Investment management
Kotebe University of Education Faculty of Business
and Economics Department of Accounting and
Finance Individual Assignment
Name: Kaleab Tadesse
ID: PGE/25665/14
Submitted to: Daniel Tolesa
February, 2022
1. Stock X and Y have the following probability distributions for expected future returns:
Probability Return rate of x Return rate of y
0.1 10% 20%
0.2 2% 12%
0.4 12% 5%
0.2 20% 0%
0.1 38% (4%)
a) Calculate the expected rate of return for stock X and Y
𝐸𝑅𝑥 = ∑ 𝑝𝑥𝑟𝑥, .1*10%+.2*2%+.4*12%+.2*20%+.1*38
𝐸𝑅𝑥 = 12%
b) Calculate the standard deviation for stock X and Y
𝐸𝑅𝑦 = ∑ 𝑝𝑦𝑟𝑦, 0.1*20+.2*12+.4*5+0+.1*-4 = 6
Coefficient of variance of sock x
Rx ERx (r-ERx) (r-ERx)2 Pi (r-ERx)2*pi
- 12 (22) 484 0.1 48.4
10%
2% 12 (10) 100 0.2 20
12% 12 0 0 0.4 O
20% 12 8 64 0.2 12.8
38% 12 26 676 0.1 67.6
Variance stock x=148.8
𝑆𝐷(𝛿)𝑥 = √𝑉𝐴𝑅𝐼𝐴𝑁𝐶𝐸, √148.8 = 12.2=
Variance of stocky
ry ERy ry- (ry- Pi (ry-
ERy ERy)2 ERy)2pi
20% 6 14 196 0.1 19.60
12% 6 6 36 0.2 7.2
5% 6 -1 1 0.4 0.4
0% 6 -6 36 0.2 7.20
(4%) 6 -10 100 0.1 10
Variance 44.4
Standard deviation of stock y
ẟy=√𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒, √44.4=6.66
c) Calculate the coefficient of variation of stock X and Y
𝜎 12.2
𝑐𝑜𝑣 𝑋 = 𝐸𝑅𝑥 , =1.02
12
𝜎 6.66
𝑐𝑜𝑣 𝑦 = 𝐸𝑅𝑦 , =1.11
6
d) Which stock is more risky based on the result in requirement b and c.
✓ To measure the variety of return required to persuade investors to postpone
present expenditure, standard deviation is utilized. The higher the standard
deviation, the more unpredictable the returns, and the riskier the
investment,.
✓ Stock x is the risky asset because the coefficient of variation measures
relative measurement risk per unit of variation.
e) If an investor decides to hold a portfolio of the two stocks ( i.e 2/3 of X and 1/3 of Y)
and the correlation coefficient between the return of stock A and B is -0.50
i. Calculate the weighted average rate of return for the portfolio
rp=∑ 𝐸𝑅𝑖𝑤𝑖 , 12*0.6667+6*0.3333=8+2=10
ii. Calculate the portfolio standard deviation
𝜎𝑝 = √𝜎𝑥2𝑤𝑥2 + 𝜎𝑦2𝑤𝑦2 + 2𝑤𝑥𝑤𝑦. 𝑐𝑜𝑟𝑟𝑥𝑦𝜎𝑥𝜎𝑦
(12.2)2(2/3)2+(6.66)2(1/3)2+2(2/3*1/3*-0.5*12.2*6.66)
𝜎𝑝 = √71.14 + (−18.04)=53.09
iii. Did the diversification reduced risk?
Covariance and correlation are related and they generally measure the same
phenomena’s. From the given let me explain this question based on correlation coefficient
give above.
The degree of correlation between a portfolio of assets, or correlation coefficient, only
ranges from -1 to +1. It is ideal to combine assets that have low or negative
correlations.
2. As a result, the risk is reduced through diversification as the correlation coefficient is -
0.5. Assume you are planning to invest Br, 800,000. Two securities are available, X &
Y, and you can invest in either of them or in a portfolio with some of each. You
estimate that the following probability distributions of returns are applicable for
investment in X &Y:
Probability X Y
0.1 (20%) (60%)
0.2 10% 0%
0.4 24% 40%
0.2 52% 80%
0.1 80% 140%
Required:
a) Calculate the coefficient of variation for the two securities and determine which of
the two is more risky
ERx=∑ 𝑟𝑖𝑝𝑖, ERx= (20)*0.1+10*0.2+24*0.4+52*0.2+80*0.1=28%
𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = ∑𝑛𝑖=1(𝑟𝑖 − 𝐸𝑅𝑖)2 ∗ 𝑝𝑖
𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒𝑥 =(-20-28)2*0.1+(10-28)2*0.2+(24-28)2*0.4+(52-28)2*.2+(80-
28)2*0.2=422
𝜎𝑥(𝑆𝐷) = √𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = √422 = 𝟐𝟎. 𝟓𝟒
CV=σx/ERx=20.54/28=0.73
ERy=∑ 𝑟𝑖𝑝𝑖, ERy= (60)*0.1+0+40*0.4+80*0.2+140*0.1=40%
𝑣𝑎𝑟𝑖𝑎𝑎𝑛𝑐𝑒𝑦 = (−60 − 40)2*0.1+(0-40)2*.2+0+(80-40)2*0.2+(140-40)2*.1=2640
𝜎𝑦(𝑆𝐷) = √𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = √2640 = 𝟓𝟏. 𝟑𝟖
CV=σy/ERy=51.38/40=1.28
𝒃𝒂𝒔𝒆𝒅 𝒐𝒏 𝒕𝒉𝒆 𝒄𝒐𝒎𝒑𝒖𝒕𝒂𝒕𝒊𝒐𝒏; Security y is more risky b/c its variance is higher
than security x
b) Calculate the portfolio return and standard deviation if equal amount of the two
securities are combined assuming the correlation between the returns of the two
securities is -0.50
𝑟𝑝 = 𝑤1𝐸𝑅1 + 𝑤2𝐸𝑅2 … 𝑤𝑛𝐸𝑅𝑛
rp=0.5*28+0.5*40=34%
𝜎𝑝 = √𝜎𝑥2𝑤𝑥2 + 𝜎𝑦2𝑤𝑦2 + 2𝑤𝑥𝑤𝑦. 𝑐𝑜𝑟𝑟𝑥𝑦𝜎𝑥𝜎𝑦
= (20.54)2*(.5)2+(51.38)2*(0.5)2+2(0.5*0.5*-.50*20.54*51.38)= √501.6 = 22.4
c) In what proportion should the two securities combined to have the least risky
portfolio?
𝜎𝑦(𝜎𝑦−𝜌𝑥𝑦𝜎𝑦)
𝑊𝑥 = 𝜎𝑥2+𝜎𝑥2−2𝜌𝑥𝑦𝜎𝑥𝜎𝑦
=51.38(51.38-(-0.5)*51.38)/(20.54)2+(51.38)2-2(-0.5*20.54*51.38)
Wx=3959.85/4117.14=0.96 6%
Wy=0.04=4%
3. You are given the following historical data for stock A and market portfolio
Year Required rate of return for Market rate of return
stock A (%) (%)
2003 2 -4
2004 6 10
2005 7 12
2006 10 20
Assuming that the risk free rate is 5%,
Compute
a) The beta coefficient for stock A and interpret the result
M K MK M2
-9 -3 27 81
5 1 5 25
7 2 14 49
15 5 75 225
Total 18 5 121 380
Mean 4.5 1.25 5.625
𝛽 = ∑(𝑀𝐾 − 𝑛𝑀𝐾)/(∑ 𝑀2 − 𝑛𝑀2) =
121−4∗4.5∗1.25
= =98.5/299=0.33
380−4∗(4.5)2
𝒊𝒏𝒕𝒆𝒓𝒑𝒓𝒆𝒕𝒂𝒕𝒊𝒐𝒏
The asset is less riskier than market index, because its β<1
b) The required rate of return to be used for capital budgeting in 2007 when the market
rate of return is expected to be
i) 15%, r=5%+0.33(15%-5%) =8.3%
ii) 10%, r=5%+.33(10%-5%) =6.65%
iii) 30%, r=5%+0.33(30%-5%) =13.25%
7. Assume the following data for four investment managers
Portfolio Risk free Return Standard Beta
rate deviation
A 5% 15% 0.2 0.9
B 5% 13% 0.15 1.5
C 5% 20% 0.3 2.0
D 5% 10% 0.1 0.5
Market 5% 14% 0.2 1.0
Evaluate the performance of the four portfolio managers using
a. Treynor composite performance measure
𝑅𝑖−𝑅𝑓
𝑇= ,
𝛽𝑖
14−5
𝑇𝑚 = 1
= 9%, = 0.09
.15−0.05
𝑇𝐴 = = 0.11,
0.9
TB= (0.13-0.05)/1.5=0.0533,
0.2−0.05
𝑇𝑐 = = 0.075
2
0.1−0.05
TD= = 0.1
.5
Explanation
Portfolio A is better that all other because Treynor measure is more than other
three ie 0.11. Base on their performance ranked from higher to lower as; TA, TD,
TC, TB. The lowest or worst is portfolio TB.
b. Sharp composite performance measure
𝑅𝑖−𝑅𝑓
𝑠= 𝜎𝑖
𝑆𝑚 = (𝑟𝑚 − 𝑟𝑓)/𝜎𝑚 , (0.14-.05)/.2=0.45
.15−.05
𝑆𝐴 = = 0.5,
.2
0.13−0.05
𝑆𝐵 = = 0.533
.15
Sc=(.2-0.05)/0.3=0.5,
.1−0.05
SD= =0.5
.1
Explanation
Portfolio SA has a superior performance than the other three when calculated or
measured using sharp measures. The performance of the other three is 0.5.Jensen
performance measure
𝛼 = (𝑅𝑖 − 𝑅𝑓) − 𝛽(𝑅𝑖 − 𝑅𝑓),
𝛼𝐴 = (. 15 − .05) − 0.9(. 14 − .05) = 0.019
𝛼𝐵 = (0.13 − 0.5) − 1.5(. 14 − .05) = −0.055
𝛼𝐶 = (. 2 − .05) − 2(. 14 − .05) = −0.03
𝛼𝐷 = (. 1 − .05) − .5(. 14 − .05) = 0.005
Explanation
If a portfolio manager performs better, the portfolio's alpha will be positive,
according to Jonsen performance measures.
• Managers A and D outperform expectations by 1.9% and 0.5%,
respectively, showing that they are superior.
• Managers C and B produced returns that were 3% and 5.5% lower than
what the market had anticipated.
• Therefore, portfolio A&B is worst than other.
