5
Risk Chapter and Return:
Past and
Prologue
McGraw-Hill/Irwin
Copyright 2010 by The McGraw-Hill Companies, Inc. All rights reserved.
5.1 Rates of Return
5-2
Measuring Ex-Post (Past) Returns
One period investment: regardless of the length of the
period.
Holding period return (HPR):
HPR = [PS - PB + CF] / PB
where
PS
= Sale price (or P1)
PB
= Buy price ($ you put up) (or P0)
CF
= Cash flow during holding period
Q: Why use % returns at all?
What are we assuming about the cash flows in the
Q:
HPR calculation?
5-3
Annualizing HPRs
Q: Why would you want to annualize returns?
1. Annualizing HPRs for holding periods of greater
than one year:
Without compounding (Simple or APR):
HPRann = HPR/n
With compounding: EAR
HPRann = [(1+HPR)1/n]-1
where n = number of years held
5-4
Measuring Ex-Post (Past) Returns
An example: Suppose you buy one share of a stock today for
$45 and you hold it for two years and sell it for $52. You also
received $8 in dividends at the end of the two years.
(PB = $45, PS = $52 CF = $8):
,
HPR = (52 - 45 + 8) / 45 = 33.33%
HPRann = 0.3333/2 = 16.66%
Annualized w/out compounding
The annualized HPR assuming annual compounding is (n = 2):
1/2
HPRann = (1+0.3333) - 1 = 15.47%
5-5
Measuring Ex-Post (Past) Returns
Annualizing HPRs for holding periods of less than one
year:
Without compounding (Simple): HPRann = HPR x n
With compounding: HPRann =
[(1+HPR)n]-1
where n = number of compounding periods per year
5-6
Measuring Ex-Post (Past) Returns
An example when the HP is < 1 year:
Suppose you have a 5% HPR on a 3 month
investment. What is the annual rate of return with and
without compounding?
Without: n = 12/3 = 4 so HPRann = HPR*n = 0.05*4 = 20%
With:
HPRann = (1.054) - 1 = 21.55%
Q: Why is the compound return greater than the
simple return?
5-7
Arithmetic Average
Finding the average HPR for a time series of returns:
i. Without compounding (AAR or Arithmetic Average
Return):
n
HPR avg =
HPR T
n
T =1
n = number of time periods
5-8
Arithmetic Average
An example: You have the following rates of return on a stock:
2000 -21.56%
2001 44.63%
2002 23.35%
2003 20.98%
2004
3.11%
2005 34.46%
2006 17.62%
n
HPR T
n
T =1
HPR avg =
HPR avg =
(-.2156 + .4463 + .2335 + .2098 + .0311 + .3446 + .1762)
= 17.51%
7
AAR = 17.51%
5-9
Geometric Average
An example: You have the following rate s of return on a stock:
2000 -21.56%
2001 44.63%
2002 23.35%
2003 20.98%
2004
3.11%
2005 34.46%
2006 17.62%
With compounding (geometric average or GAR:
Geometric Average Return):
1/ n
HPR avg
n
=
(1 + HPR T )
T =1
1
HPR avg = (0.7844 1.4463 1.2335 1.2098 1.03111.3446 1.1762)1/7 1 = 15.61%
GAR = 15.61%
5-10
Measuring Ex-Post (Past) Returns
Finding the average HPR for a portfolio of assets for a
given time period:
J
HPR avg
VI
=
HPRI TV
I=1
where VI = amount invested in asset I,
J = Total # of securities
and TV = total amount invested;
thus VI/TV = percentage of total investment invested in
asset I
5-11
Measuring Ex-Post (Past) Returns
For example: Suppose you have $1000 invested in a stock
portfolio in September. You have $200 invested in Stock A,
$300 in Stock B and $500 in Stock C. The HPR for the month of
September for Stock A was 2%, for Stock B the HPR was 4%
and for Stock C the HPR was - 5%.
The average HPR for the month of September for this portfolio
is:
J
VI
HPR avg =
HPRI TV
I=1
HPR avg = (.02 (200/1000) ) + (.04 (300/1000) ) + (-.05 (500/1000) ) =-0.9%
5-12
Measuring Ex-Post (Past) Returns
Measuring returns when there are investment
changes (buying or selling) or other cash flows
within the period.
An example: Today you buy one share of stock
$50
$2
costing ___. The stock pays a __ dividend one year
from now.
Also one year from now you purchase a second
$53
share of stock for ____.
$2
Two years from now you collect a ___ per share
dividend and sell both shares of stock for $54 a
___
share.
Q: What was your average (annual) return?
A: It depends. There are different ways to measure
this.
5-13
Dollar-Weighted Return
i. Dollar-weighted return procedure (DWR):
Find the internal rate of return for the cash
flows (i.e. find the discount rate that makes the
NPV of the net cash flows equal zero.)
5-14
Tips on Calculating Dollar
Weighted Returns
This measure of return considers both changes in investment
and security performance
Initial Investment is an _______
outflow
Ending value is considered as an ______
inflow
Additional investment is an _______
outflow
Security sales are an ______
inflow
5-15
Measuring Ex-Post (Past) Returns
i. Dollar-weighted return procedure (DWR):
Find the internal rate of return for the cash
flows (i.e. find the discount rate that makes
the NPV of the net cash flows equal zero.)
Total Cash Flows Each Year
Year
0
1
2
-$50
$2
$4
-$53
$108
Net
-$50
-$51
$112
NPV = $0 = -$50/(1+IRR)0 - $51/(1+IRR)1 + $112/(1+IRR)2
Solve for IRR:
IRR = 7.117% average annual dollar weighted return
The DWR gives you an average return based on the stocks
performance and the dollar amount invested (number of
shares bought and sold) each period.
5-16
Measuring Ex-Post (Past) Returns
Total Cash Flows Each Year
Year
0
1
2
-$50
$2
$4
-$53
$108
Net
-$50
-$51
$112
Q: You are paying somebody to advise you which assets to
buy, but you are deciding when to buy and sell shares.
If you want to evaluate the quality of the investment advice
you are getting, should you use dollar weighted returns to
evaluate the quality of the investment advice?
5-17
Time-Weighted Returns
ii. Time-weighted returns (TWR):
TWRs assume you buy one share of
___
the stock at the beginning of each
one
interim period and sell ___ share at
the end of each interim period. TWRs are thus
___________ of the amount invested in a given period.
independent
To calculate TWRs:
Calculate the return for each time period, typically a year.
either an arithmetic (AAR) or a geometric
Then calculate of the returns.
average (GAR)
5-18
Time-Weighted Returns
TWR Cash Flows
Year 0-1
Year 1-2
0
1
-$50
1
$2
-$53
+$53
2
$2
+$54
Same example as before, initially buy one share at $50,
in one year collect a $2 dividend, and you buy another
share at $53. In two years you sell the stock for $54,
after collecting another $2 dividend per share.
TWRs assume you buy one share of the stock
at the beginning of each period and sell it at the
end of each period after collecting any cash
flow.
5-19
Measuring Ex-Post (Past) Returns
TWR Cash Flows
Year 0-1
Year 1-2
Year 0-1
Year 1-2
0
1
0
1
1
-$50
$2
2
-$53
+$53
1
-$50
$2
$2
2
-$53
$2
+$53
+$54
$54
Same example as before, initially buy one share at $50,
in one year collect a $2 dividend, and you buy another
share at $53. In two years you sell the stock for $54,
after collecting another $2 dividend per share.
Year 0-1
0
Year 0-1
0
-$50
1
$2
+$53
Year 1-2
Year 1-2
1
-$50
1
$2
2
-$53
+$53
1
-$53
2
$2
+$54
$2
$54
TWR Cash Flows
Year 0-1
Year 1-2
0
1
-$50
1
$2
+$53
-$53
2
$2
+$54
5-20
Measuring Ex-Post (Past) Returns
TWR Cash Flows
Year 0-1
Year 1-2
0
1
-$50
1
$2
-$53
+$53
2
$2
+$54
HPR for year 1:
[$53 + $2 - $50] / $50 = 10%
HPR for year 2:
[$54 - $53 +$2] / $53 = 5.66%
a) Calculating the arithmetic average TW return:
Arithmetic Average Return (AAR): Calculate the
arithmetic average
AAR = [0.10 + 0.0566] / 2 = 7.83%
5-21
Measuring Ex-Post (Past) Returns
TWR Cash Flows
Year 0-1
0
HPR1=10%
Year 1-2
1
-$50
HPR2=5.66%
1
$2
-$53
+$53
2
$2
+$54
b) Calculating the geometric average TW return (GAR):
1/ n
n
HPR avg =
(1 + HPR T )
1
T =1
HPR avg = (1.10 1.0566)1/2 1 = 7.81%
GAR =
7.81%
5-22
Measuring Ex-Post (Past) Returns
Q: When should you use the GAR and when should you use
the AAR?
A1: When you are evaluating PAST RESULTS (ex-post):
Use the AAR (average without compounding) if you ARE
NOT reinvesting any cash flows received before the end of
the period.
Use the GAR (average with compounding) if you
ARE reinvesting any cash flows received before the end of
the period.
A2: When you are trying to estimate an expected return (exante return):
Use the AAR
5-23
5.2 Risk and Risk
Premiums
5-24
Measuring Mean:
Scenario or Subjective Returns
a. Subjective or Scenario
Subjective expected returns
E(r) = p(s) r(s)
s
E(r) = Expected Return
p(s) = probability of a state
r(s) = return if a state occurs
1 to s states
5-25
Measuring Variance or
Dispersion of Returns
a. Subjective or Scenario
Variance
2 = p(s) [rs E(r)] 2
s
= [ 2]1/2
E(r) = Expected Return
p(s) = probability of a state
rs = return in state s
5-26
Numerical Example: Subjective or
Scenario Distributions
State Prob. of State Return
1
.2
- .05
2
.5
.05
3
.3
.15
E(r) =
2 =
(.2)(-0.05) + (.5)(0.05) + (.3)(0.15) = 6%
p(s) [rs E(r)] 2
s
2 = [(.2)(-0.05-0.06)2 + (.5)(0.05- 0.06)2 + (.3)(0.15-0.06)2]
2 = 0.0049%2
= [ 0.0049]1/2 = .07 or 7%
5-27
Expost Expected Return &
n HPR
T
r=
T =1 n
Expost Variance : 2
r = average HPR
n = # observatio ns
n
1
=
( ri r ) 2
n 1 i =1
Expost Standard Deviation : = 2
Annualizing the statistics:
rannual = rperiod # periods
annual = period # periods
5-28
Obs
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Monthly Source Yahoo finance
HPRs
2
(r - ravg)
DIS
-0.035417 0.002212808 9/3/2002
0.093199 0.006654508 10/1/2002
0.15756 0.021297275 11/1/2002
-0.200637 0.045054632 12/2/2002
0.068249 0.00320644 1/2/2003
-0.026188 0.001429702 2/3/2003
-0.00183 0.000181016 3/3/2003
0.087924 0.005821766 4/1/2003
0.050211 0.001489002 5/1/2003
0.004734 4.74648E-05 6/2/2003
0.099052 0.00764371 7/1/2003
-0.068896 0.006483384 8/1/2003
-0.016478 0.000789704 9/2/2003
0.109174 0.009516098 10/1/2003
0.019343 5.95893E-05 11/3/2003
0.019409 6.06076E-05 12/1/2003
0.02829 0.000277753 1/2/2004
0.095035 0.00695741 2/2/2004
-0.061342 0.005324028 3/1/2004
-0.085344 0.00940277 4/1/2004
0.018851 5.22376E-05 5/3/2004
0.079128 0.004556811 6/1/2004
-0.103832 0.013330149 7/1/2004
-0.028414 0.001603051 8/2/2004
0.004562 4.98687E-05 9/1/2004
0.105671 0.008844901 10/1/2004
0.061998 0.002537528 11/1/2004
0.041453 0.000889761 12/1/2004
0.028856 0.000296963 1/3/2005
-0.024453 0.001301505 2/1/2005
Obs
31
1
32
2
33
3
34
4
35
5
36
6
37
7
38
8
39
9
40
10
41
11
42
12
43
13
44
14
45
15
46
16
47
17
48
18
49
19
50
20
51
21
52
22
53
23
54
24
55
25
56
26
57
27
58
28
59
29
60
30
Monthly
HPRs
DIS
0.027334
-0.035417
-0.088065
0.093199
0.037904
0.15756
-0.089915
-0.200637
0.0179
0.068249
-0.017814
-0.026188
-0.043956
-0.00183
0.010042
0.087924
0.022495
0.050211
-0.029474
0.004734
0.05303
0.099052
0.09589
-0.068896
-0.003618
-0.016478
0.002526
0.109174
0.083361
0.019343
-0.016818
0.019409
-0.010537
0.02829
-0.001361
0.095035
0.04081
-0.061342
0.01764
-0.085344
0.047939
0.018851
0.044354
0.079128
0.02559
-0.103832
-0.026861
-0.028414
0.005228
0.004562
0.015723
0.105671
0.01298
0.061998
-0.038079
0.041453
-0.034545
0.028856
0.017857
-0.024453
2
(r - ravg)
0.000246811
0.002212808
0.009937839
0.006654508
0.000690654
0.021297275
0.010310121
0.045054632
3.93874E-05
0.00320644
0.000866572
0.001429702
0.003089121
0.000181016
2.50266E-06
0.005821766
0.00011818
0.001489002
0.001689005
4.74648E-05
0.001714497
0.00764371
0.007100858
0.006483384
0.000232311
0.000789704
8.27674E-05
0.009516098
0.005146208
5.95893E-05
0.000808939
6.06076E-05
0.000491104
0.000277753
0.000168618
0.00695741
0.000851813
0.005324028
3.61885E-05
0.00940277
0.001318787
5.22376E-05
0.001071242
0.004556811
0.000195054
0.013330149
0.001481106
0.001603051
4.09065E-05
4.98687E-05
1.68055E-05
0.008844901
1.83836E-06
0.002537528
0.002470321
0.000889761
0.002131602
0.000296963
0.000038854
0.001301505
3/1/2005
9/3/2002
4/1/2005
10/1/2002
5/2/2005
11/1/2002
6/1/2005
12/2/2002
7/1/2005
1/2/2003
8/1/2005
2/3/2003
9/1/2005
3/3/2003
10/3/2005
4/1/2003
11/1/2005
5/1/2003
12/1/2005
6/2/2003
1/3/2006
7/1/2003
2/1/2006
8/1/2003
3/1/2006
9/2/2003
4/3/2006
10/1/2003
5/1/2006
11/3/2003
6/1/2006
12/1/2003
7/3/2006
1/2/2004
8/1/2006
2/2/2004
9/1/2006
3/1/2004
10/2/2006
4/1/2004
11/1/2006
5/3/2004
12/1/2006
6/1/2004
1/3/2007
7/1/2004
2/1/2007
8/2/2004
3/1/2007
9/1/2004
4/2/2007
10/1/2004
5/1/2007
11/1/2004
6/1/2007
12/1/2004
7/2/2007
1/3/2005
8/1/2007
2/1/2005
Average
0.011624
Variance
Source Yahoo finance
0.003725
Stdev
0.219762458
0.061031
(r - ravg)2 =
n
60
n-1
59
Annualized
Average
0.139486
Variance
0.044697
Stdev
0.211418
n
r=
HPR T
r = average HPR n = # observatio ns
n
T =1
Expost Variance :
n
2
1
=
( ri r ) 2
n 1 i =1
Expost Standard Deviation : = 2
Annualizing the statistics:
rannual = rmonthly 12
annual = monthly 12
5-29
Using Ex-Post Returns to estimate
Expected HPR
Estimating Expected HPR (E[r]) from ex-post data.
Use the arithmetic average of past returns as a
forecast of expected future returns as we did and,
Perhaps apply some (usually ad-hoc) adjustment to
past returns
Which historical time period?
Problems?
Have to adjust for current economic
situation
Unstable averages
Stable risk
5-30
Characteristics of Probability
Distributions
Arithmetic average & usually most likely
1. Mean: __________________________________ _
2. Median: Middle observation
_________________
3. Variance or standard deviation:
Dispersion of returns about the mean
Long tailed distribution, either side
4. Skewness:_______________________________
Too many observations in the tails
5. Leptokurtosis: ______________________________
If a distribution is approximately normal, the distribution
is fully described by Characteristics 1 and 3
_____________________
5-31
Normal Distribution
Risk is the
Risk
possibility of getting
returns different
from expected.
from
measures deviations
above the mean as well as
below the mean.
Returns > E[r] may not be
considered as risk, but with
symmetric distribution, it is
ok to use to measure risk.
I.E., ranking securities by
will give same results as
ranking by asymmetric
measures such as lower
partial standard deviation.
partial
Average = Median
E[r] = 10%
= 20%
5-32
Skewed Distribution:
Large Negative Returns
Possible (Left Skewed)
Implication?
r = average
is an incomplete
risk measure
Median
Negative
r
Positive
5-33
Skewed Distribution:
Large Positive Returns
Possible (Right Skewed)
r = average
Median
Negative
r
Positive
5-34
Implication?
an is incomplete
risk measure
risk
Leptokurtosis
5-35
Value at Risk (VaR)
Value at Risk attempts to answer the following question:
How many dollars can I expect to lose on my portfolio in a
given time period at a given level of probability?
The typical probability used is 5%.
We need to know what HPR corresponds to a 5%
probability.
If returns are normally distributed then we can use a
standard normal table or Excel to determine how many
standard deviations below the mean represents a 5%
probability:
From Excel: =Norminv (0.05,0,1) = -1.64485 standard
deviations
5-36
Value at Risk (VaR)
From the standard deviation we can find the corresponding
level of the portfolio return:
VaR = E[r] + -1.64485
For Example:
A $500,000 stock portfolio has an annual expected return of
12% and a standard deviation of 35%. What is the portfolio
VaR at a 5% probability level?
VaR = 0.12 + (-1.64485 * 0.35)
VaR = -45.57%
(rounded slightly)
VaR$ = $500,000 x -.4557 = -$227,850
What does this number mean?
5-37
Value at Risk (VaR)
VaR versus standard deviation:
For normally distributed returns VaR is equivalent to
standard deviation (although VaR is typically
reported in dollars rather than in % returns)
VaR adds value as a risk measure when return
distributions are not normally distributed.
Actual 5% probability level will differ from 1.68445
standard deviations from the mean due to
kurtosis and skewness.
5-38
Risk Premium & Risk Aversion
The risk free rate is the rate of return that can be
earned with certainty.
The risk premium is the difference between the
expected return of a risky asset and the risk-free rate.
Excess Return or Risk Premiumasset = E[rasset] rf
Risk aversion is an investors reluctance to accept
risk.
How is the aversion to accept risk overcome?
By offering investors a higher risk premium.
5-39
5.3 The Historical Record
5-40
Frequency distributions of annual HPRs,
1926-2008
5-41
Rates of return on stocks, bonds and
bills, 1926-2008
5-42
Annual Holding Period Returns Statistics 1926-2008
From Table 5.3
Geom.
Excess
Mean%
Series
Arith.
Mean%
Return%
Kurt.
Skew.
World Stk
9.20
11.00
7.25
1.03
-0.16
US Lg. Stk
9.34
11.43
7.68
-0.10
-0.26
11.43
17.26
13.51
1.60
0.81
5.56
5.92
2.17
1.10
0.77
5.60
1.85
0.80
0.51
Sm. Stk
World Bnd
LT Bond
5.31
Geometric mean:
Best measure of
compound historical
return
Deviations from
normality?
Arithmetic Mean:
Expected return
5-43
Deviations from Normality: Another Measure
Portfolio
World Stock
US Small Stock
US Large Stock
Arithmetic Average
.1100
.1726
.1143
Geometric Average
.0920
.1143
.0934
Difference
.0180
.0483
.0209
Historical Variance
.0186
.0694
.0214
If returns are normally distributed then the following
relationship among geometric and arithmetic averages
holds:
Arithmetic Average Geometric Average = 2
The comparisons above indicate that US Small Stocks may
have deviations from normality and therefore VaR may be
an important risk measure for this class.
5-44
Actual vs. Theoretical VaR 1926-2008
Series
World Stk
US Lg. Stk
US Sm. Stk
World Bnd
US LT Bond
Actual
VaR%
VaR% if Normal
-21.89
-29.79
-46.25
-6.54
-7.61
-21.07
-22.92
-44.93
-8.69
-7.25
These comparisons indicate that the U.S. Large
Stock portfolio, the US small stock portfolio and the
World Bond portfolio may exhibit differences from
normality.
5-45
Annual Holding Period Excess Returns
1926-2008 From Table 5.3 of Text
Series
World Stk
US Lg Stk
US Sm Stk
World Bonds
US LT Bonds
Arith.
Avg%
7.25
7.68
13.51
2.17
1.85
Required
Return%
10.25
10.68
16.51
5.17
4.85
If the risk free rate is currently 3%, then what return
should an investor require for each asset class?
Problems with this approach?
Historical data
Assumes all securities in the
category are equally risky
5-46
5.4 Inflation and Real Rates
of Return
5-47
Inflation, Taxes and Returns
The average inflation rate from 1966 to 2005 was _____.
4.29%
This relatively small inflation rate reduces the terminal
value of $1 invested in T-bills in 1966 from a nominal
value of ______ in 2005 to a real value of $1.63
_____.
$10.08
Taxes are paid on _______ investment income. This
nominal
real
reduces _____ investment income even further.
6%
You earn a ____ nominal, pre-tax rate of return and you
15%
4.29%
are in a ____ tax bracket and face a _____ inflation rate.
What is your real after tax rate of return?
rreal [6% x (1 - 0.15)] 4.29% 0.81%; taxed on
nominal
5-48
Real vs. Nominal Rates
Fisher effect: Approximation
real rate nominal rate - inflation rate
rreal rnom - i
r = real interest rate
Example rnom = 9%, i = 6%
rreal 3%
real
rnom = nominal interest rate
i = expected inflation rate
Fisher effect: Exact
rreal = [(1 + rnom) / (1 + i)] 1
r = (rnom - i) / (1 + i)
or
real
rreal = (9% - 6%) / (1.06) = 2.83%
The exact real rate is less than the approximate
real rate.
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Exact Fisher Effect Explained
1) I want to be able to buy more Quantity or
Qnew = Qold x (1 + rreal)
BUT
2) The Price, P, is also rising
Pnew = Pold x (1 + i)
i = inflation
Total $ spent = Pnew x Qnew
Pnewx Qnew = Pold x Qold x [(1 + rreal) x (1 + i)]
or (1 + rnom)= (1 + rreal) x (1 + i)
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Nominal and Real interest rates and
Inflation
5-51
Historical Real Returns & Sharpe
Ratios
Series
World Stk
US Lg. Stk
Sm. Stk
World Bnd
LT Bond
Real Returns%
6.00
6.13
8.17
Sharpe Ratio
0.37
0.37
0.36
2.46
2.22
0.24
0.24
Real returns have been much higher for stocks than for bonds
Sharpe ratios measure the excess return to standard deviation.
The higher the Sharpe ratio the better.
Stocks have had much higher Sharpe ratios than bonds.
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5.5 Asset Allocation Across
Risky and Risk Free
Portfolios
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Allocating Capital Between Risky &
Risk-Free Assets
Possible to split investment funds between safe and
risky assets
Risk free asset rf : proxy; T-bills or money market fund
________________________
risky portfolio
Risky asset or portfolio rp: _______________________
Example. Your total wealth is $10,000. You put $2,500
in risk free T-Bills and $7,500 in a stock portfolio
invested as follows:
Stock A you put $2,500
______
Stock B you put $3,000
______
Stock C you put $2,000
______
$7,500
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Allocating Capital Between Risky &
Risk-Free Assets
Stock A $2,500
Weights in rp
WA = $2,500 / $7,500 = 33.33%
WB = $3,000 / $7,500 = 40.00%
WC = $2,000 / $7,500 = 26.67%
100.00%
Stock B $3,000
Stock C $2,000
The complete portfolio includes the riskless $2,500 in risk free
Your total wealth is $10,000. You put
investment and rp. T-Bills and $7,500 in a stock portfolio invested as follows
Wrf = 25% ; Wrp = 75%
In the complete portfolio
WA = 0.75 x 33.33% = 25%; WB = 0.75 x 40.00% = 30%
WC = 0.75 x 26.67% = 20%; Wrf = 25%
5-55
Allocating Capital Between Risky &
Risk-Free Assets
Issues in setting weights
risk & return tradeoff
Examine ___________________
Demonstrate how different degrees of risk
allocations
aversion will affect __________ between risky and
risk free assets
5-56
Example
rf = 5%
rf = 0%
E(rp) = 14%
rp = 22%
y = % in rp
(1-y) = % in rf
5-57
Expected Returns for Combinations
E(rC) = yE(rp) + (1 - y)rf
c = y rp + (1-y) rf
E(rC) = Return for complete or combined portfolio
rf = 5%
0.75
For example, let y = ____
E(rC) = (.75 x .14) + (.25 x .05)
E(rC) = .1175 or 11.75%
C = y rp + (1-y) rf
rf = 0%
E(rp) = 14%
rp = 22%
y = % in rp
(1-y) = % in rf
C = (0.75 x 0.22) + (0.25 x 0) = 0.165 or 16.5%
5-58
Complete portfolio
E(rc) = yE(rp) + (1 - y)rf
c = y rp + (1-y) rf
linear
Varying y results in E[rC] and C that are ______
combinations of E[rp] and rf and and
___________
rp
rf
respectively.
This is NOT generally the case for
the of combinations of two or
more risky assets.
5-59
E(r)
Possible Combinations
E(rp) = 14%
P
E(rp) = 11.75%
y=1
y =.75
rf = 5%
F
y=0
0
16.5%
22%
5-60
E(r)
Possible Combinations
E(rp) = 14%
P
E(rp) = 11.75%
y=1
y =.75
rf = 5%
F
y=0
0
16.5%
22%
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Combinations Without Leverage
rf = 5%
rf = 0%
E(rp) = 14%
rp = 22%
y = % in rp
(1-y) = % in rf
Since rf = 0
E(rc) = yE(rp) + (1 - y)rf
c= y p
y = .75
If y = .75, then
(.75)(.14) + (.25)(.05) = 11.75%
75(.22) = 16.5% E(rc) =
c=
If y = 1
c= 1(.22) = 22%
If y = 0
0(.22) = 0%
c=
y=1
E(rc) = (1)(.14) + (0)(.05) = 14.00%
y=0
E(rc) = (0)(.14) + (1)(.05) = 5.00%
5-62
Using Leverage with Capital
Allocation Line
Borrow at the Risk-Free Rate and invest in stock
Using 50% Leverage
y = 1.5
E(rc) = (1.5) (.14) + (-.5) (.05) = 0.185 = 18.5%
(1.5) (.22) = 0.33 or 33%
rf = 5%
rf = 0%
c =
E(rp) = 14%
rp = 22%
y = % in rp
(1-y) = % in rf
E(r)
Possible Combinations
E(rC) =18.5%
E(rp) = 14%
y = 1.5
P
P
E(rp) = 11.75%
y=1
y =.75
y = 0 5%
r=
f
0
F
16.5%
33%
22%
5-63
Risk Aversion and Allocation
Greater levels of risk aversion lead investors to
choose larger proportions of the risk free rate
Lower levels of risk aversion lead investors to
choose larger proportions of the portfolio of risky
assets
Willingness to accept high levels of risk for high
levels of returns would result in
Possible Combinations
leveraged combinations
E(r)
E(rC) =18.5%
y = 1.5
E(rp) = 14%
P
E(rp) = 11.75%
y=1
y =.75
r=
y =f 0 5%
0
F
16.5%
22%
33%
5-64
E(r)
E(r)
P or combinations of
or
P & Rf offer a return
per unit of risk of
9/22.
9/22.
CAL
(Capital
Allocation
Line)
P
E(rp) = 14%
E(rp) - rf = 9%
r f = 5%
0
) Slope = 9/22
Slope
F
rp = 22%
22%
5-65
Quantifying Risk Aversion
E ( rp ) rf = 0.5 A p
2
E(rp) = Expected return on portfolio p
rf = the risk free rate
0.5 = Scale factor
A x p2 = Proportional risk premium
The larger A is, the larger will be the
investors added return required to bear risk
_________________________________________
5-66
Quantifying Risk Aversion
Rearranging the equation and solving
for A
E ( rp ) rf
A=
0.5 2
p
Many studies have concluded that
investors average risk aversion is
between _______
2 and 4
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Using A
E ( rp ) rf
A=
0.5 2
p
What is the maximum
A that an investor
could have and still
choose to invest in the
risky portfolio P?
A=
0.14 0.05
0.5 0.22
2
=
3.719
CAL
(Capital
Allocation
Line)
E(r)
E(r)
P
E(rp) = 14%
E(rp) - rf = 9%
rf = 5%
0
) Slope = 9/22
F
F
rp = 22%
Maximum A = 3.719
5-68
A and Indifference Curves
The A term can used to create indifference curves.
Indifference curves describe different combinations of
return and risk that provide equal utility (U) or
satisfaction.
U = E[r] - 1/2A p2
Indifference curves are curvilinear because they exhibit
diminishing marginal utility of wealth.
The greater the A the steeper the indifference curve and all
else equal, such investors will invest less in risky assets.
The smaller the A the flatter the indifference curve and all
else equal, such investors will invest more in risky assets.
5-69
Indifference Curves
I3
I2
I1
I3 I 2 I1
Investors want
the most
return for the
least risk.
Hence
indifference
curves higher
and to the left
are preferred.
U = E[r] - 1/2A p2
5-70
A=3
A=3
E(r)
CAL
(Capital
Allocation
Line)
P
S
r f = 5%
0
Q
F
5-71
A=3
E(r)
A=2
P
S
r f = 5%
0
T
CAL
(Capital
Allocation
Line)
F
5-72
5.6 Passive Strategies and
the Capital Market Line
5-73
A Passive Strategy
Investing in a broad stock index and a risk
free investment is an example of a passive
strategy.
The investor makes no attempt to actively find
undervalued strategies nor actively switch
their asset allocations.
The CAL that employs the market (or an index
that mimics overall market performance) is
called the Capital Market Line or CML.
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Excess Returns and Sharpe Ratios
implied by the CML
Excess Return or Risk
Premium
Time
Period
1926-2008
1926-1955
1956-1984
1985-2008
Average
7.86
11.67
5.01
5.95
20.88
25.40
17.58
18.23
Sharpe
Ratio
0.37
0.46
0.28
0.33
The average risk premium implied by the CML for
large common stocks over the entire time period is
7.86%.
How much confidence do we have that this
historical data can be used to predict the risk
premium now?
5-75
Active versus Passive Strategies
Active strategies entail more trading costs than
passive strategies.
Passive investor free-rides in a competitive
investment environment.
Passive involves investment in two passive
portfolios
Short-term T-bills
Fund of common stocks that mimics a broad
market index
Vary combinations according to investors
risk aversion.
5-76
Selected Problems
5-77
Problem 1
V(12/31/2004) = V (1/1/1998) x (1 + GAR)7
= $100,000 x (1.05)7
=
$140,710.04
5-78
Problem 2
a. The holding period returns for the three scenarios are:
(50 40 + 2)/40 = 0.30 =
Boom:
30.00%
(43 40 + 1)/40 = 0.10 = 10.00%
Normal:
(34 40 + 0.50)/40 = 0.1375 =
Recession:
13.75%
[(1/3) x 30%] + [(1/3) x 10%] + [(1/3) x (13.75%)] = 8.75%
E(HPR) =
2 (HPR) = [(1/3) x (30% 8.75%) 2 ] + [(1/3) x (10% 8.75%) 2 ] + [(1/3) x (13.75% 8.75%) 2 ] = 0.031979
2
(HPR)
(HPR) = 17.88%
5-79
Problem 2 Cont.
Risky E[rp] = 8.75%
Risky p = 17.88%
b. E(r) = (0.5 x 8.75%) + (0.5 x 4%) = 6.375%
= 0.5 x 17.88% = 8.94%
5-80
Problems 3 & 4
3. For each portfolio: Utility = E(r) (0.5 4 2 )
Investment
U
1
0.12
0.30
-0.0600
2
0.15
0.50
-0.3500
3
0.21
0.16
0.1588
4
value,
E(r)
0.24
0.21
0.1518
We choose the portfolio with the highest utility
which is Investment 3.
5-81
Problems 3 & 4 Cont.
4. When an investor is risk neutral, A = 0 so that the portfolio with the
_
highest expected return
highest utility is the portfolio with the _______________________.
Investment 4
So choose ____________.
5-82
Problem 5
a. TWR
Year
b. DWR
Return = [(capital gains + dividend) /
price]
(110 100 + 4)/100 = 14.00%
Time
Cash
flow
2003-2004
(90 110 + 4)/110 = 14.55%
0
-300
Purchase of three shares at $100
per share
2004-2005
(95 90 + 4)/90 = 10.00%
1
-208
Purchase of two shares at $110,
plus dividend income on three
shares held
2
110
Dividends on five shares,
plus sale of one share at $90
396
Dividends on four shares,
plus sale of four shares at $95 per
share
RW.a
T
2002-2003
AAR =
14.00% + 14.55% + 10.00%
= 3.15%
3
GAR = [1.14x(1 0.1455)x1.10]1/3 1 = 2.33%
$0 =
3
Explanation
$300
$208
$110
$396
+
+
+
=
0
1
2
3
(1 + IRR) (1 + IRR) (1 + IRR) (1 + IRR)
-0.1661%
5-83