MANAGERIAL FINANCE 410 (Rev Sp’15)
STUDENT LECTURE NOTE 4
I. Fundamentals of Risk and Return
A. Risk and Risk-Types
1. _____ is THE four-letter word in Finance.
Everything has something to do with risk.
2. Investment risk: the possibility that the return on
your investment does not match your expectations.
You should understand the difference between exante (expected) risk versus ex-post (realized/
historical) risk.
3. Interest-rate risk: Rising or falling interest rates
due to market conditions impact the rates you pay
or earn when you borrow or invest, as well as the
value of the investments in your portfolio.
4. Inflation risk: Changes in the general price level
mean that the purchasing power of your money may
_______ (rising prices) or _______ (falling prices).
5. Liquidity risk: describes how easily (or
inexpensive) it is to convert a given asset
(something you own) into cash. Different types of
assets have varying degrees of liquidity. Cash (or
savings/checking accounts) by definition is the most
liquid asset. Figure 4.1 below provides a general
idea of potential investments and their relative
liquidity.
Finance Elective: MV Analysis
1
Assets
FIGURE 4.1
Liquidity
Comment
Publicly-traded
Stocks, High
through
Stock and Bond Mutual secondary markets.
Funds and REITs
Bank CDs and T-Bills
Medium High
Corporate, Treasury and
Municipal Bonds
Real Estate and Used
Vehicles (not including
classic cars)
Collectibles like Antiques,
Art, Classic Cars, Comic
Books, Firearms, Jewelry,
etc.
the Easy to sell through broker
or account manager at
market price.
CDs are easy to cash out
prior to maturity however
interest penalty may be
substantial.
Medium High as the Bonds trading at a discount
secondary market is not mean that you will not
nearly as active as for receive face value if sold
stocks.
before maturity.
Medium to Low as market The price concession to
conditions
are
very liquidate quickly may be
important.
substantial. The keys are
demand and desirability.
Generally Low
To sell these items without
a large price concession it
is critical to find the right
buyer.
6. Income risk: Losing one’s job is never a good
thing whether due to the economic situation,
company factors or for personal reasons. Ideally,
experts recommend you have savings equal to ___
months of income for such a job-loss (or medical)
emergency.
7. Personal risk: describes risks that are basically due
to the decisions you make in life, albeit some may
be less in your control than others. These may
include purchase, career, health and safety
decisions.
B. Return Types and Calculating the Rate of Return
1. Return: An economist would say the return gained
through an investment is a reward earned for
Finance Elective: MV Analysis
2
deferring consumption. A financial economist would
say that return is the reward earned for bearing ____.
2. Again ex-ante (expected) return versus ex-post
(realized/historical) return. There is no risk in
realized returns.
3. Rate of return is typically measured in relation to the
initial investment and is stated on an annual
percentage basis. In its simplest form, the ________
(annual) percentage return (PR1) is calculated as in
(4.1) below. This is also the most basic formula to
calculate percentage change in general.
P  P   P 
PR1 =  1 0  =  1   1,
 P0   P0 
(4.1)
Where: P0 is the price at time zero, and
P1 is the price one year later.
Ex. 4.1
Assume today is April 12th, 2004 and the price of a gallon
of regular, unleaded gasoline at the Hammond Racetrac
station is $1.769. If the gas price at this same time last
year (April 2003) was $1.479, by what percentage have
gas prices changed over the past year?
 $1.769  $1.479   $1.769 
A4.1. PR1(%) = 
 =  $1.479   1
$1.479

= _______ = _______.
Finance Elective: MV Analysis
3
4. Return on investment usually is going to be
measured over multi-year investment periods.
However, what investors should be interested in for
comparative purposes will be __________ rates of
return. When an asset’s beginning (present) value
(PV0), ending (future) value (FVt) and the term of
investment (t) are known (and there are no interim
cash flows) then the following equation can be used
to calculate the (implicit) percentage rate of return
earned on the investment (r%).
1/t
 FV 
r(%) =  t   1,
 PV0 
(4.2)
Ex. 4.2
Assume again that today is April 12th, 2004 and the
unleaded gas price (per gallon) at the Hammond Racetrac
station is $1.769.
a) If the gas price on this same date in 2009 (five years
later) was $1.899, what is the annualized percentage
price change over this period?
b) Does this small average price increase surprise you?
Given what you know about gas prices over this
period, especially Summer and Fall of 2008, what does
this knowledge make you realize about risk (compared
to the average annualized return)?
Finance Elective: MV Analysis
4
1
A4.2a) r(%)
 $1.899  5
= 
 1 = _______ = _______.

 $1.769 
A4.2b) Clearly average price changes do not tell the whole
story. From $1.769 (mid-2004) to over $4.00 (in
early July 2008) petrol prices have been very
volatile. See graph below. Thus, one must also
specifically assess risk when considering potential
investments.
Ave rage U.S. Gas oline Price s
Cents per gallon
450
400
350
300
250
200
150
Jan-04
May-05
Oct-06
Feb-08
Jul-09
C. A Look at Historical Risk and Return for the Major
Asset Classes
1. Cash: Does not literally mean just the (paper)
money stuffed under your mattress, in your purse or
wallet. It means very liquid, very ___ risk and very
short-term debt securities. Also termed money
market instruments these also include other
investments like Treasury-Bills, savings deposits,
CDs and commercial paper.
2. Bonds: Debt instruments that have a _____ maturity
Finance Elective: MV Analysis
5
than cash. These assets are represented by long-term
Treasury Notes and Bonds, Municipal, Corporate
and Foreign bonds.
3. Stocks: There a few hundred stocks issued by the
largest corporations in America that are quite visible
and trade very frequently. Although this is not quite
exact, the stocks in the S&P 500 Index may be
thought of as comprising the _________ stock class.
There are a few thousand other publicly-traded midcap and small-cap stocks. Stocks are also sometimes
categorized as “value stocks” or “growth stocks”.
4. Other Assets: There are other asset classes like
mutual funds, ETFs, hedge funds, derivatives,
foreign investments, real estate (also REITs), art,
collectibles, etc.
5. Figure 4.2 below depicts the historical risk-return
relationship for the periods of 1926-2007 and the
more recent period of 1970-2007.
Figure 4.2
Asset Class
U.S. Inflation
30-Day T-Bills
Inter Gov’t Bonds
L-T Gov’t Bonds
L-T Corp Bonds
S&P 500 Stocks
Small-Cap Stocks
1926-2007
Return (%)
Risk (%)
Geom
Arith
St Devn
3.0
3.1
4.2
3.7
3.8
3.1
5.3
5.5
5.7
5.5
5.8
9.2
5.9
6.2
8.4
10.4
12.3
20.0
12.5
17.1
32.6
1970-2007
Return (%)
Risk (%)
Geom Arith
St Devn
4.6
4.7
3.1
6.0
6.0
2.9
8.2
8.4
6.6
8.9
9.4
11.2
8.9
9.4
10.5
11.1
12.4
16.6
13.4
15.6
22.6
Source: Ibbotson Stocks, Bonds, Bills and Inflation®, SBBI® Valuation Yearbook,
© Morningstar 2008.
Finance Elective: MV Analysis
6
II. Measuring Return and Risk for Individual Securities
A. Statistical Risk and Return Measures
1. Expected Return: E(R) is the probability-weighted,
arithmetic average return. In other words, it is the
____-_____ return given the different possible states
of nature, the associated returns and their respective
probabilities of occurrence.
2. The Expected Return may be calculated as follows
when the probabilities of the states of nature are
different:
n
E(R) = (R) =  [Ri * Probi]
i 1
= (R1Prob1)+(R2Prob2)+…+(RnProbn).
(4.3)
3. Formula (4.3) may be reformulated when the
probability of each state of nature is ____-______ as
is given below. This formula is usually applied to
asset-return calculations that are based on annual,
monthly, or daily data, etc.
n 
  Ri 
E(R’) =  i 1  .
 n 


(4.3’)
4. The Excel Function used to calculate the mean
return (assuming equi-likely probabilities as in
Finance Elective: MV Analysis
7
(4.3’)) is AVERAGE. Its form is given below. In
this Excel equation (and also the next two) the
values used in the calculation will typically be in the
form of an ARRAY, i.e., (aa:zz) which refers to the
range of values from cell aa to cell zz.
=AVERAGE(number1,[number2],… )
(E4.1)
5. When monthly prices are used to calculate
(monthly) returns, the average needs to be multiplied
by 12 to annualise it as in (E4.1A).
=AVERAGE(number1,[number2],… ) * 12 (E4.1A)
B. Standard Deviation: A way to measure the dispersion
of possible outcomes. It is found as the square root of
the probability-weighted, summed, squared deviation of
each return from the mean. Thus, the ____ concentrated
the observations are around the mean, the lower the SD
will be, and vice versa.
1. It is calculated on the basis of the Variance as
follows:
n


Var(R) = 2(R) =  ( Ri   ) 2 * Probi .
i 1
SD = (R) =
Finance Elective: MV Analysis
Var(R) .
8
(4.4)
(4.5)
2. If the probabilities of each state of nature are __________ or, for example, returns are yearly, monthly,
daily, etc., then the “population” variance and
standard deviation formulas may be simplified as
shown below.1
n
2
(
R
i  )



i

1
2
 (R’) = 
 . (4.4’)
n




(R’) =
 2 ( R' ) . (4.5’)
3. The Excel functions to calculate the population
variance and standard deviation (again, assuming
equi-likely probabilities) are shown in (E4.2) and
(E4.3), respectively.
=VARP(number1,[number2],… )
(E4.2)
=STDEVP(number1,[number2],… )
(E4.3)
To annualise the variance it also needs to be
multiplied by 12. So, that (E4.2) is annualised as in
(E4.2A) as shown below.
=VARP(number1,[number2],… ) * 12 (E4.2A)
Because each deviation from the mean is squared to
find the variance, and the square root is taken to
calculate the standard deviation, the annualisation
1
Note: To find the “population” measures you need to use the VARP and STDEVP
functions. The Excel VAR and STDEV functions calculate the “sample” measures.
Finance Elective: MV Analysis
9
approach used for the standard deviation is to
multiply the monthly STDEVP by the square root of
12 as in (E4.3A).
=STDEVP(number1,[number2],… ) * 12
(E4.3A)
Ex. 4.3
George Bekos (Fin 304-Fall’84, Fin 305-Win’85, Fin
348-Spr’85) is considering an investment in an 8% (RN)
fixed-coupon bond. The real return he will earn on the
bond will depend on the rate of inflation over the
investment period. The inflation rates, real returns, the
states of nature and the probability of that state
occurring are given below. The Fisher Approximation
(i.e., the equation shown below) is used to find the real
interest rate. Based on this information determine the
bond’s expected return and standard deviation.
RR = RN – IE.
Inflation Rate
Very High
= 9%
High
= 7%
Moderate
= 5%
Low
= 3%
Probability
15%
25%
50%
10%
Real Return
- 1%
+1%
+3%
+5%
A4.3. (R) = (.15  -.01) + (.25  .01) + (.50  .03)
+ (.10  .05) = ____ = _____.
2(R)
= [(-.01-.021)2 (.15) + (.01-.021)2 (.25)
+ (.03-.021)2 (.50) + (.05-.021)2 (.10)]
Finance Elective: MV Analysis
10
= _________.
(R) = .0002991 = _______ = ________.
Ex. 4.3 extended
Re-do Ex. 4.3 using formulas (4.3’), (4.4’) and (4.5’)
under the assumption that the states of nature are equilikely. In general what should be expected to happen to
these measures under this new assumption?
A4.3. Ext. Expect mean return to go down as the lowest
returns will get heavier weight and higher returns get
less (or same) weight. Standard deviation goes up
because returns which are farthest from mean get
relatively greater weight.
(R’) = [( -.01 + .01 + .03 + .05)/4]
= ____ = _____.
2(R’)
= (¼)[(-.01-.02)2 + (.01-.02)2
+ (.03-.02)2 + (.05-.02)2]
= _____.
(R’) = .0005 = ________ = ______.
C. Coefficient of Variation
1. The CV is a measure that may be used to quantify the
Risk/Return Tradeoff. It measures the amount of risk
borne per unit of expected return. Do simple example.
Finance Elective: MV Analysis
11
2. It is calculated as follows:
 ( R) 
Coefficient of Variation (CV) = 
.

 ( R) 
(4.6)
3. Given the form of the measure in (4.6), it is useful to
consider what makes for a superior investment: high
expected project return; low standard deviation. Use
ceteris paribus analysis.
4. Decision Criteria: Choose the project with the CV
which has an [absolute] value closest to zero. Note: the
technical aspect of this rule, i.e., an asset with a negative
CV close to zero would technically be preferred to one
with a high positive CV.
Ex. 4.4
Kathleen Hendrix (F381 Su’08, F382 Fa’08) has
collected monthly return data for the five-year period
from 7/01/02 to 7/02/07 for the stock of Abbott Labs and
seven Vanguard mutual funds. She has calculated the
annualised average returns and standard deviations shown
in the exhibit below. Your task is to calculate the
coefficients of variation and to rank the risk-return tradeoffs for these eight assets, where rank=1, represents the
best trade-off.
Stock/Mutual Fund
SYM
Abbott Labs
ABT
GNMA Fund Inv Shares
VFIIX
Interm-Term Bond Index Inv VBIIX
Shares
Finance Elective: MV Analysis
Retn(%) StDev(%)
12.346
17.851
3.691
2.839
5.110
5.438
12
CV
1.4459
0.7692
1.0642
Rank
8
1
6
Euro Stock Index Inv Shares
Explorer Fund Inv Shares
Mid-Cap Index Fund Inv Shares
Tax-Managed Cap Appr Inv
Shares
Tax-Managed Cap Appr Adm
Shares
VEURX
VEXPX
VIMSX
VMCAX
18.517
14.247
15.516
12.162
14.696
15.697
12.681
11.908
0.7936
1.1018
0.8173
0.9791
2
7
3
4
VTCLX
12.016
12.027
1.0009
5
A4.4. A sample calculation is shown for ABT below.
 St Dev  17.851% 
CVABT = 
=
= ______.
 Retn  12.346% 
Figure 4.3:
Annualization of Return & Risk Excel Functions
Using Monthly Data
Monthly Returns
=AVERAGE * 12
=VARP * 12
=STDEVP * SQRT(12)
=COVAR * 12
=CORREL
Annualized Returns
=AVERAGE
=VARP/12
=STDEVP/SQRT(12)
=COVAR/12
=CORREL
Ex. 4.5
Jessica Poret (F382 Sp’09) is trying to determine the risk
and return characteristics for three assets over the past
five-year period of June 2004-2009. Two assets are for
comparative purposes and these are the 13-week T-Bill
(^IRX) interest rates and the S&P 500 (^SPSC) stock
index. The third asset is Abbott Labs (ABT) common
stock. Use the data provided in the Ex. 4.5 Worksheet of
SS #2. Assuming each monthly return is equi-likely
Finance Elective: MV Analysis
13
Jessica will determine the following return and risk
measures for these three assets over the period given.
a) Convert the T-Bill (percentage) rates into a decimal.
Calculate the monthly returns for the S&P 500
Index and the Abbott Labs stock.
b) Use the COUNT function to find the number of
returns.
c) Calculate the annualised Average percentage return
using the function in (E4.1A).
d) Calculate the annualised population Variance using
the function given in (E4.2A).
e) Calculate the annualised population Standard
Deviation using the function given in (E4.3A).
f) Calculate the Coefficients of Variation for each
asset.
g) Rank the assets on the basis of the risk-return tradeoff (where best trade-off is ranked 1).
A4.5a) Convert T-Bill (%) rate to decimal by dividing by
100. Note: to have a number of returns that is
consistent with the stocks ignore the first T-Bill
rate.
Sample Monthly Return (ABT 6/1/2004-7/1/2004)
 $34.83 
 1 = _______.
R6/04 – 7/04 = 

$
35
.
85


Excel Eqn.(H9): =(G9/G8)-1
Finance Elective: MV Analysis
14
A4.5b) There are 61 months of data so there are 60 monthly
returns (for ABT).
Excel Eqn.(H70): =COUNT(H$9:H$68)
A4.5c) Annualised Average (for ABT), (E4.1A) is as
follows:
Excel Eqn.(H71): =AVERAGE(H$9:H$68)*12
For T-Bill: Excel Eqn.(D71): =AVERAGE(D$8:D$68)
A4.5d) Annualised population Variance (for ABT),
(E4.2A) is as follows:
Excel Eqn.(H72): =VARP(H$9:H$68)*12
For T-Bill: Excel Eqn.(D72): =VARP(D$8:D$68)/12
A4.5e) Annualised population Standard Deviation (for
ABT), (E4.3A) is as follows:
Excel Eqn.(H73): = STDEVP(H$9:H$68)*SQRT(12)
For T-Bill: Excel Eqn.(D73): =STDEVP(D$8:D$68)/SQRT(12)
A4.5f) Coefficient of Variation (for ABT), using (4.6) is as
follows:
18.014% 
CVABT = 
= _________  ______.

 7.506% 
Excel Eqn.(H74): = H73/H71
Finance Elective: MV Analysis
15
A4.5g) Rank Coefficients of Variation (closest to zero is
best risk-return trade-off):
Asset
13wk T-Bills
S&P 500 Index
Abbott Labs
7
8
9
10
11
…
64
65
66
67
68
69
70
71
72
73
74
75
76
77
B
Date
6/1/2004
7/1/2004
8/2/2004
9/1/2004
…
2/2/2009
3/2/2009
4/1/2009
5/1/2009
6/1/2009
CV
0.5733
-5.0285
2.4000
C
13w T-Bill
1.30
1.41
1.57
1.67
…
0.25
0.20
0.12
0.13
0.17
D
Decimal
0.0141
0.0157
0.0167
…
0.0025
0.0020
0.0012
0.0013
0.0017
COUNT
AVERAGE
VARP
STDEVP
Coef Varn
Rank
60
2.890%
0.00002
0.478%
0.1655
1
E
S&P 500
1140.84
1101.72
1104.24
1114.58
…
735.09
797.87
872.81
919.14
920.26
Rank
1
3
2
F
%Chg
-0.0343
0.0023
0.0094
…
-0.1099
0.0854
0.0939
0.0531
0.0012
G
ABT
35.85
34.83
36.91
37.50
…
46.91
47.27
41.85
45.06
48.02
60
-3.058%
0.0236
15.376%
-5.0285
3
Will come back later to calculate BETA.
H
%Chg
-0.0285
0.0597
0.0160
…
-0.1462
0.0077
-0.1147
0.0767
0.0657
60
7.506%
0.0324
18.014%
2.4000
2
BETA
0.214046
III. Expected Return, Variance, Standard Deviation,
Covariance and Correlation Coefficients for/between
Individual Assets
A. Expected Return: Covered above.
B. Variance and Standard Deviation: Covered above.
C. Covariance: The covariance between two assets
measures the extent to which the asset prices (or
Finance Elective: MV Analysis
16
returns) ____ ________, or covary. If the returns
between two assets tend to rise and fall together their
covariance is positive. Conversely, negative covariances
imply that if one asset’s return is above its mean, the
other asset’s return tends to be below its mean. A
covariance of zero implies no systematic relationship
between the returns of the two assets.
The general formula is given in (4.7) below when the
probabilities of each state are different.
 n

Covij = ij =   [(Ri  i) * (Rj  j) * Pri  j ] , (4.7)
i  j 1

where Rj = the return on asset j, and
j = the mean return for asset j.
The formula when the probabilities of each state are
assumed to be equi-likely, i.e., 1/n, is then given in
(4.7’).
n
1

Covij’ = ij =  *   (Ri   i )(Rj   j )  .
 
 n i 1
(4.7’)
The covariance calculation is similar to that of
variance except that instead of squaring each asset’s
deviation from its mean (the demeaned return), the
two demeaned returns are multiplied together. In fact,
an asset’s covariance with itself equals its ________.
Finance Elective: MV Analysis
17
The Excel function to find the covariance assumes
equi-likely probabilities and is given in (E4.7). Again
the values used in the calculation will (necessarily) be
in the form of an ARRAY.
=COVAR(array1,array2)
(E4.7)
D. Correlation Coefficient: The correlation coefficient
is a measure, which like the covariance, describes
how the returns on two assets are related to each
other. However, unlike the covariance which could
vary from - to +, it is scaled to vary between -1
and +1.
The correlation coefficient is found by dividing the
covariance by the product of the asset standard
deviations as shown in (4.8) below.
 ij 
Correlation Coefficient = ij = 
.

i
*

j


(4.8)
Again, when the probabilities are equi-likely, the
Excel correlation function can be used. Its form is
given in (E4.8) below.
=CORREL(array1,array2)
(E4.8)
Knowledge of asset mean returns and standard
deviations are useful when assessing an asset’s
risk/return tradeoff characteristics. Pairwise covariances and correlations provide information on the
Finance Elective: MV Analysis
18
potential diversification benefits which may be
achieved when assets are combined into portfolios.
For example, combining two assets that are perfectly
positively correlated, i.e. ij = +1, would not be
expected to yield diversification benefits. On the other
hand, combining two assets which are perfectly
negatively correlated (ij = -1), will yield the
maximum benefits in risk reduction.
As might be expected, most assets (of the same type,
eg. all shares) in a given market are generally,
positively correlated to at least some degree.
However, when different types of assets in the same
(domestic) market, for example, shares, bonds and
property, or similar assets from different
(international) markets are combined, significant
diversification benefits are achievable.
The following example is meant to illustrate the use of
the preceding formulas and provide students with an
introduction to these calculations.
Ex. 4.6
As a financial analyst you have projected the probabilities
of three possible future states of (economic) nature and
the relevant returns for stocks Alpha and Beta in each
state. These are given in the table below. Use the
Ex.4.6_Ex.4.7 Worksheet in Spreadsheet #2 to determine the:
a) Expected Mean Return for each Asset, and
Finance Elective: MV Analysis
19
b) Deviation from the Mean in each state for both
assets.
c) Use these deviations to find the Variances for both
assets, and the
d) Standard Deviation for each share.
e) Also calculate the Covariance between the two
shares using the deviations from part b), and
f) Find the Correlation Coefficient.
State of
Economy Probability
Expansion
0.25
Slow Growth
0.45
Contraction
0.30
E(RAi)
Alpha
15.0%
5.5%
-3.0%
E(RBi)
Beta
1.0%
4.0%
8.0%
A4.6. The answers are provided in the spreadsheet results
shown below.
A
9
10
11
12
13
14
15
16
17
18
19
20
B
State of
Economy
Expansion
Slow Growth
Contraction
C
Probability
0.25
0.45
0.30
Mean
Variance
Stnd Devn
D
E(Rtn)
Alpha
15.0%
5.5%
-3.0%
E
E(Rtn)
Beta
1.0%
4.0%
8.0%
5.3250%
0.0044207
6.649%
4.4500%
0.000685
2.617%
-0.001725
-0.991251
Covariance
CorrCoef
Some sample calculations are shown below:
Finance Elective: MV Analysis
20
F
G
RA - A
0.09675
0.00175
-0.08325
RB - B
-0.03450
-0.00450
0.03550
A4.6a) E(RA) = (.25*.15)+(.45*.055)+(.30*-.03) = ______.
Excel Eqn.(D15): =($C11*D11)+($C12*D12)+($C13*D13)
A4.6b) Deviations are shown in the spreadsheet above.
Excel Eqn.(F11): =D11-$D$15
A4.6c) σ2(RA) = [(0.15 – 0.05325)2 * (.25)]
( = 0.0023401)
+ [(0.055 - 0.05325)2 * (.45)] ( = 0.0000014)
+ [(-0.03 - 0.05325)2 * (.30)] ( = 0.0020792)
= 0.0044207
( = 0.0044207)
Note: I realize that you are all capable of doing this calculation in one go (step), using
your calculator, using the parentheses as necessary. However, as a practical matter I would
suggest that you do each of the three (grouped) calculations separately, as I have shown
them above. The problem with a one-step approach is (very obviously) that if you make a
mistake and do not see it, you will end up with the wrong answer. Further, you will not
know where the calculation has gone wrong and cannot check it without doing the whole
calculation again.
Excel Eqn.(D16):
=((F11^2)*$C$11)+((F12^2)*$C$12)+((F13^2)*$C$13)
A4.6d) σ(RA) =
0.0044207 = _________ = ______.
A4.6e) σAB = [(0.15-0.05325)*(0.01-0.0445)*(0.25)]
+ [(0.055-0.05325)*(0.04-0.0445)*(0.45)]
+ [(-0.03-0.05325)*(0.08-0.0445)*(0.30)]
= (-0.0008345)+(-0.0000035)+(-0.0008866)
= ___________.
Finance Elective: MV Analysis
21
Excel Eqn.(E19):
=(F11*G11*C11)+(F12*G12*C12)+(F13*G13*C13)
 σ
   0.0017246 
A4.6f) ρAB =  AB  = 
= ________.

0
.
06649
*
0
.
02617
σ
*
σ

 A B 
Excel Eqn.(E20): =E19/(D17*E17)
IV. Calculating Markowitz Portfolio Return, Variance
and Standard Deviation
A. Portfolio Expected Return
In general, portfolio E(Rp) is calculated by weighting
the E(Ri)'s of the portfolio's component securities by
the percentage of total portfolio market value for
which they account. As a practical matter, the issue of
how to appropriately calculate the weights is rather
important from an investor’s viewpoint. <EXPLAIN>
The general equation is:
E(Rp) =
where: E(Ri)
wi
N
N
 [wi * E(Ri)],
i 1
= expected return on asset i,
= market value proportion of portfolio's
total market value for which security
i accounts, and
= total number of securities in portfolio.
B. Portfolio Variance and Standard Deviation
Finance Elective: MV Analysis
(4.9)
22
Unfortunately, calculation of portfolio variance (and
standard deviation) is not quite as simple as before
because ___________ between securities must also be
taken into account.
The general formula for the Markowitz variance of a
portfolio is:
N N

Var(Rp) =    wi * wj * ij  ,
i 1 j1

where: wj
ij
(4.10)
= proportion invested in security j, and
= covariance between assets i and j.
Recall from statistics that covariance may also be
found given knowledge of the correlation coefficient
as:
ij = ij i j,
(4.11)
and also that the covariance of any asset with itself is
equal to its ________.
In long-hand, the Markowitz variance for a two-asset
portfolio is
Var(R2Ap) = (wi2i2) + (2wiwjij)+ (wj2j2). (4.12)
With a three-asset portfolio, there are three variances
and three covariances, so that this Markowitz portfolio
variance has six terms.
Finance Elective: MV Analysis
23
Var(R3Ap) = (wa2a2) + (2wawbab) + (2wawcac)
+ (wb2b2) + (2wbwcbc) + (wc2c2). (4.13)
Beyond three assets, the variance calculation gets to be
complicated fairly quickly. With four assets, there are
four variance terms and six covariance terms (10 total).
With five assets there are five variances and ten
covariance terms (15 total). In fact, a ten-asset
portfolio variance (10 variance terms and 45
covariance terms) is the largest that can be calculated
in a single cell using Excel. F.Y.I., the formula to find
the total number of terms equals [[(N*(N-1))/2]+N].
Ex. 4.7
Assume that on January 4, 1982, Mr. John Ross Ewing,
Jr. (more popularly known as J.R.), President of Ewing
Oil Company made an investment of approximately
$10,000 in a three-asset stock portfolio to accumulate
tuition money for his son John Ross’s college education at
the University of Texas. Today is now July 1, 1997 and
J.R. has you assigned you (as a financial analyst at Ewing
Oil) the task of evaluating the portfolio’s investment
performance.
Because J.R. is aware of the benefits of portfolio
diversification the three stocks he had chosen for
investment are all in different industries, and are unrelated
to the oil business. Namely, he has chosen Aetna
Insurance (AET), Boeing Co. (BA) and ColgatePalmolive Co. (CL). The number of shares initially
Finance Elective: MV Analysis
24
purchased as well as the initial prices for each stock are
given as follows.
Name
Aetna
Boeing
Colgate
#Shares
2250
1500
2500
Initial
$2.38
$1.83
$0.50
Use the format given in the Ex.4.6_Ex. 4.7 Worksheet
of Spreadsheet #2 as well as the monthly price data
provided to determine the items below. Note, comparative
data for the S&P 500 Index (^GSPC) is also provided.
For Individual Assets:
a) Geometric, monthly (unannualised holding period
return (HPRM), and the monthly-annualised HPR
(HPRMA) for each asset;
b) Use the COUNT function to determine the number of
monthly returns and convert it to years;
c) Geometric HPR (annualised over the entire period)
from 1/4/1982 to 7/1/1997 for each asset;
d) Arithmetic mean return using both monthly HPRs then
annualised (termed Mean(MonAnn)), and the monthlyannualised HPRs (termed Mean(AnnMon)), for each
asset;
e) Annualised (population) variances for each asset,
using the:
i) monthly returns, annualised; and
ii) annualised monthly returns.
f) Annualised (population) HPR standard deviations for
each asset, using the:
i) monthly returns, annualised; and
Finance Elective: MV Analysis
25
ii) annualised monthly returns.
g) Asset coefficients of variation using geometric mean
as the return measure.
h) Pairwise covariances (Annualised) between the three
assets using the monthly HPRs.
i) Correlation coefficients (Annualised) between the
three assets using the monthly HPRs.
For the Three-Asset Portfolio
j) Calculate the beginning market value of the portfolio
on 1/4/1982 and then the ending value on 7/1/1997.
k) The portfolio’s geometric, annualised return based on
ending, versus beginning total portfolio market value.
Note: use the value for “t” previously determined in
b) in the annualisation factor.
l) The market value weights based on Beginning-ofperiod weights (BOP)w (use ROUND fn to 5 places);
m) The Market-Value Weighted, Portfolio Mean Return
using Asset Geometric Returns (w/ BOP weights);
n) The three-asset Markowitz portfolio variance (w/ BOP
weights);
o) The Markowitz portfolio standard deviation;
p) The weighted-average portfolio standard deviation
(under the assumption that all assets are perfectly,
positively correlated); and
q) What can you conclude about the diversification
benefits from the combination of three assets with
correlations that are positive, but are relatively close
to zero?
Finance Elective: MV Analysis
26
r) The portfolio coefficient of variation. How does the
portfolio CV compare to the individual asset CVs as
well as the CV for the SPX?
Finance Elective: MV Analysis
27
S
1
T
U
V
^GSPC
W
X
Y
Aetna
Z
AA
AB
Boeing
AC
AD
AE
Colgate
2
3
Date
1/4/1982
Adj Close
120.40
%Chg
-
AnnChg
-
Adj Close
%Chg
-
AnnChg
-
Adj Close
1.83
%Chg
-
AnnChg
-
Adj Close
2.38
0.50
%Chg
-
AnnChg
-
4
2/1/1982
113.11
-0.0605
-0.7266
2.34
-0.0168
-0.2017
1.62
-0.1148
-1.3770
0.52
0.0400
0.4800
5
3/1/1982
111.96
-0.0102
-0.1220
2.36
0.0085
0.1026
1.51
-0.0679
-0.8148
0.53
0.0192
0.2308
6
4/1/1982
116.44
0.0400
0.4802
2.22
-0.0593
-0.7119
1.65
0.0927
1.1126
0.56
0.0566
0.6792
…
…
…
…
…
…
…
…
…
…
…
…
…
…
186
4/1/1997
801.34
0.0584
0.7009
10.88
0.0635
0.7625
42.12
0.0000
0.0000
22.74
0.1257
1.5089
187
5/1/1997
848.28
0.0586
0.7029
12.06
0.1085
1.3015
45.13
0.0715
0.8575
25.40
0.1170
1.4037
188
6/2/1997
885.14
0.0435
0.5214
12.22
0.0133
0.1592
45.45
0.0071
0.0851
26.73
0.0524
0.6283
189
7/1/1997
954.31
0.0781
0.9377
13.63
0.1154
1.3846
50.27
0.1061
1.2726
31.27
0.1698
2.0382
190
191
192
Count/12
193
GeomMn
194
ArithMn
195
Variance
196
StdDevn
197
CoefVarn
^GSPC
Aetna
Boeing
Colgate
15.5
14.29%
14.49%
0.02090
14.46%
1.0118
15.5
11.92%
14.39%
0.06182
24.86%
2.0862
15.5
23.83%
25.60%
0.08556
29.25%
1.2275
15.5
30.58%
29.23%
0.04551
21.33%
0.6976
14.49%
0.02090
14.46%
14.39%
0.06182
24.86%
25.60%
0.08556
29.25%
198
199
VAR/COV
^GSPC
Aetna
Boeing
Colgate
CORRELS
^GSPC
Aetna
Boeing
Colgate
200
^GSPC
0.02090
0.58383
1.00000
202
Boeing
0.58488
0.29555
1.00000
203
Colgate
0.01916
0.01657
0.02024
0.04551
1.00000
Aetna
0.02476
0.02150
0.08557
^GSPC
201
0.02099
0.06182
0.62102
0.31235
0.32441
1.00000
Finance Elective: MV Analysis
Aetna
Boeing
Colgate
28
29.23%
0.04551
21.33%
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
J
K
L
Name
AET
BA
CL
Initial
#Shares
2250
1500
2500
1/4/1982
Initial
$2.38
$1.83
$0.50
Wtd Price
M
O
Initial
Weights
0.57273
0.29358
0.13369
1.00000
MV
$5,355.00
$2,745.00
$1,250.00
$9,350.00
$1.97
Portfolio Return
Corr Coefs
0.2956
A&B
0.3124
A&C
0.3244
B&C
N
7/1/1997
Reverse
$13.63
$50.27
$31.27
Wtd Price
Geometric
21.205%
3 ASSET
MEAN
(R3A)
AET
BA
CL
Variance/Covariance Matrix
Aetna
Boeing
Colgate
0.06182
0.02149
0.01657
0.02149
0.08556
0.02024
0.01657
0.02024
0.04551
AET
BA
CL
Matrix of Weights
Boeing
Colgate
0.16814
0.07657
0.08619
0.03925
0.03925
0.01787
Aetna
0.32802
0.16814
0.07657
MRKWTZ
0.039819447
ST DEV
MRKWTZ
25.679%
PORTFOLIO COEF of VARN
ASSET Coef of Variations
3 ASSET
PORT VAR 2(R3A)
17.910%
WGHTD AVERAGE
3 ASSET
MV
$30,667.50
$75,405.00
$78,175.00
$184,247.50
$36.11
Wtd Price Geometric
20.652%
3-A Portfolio Variance (Matrix Multiplication)
PORT
P
3 ASSET
PORT ST DV (R3A)
3 ASSET
AET
BA
CL
^GSPC
29
0.039819
1.1142
2.0862
1.2275
0.6976
1.0118
19.955%
Q
Reversal
Weights
0.16645
0.40926
0.42429
1.00000
A4.7. Two excerpts from Spreadsheet #2 are shown above.
Note: The individual asset solutions and sample
calculations apply to the SPX (the Standard & Poor’s
500 Index) data.
For Individual Assets:
 P  P  113.11  120.40 
A4.7a) HPRM(SPX) =  1 0  = 

120.40
 P0  
= ___________.
Excel Eqn.(U4): =(T4-T3)/T3
HPRMA = HPRM * 12 = -0.06054817 * 12
= ___________.
Excel Eqn.(V4): =U4*12
A4.7b) Using the COUNT function to find the number of
returns there are 187 prices (15 years and 7
months), there are 186 monthly price changes.
Thus, 186 converted to years would be (186/12=)
15.5 years.
Excel Eqn.(U192): =(COUNT(U4:U189))/12
A4.7c) Geometric, annualised HPR from 1/4/82 to 7/1/97:
P 
HPRG =  t 
 P0 
1
t
1
 954.31 15.5
1 = 
120.40 
30
1
= __________ = ________.
Excel Eqn.(U193): =(T189/T3)^(1/U192)-1
A4.7d) Arithmetic Mean Monthly (Annualised) HPR:
n 
Ri 
 i
 2.2460412 
*12
Mean(MonAnn) =  1  *12 = 

186
n






= __________ = ______.
Excel Eqn.(U194): =AVERAGE(U4:U189)*12
Arithmetic Mean Monthly-Annualised HPR:
n

(R
*
12
)

 i 1 i
  26.9524940 
Mean(AnnMon) = 
=

186
n

 


= __________ = ______.
Excel Eqn.(V194): =AVERAGE(V4:V189)
A4.7e) Annualised (population) HPR Variance
i) Monthly returns, annualised VARP:
31
 n (R  μ ) 2 
i
i
 *12 = _________.
VARP Mon. HPR (Ann) =  i 1


n


Excel Eqn.(U195): =VARP(U4:U189)*12
ii) Annualised monthly returns VARP:
n
2
(12
*
R

12
*
μ
)

i
i 
 i 1
VARP Ann.-Mon. HPR = 
 /12
n




= _________.
Excel Eqn.(V195): =VARP(V4:V189)/12
A4.7f) Annualised (population) HPR Standard Deviations
i) Monthly returns, annualised STDEVP:
STDEVP Mon. HPR (Ann) =
 n
2
(R

μ
)
 
i
i 
i  1
 * 12


n




= _________.
Excel Eqn.(U196): =STDEVP(U4:U189)*(12^0.5)
ii) Annualised monthly returns STDEVP:
32
STDEVP Ann. =
Mon. HPR
 n (12 * R  12 * μ ) 2 
i
i 
 i 1

 / 12
n




= _________.
Excel Eqn.(V196): =STDEVP(V4:V189)/(12^0.5)
A4.7g) Coefficients of Variation based on Geometric
Means:
STDEVP  14.45781% 
CVSPX = 
 =  14.2889%  = ______.
HPR



G 
Excel Eqn.(U197): =U196/U193
A4.7h) Covariances (annualised) based on the monthly
returns:
 n

(R

μ
)
*
(R

μ
)

i
i
j
j
 i  j1

 * 12 = ________.
σSPX,AET = 
n




Excel Eqn.(U200): =COVAR($U4:$U189,X4:X189)*12
A4.7i) Correlation coefficients (annualised) based on the
monthly returns; Note, as shown in the two
formulations given below, to calculate the
33
correlations the CORREL function does not
actually need to be based on annualised data.

  σ ij 
σ ij *12
ρSPX,AET = 
=

(σ
*
σ
)
(σ
*
12
)
*
(σ
*
12
)
 i
  i
j 
j
= _________.
Excel Eqn.(AB200): =CORREL($U$4:$U$189,X4:X189)
For the Three-Asset Portfolio:
A4.7j) Beginning Market Value of the Portfolio: 1/4/1982:
Name
AET
BA
CL
#Shares
2250 *
1500 *
2500 *
Initial Price
$2.38 =
$1.83 =
$0.50 =
MV
$5,355.00
$2,745.00
$1,250.00
$9,350.00
Ending Market Value of the Portfolio: 7/1/1997:
Name
AET
BA
CL
#Shares
2250 *
1500 *
2500 *
Initial Price
$13.63 =
$50.27 =
$31.27 =
MV
$30,667.50
$75,405.00
$78,175.00
$184,247.50
A4.7k) The (geometrically annualised) Portfolio Holding
Period Return:
 $184,247.50 
Port HPRG = 

 $9,350.00 
1
15.5
34
 1 = ________ = _______.
Excel Eqn.(N19): =(P15/M15)^(1/U192)-1
A4.7l) To find the beginning-of-the-period (BOP) weights
we need to first determine the initial market value
(MV) of each position and then the initial MV of
the portfolio:
 $5,355.00 
(BOP)wAET = 
= _________  _______.

 $9,350.00 
Excel Eqn.(N12): =ROUND(M12/$M$15,5)
A4.7m) MV Wtd 3-Asset Mean Portfolio Return, μ(R3A)
using equation (3.3) written out:
μ(R3A) = (0.57273*0.1192)+(0.29358*0.2383)+(0.13369*0.3058)
= ________.
Excel Eqn.(L36): =(N12*X193)+(N13*AA193)+(N14*AD193)
A4.7n) The 3-asset variance equation (3.7) written out with
the specific notation for these three particular
assets is shown below:
2 (R3A) = (w2AET2AET) + (2wAETwBAAET,BA)
+ (2wAETwCLAET,CL) + (w2BA2BA)
+ (2wBAwCLBA,CL) + (w2CL2CL).
Here:
35
2(R3A) = (.57273)2*(.06182) + (2*(.57273)*(.29358)*(.02149))
<0.020278175>
<0.007226746>
+ (2*(.57273)*(.13369)*(.01657)) + (.29358)2*(.08556)
<0.002537473>
<0.007374349>
+ (2*(.29358)*(.13369)*(.02024)) + (.13369)2*(.04551)
<0.001588788>
<0.000813401>
= 0.039818932.
Excel Eqn.(P36): =(N12^2*N23)+(2*N12*N13*O23)
+(2*N12*N14*P23)+(N13^2*O24)
+(2*N13*N14*P24)+(N14^2*P25)
A4.7o)
(R3A) =
0.039818932 = __________ = _______.
Excel Eqn.(P39): =(P36^0.5)
A4.7p) The average standard deviation (WA-SD) is
essentially calculated just like the market-value,
weighted portfolio return except that the asset
standard deviations replace the asset returns. Prior
to Markowitz, this approach was used to find
portfolio standard deviation. It will always
overstate portfolio SD whenever the assets being
combined are not all perfectly positively
correlated.
WA-SD = (wAET*σAET)+(wBA*σBA)+(wCL*σCL).
36
WA-SD = (0.57273 * 0.2486) + (0.29358 * 0.2925)
+ (0.13369 * 0.2133) = _______.
Excel Eqn.(L39): =(X196*N12)+(AA196*N13)+(AD196*N14)
A4.7q) The correlation coefficients between the three
assets (shown in cells L23:L25) are all positive
and range from 0.2956 up to 0.3244. Thus, we
would expect marked diversification benefits.
Indeed comparison of the Markowitz standard
deviation to the WA S-D shows the Markowitz SD
to be 28.68% lower [(19.955-25.679)/19.955].
This difference is due totally to diversification
benefits.
 σ(R 3A )   0.19955 
A4.7r) CV3-APort = 
=
= ________.


 μ(R 3A )   0.1791 
Excel Eqn.(O41): =P39/L36
CVAET
CVBA
CVCL
CVSPX
= 2.0862;
= 1.2275;
= 0.6976;
= 1.0118.
The CV for Colgate is the only individual asset that
yields a better risk-return trade-off compared to the
portfolio although the SPX (a well-diversified portfolio
containing 500 stocks) does too. Indeed this three-asset
portfolio nearly has a CV that is as low as the SPX.
37
PRINCIPAL INVESTMENT RISKS GLOASSARY2
Active Management Risk: A fund may underperform because of
the portfolio manager’s allocation decisions or individual security
selections.
Asset Allocation Risk: A fund may not maintain its target asset
allocations. There is also the risk that those allocations may not
achieve the desired risk-return characteristic or that the selection of
underlying funds and/or the allocations among them will cause a
fund to underperform similar funds or lose money.
Call Risk: During periods of falling interest rates, an issuer may call
(or repay) a fixed-income security prior to maturity, resulting in a
decline in a fund’s income.
Company Risk: The financial condition of a company may
deteriorate, causing a decline in the value of the securities it issues.
Credit Risk: The issuers of individual securities may default.
Current Income Risk: The income a fund receives may fall as a
result of a decline in interest rates.
Derivatives Risk: A fund’s use of futures and options, and more
complex derivatives such as swaps, may present liquidity, credit and
counterparty problems. The risks associated with derivatives may be
different and greater than the risks associated with directly investing
in the underlying securities and other instruments.
Downgrade Risk: A security may lose value because its rating is
downgraded.
Emerging Markets Risk: The risk of foreign investment may
increase in countries with emerging markets where there is greater
potential for political, currency and economic volatility. Securities
issued in emerging market nations may be less liquid than those
issued in more developed economies.
Enhanced Index Risk: A fund may underperform its benchmark
index due to differences between the fund and the benchmark index.
Interest Rate Risk: Increases in interest rates can cause the price of
fixed-income securities to decline.
Large-Cap Risk: Large companies may grow more slowly than the
overall market.
Market Risk: The price of securities may fall in response to
economic conditions.
Market Volatility, Liquidity and Valuation Risk: Volatile trading
activity may make it difficult for a fund to value its portfolio
securities and the fund may not be able to purchase or sell a security
at an attractive price, if at all.
Mid-Cap Risk: The stocks of mid-capitalization companies may have
greater price volatility, lower trading volume and less liquidity than
the stocks of larger, more established companies.
Mortgage Roll Risk: A fund’s managers may not correctly predict
mortgage prepayments and interest rates, thus reducing fund return.
Non-Investment-Grade Securities Risk: Non-investment-grade
securities, which are usually called “high-yield” or “junk bonds” and
whose issuers are typically in weak financial health and may be harder
to value and sell.
Prepayment Risk: The issuers of individual may prepay them at a
time when interest rates have declined.
Quantitative Analysis Risk: Stocks selected by the fund’s
investment advisor using quantitative modelling and analysis could
perform differently from the market as a whole.
Real Estate Investing Risks: These are the various risks associated
with the ownership of real estate including fluctuations in property
values, higher expenses or lower income than expected, and potential
environmental problems and liabilities.
Extension Risk: The value of individual securities may decline
because principal payments are not made as early as possible.
Small-Cap Risk: Smaller company securities may be more volatile
than those of larger ones. Securities of small-cap companies are often
less liquid than securities of larger companies as a result of there
being a smaller market for their securities.
Foreign Investment Risk: Securities of foreign issuers may lose
value because of erratic market conditions, economic and political
instability, or fluctuations in currency exchange rates. This risk may
be heightened in emerging or developing markets.
Social Criteria Risk: Because a fund’s social criteria exclude
securities of certain issuers for non-financial reasons the fund may
forgo some market opportunities available to funds not using these
criteria.
Illiquid Securities Risk: Illiquid securities may be difficult to sell at
their fair market value.
Special Risks for Inflation-Indexed Bonds: Interest payments on, or
market values of, inflation-indexed bonds may decline because of a
change in inflation (or deflation) expectations.
Income Volatility Risk: The income from a portfolio of securities
may decline in certain interest rate environments.
Index Risk: A fund’s performance may not match that of its
benchmark index.
Industry Concentration Risk: Because a fund’s investments are
concentrated in a single industry, the value of its portfolio may
fluctuate more and be subject to greater risk of loss than those of other funds.
2
Special Situation Risk: Stocks of companies involved in
reorganizations, mergers and other special situations can involve more
risk or display increased volatility than would ordinary securities.
Style Risk: A fund’s investing style may lose favour in the marketplace.
Underlying Fund Risk: A fund may invest in an underlying fund that fails to
achieve its investment objectives.
Source: 2010 Annual Review – TIAA-CREF Funds, September 30, 2010.
38