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Lecture 3
Investment and Portfolio Analysis (Auckland University of Technology)
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Lecture 3
Risk and Return
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Rates of Return
The total holding-period return (HPR) measures by how much the
value of initial investment has grown over the investment period:
Dollar Return
HPR
Beginning Price
Ending Price (P1 ) - Beginning Price (P0 ) Dividend ( D1 )
Beginning Price (P0 )
Capital Gains Yield Dividend Yield
Example: (Single period)
P1=$24, P0=$20, D1=$1
HPR=(24-20+1)/20
=0.25=25%
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Returns Over Multiple Periods
Average returns per period:
Arithmetic average
ra = (r1 + r2 + r3 + ... rn) / n
Geometric average
rG={[(1+r1)(1+r2)....(1+rn)]}1/n-1
Note: ri is the return for period i,
n= number of periods.
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Should we care about the
difference between two formulas?
Consider returns for three periods:
r1 =200%, r2 = -90%, r3 =100%
ra =?
ra = (r1 + r2 + r3 + ... rn) / n
(200%-90%+100%)/3 = 70%
rG =?
rG={[(1+r1)(1+r2)....(1+rn)]}1/n-1
{[(1+200%)(1-90%)(1+100%)]}1/3-1
=-15.7%
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Returns Over Multiple Periods:
Example, Data from Table 5.1
1stQ 2ndQ 3rdQ 4thQ
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
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Returns Over Multiple Periods:
Example
In the example, n=4, r1 =.10, r2 =.25,
r3 =-.20, r4 =.25
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
= (1.375) 1/4 -1 = .0829 = 8.29%
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Returns Over Multiple Periods
Dollar-weighted Return:
The dollar-weighted return is essentially an internal
rate of return (IRR), which in the following example
translates into:
0.1 0.5 0.8 1.0
1
(1 IRR) (1 IRR) 2 (1 IRR)3 (1 IRR) 4
Example: Time (quarter) NCF
Financial calculator (BAII Plus): 0 -1
Press CF; C01=-1; C01=-0.1; 1 -0.1
C02=-0.5;C03=0.8; C04=1; 2 -0.5
Press CPT + IRR; Get the result 3 0.8
of 4.1744% 4 1
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Returns Over Multiple Periods
Dollar-weighted Return:
The dollar-weighted return is essentially an internal
rate of return (IRR), which in the following example
translates into:
0.1 0.5 0.8 1.0
1
(1 IRR) (1 IRR) 2 (1 IRR)3 (1 IRR) 4
Example: Time (quarter) NCF
The =IRR() function in Excel can 0 -1
be used to compute the dollar- 1 -0.1
weighted return. The solution 2 -0.5
for the equation above is 3 0.8
IRR=4.17%. 4 1
4.17% 8
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Annualizing HPRs
Annualizing HPRs for holding periods of
greater than one year:
Without compounding:
HPRann = HPR/n
With compounding:
1/n
HPRann = [(1+HPR) ]-1
where n = number of years held
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Annualizing HPRs: 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. Find HPR over two years and annualized
HPR with and without compounding.
P1 =$52, P0 =$45, D1 =$8
HPR = (52-45+8)/45 = 33.33%
Annualized without compounding
HPRann =33.33%/2 = 16.66%
The annualized HPR assuming annual compounding is
(n = 2 ):
HPRann = (1+0.3333)1/2 -1=15.47%
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Quoting Conventions
APR = annual percentage rate
(periods in year) X (rate per period)
EAR = effective annual rate
( 1+ rate per period)Periods per year - 1
Example: For a monthly return of 1%,
APR = 1% X 12 = 12%
(Assuming monthly compounding)
EAR = (1+1%)12 - 1 = 12.68%
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Quoting Conventions
The general formula for the relationship
between APR and EAR is
æ APR ÷ö æ APR ÷ö
n n
ç
1 + EAR = çç1 + ÷÷ or EAR = çç1 + ÷÷ - 1
çè n ÷ø ç
èç n ÷ø
where n is the number of compounding
periods per year.
Given EAR, we can get APR:
( )
1/n
APR = [ 1 + EAR - 1]´ n
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Measuring Expected Return:
Scenario Analysis
The expected return based on scenario
analysis is determined by:
k
E (r ) = å p(s )r (s )
s =1
E(r) = Expected Return
p(s) = probability that State s occurs
r(s) = return if State s occurs
k = total number of possible states
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Measuring Variance or Standard Deviation
of Returns: Scenario Analysis
The variance of returns based on
scenario analysis is determined by:
k
s =
2
å p(s )[r (s ) - E (r )] 2
s =1
= [2]1/2
= Standard deviation of returns
2 = Variance of returns
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Numerical Example: Scenario Analysis
for investment returns
State P(s) Return r(s)
1 .2 - 0.05
2 .5 0.05
3 .3 0.15
E(r) =(.2)(-0.05) + (.5)(0.05) + (.3)(0.15)
= 6%
2 = [(.2)(-0.05-0.06)2 + (.5)(0.05- 0.06)2 + (.3)(0.15-0.06)2]
= 0.0049%2
= [ 0.0049]1/2 = .07 or 7%
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Expected return and standard deviation
based on historical returns
n
ri r average HPR
r
i 1 n n # of observations
n
1
Ex-post Variance: s 2 = å (r
n - 1 i =1 i
- r )2
Expost Standard Deviation : σ σ 2
Annualizing the statistics:
rannual rperiod # periods
annual period # periods
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Monthly Source Yahoo finance Average 0.011624 0.219762458
HPRs Monthly Source Yahoo finance
2 HPRs
Obs DIS (r - ravg)
Obs DIS (r - ravg)
2 Variance 0.003725 (ri - ravg)2
1 -0.035417 0.002212808 9/3/2002 31 0.027334 0.000246811 3/1/2005 Stdev 0.061031 n 60
2 0.093199 0.006654508 10/1/2002 32 -0.088065 0.009937839 4/1/2005
3 0.15756 0.021297275 11/1/2002 33 0.037904 0.000690654 5/2/2005 n-1 59
4 -0.200637 0.045054632 12/2/2002 34 -0.089915 0.010310121 6/1/2005
5 0.068249 0.00320644 1/2/2003 35 0.0179 3.93874E-05 7/1/2005 Annualized
6 -0.026188 0.001429702 2/3/2003 36 -0.017814 0.000866572 8/1/2005
7 -0.00183 0.000181016 3/3/2003 37 -0.043956 0.003089121 9/1/2005 Average 0.139486
8 0.087924 0.005821766 4/1/2003 38 0.010042 2.50266E-06 10/3/2005
9 0.050211 0.001489002 5/1/2003 39 0.022495 0.00011818 11/1/2005 Variance 0.044697
10 0.004734 4.74648E-05 6/2/2003 40 -0.029474 0.001689005 12/1/2005
11 0.099052 0.00764371 7/1/2003 41 0.05303 0.001714497 1/3/2006
Stdev 0.211418
12 -0.068896 0.006483384 8/1/2003 42 0.09589 0.007100858 2/1/2006
n
13 -0.016478 0.000789704 9/2/2003 43 -0.003618 0.000232311 3/1/2006 HPRT
14 0.109174 0.009516098 10/1/2003 44 0.002526 8.27674E-05 4/3/2006 r r averageHPR n # observations
15 0.019343 5.95893E-05 11/3/2003 45 0.083361 0.005146208 5/1/2006 T 1
n
16 0.019409 6.06076E-05 12/1/2003 46 -0.016818 0.000808939 6/1/2006
17 0.02829 0.000277753 1/2/2004 n
47 -0.010537 0.000491104 7/3/2006 1
18 0.095035 0.00695741 2/2/2004 48 -0.001361 0.000168618 8/1/2006 Expost Variance: 2 (ri r )2
19 -0.061342 0.005324028 3/1/2004 49 0.04081 0.000851813 9/1/2006 n 1 i 1
20 -0.085344 0.00940277 4/1/2004 50 0.01764 3.61885E-05 10/2/2006
21 0.018851 5.22376E-05 5/3/2004 51 0.047939 0.001318787 11/1/2006
22 0.079128 0.004556811 6/1/2004 52 0.044354 0.001071242 12/1/2006 Expost Standard Deviation: σ σ 2
23 -0.103832 0.013330149 7/1/2004 53 0.02559 0.000195054 1/3/2007
24 -0.028414 0.001603051 8/2/2004
25 0.004562 4.98687E-05 9/1/2004
54 -0.026861 0.001481106 2/1/2007 Annualizing the statistics:
55 0.005228 4.09065E-05 3/1/2007
26 0.105671 0.008844901 10/1/2004 56 0.015723 1.68055E-05 4/2/2007 rannual rmonthly 12
27 0.061998 0.002537528 11/1/2004 57 0.01298 1.83836E-06 5/1/2007
28 0.041453 0.000889761 12/1/2004 58 -0.038079 0.002470321 6/1/2007
29 0.028856 0.000296963 1/3/2005 59 -0.034545 0.002131602 7/2/2007 annual monthly 12
30 -0.024453 0.001301505 2/1/2005 60 0.017857 0.000038854 8/1/2007
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Normal Distribution
measures deviations above
Risk is the possibility of the mean as well as below the
getting returns different mean.
from expected return. 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.
E[r] = 10%
Average = Median
= 20%
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Value at Risk (VaR)
Value at Risk: measure of downside risk, worst
loss that will be suffered given probability,
usually α=5%.
If the probability distribution of HPRs is
normal, VaR at α=5% is given by
VaR = E[r] + (-1.64485)×
where E[r] and are the expected return and
standard deviation for an investment, and (-
1.64485) is the 5th percentile of standard
normal distribution, calculated by Excel
function =Norminv (0.05,0,1).
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Value at Risk (VaR)
Example: A stock portfolio with current market value
of $500,000 has an annual expected return of 12%
and a standard deviation of 35%. What is the
portfolio VaR at a 5% probability level over one-year
period?
VaR = 0.12 + (-1.64485 * 0.35) = -45.57%
95% of the time, the biggest possible loss of
the portfolio over a year will not exceeds the
amount of $227,850, determined by
$500,000 x -.4557 = -$227,850
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Risk Premium & Risk Aversion
The risk free rate is the rate of return that can be
earned with certainty, denoted as rf.
The risk premium is the difference between the
expected return of a risky asset and the risk-free
rate, i.e.,
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.
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Asset Allocation across Portfolios
The Asset Allocation
Portfolio choice among broad investment
classes
Complete Portfolio
Entire portfolio, including risky and risk-free
assets
Capital Allocation
Choice between risky and risk-free assets
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Allocating Capital Between Risky &
Risk-Free Assets
An investor splits investment fund between risk-free
and risky assets according his/her level of risk
aversion, which is a basic asset allocation decision.
Example. Suppose 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_
What percentage of your holdings are in the risk-free
asset? What’s the fraction of your portfolio invested
in each of the three stocks.
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Allocating Capital Between
Risky & Risk-Free Assets
Weights in rp
WA = $2,500 / $7,500 = 33.33% Stock A $2,500
WB = $3,000 / $7,500 = 40.00% Stock B $3,000
WC = $2,000 / $7,500 = 26.67%
Stock C $2,000
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%
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Investment Opportunity set with a
Risk-Free Asset
For a complete portfolio consisting of risky and
risk-free assets, its expected return and
standard deviation of returns for the complete
portfolio are:
E (rc ) yE (rp ) (1 y )rf
c y p
Sharpe ratio: reward-to-variability ratio =
portfolio risk premium/portfolio standard
deviation, defined as:
Sharpe Ratio = (E(rp) - rf ) / p
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Investment Opportunity set with a
Risk-Free Asset
Example: Suppose that a risk asset has E(rp)
=14% and σp=22%, and rf =5%. If you invest
$7,500 and $2,500 in the risky and risk-free
assets respectively, what is the expected return
and standard deviation of your total investment?
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Investment Opportunity set with a
Risk-Free Asset
rf = 5% rf = 0%
E(rp) = 14% rp = 22%
y = % in rp (1-y) = % in rf
In this example, y = ____
0.75
E(rC) =(.75 x .14) +(.25 x .05)
E(rC) = .1175 or 11.75%
C = yrp + (1-y)rf
C = (0.75 x 0.22) + (0.25 x 0) = 0.165 or 16.5%
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Capital Allocation Line
o Capital Allocation Line (CAL)
Plot of risk-return combinations available by
varying allocation between risky and risk-
free
o Risk Aversion and Capital
Allocation
y: Preferred capital allocation
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E(r)
Possible Combinations
E(rp) = 14%
E(rc) = 11.75% P
y=1
y =.75
rf = 5%
F
y=0
0 16.5% 22%
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Risk Aversion and Asset 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%
E(rp) = 14% y = 1.5
E(rp) = 11.75% P
y=1
y =.75
rf = 5%
F
y=0
0 16.5% 22%
33%
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Passive Investment Strategy
Passive Strategy
Investment policy that avoids security
analysis
Investing in a stock index fund for the
whole investment holding period is a
typical example of a passive investment
strategy for stock investments.
Capital Market Line (CML)
Capital allocation line using market-index
portfolio as risky asset
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