THE MOT AND VENTURE BUSINESS

Prof. Takao Ito, Doctor of Economics, PH.D. of Engineering, Graduate School of Engineering, Hiroshima University

Thursday, October 16, 2014

TOPIC 8 PORTFOLIO MANAGEMENT

TOPIC 8 CAPM BASICS The ROI A performance measure used to

evaluate the efficiency of an investment or to compare the efficiency of a number of different investments.

The P/E Ratio of a stock (price-to-earnings ratio, "P/E", "PER", "earnings multiple," or simply "multiple")

It is a measure of the price paid for a share relative to the annual net income or profit earned by the firm per share.

Basic formula of the ratio of a stock return

0

01

P

DPPR

P0……Initial priceP1……trading priceD…….Dividend yield

Problem: Suppose you invest $10,000 in Toyota, and $30,000 in Nissan international stock. You expect a return of 10% for Toyota and 16% for Nissan. What is the expected return for your portfolio?

Solution: You have $40,000 invested in total, so your portfolio weight are 10,000/40,000=25% in Toyota and 30,000/40,000=75% in Nissan. Therefore, the expected return on your portfolio is

%5.1416.0%7510.0%25

][][)(

NNTTP RExRExRE

In the case of Stock A

Events Probability Ratio of Return

BetterNormalWorse

0.250.5

0.25

20%12%

4%

Expectation value of stock A

l

kkiki PRRE

1

)(

Expectation value of stock AExpectation value of stock A

l

kkiiki PRERR

1

2))(()(

Then we get

%12425.0125.02025.0)( ARE

%32

)124(25.0)1212(5.0)1220(25.0)( 222

AR

Assume that stock B

Event Probability Ratio of Return

BetterNormalWorse

0.250.5

0.25

7%7%

11%

%81125.075.0725.0)( BRE

%3

)811(25.0)87(5.0)87(25.0)( 222

BR

In the case of two stocks: Stock A and stock B, the expectation value and risk

;1 BA XX

);()()( BBAAi REXREXRE

),(2)()(

)(

2222BABABBAA

i

RRCovXXRXRX

R

Then we getBAp XXRE 812)(

BABA

BABAp

XXXX

XXXXR

16332

)8(2332)(

22

22

Covariance of A and BCovariance of A and B

8

25.0)811)(124(5.0)87)(1212(25.0)87)(1220(

)]}()][({[),(

BBAABA RERRERERRCov

Correlation ratio of A and B

816.0332

8

)()(

),(

BA

BAAB RR

RRCov

Risk of stocks A and BRisk of stocks A and B

32251

)1(16)1(332

16332)(

2

22

22

AA

AAAA

BABAp

XX

XXXX

XXXXR

Best answer (differentiate)

%5.78;215.0102/22 BA XX

0)(

A

p

dX

Rd

032251

22102

2

1)(

2

AA

A

A

p

XX

X

dX

Rd

Then we getThen we get

M

E(Rp)

RF

σ(Rp)

G

H

)()1()()1()()( SFSFP REXRXREXRXERE

)()1(

)0)(()()1(

)()()1(2)()1()()(

22

2222

S

FS

SFFSSFP

RX

RRX

RRXXRXRXR

Then we getThen we get

)()(

)()( p

M

FMFP R

R

RRERRE

RFM……capital market line

STOCK J AND STOCK MARKET

)()1()()( MjjjP REXREXRE

),()1(2)()1()()( 2222MjjjMjjjP RRCovXXRXRXR

M

E(Rp)

RF

σ(Rp)

G

H

),()(

)()(

2 MjM

FMFj RRCov

R

RRERRE

)(

),(2

M

Mj

R

RRCov

])([)( FMjFj RRERRE

Let

We can easy to get

RF

COV(Rj,RM)

E(R ｊ )

),()(

)()(

2 MkM

FMFk RRCov

R

RRERRE

])([)( FMkFj RRERRE

RF

β ｋ

E(R ｋ )

Situa-tions

Prob.

Ratio of returns of market portfolio

Ratio of returns of each projects

Project １

Project ２

Project３

Very good

0.1 0.2 0.4 0.6 0.2

Better 0.2 0.15 0.3 0.4 0.15

Normal

0.4 0.1 0 0.1 0.1

Worse 0.2 0 -0.1 -0.1 0

Worst 0.1 -0.1 -0.2 -0.4 -0.05

E(RM)=0.1×0.2 ＋ 0.2×0.15 ＋ 0.4×0.1 ＋ 0.2×0 ＋ 0.1× （ -0.1 ） ＝ 0.08

σ ２（ RM ）＝ 0.1×(0.2-0.08) ２＋ 0.2×(0.15-0.08) ２ ＋ 0.4×(0.1-0.08) ２＋ 0.2×(0-0.08) ２ ＋ 0.1×(-0.1-0.08) ２ =0.0071

　 ① ② ③ ④ ⑤ ⑥

　 Situations Prob.:PRate of

returns:RK

P*R ｋ

[R ｋ－ E(Rk) ］ [RM － E(R

M) ］ ②×⑤

Project １

1 0.1 0.4 0.04 0.0408 0.00408

2 0.2 0.3 0.06 0.0168 0.00336

3 0.4 0 0 -0.0012 -0.00048

4 0.2 -0.1 -0.02 0.0128 0.00256

5 0.1 -0.2 -0.02 0.0468 0.00468

　 E(R1)= 0.06 COV(R1,RM)= 0.0142

　 　 　 　 　 　 　

Project ２

1 0.1 0.6 0.06 0.0648 0.00648

2 0.2 0.4 0.08 0.0238 0.00476

3 0.4 0.1 0.04 0.0008 0.00032

4 0.2 -0.1 -0.02 0.0128 0.00256

5 0.1 -0.4 -0.04 0.0828 0.00828

　 E(R2)= 0.12 COV(R2,RM)= 0.0224

　 　 　 　 　 　 　

Project ３

1 0.1 0.2 0.02 0.0168 0.00168

2 0.2 0.15 0.03 0.0063 0.00126

3 0.4 0.1 0.04 0.0008 0.00032

4 0.2 0 0 0.0048 0.00096

5 0.1 -0.05 -0.005 0.0198 0.00198

　 E(R3)= 0.085 COV(R3,RM)= 0.0062

20071.0

0142.0

)(

),(2

11

M

M

R

RRCov

15.30071.0

0224.0

)(

),(2

22

M

M

R

RRCov

87.00071.0

0062.0

)(

),(2

33

M

M

R

RRCov

03.005.008.0])([ FM RRE

11.0

)05.008.0(205.0])([)( 11

FMF RRERRE

1445.0

)05.008.0(15.305.0])([)( 22

FMF RRERRE

0761.0

)05.008.0(87.005.0])([)( 33

FMF RRERRE

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

1 2 3 4

THANK YOU FOR YOUR ATTENTION!