Chapter 6. Risk and Return: Past and Prologue. Rates of Return: Single Period. HPR = Holding Period Return P 1 = Ending price P 0 = Beginning price D 1 = Dividend during period one. Rates of Return: Single Period Example. Ending Price = 24 Beginning Price = 20 Dividend = 1
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Risk and Return: Past and Prologue
HPR = Holding Period Return
P1 = Ending price
P0 = Beginning price
D1 = Dividend during period one
Ending Price = 24
Beginning Price = 20
Dividend = 1
HPR = ( 24  20 + 1 )/ ( 20) = 25%
1 2 3 4
Assets(Beg.) 1.0 1.2 2.0 .8
HPR .10 .25 (.20) .25
TA (Before
Net Flows 1.1 1.5 1.6 1.0
Net Flows 0.1 0.5 (0.8) 0.0
End Assets 1.2 2.0 .8 1.0
Arithmetic
ra = (r1 + r2 + r3 + ... rn) / n
ra = (.10 + .25  .20 + .25) / 4
= .10 or 10%
Geometric
rg = {[(1+r1) (1+r2) .... (1+rn)]} 1/n  1
rg = {[(1.1) (1.25) (.8) (1.25)]} 1/4  1
= (1.5150) 1/4 1 = .0829 = 8.29%
Internal Rate of Return (IRR)  the discount rate that results present value of the future cash flows being equal to the investment amount
Net CFs 1 2 3 4
$ (mil)  .1  .5 .8 1.0
Solving for IRR
1.0 = .1/(1+r)1 + .5/(1+r)2 + .8/(1+r)3 +
1.0/(1+r)4
r = .0417 or 4.17%
APR = annual percentage rate
(periods in year) X (rate for period)
EAR = effective annual rate
( 1+ rate for period)Periods per yr  1
Example: monthly return of 1%
APR = 1% X 12 = 12%
EAR = (1.01)12  1 = 12.68%
1) Mean: most likely value
2) Variance or standard deviation
3) Skewness
* If a distribution is approximately normal, the distribution is described by characteristics 1 and 2
E
(
r
)
=
p
(
s
)
r
(
s
)
s
Measuring Mean: Scenario or Subjective ReturnsSubjective returns
p(s) = probability of a state
r(s) = return if a state occurs
1 to s states
Numerical Example: Subjective or Scenario Distributions
StateProb. of State rin State
1 .1 .05
2 .2 .05
3 .4 .15
4 .2 .25
5 .1 .35
E(r) = (.1)(.05) + (.2)(.05)...+ (.1)(.35)
E(r) = .15
2
Variance
=
p
(
s
)
[
r

E
(
r
)]
s
s
Measuring Variance or Dispersion of ReturnsSubjective or Scenario
Standard deviation = [variance]1/2
Using Our Example:
Var =[(.1)(.05.15)2+(.2)(.05 .15)2...+ .1(.35.15)2]
Var= .01199
S.D.= [ .01199] 1/2 = .1095
Fisher effect: Approximation
nominal rate = real rate + inflation premium
R = r + i or r = R  i
Example r = 3%, i = 6%
R = 9% = 3% + 6% or 3% = 9%  6%
Fisher effect: Exact
r = (R  i) / (1 + i)
2.83% = (9%6%) / (1.06)
Geom Arith Stan.
Series Mean% Mean% Dev.%
Lg Stk 11.01 13.00 20.33
Sm Stk 12.46 18.77 39.95
LT Gov 5.26 5.54 7.99
TBills 3.75 3.80 3.31
Inflation 3.08 3.18 4.49
Risk Real
Series Premiums% Returns%
Lg Stk 9.2 9.82
Sm Stk 14.97 15.59
LT Gov 1.74 2.36
TBills  0.62
Inflation  
srf = 0%
E(rp) = 15%
sp = 22%
y = % in p
(1y) = % in rf
Example Using the Numbers in Chapter 6 (pp 171173)rc = complete or combined portfolio
For example, y = .75
E(rc) = .75(.15) + .25(.07)
= .13 or 13%
Expected Returns for Combinations= .75(.22) = .165 or 16.5%
c
If y = 1
= 1(.22) = .22 or 22%
c
If y = 0
= 0(.22) = .00 or 0%
c
Combinations Without Leverages
s
s
Borrow at the RiskFree Rate and invest in stock
Using 50% Leverage
rc = (.5) (.07) + (1.5) (.15) = .19
sc = (1.5) (.22) = .33
E(rp) = Expected return on portfolio p
rf = the risk free rate
.005 = Scale factor
A x sp = Proportional risk premium
The larger A is, the larger will be the addedreturn required for risk
Rearranging the equation and solving for A
Many studies have concluded that investors’ average risk aversion is between 2 and 4
Efficient Diversification
S
W
=
1
i
i
=1
TwoSecurity Portfolio: Returnrp = W1r1 +W2r2
W1 = Proportion of funds in Security 1
W2 = Proportion of funds in Security 2
r1 = Expected return on Security 1
r2 = Expected return on Security 2
s22 = Variance of Security 2
Cov(r1r2) = Covariance of returns for
Security 1 and Security 2
TwoSecurity Portfolio: Risksp2= w12s12 + w22s22 + 2W1W2 Cov(r1r2)
Cov(r1r2) = r1,2s1s2
r1,2 = Correlation coefficient of
returns
s1 = Standard deviation of
returns for Security 1
s2 = Standard deviation of
returns for Security 2
Range of values for r1,2
1.0 <r < 1.0
If r = 1.0, the securities would be perfectly positively correlated
If r =  1.0, the securities would be perfectly negatively correlated
rp = W1r1 +W2r2 + W3r3
s2p = W12s12
+ W22s22
+ W32s32
+ 2W1W2
Cov(r1r2)
+ 2W1W3
Cov(r1r3)
+ 2W2W3
Cov(r2r3)
rp = Weighted average of the
n securities
sp2 = (Consider all pairwise
covariance measures)
E(rp) = W1r1 +W2r2
sp2= w12s12 + w22s22 + 2W1W2 Cov(r1r2)
sp= [w12s12 + w22s22 + 2W1W2 Cov(r1r2)]1/2
13%
r = 1
r = .3
r = 1
8%
r = 1
St. Dev
12%
20%
TWOSECURITY PORTFOLIOS WITH DIFFERENT CORRELATIONS
r = 0
Sec 1
E(r1) = .10
= .15
r
= .2
12
s
Sec 2
E(r2) = .14
= .20
2
Minimum Variance Combination1
s 2
 Cov(r1r2)
2
=
W1
s 2
s 2
 2Cov(r1r2)
+
2
1
= (1  W1)
W2
(.2)2  (.2)(.15)(.2)
=
W1
(.15)2 + (.2)2  2(.2)(.15)(.2)
W1
= .6733
W2
= (1  .6733) = .3267
rp = .6733(.10) + .3267(.14) = .1131
s
= [(.6733)2(.15)2 + (.3267)2(.2)2 +
p
1/2
2(.6733)(.3267)(.2)(.15)(.2)]
1/2
= [.0171]
= .1308
s
p
(.2)2  (.2)(.15)(.2)
=
W1
(.15)2 + (.2)2  2(.2)(.15)(.3)
W1
= .6087
W2
= (1  .6087) = .3913
rp = .6087(.10) + .3913(.14) = .1157
s
= [(.6087)2(.15)2 + (.3913)2(.2)2 +
p
1/2
2(.6087)(.3913)(.2)(.15)(.3)]
1/2
= [.0102]
= .1009
s
p
The minimumvariance frontier of risky assets
E(r)
Efficient
frontier
Individual
assets
Global
minimum
variance
portfolio
Minimum
variance
frontier
St. Dev.
CAL(P) dominates other lines  it has the best risk/return or the largest slope
Slope = (E(R)  Rf) / s
[ E(RP)  Rf) / s P] > [E(RA)  Rf) / sA]
Regardless of risk preferences combinations of P & F dominate
ri = E(Ri) + ßiF + e
ßi = index of a securities’ particular return to the factor
F= some macro factor; in this case F is unanticipated movement; F is commonly related to security returns
Assumption: a broad market index like the S&P500 is the common factor
Risk Prem
Market Risk Prem
or Index Risk Prem
a
= the stock’s expected return if the
market’s excess return is zero
i
(rm  rf)= 0
ßi(rm  rf)= the component of return due to
movements in the market index
ei = firm specific component, not due to market
movements
Excess Returns (i)
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Security
Characteristic
Line
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Excess returns
on market index
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Ri = ai + ßiRm + ei
si2 = bi2sm2 + s2(ei)
where;
si2 = total variance
bi2sm2 = systematic variance
s2(ei) = unsystematic variance
Total Risk = Systematic Risk + Unsystematic Risk
Systematic Risk/Total Risk = r2
ßi2 sm2 / s2 = r2
bi2sm2 / bi2sm2 + s2(ei) = r2
Capital Asset Pricing and Arbitrage Pricing Theory
M = Market portfolio rf = Risk free rate E(rM)  rf = Market risk premium E(rM)  rf = Market price of risk
= Slope of the CAPM
s
M
b = [COV(ri,rm)] / sm2
Slope SML = E(rm)  rf
= market risk premium
SML = rf + b[E(rm)  rf]
E(rm)  rf = .08 rf = .03
bx = 1.25
E(rx) = .03 + 1.25(.08) = .13 or 13%
by = .6
e(ry) = .03 + .6(.08) = .078 or 7.8%
Excess Returns (i)
SCL
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Excess returns
on market index
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Ri = ai + ßiRm + ei
Excess
GM Ret.
Excess
Mkt. Ret.
Jan.
Feb.
.
.
Dec
Mean
Std Dev
5.41
3.44
.
.
2.43
.60
4.97
7.24
.93
.
.
3.90
1.75
3.32
a
rGM  rf = + ß(rm  rf)
a
ß
Estimated coefficient
Std error of estimate
Variance of residuals = 12.601
Std dev of residuals = 3.550
RSQR = 0.575
2.590
(1.547)
1.1357
(0.309)
Arbitrage  arises if an investor can construct a zero investment portfolio with a sure profit
Current Expected Standard
Stock Price$ Return% Dev.%
A 10 25.0 29.58
B 10 20.0 33.91
C 10 32.5 48.15
D 10 22.5 8.58
Mean Stan. Correlation
Return Dev. Of Returns
Portfolio
A,B,C 25.83 6.40 0.94
D 22.25 8.58
E. Ret.
* P
* D
St.Dev.
Short 3 shares of D and buy 1 of A, B & C to form P
You earn a higher rate on the investment than you pay on the short sale