Portfolio
analysis studies the performance of different portfolios under different
circumstances (Reilly & Brown 2011).Portfolios can be grouped according to
industries, countries or sector. Each group consists of subunit. For example,
the financial sector can be made up of several banks or the Airline industry
can be made up of several airline companies.
The analysis of each portfolio
helps an investor in making a decision when investing. Most likely, a rational
investor will choose the best portfolio and screen out the ones that are not
essential based on objective criteria. A good portfolio is characterized by
high returns on investment (Reilly & Brown 2011). Portfolio analysis
requires subjective judgment as it is not easy to segment different industries.
Portfolio
analysis is a process as different financial instruments have to be evaluated
one by one. The process is time consuming and involves a lot of effort. In
spite of these odds, Markowitz the fonder of modern portfolio analysis has
simplified the process by suggesting use of expected return and variance
(Brigham & Houston
2009).
In
this report, we will discuss four steps of constructing a portfolio.
Q 1.a Construction of a portfolio
The
portfolio we are to construct consists of IBM and Shell Gas shares using weekly
data from 2007 to 2012. A
good portfolio is characterized by high returns and moderate risk. It is also
well diversified just like in our case where we have IBM and Shell Gas shares.
The first step in constructing a portfolio is an assessment of your
expectations and attitude towards risk. Basically, there are two types of
investors, the aggressive investor and the conservative investor.
An aggressive
investor is willing to take more risks by devoting larger portions to equity
and less to bond and other fixed income securities. On the contrary, the
conservative investor takes less risk as his main goal is to protect the value.
On the other hand, an aggressive investor aims at maximizing returns by
accepting more risk. A moderately good portfolio is one which satisfies the
tolerance of average risks, attracts all those people who are willing to take
in more risks in their portfolios in order to help them in the achievement of a
balance of capital growth and income.
Therefore, as an investor you should be
in a position of knowing the category where you suit. The second step is
choosing the portfolio. In our case, we consider the individual shares that
have high returns and have the outperformed the FTSE 250 midcap index (FTMC).
The portfolio will have 21 shares. Below is the portfolio f our choice:
After
choosing the portfolio, the next step is to identify, risk and return. In
identifying risk and return, we use meanvariance analysis which was derived by
Markowitz in 1952. Markowitz (2000) suggests that a portfolio with the lowest
level of risk is to be chosen for a particular rate of return. The expected
return on a portfolio is determined as the sum total of individual weights in
the portfolio. This is shown below:
E (R_{p})

=

∑ w_{i} *r_{i}

E (R_{p}) stands
for the expected return. Since weights should be used in calculating the
expected return, we are to ascertain them.
Standard deviation is
denoted by ∂^{2 }
∂^{2}

=

W^{2}_{A} *R^{2}_{A}

+

W^{2}_{B} *R^{2}_{B}

The
weights have been determined using the minimum variance method.
MINIMUN VARIANCE PORTFOLIO EXACT


X1 (WEIGHT OF R1)

0.35

X2(WEIGHT OF R2)

0.65

The
standard deviation for the portfolio is 0.001725 and the expected return is
0.1767. As you can see, the relationship
between the returns and risk is a direct one. The higher the returns of
portfolio are, the higher the risk is .
(X1)

SD(Rp)

E(Rp)

(X2)

0

0.002400

0.0004775

1

0.05

0.002216

0.0006617

0.95

0.1

0.002061

0.0008460

0.9

0.15

0.001935

0.0010302

0.85

0.2

0.001839

0.0012144

0.8

0.25

0.001772

0.0013987

0.75

0.3

0.001734

0.0015829

0.7

0.35

0.001725

0.0017671

0.65

0.4

0.001745

0.0019513

0.6

0.45

0.001795

0.0021356

0.55

0.5

0.001874

0.0023198

0.5

0.55

0.001982

0.0025040

0.45

0.6

0.002119

0.0026882

0.4

0.65

0.002286

0.0028725

0.35

0.7

0.002482

0.0030567

0.3

0.75

0.002707

0.0032409

0.25

0.8

0.002961

0.0034252

0.2

0.85

0.003244

0.0036094

0.15

0.9

0.003557

0.0037936

0.1

0.95

0.003899

0.0039778

0.05

1

0.004270

0.0041621

0

MIN VAR

0.001725

Q 1.b Performance evaluation
The expected return of
the portfolio (0.0017671) is higher than the expected return of individual
stocks over the latter period for shell Gas the expected return is 0.0014162
and IBM 0.000478. It is rational to invest in a portfolio than in individual
shares. The risk of a portfolio can be diversified and that is the reason why
the portfolio returns are greater than those of individual assets. The risk of
individual shares is nondiversifiable and as a result lower returns.
Parameters of the single index model
The
single index model relates returns on
each security to the returns on a common index, such as the FTSE 250 midcap index
(FTMC) or S&P 500 Stock Index. Single index model helps split a security’s total
risk into the market risk and unique risk. According to this model majority of
stocks have a positive covariance since they respond to similar macroeconomic factors?
Some firms are more sensitive to the macroeconomic factors than others and the
variance of these firms is denoted by beta. The covariance among securities is
as a result of the differing responses to these factors.
In
this section we will use CAPM focuses on the sensitive risk which is the
nondiversifiable risk. This risk is also referred to as the systematic risk or
the market risk. In finance, it’s represented by beta (Î˛).
Where:
Ri
is the expected return on an individual security
Rf is the expected return
on a risk free security
Rm
is the expected return on a market security and
Î˛ is the sensitive beta coefficient
Therefore, to get the CAPM model we
solve the return on an asset (Ri) as shown:
Ri = Rf Î˛ (Rm – Rf)
In finance, (Rm –Rf) are referred to as the market risk premium. On the other
hand,
Î˛ (Rm –
Rf) is the individual risk premium.
In premium terms therefore, the
market risk premium is equal to individual risk premium.
The beta
for a stock is calculated as follows:
where
 _{im} = the Covariance between the returns on the asset and the market portfolio and
 ^{2}_{m} = the Variance of the market portfolio.
Based on the table
above, the expected returns calculated using CAPM for shell gas is higher than
the return. This is because its beta is high at 1.2. On the other hand, the
expected returns of IBM shares are lower than the returns as it beta is 0.22.
You find that when expected returns are high, the risks are also high and vice
versa. Beta measures the degree of risk in the market.
To show the results of CAPM, we will use the security market line
(SML). SML is a single pricing model showing how the market
should price individual assets in relation to their security risk class. The graph shows risk on
the xaxis and expected returns on the Yaxis as shown below:
X1

SD(Rp)

E(Rp)

X2

0

0.00246

0.00077

1

0.05

0.00225

0.00067

0.95

0.1

0.00207

0.00056

0.9

0.15

0.00193

0.00045

0.85

0.2

0.00183

0.00034

0.8

0.25

0.00176

0.00024

0.75

0.3

0.00173

0.00013

0.7

0.35

0.00173

0.00002

0.65

0.4

0.00176

0.00009

0.6

0.45

0.00183

0.00019

0.55

0.5

0.00194

0.00030

0.5

0.55

0.00208

0.00041

0.45

0.6

0.00225

0.00052

0.4

0.65

0.00246

0.00063

0.35

0.7

0.00271

0.00073

0.3

0.75

0.00299

0.00084

0.25

0.8

0.00330

0.00095

0.2

0.85

0.00366

0.00106

0.15

0.9

0.00404

0.00116

0.1

0.95

0.00446

0.00127

0.05

1

0.00492

0.00138

0

MIN VAR

0.00173

Q
2.Duration
Duration measures the sensitivity of the price of a
fixed income investment to changes in interest rates (Reilly & Brown 2011).
It is usually expressed in years. The bigger the duration, the higher degree of
exposure of bonds to the interest rate risk. The relationship between bond
prices and interest rates is an inverse one and some bonds have greater
sensitivity to changes in interest rates. As the interest rates rise, the bond
prices begin to fall, and when interest prices begin to fall, bond prices begin
to rise. Therefore, with the concept of duration, one can determine the
interest rate risk (Reilly & Brown 2011). In summary, it measures how quickly a bond will recover
its true cost. Bond duration is affected by factors like time for coupon and
maturity rates.
·
Time to maturity: Two bonds will be considered in this case with each costing
$2,000 and yield 10%. A bond
maturing in a period of one year is more able to recover its true cots than a
bond which matures in a period of 10 years. As a result, the shortermaturity
bond would have a lesser the duration and price risk. On the other hand, the
longer the maturity, the higher the duration (Kevin 2006).
·
Coupon rate: the key factor in calculating bonds duration is the payment.
For instance, consider two identical bonds paying different coupons, you will
find that the bond which has higher coupon rate will have a faster pay of its
original cost than the loweryielding bond. Therefore, higher the coupon, the lesser
the duration (Kevin 2006).
Using Duration to Your Advantage
There are two advantages of Knowing
the duration of a bond, or a portfolio of bonds:
·
Speculation
of interest rates:
investors can use the bond duration for speculation purposes. For instance, if
they anticipate a decline in market interest rates they will try to increase
the average duration of their bond portfolio. Likewise, investors who expect
the interest rates to rise they will lower their average duration (Kevin 2006).
·
Matching risk to personal tastes: duration allows you to quickly
determine bonds which are most sensitive to changes in market interest rates, ,
when selecting from bonds of different
maturities and yields.
Calculating Duration
There
are different types of formulas for calculating bond duration. In our case, we
will apply the Macaulay duration (Reilly & Brown 2011). The formula was
created by Fredrick Macaulay in the year 1938. Macaulay duration is the present value (PV) of the
weightedaverage cash flows of a bond .Here; the duration is calculated by determining the bond's value
yearly cash flow, adjusted to give greater value to payments. Then this is divided
by the bond price to calculate its duration. The formula is shown below:
n = number of cash flows
t = time to maturity C = cash flow i = required yield M = maturity (par) value P = bond price 
The bond price is given by:
Therefore, the following is an expanded version of Macaulay duration (Kevin
2006):
In
our case, we have been instructed to determine the duration of a bond considering
a bond with annual coupon payment of $150, a principle payment of $1200 in 10
years and a cost of 1000. Assume a flat yield curve with a 7% to maturity.
Price

=

150*7.028

+

1,000.00

1.97


=

843.36

+

508.36


=

1,352.00

=

150*10

+

1000*10


(1.07^{)10}

(1.07^{)10}


=

1500

+

10000


0.508

1.9671


=

2953+5085


=

8037


Duration=

8037

=

5.96


1352


Approximately 6 years

If
the yield curve remains unchanged, what is the bond’s duration in four years?
In six years? In nine years?
In
4 years, the duration for the bond will be as shown below:
Price

=

150*3.388

+

1,000.00

1.31


=

508.28

+

763.35


=

1,272

=

150*4

+

1000*4


(1.07^{)4}

(1.07^{)4}


=

600

+

4000


0.7628

1.31


=

787+3537


=

4324


Duration=

4324

=

3.399


1272


Approximately 4 years

In
6 years, the duration for the bond will be as shown below:
Price

=

150*4.771

+

1,000.00

1.501


=

716

+

666


=

1,382

=

150*6

+

1000*6


(1.07^{)6}

(1.07^{)6}


=

900

+

6000


0.666

1.501


=

1352+3997


=

5349


Duration=

5349

=

3.87 Years


1382


In
9 years, the duration for the bond will be as shown below:
Price

=

150*6.52

+

1,000.00

1.84


=

978

+

543


=

1522

=

150*9

+

1000*9


(1.07^{)6}

(1.07^{)6}


=

1350

+

9000


0.5439

1.84


=

2482+4891


=

7373


Duration=

7373

=

4.84 Years


1522


Approximately 5 years

Bond convexity
It
measures the sensitivity of bond duration to changes in yield (Kevin 2006).
Convexity and duration provide an insight to investors with regard to bond
performance should the interest rates change. Therefore, they assist investors
in understanding the risk involved on fixed securities in different interest
rate environments. Bond duration is regarded as an imperfect measure of bonds
price change because of the change is linear in nature when it exhibits a
sloped shape.
When
the duration of the bond begins to rise and its yield begins to decline
declines, the bond is said to have positive (Kevin 2006). When the bond has positive convexity, prices
tend to have large increases due to the decrease in yields and not increase in
yield (Kevin 2006). Therefore, Positive convexity favors investors since the
price becomes less sensitive when yields rise than when yields decline (Kevin 2006).
Negative convexity, indicates that duration rises as yields go up. Negative convexity always works against an
investor’s interest.
For
investors it is wise to consider bonds with shorter durations, if rates are expected in increase
(Kevin 2006). This is because bond with shorter durations are less sensitive to
an increase in yields and will fall in price less often than bonds with higher
durations. On the other hand, if rates are expected to decline the investor should consider
bonds with higher durations. As yields decline and bond prices go up, higher
duration bonds stand to gain more than shorter duration bonds (Kevin 2006).
Conclusion
Our portfolio outperformed the FTSE 250 midcap index (FTMC). Therefore, higher returns
can be anticipated. The standard deviation of the portfolio is also less hence
the risk to be involved is minimal. Generally, a well diversified
portfolio best suits the longterm growth of your investments (Reilly
& Brown 2011).
It protects assets from risks that arise as a result of market fluctuations.
Our portfolio consisted of 12 stock but a well diversified portfolio should
have at least 30.It is recommendable for any investor to monitor the diversification
of his portfolio and make adjustments when necessary to increase chances of
longterm financial success.
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