Chapter 5
Risk 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?
• Q: What are we assuming about the cash flows in the
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 ):
•HPRann = (1+0.3333)1/2 - 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 av g 

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 av g 

HPR av g 
(-.2156  .4463  .2335  .2098  .0311  .3446  .1762)
 17.51%
7
HPR T
n
T 1
AAR = 17.51%
5-9
Geometric 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%
•With compounding (geometric average or GAR:
Geometric Average Return):
HPR av g
 n


(1  HPR T ) 
 T 1


1/ n
1
HPR avg  (0.7844 1.4463 1.2335 1.2098 1.03111.3446 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 av g 
VI 

HPR

I


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 av g 
HPR I  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 The stock pays a __
$2 dividend one year
costing ___.
from now.
– Also one year from now you purchase a second
$53
share of stock for ____.
$2 per share
– Two years from now you collect a ___
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 stock’s
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 share at
interim period and sell ___
the end of each interim period. TWRs are thus
___________
independent of the amount invested in a given period.
To calculate TWRs:
 Calculate the return for each time period, typically a year.
calculate either an arithmetic (AAR) or a geometric
 Then
average (GAR) of the returns.
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
-$50
1
$ 2
+$53
Year 1-2
Year 0-1
Year 1-2
0
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
HPR1 = 10%
Year 0-1
Year 1-2
0
1
-$50
HPR2 = 5.66%
1
$ 2
-$53
+$53
2
$ 2
+$54
b) Calculating the geometric average TW return (GAR):
1/ n
 n

HPR av g  
(1  HPR T ) 
1
 T 1

HPR av g  (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) = S 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
σ 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)(-0.05) + (.5)(0.05) + (.3)(0.15) = 6%
σ 2   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: σ  σ 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
HPRs
DIS
-0.035417
0.093199
0.15756
-0.200637
0.068249
-0.026188
-0.00183
0.087924
0.050211
0.004734
0.099052
-0.068896
-0.016478
0.109174
0.019343
0.019409
0.02829
0.095035
-0.061342
-0.085344
0.018851
0.079128
-0.103832
-0.028414
0.004562
0.105671
0.061998
0.041453
0.028856
-0.024453
Source Yahoo finance
(r - ravg)2
0.002212808
0.006654508
0.021297275
0.045054632
0.00320644
0.001429702
0.000181016
0.005821766
0.001489002
4.74648E-05
0.00764371
0.006483384
0.000789704
0.009516098
5.95893E-05
6.06076E-05
0.000277753
0.00695741
0.005324028
0.00940277
5.22376E-05
0.004556811
0.013330149
0.001603051
4.98687E-05
0.008844901
0.002537528
0.000889761
0.000296963
0.001301505
9/3/2002
10/1/2002
11/1/2002
12/2/2002
1/2/2003
2/3/2003
3/3/2003
4/1/2003
5/1/2003
6/2/2003
7/1/2003
8/1/2003
9/2/2003
10/1/2003
11/3/2003
12/1/2003
1/2/2004
2/2/2004
3/1/2004
4/1/2004
5/3/2004
6/1/2004
7/1/2004
8/2/2004
9/1/2004
10/1/2004
11/1/2004
12/1/2004
1/3/2005
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
Source Yahoo finance
(r - ravg)2
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
0.219762458
Variance
0.003725
S (r - ravg)2 =
Stdev
0.061031
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: σ  σ 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
4. Skewness:_______________________________
Long tailed distribution, either side
5. Leptokurtosis: ______________________________
Too many observations in the tails

If a distribution is approximately normal, the distribution
1 and 3
is fully described by Characteristics
_____________________
5-31
Normal Distribution
Risk is the
possibility of getting
returns different
from expected.
 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.
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?
 is an incomplete
risk measure
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.
E[rasset] – rf
Excess Return or Risk Premiumasset =
Risk aversion is an investor’s 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
Series
Geom.
Arith.
Excess
Mean%
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
World Bnd
5.56
5.92
2.17
1.10
0.77
LT Bond
5.31
5.60
1.85
0.80
0.51
Sm. Stk
• 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
½ Historical Variance
.0186
.0694
.0214
If returns are normally distributed then the following
relationship among geometric and arithmetic averages
holds:
Arithmetic Average – Geometric Average = ½ 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 ______
_____.
$10.08 in 2005 to a real value of $1.63
Taxes are paid on _______
nominal investment income. This
real investment income even further.
reduces _____
6% nominal, pre-tax rate of return and you
You earn a ____
15% tax bracket and face a _____
are in a ____
4.29%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
rreal = real interest rate
Example rnom = 9%, i = 6%
rnom = nominal interest rate
rreal  3%
i = expected inflation rate
Fisher effect: Exact
rreal = [(1 + rnom) / (1 + i)] – 1
or
rreal = (rnom - i) / (1 + i)
rreal = (9% - 6%) / (1.06) = 2.83%
The exact real rate is less than the approximate
real rate.
5-49
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)
5-50
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.
5-52
5.5 Asset Allocation Across
Risky and Risk Free
Portfolios
5-53
Allocating Capital Between Risky &
Risk-Free Assets
 Possible to split investment funds between safe and
risky assets
or money market fund
 Risk free asset rf : proxy; T-bills
________________________
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:
$2,500
– Stock A you put ______
– Stock B you put $3,000
______
– Stock C you put $2,000
______
$7,500
5-54
Allocating Capital Between Risky &
Risk-Free Assets
Stock A $2,500
Weights in rp
Stock B $3,000
– WA = $2,500 / $7,500 = 33.33%
Stock C $2,000
– WB = $3,000 / $7,500 = 40.00%
– WC = $2,000 / $7,500 = 26.67%
100.00%
The complete portfolio includes the riskless
investment and rp. 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
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 between risky and
aversion will affect __________
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
For example, let y = 0.75
____
rf = 5%
rf = 0%
E(rp) = 14%
rp = 22%
E(rC) = (.75 x .14) + (.25 x .05)
y = % in rp
(1-y) = % in rf
E(rC) = .1175 or 11.75%
C = yrp + (1-y)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 rp and 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%

5-61
Combinations Without Leverage
rf = 5%
rf = 0%
E(rp) = 14%
rp = 22%
y = % in rp
(1-y) = % in rf
Since σrf = 0
E(rc) = yE(rp) + (1 - y)rf
σ c= y σ p
y = .75
If y = .75, then
σc= 75(.22) = 16.5% E(rc) = (.75)(.14) + (.25)(.05) = 11.75%
If y = 1
σc= 1(.22) = 22%
y=1
E(rc) = (1)(.14) + (0)(.05) = 14.00%
If y = 0
σc= 0(.22) = 0%
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%
c =
E(r ) = 14%
p
y = % in rp
E(rC) =18.5%
rf = 0%
rp = 22%
(1-y) = % in rf
y = 1.5
y=0
33%
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
leveraged combinations
E(rC) =18.5%
y = 1.5
y=0
33%
5-64
E(r)
P or combinations of
P & Rf offer a return
per unit of risk of
9/22.
CAL
(Capital
Allocation
Line)
P
E(rp) = 14%
E(rp) - rf = 9%
) Slope = 9/22
rf = 5%
0
F
rp = 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
_________________________________________
investor’s added return required to bear risk
5-66
Quantifying Risk Aversion
Rearranging the equation and solving
for A
E ( rp )  rf
A 
0.5  σ 2
p
Many studies have concluded that
investors’ average risk aversion is
between _______
2 and 4
5-67
Using A
E ( rp )  rf
A 
0.5  σ 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
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  I2  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
Q
S
rf = 5%
0
F

5-71
A=3
E(r)
A=2
P
T
CAL
(Capital
Allocation
Line)
S
rf = 5%
0
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.
5-74
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 investor’s
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 = 30.00%
Boom:
Normal:
(43 – 40 + 1)/40 = 0.10 = 10.00%
(34 – 40 + 0.50)/40 = –0.1375 = –13.75%
Recession:
[(1/3) x 30%] + [(1/3) x 10%] + [(1/3) x (–13.75%)] = 8.75%
E(HPR) =
2
2
2
2
2(HPR) σ (HPR)  [(1/3) x (30% – 8.75%) ]  [(1/3) x (10% – 8.75%) ]  [(1/3) x (–13.75%– 8.75%) ]  0.031979
σ (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
E(r)

U
1
0.12
0.30
-0.0600
2
0.15
0.50
-0.3500
3
0.21
0.16
0.1588
4
0.24
0.21
0.1518
We choose the portfolio with the highest utility value,
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
2002-2003
a. TWR
2003-2004
2004-2005
AAR 
Return = [(capital gains + dividend) /
price]
b. DWR
(110 – 100 + 4)/100 = 14.00%
Time
Cash
flow
(90 – 110 + 4)/110 = –14.55%
0
-300
Purchase of three shares at $100
per share
(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
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