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Chapter 7
Sources of Risks and Their Determination
ByCheng Few LeeJoseph Finnerty
John LeeAlice C Lee
Donald Wort
Chapter Outline• 7.1 RISK CLASSIFICATION AND MEASUREMENT
• 7.1.1 Call Risk• 7.1.2 Convertible Risk• 7.1.3 Default Risk• 7.1.4 Interest-Rate Risk• 7.1.5 Management Risk• 7.1.6 Marketability (Liquidity) Risk• 7.1.7 Political Risk• 7.1.8 Purchasing-Power Risk• 7.1.9 Systematic and Unsystematic Risk
• 7.2 PORTFOLIO ANALYSIS AND APPLICATION• 7.2.1Expected Return on a Portfolio• 7.2.2Variance and Standard Deviation of a Portfolio• 7.2.3The Two-Asset Case• 7.2.4 Asset Allocation among Risk-Free Asset, Corporate Bond, and Equity
• 7.3 THE EFFICIENT PORTFOLIO AND RISK DIVERSIFICATION• 7.3.1 The efficient Portfolio • 7.3.2 Corporate Application of Diversification• 7.3.3 The Dominance Principle• 7.3.4 Three Performance Measures• 7.3.5 Interrelationship among Three Performance Measure
• 7.4 DETERMINATION OF COMMERCIAL LENDING RATE• 7.5 THE MARKET RATE OF RETURN AND MARKET RISK PREMIUM
2
7.1 RISK CLASSIFICATION AND MEASUREMENT
• Call Risk• Convertible Risk• Default Risk• Interest-Rate Risk• Management Risk• Marketability (Liquidity) Risk• Political Risk• Purchasing-Power Risk• Systematic and Unsystematic Risk
3
Figure 7-1
Probability Distributions Between Securities A and B
4
TABLE 7-1 Types of RiskRisk Type Description
Call risk The variability of return caused by the repurchase of the security before its stated maturity.
Convertible risk The variability of return caused when one type of security is converted into another type of security.
Default risk The probability of a return of zero when the issuer of the security is unable to make interest and principal payments—or for equities, the probability that the market price of the stock will go to zero when the firm goes bankrupt.
Interest-rate risk The variability of return caused by the movement of interest rates.
Management risk The variability of return caused by bad management decisions; this is usually a part of the unsystematic risk of a stock, although it can affect the amount of systematic risk.
Marketability risk(Liquidity risk)
The variability of return caused by the commissions and price concessions associated with selling an illiquid asset.
Political risk The variability of return caused by changes in laws, taxes, or other government actions.
Purchasing-power risk The variability of return caused by inflation, which erodes the real value of the return.
Systematic risk The variability of a single security’s return caused by the general rise or fall of the entire market.
Unsystematic risk The variability of return caused by factors unique to the individual security.
（ Convertible Risk ）
5
• Business risk refers to the degree of fluctuation of net income associated with different types of business operations. This kind of risk is related to different types of business and operating strategies.
• Financial risk refers to the variability of returns associated with leverage decisions. The question then arises as to how much of the firm should be financed with equity and how much should be financed with debt.
Types of Risk (Continued)
6
7.2 PORTFOLIO ANALYSIS AND APPLICATION
• Expected Return on a Portfolio• Variance and Standard Deviation of a Portfolio
• The Two-Asset Case• Asset Allocation among Risk-Free Asset, Corporate Bond, and Equity
7
7.2.1 Expected Return on a Portfolio
• Portfolio analysis is used to determine the return and risk for these combinations of assets.
• The rate of return on a portfolio is simply the weighted average of the returns of individual securities in the portfolio.
in which are the percentages of the portfolio invested in securities A, B, and C, respectively.
091.0
)12.0)(3.0()5.0)(3.0()1.0)(4.0(
cccbaap RWRWRWR
cba WWW and ,,
8
(7.1)
where:
n
iiip WRR
1
1
1
the proportion of the individual's investment allocated to security ;and
the expected rate of return for security .
n
ii
i
i
W
W i
R i
7.2.1 Expected Return on a Portfolio
9
7.2.2 Variance and Standard Deviation of a Portfolio
(7.2)
where:
),( Cov
1
))((
1
))((),( Cov
2121
2211
121
1
222211112211
RRWW
N
RRRRWW
N
RWRWRWRWRWRW
ttN
t
N
t
tt
. and between covariance the),( Cov
and ly;respective security, second
theandsecurity dirst for thereturn of rates average and
; periodin security second for thereturn of rate the
; periodin security first for thereturn of rate the
2121
21
2
1
RRRR
RR
tR
tR
t
t
10
The covariance as indicated in Equation (7.2) can be used to measure the covariability between two securities (or assets) when they are used to formulate a portfolio. With this measure the variance for a portfolio with two securities can be derived:
(7.3)1
)]()[()(Var
222112211
12211
N
RWRWRWRWRWRW tt
N
ttt
7.2.2 Variance and Standard Deviation of a Portfolio
11
• The general formula for determining the number of terms that must be computed (NTC) to determine the variance of a portfolio with N securities is
7.2.2 Variance and Standard Deviation of a Portfolio
2
NTC variances + covariances 2
N NN
12
Sample Problem 7.1
Security 1 Security 2
%15
20% 3
15% 2
%10 1
%40
1
1
1
R
Rt
W
t
%10
15% 3
10% 2
%5 1
%60
1
2
2
R
Rt
W
t
0025.0 portfolioVar 2
)12.017.0(
2
)12.012.0(
2
)12.007.0(
2/](0.6)(0.1))(0.4)(0.15)(0.6)(0.15)[(0.4)(0.2
2/](0.6)(0.1))(0.4)(0.15(0.6)(0.1)5)[(0.4)(0.1
2/)]1.0)(6.0()15.0)(4.0()05.0)(6.0()1.0)(4.0[(
1
)]()[()(Var
222
2
2
2
222112211
12211
N
RWRWRWRWRWRW tt
N
ttt
13
Sample Problem 7.1
14
Sample Problem 7.1
• The riskiness of a portfolio can be measured by the standard deviation of returns as:
(7.6)
where is the standard deviation of the portfolio’s return and is the expected return of the n possible returns.
2
1
( )
1
N
pt pt
p
R R
N
ppR
15
7.2.3 The Two-Asset Case• To explain the fundamental aspect of the risk-diversification process in a portfolio, consider the two-asset case:
(7.7)
where
),(Cov2)(Var)Var(W
1
]))((2)()([W
1
)(
212122
212
1
N
1t221121
222
22
211
21
1
2
RRWWRWR
N
RRRRWWRRWRR
N
RR
tt
tttt
N
tppt
p
1 2 1W W
16
7.2.3 The Two-Asset Case
• By the definitions of correlation coefficients between and , the can be rewritten:
(7.8)
Where and are the standard deviations of the first and second security, respectively.
• From Equations (7.7) and (7.8), the standard deviation of a two-security portfolio can be defined as
(7.9)
1 2 12 1 2( , )Cov R R
12
1R 2 12, ( )R 1 2( , )Cov R R
21121122
21
21
21
2211
)1(2)1(
)(Var
WWWW
RWRW ttp
17
Sample Problem 7.2
For securities 1 and 2 used in the previous example, applying Equation (7.9), we get:
Security 1 Security 2
Var portfolio = 0.0025, the same answer as for
Sample Problem 7.1.
1
4.0
0025.0
12
1
21
W 6.0
0025.0
2
22
W
0.0025
)05.0)(05.0)(1)(6.0)(4.0(2)0025.0()6.0()0025.0()4.0( 22
p
or 05.0p
18
Sample Problem 7.2 If = 1.0, Equation (7.6) can be simplified to the linear expression:
where . Since Equation (7.9) is a quadratic equation, some value of minimizes .
To obtain this value, differentiate Equation (7.9) with respect to and set this derivative equal to zero. Then we get:
(7.10a)
12
2211 WWp
)1( 12 WW 1W p
211222
21
112221 2
)(
W
1W
19
Sample Problem 7.2
If Equation (7.10a) reduces from:
To
(7.10b)
,112
])1(2)1([
][]2[
])21(2)1(22[
])1(2)1([
21121122
21
21
21
2112212112
22
211
21121221
211
2/1211211
22
21
21
21
1
WWWW
W
WWW
WWWWW
p
2 2 1 21
2 1 2 1 2 1
( )
( )( )W
20
Sample Problem 7.2• If , Equation (7.10a) reduces to:
(7.10c)• However, if the correlation coefficient between 1 and 2 is –1, then
the minimum-variance portfolio must be divided equally between security 1 and security 2—that is:
112
2 2 1 21
1 2 1 2 2 1
( )
( )( ) ( )W
5.0 05.005.0
05.01
W
21
Sample Problem 7.2As an expanded form of Equation (7.9), a portfolio can be written:
(7.11)
where:
2/1
11
2/1
1 1
1
1
21
21
|),(Cov
2
jtitji
n
i
n
j
jiijji
n
i
n
ij
n
ip
RRWW
WWW
portfolio. in the included securities ofnumber the
and ; security and security between t coefficienn correlatio the
ly;respective ,security and security toallocated investment sinvestor' the and
n
ji
jiWW
ij
ji
22
Sample Problem 7.3
Consider two stocks, A and B: (1) If a riskless portfolio could be formed from A and B, what would be the expected return of ? (2) What would the expected return be if ?
Solution
1.
If we let
So
2/12222 ))1(2)1(( BAABAABAAAp WWWW
B10%, 15%, 4, and 6.A B AR R
pR0AB
1AB[ (1 ) ], 0 4 6(1 )
3 / 5
p A A A B p A A B
A
W W W W R
W
3 2(1 ) (10%) (15%) 12%
5 5p A A A BR W R W R
23
Sample Problem 7.3
Solution
2. If we let
Then,
0AB 2 2 2 2 1/2
2 2 2 2 2 2 1/2
[ (1 ) ]
1[2 2(1 ) ( 1)][ (1 ) ] 0
2
p A A A B
pA A A B A A A B
A
W W
W W W WW
2 2
2 2 2 2 2 2
(1 ) 0
/ ( ) 6 / (4 6 ) 9 /13
9 4 (10%) (15%) 11.54%
13 13
A A A B
A B A B
P
W W
W
R
24
7.2.4 Asset Allocation among Risk-Free Asset, Corporate Bond, and Equity• The most straightforward way to control the risk of the portfolio is through the fraction of the portfolio invested in Treasury bills and other safe money market securities versus risky assets.
• The capital allocation decision is an example of an asset allocation choice — a choice among broad investment classes, rather than among the specific securities within each asset class.
• Most investment professionals consider asset allocation as the most important part of portfolio construction.
25
Sample Problem 7.4• Private fund $500,000 investing in a risk-free asset $100,000, risky equities (E) $240,000, and long-term bonds (B) $160,000.
• Current risky portfolio consists 60% of E and 40% of B, and the weight of the risky portfolio in the mutual fund is 80%.
26
Sample Problem 7.4• Suppose the fund manager wishes to decrease risk portfolio from 80% to 70%, then should sell $400,000-0.7 ($500,000)=$50,000 of risky holdings, with the proceed used to purchase more shares in risk-free asset.
• To keep the same weights of E and B (60% and 40%)in the risky portfolio, the fund manager should sell
• 0.6×50,000=$30,000 in E• 0.4×50,000=$20,000 in B
27
7.3 THE EFFICIENT PORTFOLIO AND RISK DIVERSIFICATION
• The efficient Portfolio • Corporate Application of Diversification• The Dominance Principle• Three Performance Measures• Interrelationship among Three Performance Measure
28
7.3.1 The Efficient Portfolio
• Definition: A portfolio is efficient, if there exists no other portfolio having the same expected return at a lower variance of returns, or, if no other portfolio has a higher expected return as the same risk of returns.
• This suggests that given two investments, A and B, investment A will be preferred to B if:
Where E(A) and E(B) = the expected returns of A and B, Var(A) and Var(B) = their respective variances or risk.
(A) (B) and Var A Var B
(A) (B) and Var A Var B
E E
or
E E
29
7.3.1 The efficient Portfolio
30
Sample Problem 7.5• Monthly rates of return for April, 2001 to April, 2010 for
Johnson & Johnson (JNJ) and IBM are used as examples. The basic statistical estimates for these two firms are average monthly rates of return and the variance-covariance matrix.in Table 7.3:
Variance-Covariance Matrix
JNJ IBM
JNJ 0.0025 0.0007
IBM 0.0007 0.0071
31
Sample Problem 7.5From Equation (7.10), we have:
Using the weight estimates and Equations (7.2) and (7.3):
When is less than 1.00 it indicates that the combination of the two securities will result in a total risk less than their added respective risks.
12
1
2
0.0071 .0007 0.00640.5818
0.0025 0.0071 2(0.0007) 0.011
1.0 0.5818 0.4182
W
W
2 2 2
( ) (0.5818)(0.0080) (0.4182)(0.0050)
0.0067454
(0.5818) (0.0025) (0.4182) (0.0071) 2(0.5818)(0.4182)(0.0007)
0.0024
0.0493
P
P
P
E R
32
7.3.2 Corporate Application of Diversification• The effect of diversification is not necessarily limited to
securities but may have wider applications at the corporate level.
• Instead of “putting all the eggs in one basket,” the investment risks are spread out among many lines of services or products in hope of reducing the overall risks involved and maximizing returns.
• The overall goal is to reduce business risk fluctuations of net income.
33
7.3.3 The Dominance Principle• The dominance principle has been developed as a means of conceptually
understanding the risk/return tradeoff.
• As with the efficient-frontier analysis, we must assume an investor prefers returns and dislikes risks.
34
7.3.4 Three Performance Measures The Sharpe measure (SP) (Sharpe, 1966) is of immediate concern.
Given two of the portfolios depicted in Figure 7.4, portfolios B and D, their relative risk-return performance can be compared using the equations:
and
where
D
fDD
RR
SPB
fBB
RR
SP
portfolio.each ofrisk on deviation standard respective the,
and rate; free-risk
portfolio;each ofreturn average the,
measures; eperformanc SharpeSP,SP
BD
f
BD
BD
R
RR
35
Sharpe measure (SP)If a riskless rate exists, then all investors would prefer A to B because combinations of A and the riskless asset give higher returns for the same level of risk than combinations of the riskless asset and B.
36
Sample Problem 7.6
Using the Sharpe performance measure, the risk-return measurements for these two firms are:
Jones fund has better performance based on Sharpe measure.
Table 7.4 Smyth Fund Jones Fund
Average return R (%) 18 16
Standard deviation (%) 20 15
Risk-free rate (%) 9.5fR
433.015.0
095.016.0SP
425.020.0
095.018.0SP
Jones
Smyth
37
Sample Problem 7.7The performances of portfolios A-E shown in Table 7.5.
By using Sharpe measure , assume risk-free rate is 8%, the rank of portfolios is A>B>E>C>D:
Protfolio A is the most desirable.
However, for risk-free rate 5%, the order changes to E>B>A>D>C:
Now E is the best portfolio.
Portfolio Return (%) Risk (%)
A 50 50
B 19 15
C 12 9
D 9 5
E 8.5 1
fM
M
RR SP
SP 0.84,SP 0.73,SP 0.44,SP 0.20,SP 0.50A B C D E
SP 0.90,SP 0.933,SP 0.77,SP 0.80,SP 0.35A B C D E
38
Treynor measure (TP)• Treynor measure (TP), developed by Treynor in 1965, examines
differential return when beta is the risk measure.
The Treynor measure can be expressed by the following:
(7.13)
where:
• The Treynor performance measure uses the beta coefficient (systematic risk) instead of total risk for the portfolio as a risk measure.
j
fj RR
TP
average return of th portfolio;
risk free rate; and
beta coefficient for th portfolio.
j
f
j
R j
R
j
39
Jensen’s measure (JM)
PfMfP RRRR )(
])([JM pfMfP RRRR
• Jensen (1968, 1969) has proposed a measure referred to as the Jensen differential performance index (Jensen’s measure or JM).
• JM is the differential return which can be viewed as the difference in return earned by the portfolio compared to the return that the capital asset pricing line implies should be earned.
• CAPM: • (7.14)
• (7.15)
40
Sample Problem 7.8
Rank portfolios based on JM:
(1)When RM=10% and Rf=8%,
(2)When RM=12% and Rf=8%,
Portfolio (%) (%)
A 50 50 2.5
B 19 15 2.0
C 12 9 1.5
D 9 5 1.0
E 8.5 1 0.25
JM ( ) ( )i f i M fR R R R
iR i
JM 37 %,JM 7 %,JM 1%,JM 2 %,JM 0 %
A>B>C>E>DA B C D E
JM 32%,JM 3 %,JM 2 %,JM 3 %,JM 0.5 %
A>B>E>C>DA B C D E
41
Sample Problem 7.8
Rank portfolios based on JM:
(3)When RM=8% and Rf=8%,
(4)When RM=12% and Rf=4%,
Portfolio (%) (%)
A 50 50 2.5
B 19 15 2.0
C 12 9 1.5
D 9 5 1.0
E 8.5 1 0.25
JM ( ) ( )i f i M fR R R R
iR i
JM 42%,JM 11 %,JM 4 %,JM 1 %,JM 0.5 %
A>B>C>D>EA B C D E
JM 26 %,JM 1 %,JM 4 %,JM 3 %,JM 2.5 %
A>E>B>D>CA B C D E
42
7.3.5 Interrelationship among Three Performance Measure
mppmmpmp / and / pm2
[ ] [ ] ( )JM (7.17)
[ ] [ ] SP SP (commom constant)
P f M f pm
P P m m p
P f M fP m
P m
R R R R
R R R R
[ ] [ ]JM (7.18)
TM [ ] TM commom constant
P f M f P
P P P
P M f P
R R R R
R R
Since
Since (7.16) The JM must be multiplied by in order to derive the equivalent SM:
If the JM divided by ,it is equivalent to the TM plus some constant common to all portfolios:
1/𝜎 𝑃
𝛽𝑃
43
Sample Problem 7.9 Continuing with the example used for the Sharpe performance measure in
Sample Problem 7.6, assume that in addition to the information already provided the market return is 10 percent, the beta of the Smyth Fund is 0.8, and the Jones Fund beta is 1.1. Then, according to the capital asset pricing line, the implied return earned should be:
Using the Jensen measure, the risk-return measurements for these two firms are:
1005.0)1.1)(095.010.0(095.0
099.0)8.0)(095.010.0(095.0
Jones
Smyth
R
R
0595.01005.016.0JM
081.0)099.018.0JM
Jones
Smyth
44
7.4 Determination of Commercial Lending Rate Based upon the mean and variance Equations (7.1) and (7.2) it is possible to
calculate the expected lending rate and its variance. Using the information provided in Table 7.8, the weighted average and the standard deviation can be calculated:
(0.100)(15%) (0.075)(17%) (0.075)(20%) (0.200)(13%) (0.150)(15%)
(0.150)(18%) (0.100)(11%) (0.075)(13%) (0.075)(16%)
15.1%
R
2 2 2 2 2
2 2 2 2 1/2
[(0.100)(15 15.1) (0.075)(17 15.1) (0.075)(20 15.1) (0.200)(13 15.1) (0.150)(15 15.1)
(0.150)(18 15.1) (0.100)(11 15.1) (0.075)(13 15.1) (0.075)(16 15.1) ]
2.51%
(A) (B) (C) (D) ( ) ( A + C ) Joint Lending
Economic Probability RateConditions (%) Probability (%) Probability of Occurrence (%)
Boom 12 0.25 3.0 0.40 0.100 15 5.0 0.30 0.075 17 8.0 0.30 0.075 20
Normal 10 0.50 3.0 0.40 0.200 13 5.0 0.30 0.150 15 8.0 0.30 0.150 18
Poor 8 0.25 3.0 0.40 0.100 11 5.0 0.30 0.075 13 8.0 0.30 0.075 16
DB
fR pR
45
According to lending rates in Table 7.8The weighted average and the standard deviation are:
15.1%, 2.51%R
46
7.5 The Market Rate Of Return And Market Risk Premium
• The market rate of return is the return that can be expected from the market portfolio.
• The market rate of return can be calculated using one of several types of market indicator series, such as the Dow-Jones Industrial Average or the Standard and Poor (S&P) 500 by using the following equation:
(7.19)
where:
mtt
tt RI
II
1
1
.1at index market
and ;at index market
; at timereturn of ratemarket
1
tI
tI
tR
t
t
mt
47
7.5 The Market Rate Of Return And Market Risk Premium • A risk-free investment is one in which the investor is sure
about the timing and amount of income streams arising from that investment.
• The reasonable investor dislikes risks and uncertainty and would, therefore, require an additional return on his investment to compensate for this uncertainty. This return, called the risk premium, is added to the nominal risk-free rate.
• Table 7.9 illustrates the concept of risk premium by using the market rate of return of S&P 500 index.
48
7.5 The Market Rate Of Return And Market Risk Premium
TABLE 7.9 Market Returns and T-bill by Quarters
Quarter S&P 500
(A)MarketReturn
(percent)
(B)T-BillRate
(percent)
(A) － (B)Risk
Premium(percent)
1980
IV 135.76
1981
I 136.00 + 0.18 13.36 -13.18
II 131.21 - 3.52 14.73 -18.25
III 116.18 -12.94 14.70 -27.64
IV 122.55 + 5.48 10.85 - 5.37
49
7.5 The Market Rate Of Return And Market Risk Premium
Quarter S&P 500
(A)MarketReturn
(percent)
(B)T-BillRate
(percent)
(A) － (B)Risk
Premium(percent)
1983
I 152.96 + 8.76 8.35 0.41
II 168.11 + 9.90 8.79 1.11
III 166.07 - 1.22 9.00 -10.22
IV 164.93 - 0.69 9.00 - 9.69
1984
I 159.18 - 3.49 9.52 -13.01
II 153.18 - 3.77 9.87 -13.64
III 166.10 + 8.43 10.37 - 1.94
IV 167.24 + 0.68 8.06 - 7.37
TABLE 7.9 Market Returns and T-bill by Quarters (Continued)
50
7.5 The Market Rate Of Return And Market Risk Premium
Quarter S&P 500
(A)MarketReturn
(percent)
(B)T-BillRate
(percent)
(A) － (B)Risk
Premium(percent)
1985
I 180.66 + 8.02 8.52 - 0.50
II 188.89 + 4.55 6.95 - 2.4
III 184.06 - 2.62 7.10 - 9.72
IV 207.26 +12.60 7.07 + 5.53
1986
I 232.33 +12.09 6.56 + 5.53
II 245.30 + 5.58 6.21 - 0.63
III 238.27 - 2.86 5.21 - 8.07
IV 248.61 + 4.33 5.43 - 1.10
TABLE 7.9 Market Returns and T-bill by Quarters (Continued)
51
7.5 The Market Rate Of Return And Market Risk Premium
Quarter S&P 500
(A)MarketReturn
(percent)
(B)T-BillRate
(percent)
(A) － (B)Risk
Premium(percent)
1987
I 292.47 +17.64 5.59 +12.05
II 301.36 + 3.03 5.69 - 2.66
III 318.66 + 5.74 6.40 + 0.05
IV 240.96 -24.38 5.77 -30.10
TABLE 7.9 Market Returns and T-bill by Quarters (Continued)
52
7.6 SUMMARY
This chapter has defined the basic concepts of risk and risk measurement. The efficient-portfolio concept and its implementation was demonstrated using the relationships of risk and return. The dominance principle and performance measures were also discussed and illustrated. Finally, the interest rate and market rate of return were used as measurements to show how the commercial lending rate and the market risk premium can be calculated.
Overall, this chapter has introduced uncertainty analysis assuming
previous exposure to certainty concepts. Further application of the concepts discussed in this chapter as related to security analysis and portfolio management are explored in later chapters.
53
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