Prepared by Students of University of Rajshahi Pranto Karmoker Ariful Islam Tonmoy Halder Monir Hossain 1711033122 1710733119 1710833120 1711033205 Ashikur Rahman Mahfuzul Haque Jibon Rahman Sohag Miah 1710133113 1710933297 1711033210 1710933202 Siam Hossain Shammira Parvin Farhana Afrose Anjuman Ara 1710333148 1712033136 1712033209 1712433159 Shakil Hossain 1710833138 presented by Group 2 For downloading this contact- bikashkumar.bk100@gmail.com

1. To The World of
Investment

2. • Standard Deviation of a Portfolio Investment
• Three Assets Portfolio
• Estimation Issues
• The Effective Frontier
• The Effective Frontier and Investor Utility
Objectives

3. STANDARD DEVIATION OF A PORTFOLIO
 Portfolio standard deviation is the standard deviation of the rate of return
on an investment portfolio and is used to measures the inherent volatility
of an investment.
 It measures the investment’s risk and helps in analyzing the stability of
returns of a portfolio

5. Example::
Weight Expected return SD.
Stocks A .5 .20 .10
Stocks B .5 .20 .10
(0.5)2 (0.10)2 + (0.5)2 (0.10)2 + 2 (0.5) (0.5) (0.01)
(0.25) (0.01) + (0.25) (0.01) + 2 (0.25) (0.01)
(0.0025) + (0.0025) + (0.005)
0.01
0.10
=
=
=
=
=

6. When the co-relation is :: 0.5
(0.5)2
(0.10)2
+ (0.5)2
(0.10)2
+ 2 (0.5) (0.5) (0.005)
(0.0025) + (0.0025) + 2 (0.25) (0.005)
0.0075
0.0866
=
=
=
=

7. A Three Asset Portfolio Model
A demonstration of what occurs with a three assets portfolio is useful
because it shows the dynamic of the portfolio process when assets are
added. It shows the rapid growth in the computation required, which is why
we generally stop at there.
Three is a better option….
The assets are Stocks, Bonds,
Cash or any Cash equivalent………..

8. Assets classes E (𝑹𝒊) E (ɕ𝒊) 𝑾𝒊
Stocks (S) 0.12 0.20 0.60
Bonds (B) 0.08 0.10 0.30
Cash Equivalent(C) 0.04 0.03 0.10
Calculation
Here the co relation are : 𝑟𝑆,𝐵 = 0.25; 𝑟𝑆,𝐶 = -0.08; 𝑟𝐵,𝐶 = 0.15
The weights specified, the E(𝑅 𝑝𝑜𝑟𝑡) is
E(𝑅 𝑝𝑜𝑟𝑡) = (0.60) (0.12) + (0.30) (0.08) + (0.10) (0.04)
= 0.072 + 0.024 + 0.004
= .100
= 10.00%
E (𝑹𝒊) = Expected Risk
E (ɕ𝒊) = Expected Standard Deviation
𝑾𝒊 = Weight of the Investment

9. When we apply the generalized formula, the expected standard deviation of a three
asset portfolio, it is
ɕ2 𝒑𝒐𝒓𝒕 = (𝑤𝑆
2
ɕ 𝑆
2
) + (𝑤 𝐵
2
ɕ 𝐵
2
) + (𝑤 𝐶
2
ɕ 𝐶
2
) + (2 𝑤𝑆 𝑤 𝐵ɕ 𝑆ɕ 𝑩ɕ 𝑺.𝑩) + (2 𝑤𝑆 𝑤 𝐶ɕ 𝐶ɕ 𝑪ɕ 𝑺.𝑪) +
(2 𝑤 𝐵 𝑤 𝐶ɕ 𝐵ɕ 𝑪ɕ 𝑩.𝑪)
ɕ 𝒑𝒐𝒓𝒕
𝟐
= [(0.6)2
(0.20)2
+ (0.3)2
(0.10)2
+ (0.1)2
(0.03)2
+
{[2(0.6)(0.3)(0.20)(0.10)(0.25)] + 2(0.6)(0.1)(0.20)(0.3)(-0.08)] +
2(0.3)(0.1)(0.10)(0.3)(0.15)]
= [0.015309 + (0.0018) + (-0.0000576) + (0.000027)]
=(0.0170784)
1/2
= 0.1306, = 13.06%

10. ESTIMATION ISSUES
To allocate the portfolio asset we have to consider some factors, which
will effect our portfolio asset allocation, this portfolio asset allocation
depends on the accuracy of the statistical inputs.
This means that for every asset being considered for inclusion in the
portfolio, we must estimate its expected return and standard deviation.
So this factors which we have to consider before portfolio asset
allocation are called estimation issues.

11. Steps to be followed ::
 To consider the accuracy of the statistical inputs
 To calculate the correlation estimates
 To calculate the estimation risk
 To reduce the estimation risk

12. SINGLE INDEX MARKET MODEL ::
𝑅𝑖 = 𝑎𝑖 + 𝑏𝑖 𝑅 𝑚 + 𝑒𝑖
Here,
𝑅𝑖 = that part of security i’s return, which is independent of market
performance
𝑏𝑖 = the slope coefficient that relates the returns for security i to the
returns for the aggregate stock market
𝑅 𝑚 = the retunes for the aggregate stock market
𝑒𝑖 = unsystematic risk of the security of the i.

13. DOUBLE INDEX MARKET MODEL ::
rij = 𝑏𝑖 𝑏𝑗
ɕ 𝑚
2
ɕ 𝑖ɕ 𝑗
Here , rij = the correlation coefficient of returns
𝑏𝑖 = the slope coefficient that relates the returns for security i, to the
returns for the aggregate stocks market
𝑏𝑗 = the slope coefficient that relates the return for security j, to the returns
for the aggregate stock market
ɕ 𝑚
2 = the variance of returns for the aggregate stock market
ɕ𝑖 = the standard deviation of Rij
ɕ𝑗 = the standard deviation of Rji

14. THE EFFICIENT FRONTIER
The Efficient frontier is the set of optimal portfolios that offer the
highest expected return for a define level of risk or the lowest risk for a
given level of expected return. Portfolios that lie below the efficient
frontier are sub-optimal because they do not provide enough return for
the level of risk.

15. EFFICIENT FRONTIER FOR ALTERNATIVE
PORTFOLIOS

16. The Efficient Frontier and investment Utility
Objectives::
Use utility to determine the optimal risk portfolio
The efficient frontier and the utility curve recover the highest possible
utility.

18. Any
Question???