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Ch. 7 Introduction to Risk, Return, and the Opportunity Cost of Capital
━ three issues on risk:
1) How risk is defined (ch. 7)
2) What the links are between risk and the OCC (ch. 8)
3) How the financial manager can cope with risk in practical situations (ch. 9)
notes: 1) two major modern developments in finance: risk --> r and OPM
2) CAPM, APT, and three-factor model --> cross-sectional variations in stock returns
━ some concepts on risk:
The risk in investment means that future returns are unpredictable. The spread of possible
outcomes is usually measured by the standard deviation of return.
However, we will explain that the effective risk of any security can not be judged by an
examination of that security alone. Risk is best judged in a portfolio context.
We will introduce the concept of beta, the standard risk measure for individual securities. It is a
security‟s contribution to the risk of a well-diversified portfolio.
━ main themes in the chapter: diversification and portfolio risk
notes: 1) implicit assumption: Investors are diversified investors. Why?
2) Diversification practice in Philippines? STD as risk measure?
3) diversified investors => diversification => portfolio risk => beta
7-1 Over A Century of Capital Market History in One Easy Lesson
━ CRSP (University of Chicago‟s Center for Research in Security Prices) database
a note: Other well-known databases: a) bond, stock, option data - Datastream and Bloomberg
b) financial statement data - Compustat
━ a study by Dimson, Marsh, and Staunton (2002) on the historical performance of 3 portfolios
of securities for the period 1900 to 2003 (updated by Brealey and Myers):
3 portfolios: 1) Treasury bills (some uncertainty of inflation)
2) long-term government bonds (interest-rate risk)
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[3) long-term corporate bonds (default risk)]
4) common stocks of large firms (S&P 500) (business risk)
[5) common stocks of small firms (greater business risk and some „size‟ risk)]
the performance: Figures 7.1 and 7.2, Table 7.1
a note: an earlier study by Ibbotson Associates on the historical performance of 5 portfolios of
securities for the period 1926 to 2000:
Average Annual Rate of Return
Portfolio Nomial Real Average Risk Premium
Treasury Bills 3.9 0.8 0
Government Bonds 5.7 2.7 1.8
Corporate Bonds 6.0 3.0 2.1
Common Stocks (S&P 500) 13.0 9.7 9.1
Small-firm Common Stocks 17.3 13.8 13.4
a lesson: Portfolio performance (or return) coincides with our intuitive risk rankings among the
five portfolios.
━ arithmetic averages and compound annual returns (i.e., geometric averages)
T
● AM = r /T
t 1
t
T
● GM = [ (1 rt )]1 / T 1 ( AT / A0 )1 / T 1 , where A0 = the initial wealth, AT = the
t 1
ending-period wealth
Notice that the GM is the rate r such that A0 (1 r ) T AT , which indicates that the GM is a
compound return.
● comparison:
AM is biased upward if you are attempting to measure an asset‟s long-term performance. GM
is considered a superior measure of the long-term mean rate of return because it indicates the
compound annual rate of return.
footnote 5: 1) For S&P 500 in the period 1900-2003, GM = (15,579)1/75 – 1 = 0.097 while AM
= 0.117.
2) For lognormally distributed returns, GM = AM – 0.5 ╳ variance
2
● moral: If the cost of capital is estimated from historical returns or risk premiums over a very
long period, use arithmetic averages, not compound annual rates of return. [Notice that
in calculating PV, both the cash flows and the discount rate are expectations.]
━ using historical evidence to evaluate today‟s cost of capital
● case: An asset has the same risk as the market.
● market portfolio: a portfolio including all risky assets S&P Composite Index
the expected rate of return on the market portfolio: market return, r m
● estimate r m in 2004:
the fact: rf varies over time. In 1981, it is 15% => r m is not likely to be stable over time.
a more sensible procedure:
rm (2004) = rf (2004) + normal risk premium, where rf = the interest rate on Treasury bill
= 0.01 + 0.076 = 0.086
the crucial assumption here: There is a normal, stable risk premium on the market portfolio,
so that the expected future risk premium can be measured by the average past risk premium.
● the assumption above and the argument about the market risk premium:
Many financial managers and economists believe that the long-run historical risk premium
(7.6%) is the best measure available. Others argue that investors don‟t need such a large risk
premium to persuade them to hold common stocks. The argument is based on two reasons.
● an alternative measure of the risk premium: r = DIV1/P0 + g
Since 1900, DIV1/P0 = 4.7% and g = 4.7%. => r m = 9.4% or about 5.3% above the
risk-free interest rate. This is 2.3% lower than the realized risk premium reported in Table 7.1.
● Why have we talked so much about the market risk premium? Why is it so important?
The CAPM, the theory linking risk to the OCC, hinges on the market risk premium:
ri rf i market risk premium
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7-2 Measuring Portfolio Risk
━ two benchmarks we have obtained: the discount rate for safe projects and the estimate of the
rate for average-risk projects
To estimate discount rates for assets that do not fit these simple cases, however, we have to
learn (1) how to measure risk and (2) the relationship between risks borne and risk premiums
demanded.
━ Figures 7.4 and 7.5: remarkably wide fluctuations in annual rates or market return r m
━ the standard statistical measures of spread: variance and standard deviation
example: the market portfolio
~ = the actual market return (random return)
rm
rm = the expected market return (mean)
variance of ~ = m =
rm 2
E[( ~ - rm )2]
rm
standard deviation of ~ = m
rm
When variance is estimated from a sample of observed returns, the formula of variance is
N
(~
r
t 1
mt rm ) 2 /( N 1) , where N is the number of observations and ~ is the market
rmt
return in period t.
example: a game of flipping two coins (Table 7.2)
notes:
1) footnote 18: Since STD is in the same units as the rate of return, it is generally more
convenient to use STD. However, when we are talking about the proportion of risk that is
due to some factor, it is usually less confusing to work in terms of the VAR.
2) rate of return in percentage => the STD in percentage (i.e., the spread in percentage)
The measure of risk in terms of the STD in value (i.e., the spread in value) will be easily
interpreted to investors. How do we convert the STD in percentage into the STD in value?
3) value at risk (VaR)
━ observing past variability
● the annual STDs and VARs observed for the 3 portfolios over the period 1900-2003:
4
Portfolio STD(%) VAR(%)
Treasury bills 2.8 7.9
Government bonds 8.2 68.0
Common stocks (S&P 500) 20.1 402.6
a note: the annual STDs and VARs observed for the 5 portfolios over the period 1926-2000:
Portfolio STD(%) VAR(%)
Treasury bills 3.2 10.1
Government bonds 9.4 88.7
Corporate bonds 8.7 75.5
Common stocks (S&P 500) 20.2 406.9
Small-firm common stocks 33.4 1118.4
● the annual STDs of the market returns for successive periods:
Period Market STD (%)
1926-1930 21.7
1931-1940 37.8
1941-1950 14.0
1951-1960 12.1
1961-1970 13.0
1971-1980 15.8
1981-1990 16.5
1991-2003 14.8
footnote 21: VAR is proportional to the length of time interval over which a security or
portfolio return is measured, STD is proportional to the square root of the interval.
● Figure 7.6: the STDs of stock market returns in 16 countries over the period 1900-2003
Most of the countries cluster together with percentage STDs in the low 20s.
━ How diversification reduces risk.
● comparison and question raised:
individual stocks: Tables 7.3 and 7.4
market portfolio: Tables above and 7.4
5
=> Q: m < average of individual stocks
The market portfolio is made up of individual stocks, so why doesn‟t its variability
reflect the average variability of its components?
● answer: Diversification reduces variability. (Figure 7.7)
Diversification works because prices of different stocks do not move exactly together, i.e.,
stock price changes are less than perfectly positively correlated. On many occasions, a decline
in the value of one stock was offset by a rise in the price of another. Therefore there was an
opportunity to reduce our risk by diversification. (Figure 7.8 and footnotes 24 and 25)
footnote 23:
● unique risk and market risk:
unique risk (diversifiable, unsystematic, specific, or residual risk): the perils that is peculiar to
an individual company
market risk (systematic or undiversifiable risk): the perils that is economywide, threatening
all businesses
Figure 7.9
7-3 Calculating Portfolio Risk
━ point: To understand fully the effect of diversification, we need to know how the risk of a
portfolio depends on the risk of the individual stocks.
━ covariance of two random variables X and Y:
T
Var ( X ) X E[( X X ) 2 ] or s X [ ( xt X ) 2 ] /(T 1) , X , s X 0
2 2 2 2
t 1
T
Var (Y ) Y E[(Y Y ) 2 ] or sY [ ( y t Y ) 2 ] /(T 1) , Y , sY 0
2 2 2 2
t 1
T
Cov( X , Y ) XY E[( X X )(Y Y )] or s XY [ ( xt X )( y t Y )] /(T 1) ,
t 1
XY , s XY
figure for S XY > 0:
figure for S XY < 0:
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a measure free from units: the coefficient of correlation
XY XY / X Y or rXY s XY / s X sY , 1 XY , rXY 1
figure for rXY = +1 (perfectly positive correlation):
figure for 0 < rXY < +1 (positive correlation):
figure for rXY = -1 (perfectly negative correlation):
figure for -1 < rXY < 0 (negative correlation):
figure for rXY = 0 (no correlation):
━ the expected return and risk for a portfolio consisting of two stocks:
~ = stock i‟s actual return, r = stock i‟s expected return, i = 1, 2
ri i
xi = proportion invested in stock i, i = 1, 2
~ = portfolio‟s actual return = ~ x ~ x ~
rp rp 1r1 2 r2
● portfolio‟s expected return:
rp E ( ~p ) E ( x1~ x 2 ~ ) x1 E ( ~ ) x 2 E ( ~ ) x1r1 x 2 r2 , the weighted average of the
r r1 r2 r1 r2
expected return on the two stocks
● portfolio‟s variance:
p V a (~p ) x12V a (~ ) x2V a (~ ) 2 x1 x2C o v~, ~ )
2
rr rr1 2
rr2 (r1 r2
= x12 12 x2 2 2 x1 x2 12 x12 12 x2 2 2 x1 x2 12 1 2 ,
2 2 2 2
where 12 12 / 1 2
See the box in Figure 7.10.
● example for calculating p : 1 = 18.2%, 2 = 27.3%, x1 = 0.60, x 2 = 0.40
2
r1 = 10%, r2 = 15%
1) 12 = +1.0 => p = 477%, p = 21.8% = 0.60 ╳ 18.2% + 0.40 ╳ 27.3%, the
2
weighted average of the two STDs
Notice that p x1 1 x 2 2 when 12 = +1.0.
2) 12 = +0.4 => p = 333.9%, p = 18.3%
2
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3) 12 = -1.0 => p = 0%
2
notes: 1) an intuitive way to calculate the risk of the portfolios: the weighted average of the
STDs on the two stocks = 0.60 ╳18.2% + 0.40╳27.3% = 21.8%
However, we see p = weighted average if 12 = +1.0
p < weighted average if 12 < +1.0
Hence, diversification would reduce risk if individual stocks are not perfectly
positively correlated.
2) For any 12 , there is some x1 such that p is minimized. If 12 = -1.0, we can
*
easily find the optimal x1 such that p = 0. In the example, x1 = 0.60.
* *
3) a helpful diagram:
━ general formula for the portfolio consisting of N stocks:
~ x ~ x ~ ... x ~
rp 1r1 2 r2 N rN
● expected return: rp x1r1 x 2 r2 ... x N rN , a weighted average
N N
● variance: x i x j ij
2
P
i 1 j 1
N N N N N N
xi i x i x j ij or xi i 2 x x
2 2 2 2
= i j ij
i 1 i j i 1 i j
point: The variability of a portfolio reflects mainly the covariances. See Figure 7.11.
● limits to diversification:
1
assumption: equal investment ( xi )
N
1 1
portfolio variance = N 2
average variance + ( N 2 N ) 2 average covariance
N N
1 1
= average variance + (1 ) average covariance
N N
Thus, as N , portfolio variance average covariance.
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point: Most of common stocks do move together, not independently
=> positive covariances
=> average covariance > 0
Therefore, positive covariances between most of common stocks set the limit to the
benefits of diversification.
conclusion: market risk = the average covariance remaining after diversification, i.e., the
bedrock of risk remaining after diversification has done its work
7-4 How Individual Securities Affect Portfolio Risk
━ diversified investors‟ concern: the effect that each security will have on the risk of their
portfolio, i.e., the (marginal) contribution of an individual security to the risk of the portfolio
━ the risk of any security = market risk + unique risk
unique risk: the risk that is peculiar to the stock (the firm)
market risk: the risk that is associated with market-wide variations
a note: Recall that when we are talking about the proportion or component of risk that is due to
some factor, it usually works in terms of variance.
=> The risk of a well-diversified portfolio (i.e., a portfolio without unique risk) depends on the
market risk of the securities included in the portfolio.
notes: 1) a helpful diagram for the statement above:
2) There are 3 stars ☆ in the following diagram, indicating the 3 stages about the
issues discussed here:
portfolio risk p <---------- 1 ---------- systematic risk of a security i
↑ │
p │3 │2
| <------------------- i <----------------------- |
Now we have gone by the first stage (☆1).
━ Market risk (for a security) is measured by beta.
● concept: The market risk of a security indicates that how sensitive its return is to market
movements (☆1). This sensitivity is called beta.
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● mathematical expression of beta:
market portfolio portfolio of S&P 500
actual return of market portfolio:
N
rm 1r1 2r2 N rN j rj , where ~j = actual return for stock j and
~ x ~ x ~ ... x ~ x ~
j 1
r
x j = stock j‟s weight
security i‟s beta:
Cov( ~ , ~ ) im
ri rm
i 2
Var ( ~ )
rm m
where im Cov( ~ , x1~ x2 ~ ... x N ~ )
ri r1 r2 rN
N
= x1Cov( ~, ~) x2Cov( ~, ~ ) ... xN Cov( ~, ~ ) =
ri r1 ri r2 ri rN x
j 1
j ij ,
the weighted average of covariances among stock i and others in the
market (i.e., the market risk of security i)
● interpretation to the formula:
1) The formula can be rewritten as im i m : The security beta measures how the
2
stock‟s return is liable to be affected by general market movements, i.e., it measures the
degree of sensitivity of the stock‟s return to market movements.
mm
2) Think of the market portfolio as a security: m 1 . Thus, the “average” stock has a
m 2
beta of 1. i 1 : unusually sensitive to market movements
=>
i 1 : unusually insensitive to market movements
3) i = 1.77 means that on average when the market (index) rises an extra 1.0% of return,
stock i‟s return will rise by an extra of 1.77%. A diagram illustrating the empirical use
may be helpful to explain (Figure 7.12):
● Tables 7.5 and 7.6: High STDs do not always have high betas.
● summary: The market risk of a security is measured by its beta (☆2).
━ Why security betas determine portfolio risk
● security betas and portfolio risk:
10
With more securities, portfolio risk declines until all unique risk of the portfolio is eliminated
and only the bedrock of risk (i.e., the market risk or systematic risk) remains. The bedrock of
portfolio risk depends on the average beta of the securities included.
In fact, portfolio beta ( p ) equals the weighted average beta of the securities included, i.e.,
N
p xi i . [Try to prove it by yourself.]
i 1
● The risk of a portfolio ( p ) is proportional to the portfolio beta: p p m .
sketch of proof: For a well-diversified portfolio p, we will have
pm
p ,m 1 => 1 => pm p m
p m
pm p m p
By definition, p => p p m .
m 2
m
2
m
example: p = 1.2 The portfolio‟s standard deviation would be 1.2 times the market‟s
standard deviation.
N
● summary: Securities betas determine portfolio risk (☆3). [ p p m = ( xi i ) m ]
i 1
━ conclusion: For a diversified investor, the appropriate risk measure is beta rather than the
standard deviation.
━ example for calculating beta:
● assumption: The market portfolio consists of the two securities as illustrated in Section 7-3.
● calculation:
stock 1 stock 2
--------------------------------------
2
1 x12 12 x1 x2 12 1 2 => x1 x j 1 j x1 1m
j 1
(. 60 ) 2 (18 .2) 2 .60 .40
.4 18.2 27.3 => .60[.60 (18.2)2 + .40 .4 18.2 27.3] = .60 278.2
---------------------------------------
2
2 x1 x2 12 1 2 x2 2
2 2
=> x 2 x j 2 j x 2 2 m
j 1
.60 .40 (. 40 ) 2 ( 27 .3) 2
.4 18.2 27.3 => .40[ .60 .4 18.2 27.3 + .40 (27.3)2] = .40 417.4
----------------------------------------
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We see m = .60 278.2 + .40 417.4 = 333.9
2
2
= x1 1m + x2 2 m = x
i 1
i im ,
notes: 1) im x
j
j ij , the weighted average of covariances of stock i with other securities in
the market
2) m =
2
x i
i im The market risk is the weighted average of covariances among
securities. Security i‟s contribution to the market risk depends on its
relative importance (xi) and its average covariance with other
securities ( im ).
3) The proportion of stock 1‟s contribution to the market risk = 0.60 (278.2/333.9) =
im
0.60 0.83 = 0.5. For any security i, the proportion is xi xi i .
m
● any portfolio p:
1) stock i‟s contribution to portfolio risk = xi ip
ip
2) the proportion of stock i‟s contribution = xi
p
ip
3) Let ' i = , called the beta of stock i relative to the portfolio p. Then, the proportion
p
of stock i‟s contribution = xi ' i .
4) If p is well-diversified, then ' i i . The proportion of stock i‟s contribution xi i .
7-5 Diversification and Value Additivity
━ Q: Is a diversified firm more attractive to investors?
If it is, the value additivity no longer holds.
Assuming two assets or plants A and B, this means that PV(A + B) > PV(A) + PV(B).
notes: 1) the argument: diversification => the decrease in rA B
2) another argument: economies of scale or scope (such as synergies) derived from
mergers
12
━ point: If investors can diversify on their own, they will not pay any extra for firms that
diversify.
If capital markets are large and competitive, investors can diversify more easily than firms.
Diversification thus does not add to a firm‟s value. => PV(A+B) = PV(A) + PV(B).
━ The concept of value additivity is very general.
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