Portfolio Risk Management Analysis

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 sub-unit. 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 mid-cap 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 mean-variance 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 (Rp)
 =
  wi *ri



E (Rp) 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
 =

W2A *R2A
 +
W2B *R2B
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 non-diversifiable 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 mid-cap 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 non-diversifiable risk. This risk is also referred to as the systematic risk or the market risk. In finance, it’s represented by beta (β).
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
 β  is the sensitive beta coefficient
Therefore, to get the CAPM model we solve the return on an asset (Ri) as shown:
               Ri = Rf- β (Rm – Rf)
In finance, (Rm –Rf) are referred to as the market risk premium. On the other hand,
 β (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
  • 2m = 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 x-axis and expected returns on the Y-axis 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























The individual securities are then plotted on the SML graph and if a security falls above the graph, and then it’s said to be undervalued since an investor expects a higher return for an inherent risk. If it falls below the graph, it’s said to be overvalued since an investor expects less return for the risk assumed. From the graph above most of the individual stocks lie under the SML.
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 shorter-maturity 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 lower-yielding 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 weighted-average 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 mid-cap 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 long-term 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 long-term financial success.

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