Download Risk and Return

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):
T
HPR t
HPR avg  
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%
T
HPR avg  
t 1
HPR av g 
HPR t
n
(-.2156  .4463  .2335  .2098  .0311  .3446  .1762)
 17.51%
7
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):
1/ T


HPR avg   (1  HPR t )   1
 t 1

HPR avg  (0.7844 1.44631.2335 1.2098 1.03111.3446 1.1762)1/7 1  15.61%
T
GAR = 15.61%
5-10
Measuring Ex-Post (Past) Returns
•Finding the average HPR for a portfolio of assets for a
given time period:
Vi 

  HPR i 
TV 
i 1 
J
HPR avg
•where Vi = amount invested in asset I,
•J = Total # of securities
•and TV = total amount invested;
•thus Vi/TV = Wi = 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:
Vi 

  HPRi 

TV


i 1
J
HPR avg
HPR avg  (.02 (200/1000))  (.04  (300/1000))  (-.05 (500/1000)) -0.9%
5-12
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
5-13
5.2 Risk and Risk
Premiums
5-14
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
5-15
Measuring Variance or
Dispersion of Returns
a. Subjective or Scenario
Variance
σ 2   p(s)  [r(s)  E(r)] 2
s
 = [2]1/2
E(r) = Expected Return
p(s) = probability of a state
rs = return in state “s”
5-16
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)  [r(s)  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
 = [ 0.0049]1/2 = .07 or 7%
5-17
Historical Average Return & s as
proxies for R(r) and σ
T
HPR t
r
T
t 1
r  average HPR
T  # observatio ns
T
1
2
Expost Variance : S 2 
(
r

r
)
 t
T  1 t 1
Expost Standard Deviation : s  s
2
Annualizing the statistics:
rannual  rperiod  # periods
sannual  s period  # periods
5-18
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
2
(r - ravg)
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
Variance
0.003725
Stdev
0.061031
0.219762458
S (r -
ravg)2
=
T
60
T-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
5-19
Using Ex-Post Historical 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
5-20
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
_____________________
5-21
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%
5-22
Implication?
 is an incomplete
risk measure
Leptokurtosis
5-23
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
5-24
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?
5-25
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.
5-26
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 investor’s reluctance to accept
risk.
How is the aversion to accept risk overcome?
By offering investors a higher risk premium.
5-27
5.3 The Historical Record
5-28
Frequency distributions of annual HPRs,
1926-2008
5-29
5.4 Inflation and Real Rates
of Return
5-30
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 $1.93
–(nominal interest rate=5.95%)
Taxes are paid on _______
nominal investment income. This
real investment income even further.
reduces _____
You earn a5.95%
____ nominal, pre-tax rate of return and you
15% tax bracket and face a _____
are in a ____
4.29%inflation rate.
What is your real after tax rate of return?
rreal  [5.95% x (1 - 0.15)] – 4.29%  0.77%
5-31
Real vs. Nominal Rates
Fisher effect: Approximation
Nominal risk free rate  real rate + inflation rate
rnom  rreal + i
rreal = real risk free interest rate
Example rreal = 3%, i = 6%
rnom = nominal risk free interest rate
rnom  9%
i = expected inflation rate
Fisher effect: Exact
rnom = (1 + rreal) * (1 + i) – 1
= 1.03 * 1.06 - 1 = 9.18%
The exact nominal risk free required rate is
slightly higher than the approximate nominal rate.
5-32
Historical Real Returns & Sharpe
Ratios
Series
World Stk
US Lg. Stk
Sm. Stk
World Bond
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.
5-33
5.5 Asset Allocation Across
Risky and Risk Free
Portfolios
5-34
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:
– Stock A you put $2,500
______
– Stock B you put $3,000
______
– Stock C you put $2,000
______
$7,500
5-35
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
Wf = 25%; Wp =
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%
5-36
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
5-37
Example
rf = 5%
rf = 0%
E(rp) = 14%
rp = 22%
y = % in P
(1-y) = % in f
5-38
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%
5-39
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.
5-40
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%

5-41
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%

5-42
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%
c = (1.5) (.22) = 0.33 or 33%
E(rC) =18.5%
y = 1.5
y=0
5-43
33%
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%
5-44
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%

5-45
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
5-46
A=3
A=3
E(r)
CAL
(Capital
Allocation
Line)
P
Q
S
rf = 5%
0
F

5-47
5.6 Passive Strategies and
the Capital Market Line
5-48
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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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.
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Selected Problems
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Problem 1
• V(12/31/2004) = V (1/1/1998) x (1 + GAR)7
•
= $100,000 x (1.05)7 = $140,710.04
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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%
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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%
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