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Chapter 2
RISK AND RETURN
Two Sides of the Investment Coin
OUTLINE
• Return
• Risk– Unsystematic and systematic risk
•Quantifying risk and reduction of risk through
diversification
•Measurement of Systematic ( non-diversifiable) risk
• Security Market Line and its applications
RETURN
• Return is the primary motivating force that drives
investment.
Computation of
1)Single period returns and
2) Multi period returns
Measurement of
1)Historical Returns and
2) Expected Return of a security as well as portfolio
RISK
• Risk refers to the possibility that the actual outcome of
an investment will deviate from its expected outcome.
• The three major sources of risk are : business risk,
interest rate risk, and market risk.
• Modern portfolio theory looks at risk from a different
perspective. It divides total risk as follows.
Total Unique Market
risk risk risk
( Unsystematic) ( Systematic)
= +
MEASURING HISTORICAL RISK
n
 (Ri - R)2 1/2
t =1
 =
n -1
PERIOD RETURN DEVIATAION SQUARE OF DEVIATION
Ri (Ri - R) (Ri - R)2
1 15 5 25
2 12 2 4
3 20 10 100
4 -10 -20 400
5 14 4 16
6 9 -1 1
 Ri = 60  (Ri - R)2 = 536
R = 10
 (Ri - R)2
 2 = = 107.2  = [107.2]1/2 = 10.4
n -1
MEASURING EXPECTED (EX ANTE)
RETURN AND RISK
EXPECTED RATE OF RETURN
n
E (R) =  pi Ri
i=1
STANDARD DEVIATION OF RETURN
 = [ pi (Ri - E(R) )2]
Bharat Foods Stock
i. State of the
Economy pi Ri piRi Ri-E(R) (Ri-E(R))2 pi(Ri-E(R))2
1. Boom 0.30 16 4.8 4.5 20.25 6.075
2. Normal 0.50 11 5.5 -0.5 0.25 0.125
3. Recession 0.20 6 1.2 -5.5 30.25 6.050
E(R ) = piRi = 11.5 pi(Ri –E(R))2 =12.25
σ = [pi(Ri-E(R))2]1/2 = (12.25)1/2 = 3.5%
PORTFOLIO EXPECTED RETURN
n
E(RP) =  wi E(Ri)
i=1
where E(RP) = expected portfolio return
wi = weight assigned to security i
E(Ri) = expected return on security i
n = number of securities in the portfolio
Example A portfolio consists of four securities with expected returns
of 12%, 15%, 18%, and 20% respectively. The proportions of
portfolio value invested in these securities are 0.2, 0.3, 0.3, and 0.20
respectively.
The expected return on the portfolio is:
E(RP) = 0.2(12%) + 0.3(15%) + 0.3(18%) + 0.2(20%)
= 16.3%
PORTFOLIO RISK AND REDUCTION OF RISK
THROUGH DIVERSIFICATION
The risk of a portfolio is measured by the variance (or
standard deviation) of its return. Although the expected
return on a portfolio is the weighted average of the
expected returns on the individual securities in the
portfolio, portfolio risk is not the weighted average of the
risks of the individual securities in the portfolio (except
when the returns from the securities are uncorrelated).
MEASUREMENT OF COMOVEMENTS
IN SECURITY RETURNS
• To develop the equation for calculating portfolio risk we
need information on weighted individual security risks
and weighted comovements between the returns of
securities included in the portfolio.
• Comovements between the returns of securities are
measured by covariance (an absolute measure) and
coefficient of correlation (a relative measure).
COVARIANCE
COV (Ri , Rj) = p1 [Ri1 – E(Ri)] [ Rj1 – E(Rj)]
+ p2 [Ri2 – E(Rj)] [Rj2 – E(Rj)]
+
+ pn [Rin – E(Ri)] [Rjn – E(Rj)]
•
•
•
•
ILLUSTRATION
The returns on assets 1 and 2 under five possible states of nature are given below
State of nature Probability Return on asset 1 Return on asset 2
1 0.10 -10% 5%
2 0.30 15 12
3 0.30 18 19
4 0.20 22 15
5 0.10 27 12
The expected return on asset 1 is :
E(R1) = 0.10 (-10%) + 0.30 (15%) + 0.30 (18%) + 0.20 (22%) + 0.10 (27%) = 16%
The expected return on asset 2 is :
E(R2) = 0.10 (5%) + 0.30 (12%) + 0.30 (19%) + 0.20 (15%) + 0.10 (12%) = 14%
The covariance between the returns on assets 1 and 2 is calculated below :
State of
nature
(1)
Probability
(2)
Return on
asset 1
(3)
Deviation of
the return on
asset 1 from its
mean
(4)
Return on
asset 2
(5)
Deviation of
the return on
asset 2 from
its mean
(6)
Product of the
deviations
times
probability
(2) x (4) x (6)
1 0.10 -10% -26% 5% -9% 23.4
2 0.30 15% -1% 12% -2% 0.6
3 0.30 18% 2% 19% 5% 3.0
4 0.20 22% 6% 15% 1% 1.2
5 0.10 27% 11% 12% -2% -2.2
Sum = 26.0
Thus the covariance between the returns on the two assets is 26.0.
CO EFFIENT OF CORRELATION
Cov (Ri , Rj)
Cor (Ri , Rj) or ij =
 (Ri , Rj)
ij
i j
ij = ij . i . j
where ij = correlation coefficient between the returns on
securities i and j
ij = covariance between the returns on securities
i and j
i , j = standard deviation of the returns on securities
i and j
=
PORTFOLIO RISK : 2 – SECURITY CASE
p = [w1
2 1
2 + w2
2 2
2 + 2w1w2 12 1 2]½
Example : w1 = 0.6 , w2 = 0.4,
1 = 10%, 2 = 16%
12 = 0.5
p = [0.62 x 102 + 0.42 x 162 +2 x 0.6 x 0.4 x 0.5 x 10 x 16]½
= 10.7%
The average standard deviation of two securities is 13,
which is less than standard deviation of the portfolio,
which is 10. Thus diversification reduces risk.
PORTFOLIO RISK : n – SECURITY CASE
p = [   wi wj ij i j ] ½
Example : w1 = 0.5 , w2 = 0.3, and w3 = 0.2
1 = 10%, 2 = 15%, 3 = 20%
12 = 0.3, 13 = 0.5, 23 = 0.6
p = [w1
2 1
2 + w2
2 2
2 + w3
2 3
2 + 2 w1 w2 12 1 2
+ 2w2 w3 13 1 3 + 2w2 w3 232 3] ½
= [0.52 x 102 + 0.32 x 152 + 0.22 x 202
+ 2 x 0.5 x 0.3 x 0.3 x 10 x 15
+ 2 x 0.5 x 0.2 x 05 x 10 x 20
+ 2 x 0.3 x 0.2 x 0.6 x 15 x 20] ½
= 10.79%
RISK OF AN N - ASSET PORTFOLIO
2
p =   wi wj ij i j
n x n MATRIX
1 2 3 … n
1 w1
2
σ1
2
w1w2ρ12σ1σ2 w1w3ρ13σ1σ3 … w1wnρ1nσ1σn
2 w2w1ρ21σ2σ1 w2
2
σ2
2
w2w3ρ23σ2σ3 … w2wnρ2nσ2σn
3 w3w1ρ31σ3σ1 w3w2ρ32σ3σ2 w3
2
σ3
2
…
: : :
n wnw1ρn1σnσ1 wn
2
σn
2
MEASUREMENT OF SYSTEMATIC RISK
CALCULATION OF BETA
Beta– It is the share’s sensitivity to market movements. In simple words, It indicates how much the
scrip moves for the unit change in the market index. It can be positive or negative. Negative beta
indicates that the share moves in the opposite direction of the market
Rit = i + i RMt + eit
iM
i =
M
2
CALCULATION OF BETA
Period
Return on
stock A, RA
Return on
market
portfolio, RM
Deviation of
return on stock A
from its mean
(RA - RA)
Deviation of
return on market
portfolio from its
mean (RM - RM)
Product of the
deviation,
(RA - RA)
(RM - RM)
Square of the
deviation of
return on market
portfolio from its
mean
(RM - RM)2
1 10 12 0 3 0 9
2 15 14 5 5 25 25
3 18 13 8 4 32 16
4 14 10 4 1 4 1
5 16 9 6 0 0 0
6 16 13 6 4 24 16
7 18 14 8 5 40 25
8 4 7 -6 -2 12 4
9 - 9 1 -19 -8 152 64
10 14 12 4 3 12 9
11 15 -11 5 -20 -100 400
12 14 16 4 7 28 49
13 6 8 -4 -1 4 1
14 7 7 -3 -2 6 4
15 - 8 10 -18 1 -18 1
RA = 150 RM = 135 (RA - RA) (RM - RM)2
RA =10 RM = 9 (RM - RM) = 221 = 624
So Beta = 221/624 =.3541
E(RM) - Rf
E(Ri ) = Rf + CiM
M

SECURITY MARKET LINE
iM
β i =
M

E (R i ) = R f + [ E (R M)- R f ] β i
EXPECTED • P
RETURN SML
14%
8% • 0
ALPHA = EXPECTED - FAIR
RETURN RETURN
1.0 β
Important practical applications of Security Market Line –
1) Evaluation of performance of portfolio managers.
2) To test market efficiency
3) To identify undervalued securities

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Financial analysis on Risk and Return.ppt

  • 1. Chapter 2 RISK AND RETURN Two Sides of the Investment Coin
  • 2. OUTLINE • Return • Risk– Unsystematic and systematic risk •Quantifying risk and reduction of risk through diversification •Measurement of Systematic ( non-diversifiable) risk • Security Market Line and its applications
  • 3. RETURN • Return is the primary motivating force that drives investment. Computation of 1)Single period returns and 2) Multi period returns Measurement of 1)Historical Returns and 2) Expected Return of a security as well as portfolio
  • 4. RISK • Risk refers to the possibility that the actual outcome of an investment will deviate from its expected outcome. • The three major sources of risk are : business risk, interest rate risk, and market risk. • Modern portfolio theory looks at risk from a different perspective. It divides total risk as follows. Total Unique Market risk risk risk ( Unsystematic) ( Systematic) = +
  • 5. MEASURING HISTORICAL RISK n  (Ri - R)2 1/2 t =1  = n -1 PERIOD RETURN DEVIATAION SQUARE OF DEVIATION Ri (Ri - R) (Ri - R)2 1 15 5 25 2 12 2 4 3 20 10 100 4 -10 -20 400 5 14 4 16 6 9 -1 1  Ri = 60  (Ri - R)2 = 536 R = 10  (Ri - R)2  2 = = 107.2  = [107.2]1/2 = 10.4 n -1
  • 6. MEASURING EXPECTED (EX ANTE) RETURN AND RISK EXPECTED RATE OF RETURN n E (R) =  pi Ri i=1 STANDARD DEVIATION OF RETURN  = [ pi (Ri - E(R) )2] Bharat Foods Stock i. State of the Economy pi Ri piRi Ri-E(R) (Ri-E(R))2 pi(Ri-E(R))2 1. Boom 0.30 16 4.8 4.5 20.25 6.075 2. Normal 0.50 11 5.5 -0.5 0.25 0.125 3. Recession 0.20 6 1.2 -5.5 30.25 6.050 E(R ) = piRi = 11.5 pi(Ri –E(R))2 =12.25 σ = [pi(Ri-E(R))2]1/2 = (12.25)1/2 = 3.5%
  • 7. PORTFOLIO EXPECTED RETURN n E(RP) =  wi E(Ri) i=1 where E(RP) = expected portfolio return wi = weight assigned to security i E(Ri) = expected return on security i n = number of securities in the portfolio Example A portfolio consists of four securities with expected returns of 12%, 15%, 18%, and 20% respectively. The proportions of portfolio value invested in these securities are 0.2, 0.3, 0.3, and 0.20 respectively. The expected return on the portfolio is: E(RP) = 0.2(12%) + 0.3(15%) + 0.3(18%) + 0.2(20%) = 16.3%
  • 8. PORTFOLIO RISK AND REDUCTION OF RISK THROUGH DIVERSIFICATION The risk of a portfolio is measured by the variance (or standard deviation) of its return. Although the expected return on a portfolio is the weighted average of the expected returns on the individual securities in the portfolio, portfolio risk is not the weighted average of the risks of the individual securities in the portfolio (except when the returns from the securities are uncorrelated).
  • 9. MEASUREMENT OF COMOVEMENTS IN SECURITY RETURNS • To develop the equation for calculating portfolio risk we need information on weighted individual security risks and weighted comovements between the returns of securities included in the portfolio. • Comovements between the returns of securities are measured by covariance (an absolute measure) and coefficient of correlation (a relative measure).
  • 10. COVARIANCE COV (Ri , Rj) = p1 [Ri1 – E(Ri)] [ Rj1 – E(Rj)] + p2 [Ri2 – E(Rj)] [Rj2 – E(Rj)] + + pn [Rin – E(Ri)] [Rjn – E(Rj)] • • • •
  • 11. ILLUSTRATION The returns on assets 1 and 2 under five possible states of nature are given below State of nature Probability Return on asset 1 Return on asset 2 1 0.10 -10% 5% 2 0.30 15 12 3 0.30 18 19 4 0.20 22 15 5 0.10 27 12 The expected return on asset 1 is : E(R1) = 0.10 (-10%) + 0.30 (15%) + 0.30 (18%) + 0.20 (22%) + 0.10 (27%) = 16% The expected return on asset 2 is : E(R2) = 0.10 (5%) + 0.30 (12%) + 0.30 (19%) + 0.20 (15%) + 0.10 (12%) = 14% The covariance between the returns on assets 1 and 2 is calculated below :
  • 12. State of nature (1) Probability (2) Return on asset 1 (3) Deviation of the return on asset 1 from its mean (4) Return on asset 2 (5) Deviation of the return on asset 2 from its mean (6) Product of the deviations times probability (2) x (4) x (6) 1 0.10 -10% -26% 5% -9% 23.4 2 0.30 15% -1% 12% -2% 0.6 3 0.30 18% 2% 19% 5% 3.0 4 0.20 22% 6% 15% 1% 1.2 5 0.10 27% 11% 12% -2% -2.2 Sum = 26.0 Thus the covariance between the returns on the two assets is 26.0.
  • 13. CO EFFIENT OF CORRELATION Cov (Ri , Rj) Cor (Ri , Rj) or ij =  (Ri , Rj) ij i j ij = ij . i . j where ij = correlation coefficient between the returns on securities i and j ij = covariance between the returns on securities i and j i , j = standard deviation of the returns on securities i and j =
  • 14. PORTFOLIO RISK : 2 – SECURITY CASE p = [w1 2 1 2 + w2 2 2 2 + 2w1w2 12 1 2]½ Example : w1 = 0.6 , w2 = 0.4, 1 = 10%, 2 = 16% 12 = 0.5 p = [0.62 x 102 + 0.42 x 162 +2 x 0.6 x 0.4 x 0.5 x 10 x 16]½ = 10.7% The average standard deviation of two securities is 13, which is less than standard deviation of the portfolio, which is 10. Thus diversification reduces risk.
  • 15. PORTFOLIO RISK : n – SECURITY CASE p = [   wi wj ij i j ] ½ Example : w1 = 0.5 , w2 = 0.3, and w3 = 0.2 1 = 10%, 2 = 15%, 3 = 20% 12 = 0.3, 13 = 0.5, 23 = 0.6 p = [w1 2 1 2 + w2 2 2 2 + w3 2 3 2 + 2 w1 w2 12 1 2 + 2w2 w3 13 1 3 + 2w2 w3 232 3] ½ = [0.52 x 102 + 0.32 x 152 + 0.22 x 202 + 2 x 0.5 x 0.3 x 0.3 x 10 x 15 + 2 x 0.5 x 0.2 x 05 x 10 x 20 + 2 x 0.3 x 0.2 x 0.6 x 15 x 20] ½ = 10.79%
  • 16. RISK OF AN N - ASSET PORTFOLIO 2 p =   wi wj ij i j n x n MATRIX 1 2 3 … n 1 w1 2 σ1 2 w1w2ρ12σ1σ2 w1w3ρ13σ1σ3 … w1wnρ1nσ1σn 2 w2w1ρ21σ2σ1 w2 2 σ2 2 w2w3ρ23σ2σ3 … w2wnρ2nσ2σn 3 w3w1ρ31σ3σ1 w3w2ρ32σ3σ2 w3 2 σ3 2 … : : : n wnw1ρn1σnσ1 wn 2 σn 2
  • 17. MEASUREMENT OF SYSTEMATIC RISK CALCULATION OF BETA Beta– It is the share’s sensitivity to market movements. In simple words, It indicates how much the scrip moves for the unit change in the market index. It can be positive or negative. Negative beta indicates that the share moves in the opposite direction of the market Rit = i + i RMt + eit iM i = M 2
  • 18. CALCULATION OF BETA Period Return on stock A, RA Return on market portfolio, RM Deviation of return on stock A from its mean (RA - RA) Deviation of return on market portfolio from its mean (RM - RM) Product of the deviation, (RA - RA) (RM - RM) Square of the deviation of return on market portfolio from its mean (RM - RM)2 1 10 12 0 3 0 9 2 15 14 5 5 25 25 3 18 13 8 4 32 16 4 14 10 4 1 4 1 5 16 9 6 0 0 0 6 16 13 6 4 24 16 7 18 14 8 5 40 25 8 4 7 -6 -2 12 4 9 - 9 1 -19 -8 152 64 10 14 12 4 3 12 9 11 15 -11 5 -20 -100 400 12 14 16 4 7 28 49 13 6 8 -4 -1 4 1 14 7 7 -3 -2 6 4 15 - 8 10 -18 1 -18 1 RA = 150 RM = 135 (RA - RA) (RM - RM)2 RA =10 RM = 9 (RM - RM) = 221 = 624 So Beta = 221/624 =.3541
  • 19. E(RM) - Rf E(Ri ) = Rf + CiM M  SECURITY MARKET LINE iM β i = M  E (R i ) = R f + [ E (R M)- R f ] β i EXPECTED • P RETURN SML 14% 8% • 0 ALPHA = EXPECTED - FAIR RETURN RETURN 1.0 β
  • 20. Important practical applications of Security Market Line – 1) Evaluation of performance of portfolio managers. 2) To test market efficiency 3) To identify undervalued securities