Profit and Risk Profit of a Stock

CHAPTER 4: PROFIT AND RISK PROFIT OF A STOCK

CONTENT

 Profit and risk of a stock

 Covariance and correlation

R t 

P t P 0 C t P 0

 Portfolio return and risk

 Efficient investment portfolio

 Market equilibrium

 Capital Asset Pricing Model (CAPM)

 APT model

R t : Stock profit in period t P t : Stock price at time t

P 0 : Stock price at time 0

C t : Cash received from stock in period 0-t

EXPECTED RETURN OF A STOCK (1)

R p 1 r 1p 2 r 2 ... p n r n  p i r i

i  1

EXPECTED RETURN OF A STOCK (2)

- Expected profit of stock A:

Economic status	LN of CP A	LN of CP B
Crisis	-20%	5%
recession	10%	20%
Normal	30%	-12%
peak	50%	9%

Maybe you are interested!

R   0.2  0.1  0.3  0.5  0.175  17.5%

- Expected profit of stock B:

 0.05  0.20  0.12  0.09

R B 

 0.055  5.5%

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RISK OF A STOCK (1)

Concept

- Risk is a hazard that can cause losses to the person who has to bear it.

- The risk for stock investors is the possibility of

to the actual profit of the stock is lower than its expected profit.

Measuring the risk of a stock

RISK OF A STOCK (2)

Variance (standard deviation) of CP A

Economic status	R At	R At GO OUT	( R At  R A ) 2
Crisis	-20%	-0.375	0.140625
recession	10%	-0.075	0.005625
Normal	30%	0.125	0.015625
peak	50%	0.325	0.105625
Total			0.267500

n n

 ( R t R )

( R t  R )

VAR i t  1

i 

t  1

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RISK OF A STOCK (3)

Variance (standard deviation) of CP B

VAR

RISK OF A STOCK (4)

0.2675  0.066875

Economic status	R At	R At GO OUT	( R  R ) 2 At A
Crisis	5%	-0.005	0.000025
recession	20%	0.145	0.021025
Normal	-12%	-0.175	0.030625
peak	9%	0.035	0.001225
Total			0.052900

A 

0.066875  0.2586  25.86%

VAR B

0.0529  0.013225

B 

0.013225  0.1150  11.50%

COVARIOUSNESS AND CORRELATION (1)



( R At  R A )( R Bt  R B )

Cov ( R A , R B ) t  1

COVARIOUSNESS AND CORRELATION (2)

Corr ( R A

, R B ) 

SD ( R

Cov

Economic status	R At	R At GO OUT	R Bt	R Bt R B	( R At  R A )( R Bt  R B )
Crisis	-20%	-0.375	5%	-0.005	0.001875
recession	10%	-0.075	20%	0.145	-0.010875
Normal	30%	0.125	-12%	-0.175	-0.021875
peak	50%	0.325	9%	0.035	0.011375
Total					-0.0195

A ) xSD ( R B )

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COVARIOUSNESS AND CORRELATION (3)

 0.0195

PORTFOLIO RETURN AND RISK (1)

In case the portfolio consists of 2 types of stocks

Cov ( R A , R B ) 

 0.004875

- Portfolio return

R P X A R A X B R B

Corr ( R A

, R B ) 

 0.004875

0.2586 * 0.1150

 0.1639

- Portfolio variance:

Var X 2 2  2 X X X 2 2

PAAABA , BBB

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PORTFOLIO RETURN AND RISK (2)

For example: Suppose a person invests 100 million in a portfolio of 2 stocks: A and B (60 million in stock A and 40 million in stock B). The return and variance (standard deviation) of this portfolio will be:

PORTFOLIO RETURN AND RISK (3)

Calculate portfolio variance using matrix

X 2 2

X A X B  A , B

X 2 2

R p  (0.60 * 17.5%)  (0.40 * 5.5%)  12.7%

Var P  0.36 * 0.066875  2  0.6 x 0.4 * (  0.004875 ) 

 0.16 * 0.013225  0.023851

P SD P 

Var P

 0.1544

 15.44 %

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PORTFOLIO RETURN AND RISK (4)

In case the portfolio includes many types of stocks

- Portfolio return:

PORTFOLIO RETURN AND RISK (5)

Calculate portfolio variance using matrix

R p 



i  1

X i R i

- Portfolio variance:

nnn

Var  X 2 2    XX

Cov ( R , R )

P ii

i  1

CP	1	2	3	n
1	X 2 2 1 1	X 1 X 2 Cov ( R 1 , R 2 )	X 1 X 3 Cov ( R 1 , R 3 )	X 1 X n Cov ( R 1 , R n )
2	X 2X 1 Cov ( R 2, R 1 )	X 2 2 2 2	X 2X 3 Cov ( R 2, R 3 )	X 2X n Cov ( R 2, R n)
3	X 3 X 1 Cov ( R 3 , R 1 )	X 3X 2 Cov ( R 3 , R 2)	X 2 2 3 3	X 3X n Cov ( R 3 , R n)

n	X nX 1 Cov ( R n, R 1 )	X nX 2 Cov ( R n, R 2)	X n X 3 Cov ( R n , R 3 )	X 2 2 nn

i  1

ijij

j  1, j  i

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EFFECTIVE INVESTMENT PORTFOLIO (1)

In case the portfolio consists of 2 types of stocks

EFFECTIVE INVESTMENT PORTFOLIO (2)

In case the portfolio includes stocks

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EFFICIENT INVESTMENT PORTFOLIO (3)

The portfolio includes 1 government bond and 1 stock.

For example, Mr. Phuc is considering investing in Company X's shares and government bonds. The expected returns and risks (standard deviations) of these two securities are as follows:

EFFICIENT INVESTMENT PORTFOLIO (4)

Investment ratio: 35% : 65%

Profit = 0.35x14 + 0.65x10 = 11.4%

 0.35 2 x 0.20 2  0.07  7%

	Stock X	CP bonds
Profit	14%	10%
Standard deviation	0.20	0

Investment rate: 120% : -20%

Profit = 1.20x14 + -0.20x10 = 14.8%

 1.20 2 x (  0.20 ) 2

 0.24  24 %

EFFICIENT INVESTMENT PORTFOLIO (5)

EFFICIENT INVESTMENT PORTFOLIO (6)

Expected Portfolio Return (%)

120% invested in stocks

-20% invested in bonds

Portfolio consists of one government bond and n stocks

Expected profit

of the category

Road II

Capital Market Line

10 = R F

35% invested in stocks 65% invested in bonds

5 Y

4 3 Road

2 QI

20 Standard deviation of portfolio return (%)

R F X - 40% bonds

35% bonds 140% stocks 70% bonds 65% stocks represented by Q

Figure 5.3: Return and risk of a portfolio consisting of 1 bond and 1 stock

30% stock

represented by Q

Portfolio risk (standard deviation)

SYSTEMATIC RISK AND UNSYSTEMATIC RISK (1)

- Systemic risk

• Risks common to all securities are caused by macroeconomic factors.

• Systemic risk cannot be reduced by portfolio diversification

- Unsystematic risk

• Risk that occurs to one or several securities without affecting the entire market

• Unsystematic risk can be eliminated by diversifying the portfolio.

SYSTEMATIC RISK AND UNSYSTEMATIC RISK (2)

R i R i u i

R i : Actual return of stock i

R i : Expected return of stock i

u i : Outcomes of unpredictable events

R i R i  u i R i  m i  i

m i : Systematic risk (market risk) of stock i

i : Unsystematic risk of stock i

SYSTEMATIC RISK AND UNSYSTEMATIC RISK (3)

Risk (σ)

Unsystematic risk

Systemic risk

Number of shares

MARKET BALANCE (1)

Market Balance Portfolio

- Each individual may have a different estimate of expected return and variance for each stock.

- These estimates may not differ significantly if investors have the same information.

- Homogeneous expectations hypothesis : there is a market in which all investors have the same estimates of expected return, variance and covariance.

MARKET BALANCE (2)

- Under the homogeneous expectations hypothesis, all investors should hold the portfolio consisting of risky stocks represented by point A in Figure 5.4.

- Risk-averse investors can combine portfolio A with risk-free securities to achieve a score of 4.

- For investors who like to take risks, they can borrow more money to invest in portfolio A to achieve a score of 5.

MARKET BALANCE (3)

- If all investors choose the same portfolio of risky securities, what is this portfolio?

- That portfolio is a market-value-weighted portfolio.

=> This portfolio is called the market portfolio.

- An index of a large number of stocks is a very good representative of a portfolio of many stocks of investors.

MARKET BALANCE (4)

Market risk of a stock

MARKET BALANCE (5)

 Cov ( R i , R M )

- Market risk of a stock in a large portfolio

measured by the beta coefficient of that stock.

- The beta coefficient is defined as a measure of

 i

Cov ( R i , R M )

2 ( R )

: Market risk of stock i

: Covariance

Measures the response of a stock to changes in the market portfolio.

 2 ( R ): Variance of market return

M : Market risk

- For example: Company Y's beta coefficient is 1.5, this means

That is, when market returns increase by 1%, then

 M 

X i  i  1

 i : Market risk of stock i

The company's expected earnings will increase by 1.5%.

i  1

X i : Investment ratio in stock i

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MARKET BALANCE (6)

CAPITAL ASSET PRICING MODEL - CAPM (1)

Determine beta using regression method R i

R i  i R m 

Capital Market Line

R i : Expected return of stock i R m : Market return

β i : Market risk of stock i

R M

R F

1  i

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