1. FIN 3321
Investment and Portfolio Management I
Risk and Return Relationship
01/31/13 Prepared by P D Nimal 1

2. Objectives
On satisfactory completion of this topic student will be
able to:
 Understand the relationship between risk and return of
assets
 Portfolio Risk and Return
 Importance of covariance and correlation between returns of
assets
 Diversification advantage
 Mean Variance Efficient Frontier
 Capital Market Line
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3. Expected Return and Risk
 Do not use Historical Data
 Use Forecasted Data
Suppose you are considering investing in
shares of HNB. Market price is Rs. 200.
You want to hold the share for one year.
What is your expected rate of return?
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4. Expected Return and Risk cont…
This will depend on the
 Actual dividend you would receive and
 The market price at which you could sell the share
These two will decide the rate of return that you could
earn
Both dividend and the price at which you can
sell will depend on the possible state of
economic conditions.
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5. Expected Return and Risk cont…
n
E( R) =∑ i P
R i
i=1
The average dispersion of the return is measured by the
variance or standard deviation. The equation is as follows.
n
σ = ∑ [ Ri − E ( R ) ] Pi
2 2
i =1
Calculate the E(R) and the Standard Deviation of assets
given in the table.
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6. Expected Return and Risk cont…
Suppose the state of economic conditions and
the possible rates of return with probabilities of
the occurrence of each state of economic
condition are as follows
Return and Probabilities
Rate of
Economic Rate of Probability Return
Conditions Return *Probability
Growth 17.5 0.2 3.5
Expansion 11.2 0.3 3.36
Stagnation 5.4 0.25 1.35
Decline -8.9 0.25 -2.225
1 ER=5.985
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7. Expected Return and Risk cont…
n
E( R) =∑ i P
R i
i=1
E ( R ) = 17.5 × .2 + 11.2 × .3 + 5.4 × .25 − 8.9 × .25
= 5.985
The average dispersion of the return is
measured by the variance or standard deviation.
The equation is as follows.
n
σ = ∑ [ Ri − E ( R )] Pi
2 2
i=1
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8. Expected Return and Risk cont…
n
σ =∑ [ Ri −E ( R )]2 P
2
i
i=1
Variance and standard deviation of our example
σ = 90.154
2
σ = 9.495
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9. Risk and Return
Investment Alternatives
Econ. Prob. T-Bill Alta Repo Am F. MP
Bust 0.10 8.0% -22.0% 28.0% 10.0% -13.0%
Below
0.20 8.0 -2.0 14.7 -10.0 1.0
avg.
Avg. 0.40 8.0 20.0 0.0 7.0 15.0
Above
0.20 8.0 35.0 -10.0 45.0 29.0
avg.
Boom 0.10 8.0 50.0 -20.0 30.0 43.0
1.00
Calculate the Risk and Return of assets given in the table.
9

10. Expected Return versus Risk
Expected
Security return% Risk, σ%
Alta Inds. 17.4 20.0
Market 15.0 15.3
Am. Foam 13.8 18.8
T-bills 8.0 0.0
Repo Men 1.7 13.4
10

11. What is unique about the T-bill return?
 The T-bill will return 8% regardless of
the state of the economy.
 Is the T-bill riskless? Explain.
11

12. Alta Inds. and Repo Men vs.
the Economy
 Alta Inds. moves with the economy, so it is
positively correlated with the economy. This is
the typical situation.
 Repo Men moves counter to the economy.
Such negative correlation is unusual.
12

13. Stand-Alone Risk
 Standard deviation measures the stand-
alone risk of an investment.
 The larger the standard deviation, the
higher the probability that returns will be
far below/above the expected return.
13

14. Coefficient of Variation (CV)
 CV = STD/E(R)
 CVT-BILLS = 0.0 / 8.0 = 0.0.
 CVAlta Inds = 20.0 / 17.4 = 1.1.
 CVRepo Men = 13.4 / 1.7 = 7.9.
 CVAm. Foam = 18.8 / 13.8 = 1.4.
 CVM = 15.3 / 15.0 = 1.0.
14

15. Expected Return versus
Coefficient of Variation
Expected Risk: Risk:
Security return% σ% CV
Alta Inds 17.4 20.0 1.1
Market 15.0 15.3 1.0
Am. Foam 13.8 18.8 1.4
T-bills 8.0 0.0 0.0
Repo Men
1.7 13.4 7.9
15

16. Return vs. Risk (Std. Dev.):
Which investment is best?
20.0%
18.0% Alta
16.0%
Mkt
14.0% Am. Foam
Return
12.0%
10.0%
8.0% T-bills
6.0%
4.0%
2.0% Repo
0.0%
0.0% 5.0% 10.0% 15.0% 20.0% 25.0%
Risk (Std. Dev.)
16

17. Portfolio Risk and Return
The return of a portfolio is equal to the weighted average of
the returns of individual assets in the portfolio.
Two-Asset Case
State of Probability Returns
Economy X Y
1 0.10 -8.5 8.5
2 0.20 7.2 -5.4
3 0.50 6.5 4.3
4 0.20 4.2 7.5
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18. Risk and Return
Portfolio Investment
E ( RX ) = −8.5 × .1 + 7.2 × .2 + 6.5 × .5 + 4.2 × .2
= 4.68
E ( RY ) = 8.5 × .1 − 5.4 × .2 + 4.3 × .5 + 7.5 × .2
= 3.42
Suppose we invest in an equally weighted portfolio of
these two assets. i.e., 50% of the investment in X and
50% in Y.
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19. Expected Return of the Portfolio
State of Probability Portfolio Return
Economy
1 0.10 (-8.5*.5+8.5*.5)=0
2 0.20 (7.2*.5-5.4*.5)=.9
3 0.50 (6.5*.5+4.3*.5)=5.4
4 0.20 (4.2*.5+7.5*.5)=5.85
E ( R p ) = .1 ×0 +.2 ×.9 +.5 ×5.4 +.2 ×5.85 = 4.05
E ( R p ) = E ( RX ) ×0.5 + E ( RY ) ×0.5
E ( R p ) = 4.68 ×0.5 +3.42 ×0.5 = 4.05
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20. Risk of the Portfolio
Lets Compute the Standard deviation of X and
Y separately
σX = 4.51 σY = 4.67
We will consider the same example
State of Probability Portfolio Return
Economy
1 0.10 (-8.5*.5+8.5*.5)=0
2 0.20 (7.2*.5-5.4*.5)=0.9
3 0.50 (6.5*.5+4.3*.5)=5.4
4
01/31/13
0.20Prepared by P D(4.2*.5+7.5*.5)=5.85
Nimal 20

21. Risk of the Portfolio cont…
Standard deviation of the portfolio
σp = ∑( R
n
ip − E ( R p )) P
2
i
E ( R p ) = 4.05
i=1
σ P = .1( 0 − 4.05) + .2( 0.9 − 4.05) + .5( 5.4 − 4.05) + .2( 5.85 − 4.05)
2 2 2 2 2
σ p = 5.184
2
σ = 2.28
Important:
This is not the Weighted average of the standard
deviations
σ P ≠ 4.51× .5 + 4.67 × .5
≠ 4.59
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22. Risk of the Portfolio cont…
Portfolio standard deviation can be calculated as
follows n n
σP = ∑∑ kW jσkj
2
W
k = j=
1 1
When there are two stocks in the portfolio
σP =W12σ12 +W22σ2 + 2W1W2Cov1,2
2 2
According to our ex.
σ P = .52 × 20.34 + .52 × 21.86 + 2 × .52 ( − 10.73)
2
= 5.184
σ p = 2.28
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23. Covariance between two assets
σ k , j = ∑ [ Rk − E ( Rk )][ R j − E ( R j )]Pi
n
i =1
6 7
X Y P XP YP X-ERx Y-Ery P*6*7
-8.5 8.5 0.1 -0.85 0.85 -13.18 5.08 -6.69544
7.2 -5.4 0.2 1.44 -1.08 2.52 -8.82 -4.44528
6.5 4.3 0.5 3.25 2.15 1.82 0.88 0.8008
4.2 7.5 0.2 0.84 1.5 -0.48 4.08 -0.39168
1 ER 4.68 3.42 Cov -10.7316
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24. Risk Of the Portfolio cont…
This can be written in a different way
σP =W12σ12 +W22σ2 + 2W1W2Corr1,2σ1σ2
2 2
Cov1, 2 − 10.73
Corr1, 2 = = = −0.51
σ 1σ 2 4.51× 4.67
Cov1, 2 = Corr1, 2σ 1σ 2
According to our ex.
σP =.52 ×20.34 +.52 ×21.86 +2 ×.52 ( −0.51)4.51×4.67
2
= 5.184
σp = 2.28
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25. Risk-Return Relationship of
Portfolios on Correlation
When the correlation is 1, what is the standard
deviation of the portfolio?
σ = W σ + W σ + 2W1W2Corr1,2σ 1σ 2
2
P 1
2
1
2
2
2 2
2
According to our ex.
σP = .52 ×20.34 +.52 21.86 + 2 ×.52 (1)4.51 ×4.67
2
= 21.08
σp = 4.59
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26. Portfolio ER & STD when Correlation
coefficient is 1
E ( R p ) = ∑ i Ri
n
W
i=1
Wx Wy ER STD
When Correlation (1)
1 0 5.6 5.2
6 X
5
0 1 2.6 3.5 4
ER
3
0.5 0.5 4.1 4.35 2
Y
1
n n 0
σ =∑ WiW jσ
2
P∑ ij
0 2 4 6
i=1 j=1 Std
σ P = W12σ 12 + W22σ 22 + 2W1W2 ρ12σ 1σ 2
2
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27. Portfolio ER & STD when Correlation
coefficient is -1
Wx Wy ER STD
1 0 5.6 5.2
0.5 0.5 4.1 0.85 When Correlation -1
0.4 0.6 3.8 0 6
X
0.25 0.75 3.35 1.325 5
4
0 1 2.6 3.5
ER
3
Y
2
1
0
0 2 4 6
STD
01/31/13 Prepared by P D Nimal 27

28. Portfolio ER &STD when Correlation
coefficient is 0
When Correlation 0
Wx Wy ER STD
6
1 0 5.6 5.2 5
X
4
0.5 0.5 4.1 3.13
ER
3
Y
0.25 0.75 3.35 2.93 2
1
0 1 2.6 3.5 0
0 2 4 6
STD
01/31/13 Prepared by P D Nimal 28

29. Portfolio ER & STD on Correlation
Coefficient- Summary
ER Vs. STD on Corr
6
X
5 Corr=-1
Corr+0
4 Corr=1
ER
3
2 Y
1
0
0 2 4 6
STD
01/31/13 Prepared by P D Nimal 29

30. Portfolio Risk cont…
Therefore, the standard deviation of portfolio return is
dependent on the correlation or covariance structure of stocks
in the portfolio
 When the correlation of two stocks is 1, the standard deviation is the weighted
average of standard deviations of the stocks.
 When the correlation of two stocks is less than 1, the standard deviation of the
portfolio is less than the weighted average of standard deviations of the stocks.
 Since the correlations of stocks are in general less than 1, the standard deviation of
the portfolio is less than the weighted average of standard deviations of the stocks
 This effect is called diversification advantage
01/31/13 30

31. Calculate the Expected return and std of
the portfolio of 60% Alta and 40% Repo
Econ. Prob. T-Bill Alta Repo Am F. MP
Bust 0.10 8.0% -22.0% 28.0% 10.0% -13.0%
Below
0.20 8.0 -2.0 14.7 -10.0 1.0
avg.
Avg. 0.40 8.0 20.0 0.0 7.0 15.0
Above
0.20 8.0 35.0 -10.0 45.0 29.0
avg.
Boom 0.10 8.0 50.0 -20.0 30.0 43.0
1.00
31

32. Portfolio ER & STD - Mean-Variance Efficient
Frontier (Markowitz-1959)
• When we draw the efficient
frontier of all the stocks in the
EF is From B to C market, it looks like bellow.
Because, it gives Me a n - Va ria n c e Ef f ic ie n t Fro n tie r
C
• The Highest ER at a
given level of STD
Me a n = ER
and B
• The lowest STD at a
A
given level of ER
STD
32

33. Portfolio ER & STD- Mean-Variance
Efficient Frontier (Markowitz-1959)
Me an - Va ria n c e Ef f ic ie n t Fro n tie r
C
•The line from B to C is Called
the Capital Market Line
(CML) (without risk-free
Me a n = ER
B
lending & borrowing).
A
STD
33

34. Feasible and Efficient Portfolios
 The feasible set of portfolios represents all
portfolios that can be constructed from a
given set of stocks.
 An efficient portfolio is one that offers:
 the most return for a given amount of risk, or
 the least risk for a give amount of return.
 The collection of efficient portfolios is called
the efficient set or efficient frontier.
34

35. Capital Asset Pricing Model (CAPM)
Sharpe (64), Lintner (65)
Sharpe and Lintner introduced two basic assumptions
to the Markowitz’s EF.
1. Unlimited lending and borrowing at Risk-Free rate.
2. Homogeneous expectations or complete agreement
about the ER and STD of securities. This leads to
have a similar EF for all rational investors.
01/31/13 35

36. Efficient Set with a Risk-Free Asset
With risk-free lending and borrowing, the CML is as follows (Rf-M-Z).
The tangency portfolio would be the market portfolio.
Expected Z
Return, r p
. B
^
rM
M .
r RF
A . The Capital Market
Line (CML):
New Efficient Set
σM Risk, σ p 36

37. Capital Market Line (CML)
ERe
C ML
M
Rf
S TD
0 σm
01/31/13 37

38. The CML Equation
ER M - R F
ER p = RF + σ p.
σM
Intercept Slope
Risk
measur
e 38

39. What does the CML tell us?
 The expected rate of return on any
efficient portfolio is equal to the risk-free
rate plus a risk premium.
 The optimal portfolio for any investor is
the point of tangency between the CML
and the investor’s indifference curves.
39

40. What doesn’t the CML tell us?
 CML gives the ER and STD of efficient
portfolios
 The problem is that it only gives ER and
Risk (STD) of efficient portfolios.
 ER & STD of Inefficient portfolios and
individual stocks are not given
01/31/13 Prepared by P D Nimal 40

41. Capital Market Line &
Investor portfolio selection
I2
Expected
I1 CML
Return, r p
^
rM
^
r R .
R
. M 

I1-Risk Averse
I2-Risk Taker
R =
r RF Optimal
Portfolio
Risk, σ p
σR σM 41

42. Capital Market Line (CML)
cont…
• According to this analysis, the optimal portfolio of risky assets
would be the market portfolio (M).
• The portfolios from Rf to M are lending portfolios because they
lend a portion of their investment at Rf and
• The portfolios from M to upwards are borrowing portfolios
because they borrow some money at Rf and invest both their
capital and borrowed money in the market portfolio.
• Depending on the risk preference investor can choose a lending or
borrowing portfolio.
01/31/13 42

43. Lending & Borrowing Portfolios
• ER & Risk of lending and borrowing portfolios.
ER = Wrf R f + Wm ERm
σ = Wmσ m
Weight on the market portfolio is
•Less than 1 for lending portfolios and
•Greater than 1 for borrowing portfolios.
01/31/13 43

44. Lending & Borrowing Portfolios
•ER & Risk of lending and borrowing portfolios.
Rf Rm Prob. P(50:50)
3 1 0.1 2
3 0.9 0.2 1.95
3 5.4 0.5 4.2
3 5.8 0.2 4.4
Calculate the ER & STD of (0.5 Rf and 0.5 M) (a lending
portfolio) and (-0.5 Rf and 1.5 M) (a borrowing portfolio).
01/31/13 44

45. Adding Stocks to a Portfolio
 What would happen to the risk of a
portfolio as more randomly selected
stocks were added?
 σp would decrease because the added
stocks would not be perfectly
correlated.
45

46. σ1 stock ≈ 35%
σMany stocks ≈ 20%
1 st ock
2 st ocks
Many st ocks
-75 -60 -45 -30 -15 0 15 30 45 60 75 90 10
5
Ret urns ( % )
46

47. Risk vs. Number of Stock in Portfolio
σp
Company Specific
35%
(Diversifiable) Risk
Stand-Alone Risk, σ p
20%
Market Risk
0
10 20 30 40 2,000 stocks
47

48. Market risk & Diversifiable risk
 Market risk is that part of a security’s
risk that cannot be eliminated by
diversification.
 Firm-specific, or diversifiable, risk is
that part of a security’s risk that can be
eliminated by diversification.
48

49. Market risk & Diversifiable risk
Conclusions
 As more stocks are added, each new stock has a
smaller risk-reducing impact on the portfolio.
 σp falls very slowly after about 40 stocks are included.
The lower limit for σp is σM
 By forming well-diversified portfolios, investors can
eliminate about half the risk of owning a single stock.
49

50. The Problem of CML
Investment and Portfolio Management II
 CML gives the ER and STD of efficient portfolios
 The problem is that it only gives ER and Risk
(STD) of efficient portfolios.
 ER & STD of Inefficient portfolios and individual
stocks are not given
 The SML of CAPM will solve this problem which
will be discussed in the Investment and Portfolio
Management II
01/31/13 Prepared by P D Nimal 50