risk and return

RETURN

Return

Return is the income received on an investment plus any change in market price, usually expressed as a percentage of the begging market price of the investment.

Basically return is classified into two categories:

• Ex-post return (Historical Return)Ex-post return (Historical Return)• Ex-ante return (Anticipated Return/Future Return)Ex-ante return (Anticipated Return/Future Return)

Ex-post Return (Historical Return)Ex-post Return (Historical Return)The historical returns or ex-post return are derived from the cash flows The historical returns or ex-post return are derived from the cash flows received as well as the price changes that occur during the holding received as well as the price changes that occur during the holding period of that asset.period of that asset.

Rate of Return = Annual income + Ending price-Beginning price Beginning price Beginning price

Current yield Capital gains /loss yield

Ex-ante Return (Anticipated Return/Future Return)Ex-ante Return (Anticipated Return/Future Return)Ex-ante is the expected return for a future time period calculated with Ex-ante is the expected return for a future time period calculated with the help of probability, which describes the likelihood occurrence of an the help of probability, which describes the likelihood occurrence of an event.event.

nE(R) = ∑ pi Ri

i=1

RETURN OF A SINGLE ASSETRate of Return = Annual Income + Ending Price - Beginning Price

Beginning Price Beginning Price

Current Yield Capital Gains /loss Yield

Mr. A share price on February 28, 2008 was Rs. 401 and the price on December 10, 2008 was Rs. 480. Dividend received Rs. 35. What is the rate of return earned by MR. A

35 + 480 – 401

401 401

= 8.73 + 19.70

= 28.43%

Assignment

Tata Indicom share is currently market price is Rs. 250. An investor who have already purchased 100 shares of Tata Indicom last year at a rate of Rs. 425 per share got Rs. 5 as divided in this month. Calculate the holding period return of the investor.

PROBABILITY DISTRIBUTION AND EXPECTED

RATE OF RETURN

nE(R) = pi Ri

i=1The expected rate of return on Oriental Shipping stock is:

E(Ro) = (0.30) (40%) + (0.50) (10%) + (0.20) (-20%) = 13.0%

Σ

EXPECTED RETURN ON A PORTFOLIO

Portfolio is the combination of different type of the securities.

E(Rp) = wi E(Ri)

Suppose if the expected returns on two securities are 16 and 14 per cent and an investor wants to create a portfolio of two asset with equal weight age. Then the return would be

= 0.5 x 14 + 0.5 x 16 = 15 percent

Assignment

Security Return

(percentage)

Proportion of Investment

A 12 0.20

B 17 0.30

C 23 0.10

D 20 0.40

Calculate the portfolio return by using the above data.

Risk:

• Possibility of loss or injury.

• Risk refers to the possibility that the actual outcome of an investment will differ form expected outcomes. More specially, most investors are concerned about the actual outcome being less than the expected outcomes. The wider the range of possible outcomes , the greater the risk.

Sources of Risk

Interest Rate Risk Financial Risk

Market Risk Liquidity Risk

Inflation Risk Exchange Rate Risk

Business Risk Country Risk

Element of Risk

Risk

Systematic Risk Unsystematic Risk

Interest Rate Risk Business Risk Financial Risk Market RiskPurchasing Power Risk

STANDARD DEVIATION

Illustration of the Calculation of Standard Deviation

AssignmentPossible return (in percentage) Probability of occurrence

-25 .05

-10 .10

0 .10

15 .15

20 .25

30 .20

35 .15

Calculate the security expected return and risk.

DIVERSIFICATION AND PORTFOLIO RISK Probability Distribution of Returns

 State of the Probability Return on Return on Return on Economy Stock A Stock B Portfolio  1 0.20 15% -5% 5% 2 0.20 -5% 15 5% 3 0.20 5 25 15% 4 0.20 35 5 20% 5 0.20 25 35 30%

Expected Return

 Stock A : 0.2(15%) + 0.2(-5%) + 0.2(5%) +0.2(35%) + 0.2(25%) = 15%Stock B : 0.2(-5%) + 0.2(15%) + 0.2(25%) + 0.2(5%) + 0.2(35%) = 15%Portfolio of A and B : 15*.50+15*.50 = 15%

Standard Deviation  Stock A : σ2

A = 0.2(15-15)2 + 0.2(-5-15)2 + 0.2(5-15)2 + 0.2(35-15)2 + 0.20 (25-15)2 = 200

σA = (200)1/2 = 14.14% Stock B : σ2

B = 0.2(-5-15)2 + 0.2(15-15)2 + 0.2(25-15)2 + 0.2(5-15)2 + 0.2 (35-15)2

= 200 σB = (200)1/2 = 14.14% Portfolio : σ2

(A+B) = 0.2(5-15)2 + 0.2(5-15)2 + 0.2(15-15)2 + 0.2(20-15)2 + 0.2(30-15)2 = 90 σA+B = (90)1/2 = 9.49%

RELATIONSHIP BETWEEN DIVERSIFICATION AND RISK

Standard Deviation of the Two Securities Portfolio

xyyxyyxxp arwwww cov2222

SD = (14.14)2 * (.50)2 + (14.14)2 * (.50)2 + 2* .50 * .50 * -.35 .14.14 * 14.14

= 9.49 %

Assignment

Use the following data to calculate portfolio return, variance and standard deviation.

R1 = 15%

R2 = 8%

W1 = .70

W2 = .30

R12 = .65

σ1 = 13

σ2 = 10

Standard Deviation of the more than Two Securities Portfolio

Variance Covariance Matrix

A

w1σ1B

w2 σ2C

w3 σ3

A

W1σ1r11 r12 r13

B

W2σ2 r12 R22 r23

C

w3σ3r13 R23 R33

n

i

n

jijjip xx

1 1

2 .cov

Standard Deviation of the more than Two Securities Portfolio

Variance Covariance Matrix

A

w1σ1B

w2 σ2C

w3 σ3

A

W1σ1W1w1σ1σ1r11 w1

w2σ1σ2r12W1w3σ1σ3r1

3

B

W2σ2 w2 w1σ2σ1r12 w2

w2σ2σ2r22W2w3σ1σ3r2

3

C

w3σ3W3w1σ3σ1r13 W3w2σ3σ2r23 w3w3σ3σ3r3

3

AssignmentSecurity Standard

DeviationProportion of investment

A 12 .20

B 8 .30

C 16 .50

Correlation Coefficient of returns between

A and B = 0.8

B and C = 0.6

A and C = 0.5

Calculate the Portfolio Risk.

MEASUREMENT OF MARKET RISKTHE SENSITIVITY OF A SECURITY TO MARKET MOVEMENTS IS CALLED BETA .

BETA REFLECTS THE SLOPE OF A LINEAR REGRESSION RELATIONSHIP BETWEEN THE RETURN ON THE SECURITY AND THE RETURN ON THE PORTFOLIO

Relationship between Security Return and Market Return

  Security

Return

 Market return

CALCULATION OF BETA

Beta reflects the slope of the above regression relationship. It is equal to:

Cov (Rj , RM) ρjM σj σM ρjM σj

j = = = σ2

M σ2M σM

where Cov = covariance between the return on security j and the return on market portfolio M. It is equal to:

n _ _ Rjt – Rj)(RMt – RM)/n-1

i=1

11

nrrrrn

tmmrrij

CALCULATION OF BETA

Historical Market Data _ _ _ _ _

Year Rjt RMt Rjt-Rj RMt-RM (Rjt - Rj) (RMt-RM) (RMt-RM)2

1 10 12 -2 -1 2 1 2 6 5 -6 -8 48 64 3 13 18 1 5 5 25 4 -4 -8 -16 -21 336 441 5 13 10 1 -3 -3 9 6 14 16 2 3 6 9 7 4 7 -8 -6 48 36 8 18 15 6 2 12 4 9 24 30 12 17 204 289 10 22 25 10 12 120 144

_ _ _ Σ Rjt = 120 Σ RMt = 130 Σ (Rjt- Rj) (RMt - RM) = 778 Σ(RMt - RM)2 = 1022

_ _ Rj = 12 RM = 13 Cov (Rjt , RMt) = 778/= 86.4 σM = 1022/9=113.6

Cov (Rjt , RMt) 86.4 Beta : βj = = = 0.76

σ2M 113.6

- -α = Rx - βRmmi rr

AssignmentMonth ONGC NSE

1 -0.75 -0.35

2 5.45 -0.49

3 -3.05 -1.03

4 3.41 1.64

5 9.13 6.67

6 2.36 1.13

7 -0.42 0.72

8 5.51 0.84

9 6.80 4.05

10 2.60 1.21

11 -3.81 0.29

12 -1.91 1.96

With the help of the 12 month return data of ONGC and Market calculate systematic risk of ONGC. Suppose NSE return moves up by 15 percent what will be the return of ONGC.

2 = b2 2 + e2.

2 = variance= stand-alone risk of Stock j.

b2 2 = market risk of Stock j.

e2= variance of error term= diversifiable risk of Stock j.

What is the relationship between stand-alone, market, and diversifiable risk.

j j M j

j

j

j M

Minimum Variance Portfolio

ProblemHere are returns and standard deviations for four investments.

Return Standard DeviationTreasury Bills 6% 0%Stock P 10% 14%Stock Q 14% 28%Stock R 21% 26%Calculate the return and standard deviation of the following portfolios:50 per cent in Treasury Bills and 50 per cent in stock P50 per cent each in Q and R assuming the shares have i) Perfect

Positive Correlation ii) Perfect Negative Correlation iii) No Correlation

30 per cent in Q , 20 percent in R and rest 50 per cent in P by assuming RPQ is .36, RPR is .65 and RQR is .78.

Problem

Problem

Problem

ProblemThe returns of two assets under four possible state of nature are given

below:State of Nature Probability Return on Asset 1 Return on Asset 2

1 0.10 5%0%

2 0.30 10%8%

3 0.50 15%18%

4 0.10 20%16%

What is the standard deviation of the return on asset 1 and asset 2?What is the covariance between the return on assets 1 and 2?What is the coefficient of correlation between the return on assets 1

and 2?