

CD:0.21

▷


27

무위험자산(Risk-free Asset)

ㅇ 가치가 변하지 않고 일정하게 유지되어 미래에 얻게 될 소득

의 크기가 확실한 자산

- 수익률의 표준편차 rf =“0”

- 채무불이행이 없는 (default risk free) 자산

예) 미국: T-Bill; 국내: 재정증권, 통화안정증권

Asset allocation


28

위험자산(Rp, p, )과 무위험자산(Rf, rf, 1- ) 결합

ㅇ 포트폴리오 기대수익률

E(Rc)= *E(Rp)+(1- )*Rf = Rf+ [E(Rp)-Rf]

ㅇ 포트폴리오 수익률의 위험도

2c=

2*2p + (1-)

2*2f + 2**(1-)*pf =

2*2p

c= *p => = c/p

예) E(Rp)=15%, p= 22%, Rf=7%

-> E(Rc)= 0.07 + 0.08*; -> 2c= 0.22

2 * 2, c= 0.22 * * E(aX+bY)=a*E(X) + b*E(Y),

* Var(aX+bY)=a2*Var(X) + b2*Var(Y) + 2*a*b*Cov(X,Y)

Asset allocation


29

위험자산(Rp, p, )과 무위험자산(Rf, rf, 1- ) 결합(계속)

ㅇ 기대수익률과 위험을 두 축으로 한 그래프

p=0.22

E(R)

E(Rp)=0.15

Rf=0.07

자본배분선 (CAL: Capital Allocation Line): 특정 위험포트폴리오 P와 무위험자산을 결합하여 얻을 수 있는 투자기회집합(Investment Opportunity Set)

- 기울기: [E(Rp)-Rf]/ p = (0.15-0.07)/0.22 = 0.36

. 위험보상률 (RVAR: reward-to-volatility ratio)

. Sharp ratio

Asset allocation


30

위험자산(Rp, p, )과 무위험자산(Rf, rf, 1- ) 결합(계속)

ㅇ 차입포트폴리오(Levered Portfolio)

- 투자자가 무위험수익률(Rf rate)로 차입 포트폴리오 구성

ㅇ 무위험자산 수익률로 차입할 수 없는 경우

p=0.22

E(R)

E(Rp)=0.15

E(Rf)=0.07

0.09

S(1)=0.36

S(>1)=0.27

차입

Asset allocation


31

Passive investment strategy and CML

소극적 투자전략(Passive Investment Strategy)

- 시장 평균성과 이상의 수익률을 기대하기 어렵다는 전제

- 증권분석을 하지 않고 포트폴리오 결정

- 시장포트폴리오(Market Portfolio) 선택: 단순히 증권시장을

구성하는 각 자산의 구성비에 따라 포트폴리오 구성

- 현실적으로 상장지수펀드(ETF) 등 지수연동 자산에 투자

=> CML선상에서 투자의사 결정: 무위험자산과 시장포트폴리오에

분산투자


32

0.1

):()()(

2

i

iiP

ijjiP

w

kkREwREtosubject

wwMinimize

상수

E(Rc)

c

k

efficient frontier

minimum variance portfolio

최적CAL=CML

Rf

CAL

Global minimum variance



Optimal CAL

Optimal risky portfolio =>Market portfolio

33

Capital Market Line (CML: 자본시장선)

- 무위험자산과 시장포트폴리오를 연결하는 직선으로 가장 효율

적인 포트폴리오의 기대수익률과 위험과의 관계 설명


E(R) CML

E(Rm)

Rf

m

M

Cm

fm

fC

RRERRE

)()(

현재소비의 희생에 대한 시간보상(reward for waiting)

Reward per unit of risk borne (Market price of risk)

※ Market portfolio - all the assets traded

in the market - proportion :mkt cap.(i)

/total market cap.


part ii : portfolio theorycontents.kocw.net/kocw/document/2014/hanyang/chunghyunch... ·...

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