

Financial
Management MGT201
VU
Lesson
22
PORTFOLIO
RISK ANALYSIS AND EFFICIENT
PORTFOLIO MAPS
Learning
Objectives:
After
going through this lecture,
you would be able to have an
understanding of the following
topics
·
Portfolio
Risk Analysis & Efficient
Portfolio Maps
Before
starting the new concepts we
should recap what we have
studied in the previous
lecture.
Recap:
Portfolio
is a Collection of Investments in different
Stocks, Bonds, other Securities or a mix
of
all.
Its objective is to invest in
Different UnCorrelated Stocks in
order to minimize overall
Risk &
Maximize
Portfolio Return. It is mentioned
that individuals and
companies maintain the portfolio
in
order
to reduce to reduce the
risk
There
are 2 Types of Stock
Risk
Total
Stock Risk = Diversifiable + Market
Risk
Diversification
means expanding the number of investments
which cover different kinds
of
stocks.
We can reduce the risk as random
events in one industry can
be off set by the random effects
in
the
other industry. This way
you can reduce the company
pacific or unique risk. The
market risk arises
because
of micro economic or large scale factors
such as market interest rate, inflation
etc. These factors
have
virtually identical effect on the
share prices. For example, in
event of a war stick market
go down
in
value which means almost
every share went
down.7
Stocks
are a good number for
diversification. 40
Stocks
are enough for Minimizing
Total Risk
Calculating
Expected 2Stock Portfolio Return &
Risk
Expected
Portfolio Return = rP * = xA rA + xB
rB
Portfolio
Risk is generally not a
simple weighted average.
Up
to this point we only look
at the portfolio which has
only two stocks.
Interpreting
2Stock Portfolio Risk
Formula:
=
√
XA2
σ
A 2
+XB2
σ
B 2
+ 2
(XA
XB
σ A
σ
B
AB)
is
coefficient of correlation which
states that how muck the
investments are correlated.
Here,
The
risk of investing in any one
share can be reduced if we
invest in other shares also.
There have been
several
experiment studies that show
that if you invest in
approximately 40 different
uncorrelated
different
shares of different companies
then you can entirely
eliminate the company pacific portion
of
the
risk. Even if you can
not diversify across 40
different companies but if
you diversify just across
7
different
shares from different
companies then you can
still you can reduce
most of the diversifiable
risk.
No
matter what we do we can not
eliminate the market risk
that market risk become the
minimum risk
we
have to live with in our
portfolio. The important
thing then to remember is
that how this risk
will
effected
when we talk about portfolio
of two stocks or more. The
Correlation coefficient needs to
be
understood
in order to understand the risk and
return.
Correlation
Coefficient (
AB
or "Ro"):
Risk
of a Portfolio of only 2 Stocks A & B
depends on the Correlation between those
2 stocks.
The
value of Ro is between 1.0 and
+1.0
If
Ro = 0 then Investments are Uncorrelated
& Risk Formula simplifies to
Weighted Average
Formula.
If
Ro = + 1.0 then Investments are
Perfectly Positively Correlated and
this means that
Diversification
does not reduce
Risk.
If
Ro
=  1.0, it
means that Investments are
Perfectly Negatively Correlated and the
Returns (or Prices
or
Values) of the 2 Investments move in
Exactly Opposite directions. In this
Ideal Case, All Risk can
be
diversified
away. For example, if the
price of one stock increases by
50% then the price of another
stock
goes
down by 50%.
In
Reality, Overall Ro for most
Stock Markets is about Ro = +
0.6.it
is very rough rule of thumb.
It
means
that correlations are not
completely perfect and you
should remember that if the
correlation
coefficient
is +1.0 then it is not possible to
reduce the diversifible
risk.
This
means that increasing the number of Investments in the
Portfolio can reduce some
amount of risk
but
not all risk
94
Financial
Management MGT201
VU
Portfolio
Risk  Example
Recap
Complete
2Stock Investment Portfolio
Data:
Value
(Rs) Exp Return (%)
Risk (Std Dev)
Stock
A
30
20
20%
Stock
B
70
10
5%
Total
Value = 100
Correlation
Coeff Ro = + 0.6
2Stock
Portfolio Risk
Calculation:
=
√
XA2
σ
A 2
+XB2
σ
B 2
+ 2
(XA
XB
σ A
σ
B
AB)
=
{0.0036 + 0.001225 + 0.00252}
0.5
=
0.0857= 8.57%
·
2Stock Portfolio Return
Calculation:
rP* = x
A
r
A
+ x
B
r
B
= 6 + 7 =
13%
Interpretation
of Result:
The
Portfolio Risk for our
Basket of 2 Investments is
+8.57
% (if
Ro = + 0.6). What does
this
mean?
Bell
Curve Assumption: If we
assume a Normal Probability
Distribution, then there is a
68.26%
chance
that our future Portfolio
Return will be somewhere between
(rP* σ
)
and (rP*+ σ
)
i.e.
between
(13% 8.57%) and (13%
+8.57%) or between +4.43% and
+21.57%
Portfolio
Risk lies between the Individual Risks of
the 2 Investments i.e
σ
Stock
B < σ
P
<
σ
Stock
A or 5% < 8.57% < 20% (if
Ro = +0.6)
You
can also come up with more
accurate outcome about the actual value
of the return on the portfolio
after
1 year if you take a larger
range for the standard
deviation. So, if you are
taking about the range
from
2 sigma to +2 sigma towards then there
is likelihood that actual rate of return
of the portfolio is
somewhere
in between the two standard
deviation.
Note:
If Ro = 
0.6 (Negative Correlation)
then Portfolio Risk = + 4.8%
which is lower than
both
Individual
Investments!!
Now,
we consider the case of negatively
correlated investments.
Negatively
Correlated Investments
2Stock
Investment Portfolio
Data:
Exp
Indiv Return (ri)
Indiv Risk (Std
Devi
)
Stock
A 20%
20%
Stock
B
10%
5%
Correlation
Coeff Ro =  0.6
Portfolio
Risk & Return Table (for
Different Portfolio
Mixes):
Fraction
of Stock A
Portfolio
Risk Exp Portfolio Return
(rP*)
100%
20%
20%
80%
15%
18% = 0.8(20) +
0.2(10)
50%
9%
15%
= 0.5(20) + 0.5(10)
30%
4.8%
13%
15%
3.4%
11.5%
0%(i.e.
100% Stock B) 5%
10%
Efficient
Portfolio Map
95
Financial
Management MGT201
VU
Efficient
Portfolio Map
Shows
All Combinations of 2Stock
Portfolio
Negative
Correlation (Ro =
0.6)
rP*
Point
of Minimum
Portfolio
20%
Risk
Stock
A
Return
(100%
A &
15%
80%A
50%A
0%
B)
13%
30%A
11.5%
15%A
10%
Stock
B
(0%
A &
100%
B)
3.4%
5%
20%
9%
15%
P
Portfolio
Risk
4.8%
Efficient
Portfolio Interpretation
Efficient
Portfolio Map for 2Stock
Portfolio shows all possible
Efficient Combinations
(Mixes)
of
stocks.
Efficient
Portfolios:
Efficient
Portfolios are those whose
Risk & Return values match the
ones computed using
Theoretical
Probability Formulas. The Incremental
Risk Contribution of a New
Stock to a Fully
Diversified
Portfolio of 40 UnCorrelated Stocks
will be the Market Risk
Component of the New
Stock
only.
The Diversifiable Risk of the
New Stock would be entirely
offset by random movements in the
other
40 stocks. Adding a New
Stock to the existing Portfolio
will create more Efficient
Portfolio
Curves.
The New Stock will
contribute its own
Incremental Risk and Return to the
Portfolio.
rP
* = xA rA + xB rB +
xC
rC (3
Stocks)
Efficient
Portfolio Maps
3Stock
Portfolio
Negative
Correlation
rP*
Efficient
Frontier for
Portfolio
30%
3Stock
Portfolio
Stock
C
Return
20%
Stock
A
10%
Stock
B
Old
Efficient Frontier
for
2Stock
Portfolio of A & B
40%
2.5%
5%
20%
P
Portfolio
Risk
3.4%
96
Financial
Management MGT201
VU
Now,
if we add another stock in the portfolio we can
take a look
3Stock
Portfolio Risk
Formula
3x3
Matrix Approach
Stock
A
Stock
B
Stock
C
XA2
2
Stock
XA
XB
XA
XC
A
A
B
AB
A
C
AC
A
XB2
2
Stock
XB
XA
XB
XC
B
A
BA
B
B
C
BC
B
XC2
2
Stock
XC
XA
XC
XB
C
A
CA
C
B
CB
C
C
To
compute the Portfolio
Variance for a
3Stock Portfolio, just add up
all the terms in
every
box.
To compute the Portfolio Risk (Standard
Deviation), simply take the
Square Root of the
Variance.
You
can extend this Matrix
Approach to calculate the Risk
for a Portfolio consisting of
any
number
of stocks.
Terms
in Boxes on Diagonal (Top Left to
Bottom Right) are called
"VARIANCE"
terms
associated
with
individual magnitude of risk
for each stock.
Terms
in all other (or
NONDIAGONAL) Boxes are called
"COVARIANCE"
terms
which account for
affect
of one stock's movement on another stock's
movement.
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