Chapter-9: Risk and Rate of Return
09
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Risk and Rate of Retrun
Board Question
Index
Particulars
Knowledge Based Questions & Answers
Board Short Questions and Answers
Additional Short Questions and Answers
Comprehensive Questions & Answers
Board Comprehensive Questions and Answers
Additional Comprehensive Questions and Answers
Board Creative Questions & Answers
Standard Deviation Related Math
Average/Expected Rate of Return and Standard Deviation
related math
Average/Expected Rate of Return and Coefficient of
variation related math: Method-1
Average/Expected Rate of Return and Coefficient of
variation related math: Method-2
Average/Expected Rate of Return and Coefficient of
variation related math: Method-3
Standard Deviation and Coefficient of variation related
math: Method-1
Standard Deviation and Coefficient of variation related
math: Method-2
Standard Deviation and Coefficient of variation related
math: Method-3
Portfolio Return Related Math
Portfolio Return And Portfolio Risk Related Math
Other Questions
Page No.
Question
No.
2-3
3-4
1-15
1-14
4-10
10-13
1-33
1-18
13-20
1-6
20-27
7-11
27-33
12-15
33-43
16-22
43-55
23-30
55-73
31-42
73-93
43-57
94-97
58-59
97-104
104-115
115-118
60-66
67-75
76-77
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Knowledge Based Questions & Answers:
Board Short Questions and Answers
1. What is uncertainty?
Answer: When the probability of whether an event will occur in future or not cannot be
determined mathematically, it is called uncertainty. Or,
Uncertainty is such a situation where an event may or may not take place in future.
2. What is risk?
Answer: The uncertainty of probable difference between actual return & expected return of
an investment – is called risk.
3. What is financial risk?
Answer: The risk that arises from the inability to repay debt – is called financial risk. Or,
The probable inability of an organization of paying its debt and interest timely - is called
financial risk.
4. What is liquidity risk?
Answer: The risk of being unable to convert an asset into cash easily - is liquidity risk. Or,
The risk associated with quick conversion of invested securities into cash is known as
liquidity risk.
5. What is avoidable risk?
Answer: The risk that can be avoided adopting different techniques – is called avoidable risk.
6. What is portfolio?
Answer: Portfolio means diversification. When an investor invests in more than one
companies or assets to minimize his risk, it is called portfolio.
7. What is portfolio risk?
Answer: Portfolio risk is the risk of a portfolio that is formed by investing in different assets
or projects.
8. What is company risk?
Answer: The risk that is related particularly to an organization or business institution - is
called company risk.
9. What is rate of return?
Answer: The profit that the investors expect on their investment is expected rate of return.
Or,
When summation of total return from a particular investment for a certain period and change
in its market value is expressed as percentage on initial cost of the investment, it is called rate
of return.
10. What is risk free rate of return? Or,
What is risk free return?
Answer: The rate at which return comes from risk-free investment – is called the risk-free
rate of return. Or,
The certain minimum return that comes from portfolio investment - is known as risk free
return.
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11. What is probability distribution?
Answer: The chart or table by which different values and probabilities of variable are
expressed – is probability distribution.
12. What is CAPM?
Answer: CAPM or Capital Asset Pricing Model is a model that analyzes risk and return of
an investment establishing a relationship between the two.
13. What is the standard value market beta?
Answer: The standard value market beta is 1 (one).
14. What is capital gain?
Answer: If selling price of an asset is more than its purchasing price, the proceeds is called
capital gain. Or,
The profit made on the sale of securities when their value increases – is called capital gain.
15. What is mixed cash flow?
Answer: If there is more than one payment or receipt of different amount in a cash flow, it is
called mixed cash flow. Or,
A series of unequal cash flows for a specific period at a certain interval – is called mixed cash
flow.
Additional Short Questions and Answers
1. What is business risk?
Answer: The fear of not being able to meet company’s operating expenses from its income –
is called business risk.
2. What is interest rate risk?
Answer: The risk derived from fluctuation of interest rate in the market – is interest rate risk.
3. What is inflation risk?
Answer: The risk an organization or investment may face due to inflation in the market – is
inflation risk.
4. What is diversifiable risk?
Answer: The risk we get deducting market risk from total risk – is diversifiable risk.
5. What is risk premium?
Answer: The difference between market rate and risk free rate is risk premium.
6. What is risk management?
Answer: The process of identifying, analyzing and reducing risk of an investment decision –
is called risk management.
7. Which risk is related to capital structure of a business?
Answer: Financial risk is related to capital structure of a business.
8. What is expected return?
Answer: The rate of return an investor expects from his investment is called expected return.
9. Write the formula of rate of return.
Answer: The formula of rate of return is:
Chapter-9: Risk and Rate of Return
Rate of return, R =
2/4
𝐷1 +(𝑃1 −𝑃0 )
𝑃0
× 100
10. What is standard deviation?
Answer: Standard deviation is such a process of risk measurement of an asset or investment
that measures deviation from expected return.
11. What is coefficient of variation?
Answer: Coefficient of variation is such a tool of risk measurement that expresses standard
deviation as percentage of average return or expected return.
12. What is beta coefficient?
Answer: The coefficient by which market risk is measured – is beta coefficient.
13. Who introduced the theory of portfolio diversification?
Answer: Harry Markowitz introduced the theory of portfolio diversification.
14. What is dividend?
Answer: The portion of net profit of a company that is distributed among shareholders – is
dividend.
Comprehensive Questions & Answers:
Board Comprehensive Questions and Answers
1. What is uncertainty?
Answer: When the probability of whether an event will occur in future or not cannot be
determined mathematically, it is called uncertainty.
Uncertainty is such a situation where an event may or may not take place in future.
Uncertainty can neither be measured nor avoided. Besides, as it cannot be measured, it cannot
be known whether this same incident will be repeated in future. As uncertainty is not
measurable, the relationship between uncertainty and return can also not be determined.
2. Explain the relationship between uncertainty and risk? Or,
What is the relation between uncertainty and risk?
Answer: There is a probable positive relationship between uncertainty and risk.
Uncertainty is such a situation where an event may or may not take place in future. On the
other hand, the probability of difference between expected return and actual return of an
investment – is called risk. Uncertainty creates risk. The more the uncertainty, the more the
risk.
3. Distinguish between risk and uncertainty.
Answer: The main difference between risk and uncertainty is that all risks are uncertainty,
but all uncertainties are not risk.
There is an inverse relationship between risk and uncertainty. Risk is insurable, but
uncertainty is not. Risk is mathematically measurable, uncertainty is not measurable. Risk can
be reduced by adopting various strategies, but uncertainty cannot be reduced. Risk can be
avoided whereas uncertainty cannot be avoided.
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4. Is it possible to earn profit without risk? Explain.
Answer: The probability of difference between expected return and actual return of an
investment – is called risk. In general sense, it is not possible to earn profit without risk.
But there are some govt. securities like treasury bill, treasury bond where there is not risk as
expected return equals actual return. Such securities are risk free as the govt. cannot be
bankrupt. Investors get a certain rate of profit from such securities. In this sense, it is possible
to earn profit without risk.
5. What do you mean by market risk?
Answer: The risk that is originated from outer elements of a company and out of control of
the company – is market risk.
Market risk derives from the change in economic condition, govt. policy and political
instability etc. Normally market risk is out of control of a business organization. As a result,
this risk cannot be avoided adopting diversification. Market risk is also known as systematic
risk.
6. What is meant by avoidable risk?
Answer: The risk that can be avoided adopting different techniques – is called avoidable risk.
If market risk is deducted from total risk, the risk that remains is avoidable risk. Avoidable
risk originates from different internal issues of an organization like inefficient management,
lack of skill, laborer unrest etc. An organization can avoid these situations if it wants.
7. Why is market risk called unavoidable risk?
Answer: As an investor cannot avoid market risk, it is called unavoidable risk.
The risk that is originated from outer elements of a company and out of control of the
company – is market risk. Normally market risk is originated from change in economic
condition, change in govt. policy, economic downturn, inflation, war, high interest rate etc.
As these are out of control of a company, an investor cannot avoid market risk. Besides, as
this risk affect all sectors, it cannot be avoided even through portfolio diversification. That is
why market risk is called unavoidable risk.
8. Why market risk cannot be controlled?
Answer: As market risk cannot be avoided or lessened using any tool, it cannot be controlled.
The risk that is originated from outer elements of a company and out of control of the
company – is market risk. Normally market risk is originated from change in economic
condition, change in govt. policy, economic downturn, inflation, war, high interest rate etc.
As these are out of control of a company, an investor cannot avoid market risk. Besides, as
this risk affect all sectors, it cannot be avoided even through portfolio diversification. That is
why market risk cannot be controlled.
9. Is market risk unavoidable? Explain. Or,
Why market risk cannot be avoided? Explain.
Answer: As market risk cannot be controlled, it cannot be avoided.
The risk that is originated from outer elements of a company and out of control of the
company – is market risk. Normally market risk is originated from change in economic
condition, change in govt. policy, economic downturn, inflation, war, high interest rate etc.
As these are out of control of a company, an investor cannot avoid market risk. Besides, as
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this risk affect all sectors, it cannot be avoided even through portfolio diversification. That is
why market risk cannot be avoided.
10. Can business risk be avoided?
Answer: Business risk cannot be avoided.
The fear of not being able to meet company’s operating expenses from its income – is called
business risk. A firm may not be able to attain its target profit due to excess fixed operating
income and fluctuation in demand and sales. As a result, the firm may not be able to bear its
operating expenses. That is, it may face business risk. As this risk is created from external
factors, it cannot be avoided, but can be lessened through efficient management.
11. Why business risk be considered in case of investment decision?
Answer: The fear of not being able to meet company’s operating expenses from its income –
is called business risk. Again, the fear that business profit will decrease due to competition is
also business risk.
A firm’s investment in a project depends on the project’s ability to survive in perfectly
competitive market. If the firm perceives that the business risk of the project is too high i.e.
the probability of survival is too low, then it will not invest in the project. That is why business
risk is considered in investment decision.
12. “Business risk decreases profit earning ability” – Explain.
Answer: The fear of not being able to meet company’s operating expenses from its income –
is called business risk.
The investors measure risk very carefully while investing in a business. The investors will not
be interested to invest in the business if there is excess risk. As a result, the business won’t be
able to collect its required capital. Consequently its production capacity will fall. This will
decrease its ability to earn profit. So we can say that, excess business risk decreases profit
earning ability.
13. How is “Liquidity Risk” created?
Answer: The risk associated with quick conversion of invested securities into cash is known
as liquidity risk.
Some securities can be easily sold and converted into cash, for example: Treasury bill. Again,
some securities are very hard to sell. Those assets that cannot be easily sold and converted
into cash have liquidity risk. Lack of goodwill of the issuer, non-familiarity, financial
incapability, past bad performance and inactive secondary market etc. creates liquidity risk.
14. What do you mean by financial risk?
Answer: The probable inability of an organization of paying its debt and interest timely - is
called financial risk.
The fear of not being able to achieve sufficient return form an investment to pay principal
amount and interest of a loan taken for that investment is financial risk. The more debt capital
an organization uses, the more its financial risk is.
15. How is financial risk created? Explain. Or,
How does financial risk arise? Explain.
Answer: The probable inability of an organization of paying its debt and interest timely - is
called financial risk.
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Financial risk depends on the amount of debt capital of an organization. An organization is
obliged to give interest to its lenders at a certain rate. Those organizations who have more
borrowed capital have to pay more interest, hence their risk is more. On the other hand, those
organizations that have no or a little borrowed capital have to pay little interest. So, their risk
is also less. Extensive use of borrowed capital creates more financial risk. Therefore, financial
risk arises from debt capital.
16. How does financial risk assist in capital formation?
Answer: The probable inability of paying debt and interest of an organization timely– is
called financial risk.
The organization which has debt capital in its capital structure has financial risk. If an
organization wants to avoid financial risk, it has to stop financing through debt capital and
raise own capital. That is how financial risk assists in capital formation.
17. What is meant by single risk?
Answer: The risk an investor has to bear when he invests in a single organization or on a
single asset – is called single risk.
While investing on a single asset, the investor should systematically analyze return of the
asset in future periods and continuously monitor the situation to minimize risk. In this case,
the return totally depends on a single organization. If that organization functions well, the
total investment turns out to be a profitable venture. If not, then total investment becomes a
losing one. Single risk can be minimized through multiple investments or portfolio
diversification.
18. What do you mean by risk premium?
Answer: The difference between market rate and risk free rate is risk premium.
Risk premium depends on market portfolio return. Risk premium arises if market portfolio
return is more than risk free return. The formula to determine market risk premium is: Market
risk premium = (Rm - Rf). Here, Rm is market rate and Rf is risk free rate. For example: If
market rate is 15% and risk free rate is 8%, market risk premium will be = (15% - 8%) = 7%.
19. ‘If risk increases, return also increases’ – Explain.
Answer: The uncertainty of earning this profit is called risk. And the proceeds of income
over expense in an investment is profit or return.
There is a positive relation between risk and return. The more the risk, the more the return. If
an investor invests in a riskier sector, there is a probability of higher rate of return. Because,
the investor gets risk premium taking more risk. So, the statement ‘If risk increases, return
also increases’ is right.
20. “It is not always unprofitable to take a decision in risky environment” - Explain.
Answer: As there is a positive relationship between risk and return, it is not always
unprofitable to take a decision in risky environment.
If risk is high, rate of return is also higher. Again, if risk is less, rate of return also gets lower.
So, taking decision is risky environment might be profitable too.
21. What is meant by portfolio principle?
Answer: When an investor invests in more than one companies or assets to minimize his risk,
it is called portfolio.
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An investor creates an investment set combining different tools to make his investment
efficient and profitable. These different investment set combining tools are known as
portfolio. An investor can minimize his risk by portfolio.
22. What do you mean by portfolio diversification?
Answer: Instead of investing the whole capital in a single company or asset, when an investor
invests multiple companies or assets to minimize his risk, it is called portfolio diversification.
Business risk is spread and reduced if the product or service is as much diversified as
possible. According to theory, entire capital is not invested in a single project, rather it is
invested in multiple projects. Thus it becomes possible to earn expected return at any
situation.
23. Why is portfolio formed?
Answer: When an investor invests in more than one companies or assets to minimize his risk,
it is called portfolio. That is, portfolio is formed to minimize risk.
As portfolio is a set of different investments, the loss of one investment can be recovered from
the profit of other investments. Again, an investor can minimize his risk by portfolio. So,
portfolio is formed to minimize risk by diversification.
24. Why is principle of portfolio diversification followed? Or,
Why is portfolio theory followed? Or,
What is the purpose of following portfolio theory?
Answer: Portfolio theory is followed to distribute and reduce risk.
Portfolio means diversification. When an investor invests in more than one companies or
assets to minimize his risk, it is called portfolio. According to portfolio theory, entire capital
is not invested in a single project, rather it is invested in multiple projects. Thus it becomes
possible to adjust the loss of one project with the profit of another project. Therefore, portfolio
theory is followed to distribute and reduce risk through investing in multiple projects.
25. "Risk can be reduced by investment diversification." – Explain.
Answer: Portfolio means diversification. When an investor invests in more than one
companies or assets to minimize his risk, it is called portfolio diversification.
Business risk is spread and reduced if products or services are as much diversified as possible
in terms of investment. That is, if the sales of one product decrease in any situation, it is
possible to make up for the reduced sales of the business with the sales of other products,
thereby achieving the desired amount of profit in any situation. So, it can be said that risk can
be reduced through portfolio diversification.
26. What does it mean by portfolio risk?
Answer: Portfolio risk is the risk of a portfolio that is formed by investing in different assets
or projects.
Portfolio is formed to minimize risk. In portfolio, entire money is not invested in a single
sector; rather an investment set of multiple investments is created. As a result, loss derived
from one project can be recovered by profit of other projects. But portfolio itself has also
minimum risk known as portfolio risk.
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27. What is meant by expected rate of return?
Answer: The profit that the investors expect on their investment is expected rate of return.
Expected return is totally uncertain. It is measured based on anticipated information. There is
a direct and parallel relationship between expected return and risk. That means the more the
risk, the more the expected return will be.
28. Explain the relationship between risk and rate of return. Or,
How is the relationship between risk and return? Explain. Or,
What type of relationship does exist between return and risk? Explain.
Answer: The uncertainty of earning this profit is called risk. And the proceeds of income
over expense in an investment is profit or return.
Risk and rate of return are closely related. The more the risk, the more the return. One has to
take more risk to earn more profit. Again if there is less risk, so will be the profit. That is why
an organization has to keep a tolerable balance between risk and return. So, it can be said that
there is a positive relation between risk and rate of return.
29. Explain the effect of beta on required rate of return.
Answer: The relationship of beta with required rate of return is positive.
Beta (β) is a systematic risk measuring index that measures the change in income rate of a
security compared to the change in market rate. If all other factors remains constant, required
rate of return increases if beta increases. For, risk premium of security increases with the
increase of the value of beta. And required rate of return is the sum of risk free return and
security risk premium. On the other hand, if the value of beta decreases, required rate of return
also decreases.
30. What is meant by capital asset pricing model?
Answer: CAPM or Capital Asset Pricing Model is a model that analyzes risk and return of
an investment establishing a relationship between them. It is a market model of making
financial decision.
There are two elements, risk and return, in investment decision. A financial decision should
be taken in such a way maintaining a proper combination between risk and return that it can
be helpful in wealth maximization of a firm. Ultimately, CAPM is the established relationship
between risk and return.
31. Is CAPM model essential for taking investment decision? Explain.
Answer: CAPM model is an essential issue in taking investment decision. For, expected
return is determined through this model.
It is important to know cost of capital in case of investing in a project. Investment is to be
made after comparing cost of capital of a project and expected return from it. In this case,
expected return from investment is determined through CAPM. So, CAPM model is an
essential issue in taking investment decision.
32. Why there is no risk in investing government bond?
Answer: Long-term bond issued by the government is called treasury bond or government
bond.
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Investment risk refers to the inability to pay bond's interest and principal. Such risk is not
applicable to government bond, as the government cannot be bankrupt. Moreover, the
government can collect taxes compulsorily if necessary.
33. What kind of risk do the investors of bond have to bear? Explain.
Answer: Investors in bond have to face both interest rate risk and liquidity risk.
The risk derived from fluctuation of interest rate in the market – is interest rate risk. If the
interest rate in the market increases, the market value of bond decreases, resulting in interest
rate risk. On the other hand, the risk associated with quick conversion of invested securities
into cash is known as liquidity risk. If an investor cannot sell bond in the market at expected
price, liquidity risk arises; which bond investors have to bear too.
Additional Comprehensive Questions and Answers
1. Write about different types of risk.
Answer: The uncertainty of probable difference between actual return & expected return of
an investment – is called risk.
Risk can be of different types. The classification is shown here:
a) Unsystematic risk
i) Business risk
ii) Financial risk
b) Systematic risk
i) Interest rate risk
ii) Purchasing power risk
iii) Market risk
2. What do you mean by business risk?
Answer: The fear of not being able to meet company’s operating expenses from its income –
is called business risk.
A firm has to bear different operating cost while running a business. For example: purchase
of raw material, payment of salary, office rent and insurance expense etc. Business risk
derives from a company’s inability to meet these expenses with its income.
3. Level of financial risk depends on what?
Answer: The probable inability of an organization of paying its debt and interest timely - is
called financial risk.
Financial risk depends on nature of asset and amount of debt capital in a company’s capital
structure. Amount of risk may be different even if two companies of identical size take same
amount of loan. Because an organization can reduce its financial risk changing nature of asset
and using more debt instead of equity.
4. Why is business risk different from financial risk?
Answer: Risk means future uncertainty. The fear of not being able to meet company’s
operating expenses from its income – is called business risk. On the other hand, the probable
inability of an organization of paying its debt and interest timely - is called financial risk.
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Business risk is different from financial risk. Business risk is business operation related risk,
where financial risk is related to inability of paying debt.
5. Why is market risk called non-diversifiable risk?
Answer: The risk derived from change in economic condition, govt. policy and political
instability etc. is called market risk.
Market risk is a type of systematic risk that specially affects capital market. Due to many
factors, share price in capital market faces bullish trend (sudden price hike) or bearish trend
(a general fall). Normally this fluctuation in share price is out of control of an investor. He
cannot avoid this risk even if he tries. That is why market risk is called non-diversifiable risk.
6. How is market risk premium determined?
Answer: The difference between market rate and risk free rate is called market risk premium.
The formula to determine market risk premium is: Market risk premium = (R m - Rf). Here, Rm
is market rate and Rf is risk free rate. For example: If market rate is 15% and risk free rate is
8%, market risk premium will be = (15% - 8%) = 7%.
7. What do you mean by inflation risk?
Answer: Reduction in purchasing power due to price hike is inflation. The risk an
organization or investment may face due to inflation in the market – is inflation risk.
Inflation increases cost of production. Consequently, price of product or service also increases
automatically. In this situation, if the consumers start to use alternative products or service, a
business faces inflation risk.
8. Why is actual return different from expected return?
Answer: Uncertainty is such a situation where an event may or may not take place in future.
Actual return is more or less than expected return because of uncertainty. This risk of probable
difference between actual return and expected return is actually financial risk.
9. Why does return of risk free sector differ from return of risky sector?
Answer: An investor expects additional profit or risk premium for bearing risk.
Expected return is determined adding risk premium with risk free return. That is why return
of a risk free sector is different from return of a risky sector.
10. What is portfolio? Explain.
Answer: Portfolio means diversification. When an investor invests in more than one company
or asset, it is called portfolio.
An investor creates an investment set combining different tools to make his investment
efficient and profitable. These different investment set combining tools are known as
portfolio. An investor can minimize his risk adopting portfolio.
11. Why is portfolio formed? Explain.
Answer: When an investor invests in more than one company or asset, it is called portfolio.
In other word, portfolio is a set of different investments.
As there are different types of investments in portfolio, loss of one investment can be
recovered by profit of other investments. Again, an investor can minimize his risk by
portfolio. So, portfolio is formed to minimize risk by diversification.
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12. What do you mean by portfolio investment?
Answer: When an investor invests in more than one companies or assets to minimize his risk,
it is called portfolio.
Portfolio investment is a set of different investments. It is a strategy of minimizing risk. It
reduces risk combining different investments of opposite characteristics together. In portfolio,
entire money is not invested in a single sector; rather an investment set of multiple
investments is created. If an investor invests entire money in one company, the company may
become bankrupt. But if he invests in multiple companies, all the companies will not become
bankrupt at the same time. Thus portfolio investment minimizes risk.
13. Why is portfolio investment more efficient?
Answer: When an investor invests in more than one company or asset, it is called portfolio.
There is more than one investment in portfolio. As a result, loss derived from one investment
can be recovered by profit of other investments. A firm can keep its risk at minimum level
through portfolio diversification. That is why portfolio investment is more efficient than
single investment.
14. What are the assumptions of CAPM?
Answer: CAPM or Capital Asset Pricing Model is a model that analyzes risk and return of
an investment establishing a relationship between them.
The assumptions of CAPM are:
• There is no tax,
• There is no transaction cost in money market,
• All investors behave logically,
• Capital market is efficient,
• There is no inflation in economy,
• All assets are completely divisible etc.
15. What is meant by ‘Correlation between two securities is –1’?
Answer: ‘Correlation between two securities is –1’ means that the securities are totally
negatively correlated.
If there is negative correlation between two securities, loss in one security will result in profit
in another. As a result, we can adjust loss of one security by the profit of another.
16. How is beta (β) determined?
Answer: Beta (β) is a systematic risk measuring index that measures change in income rate
of a security compared to change in market rate. Mainly non-diversifiable market risk is
measured by beta.
Beta is the combination of company and market coefficient and market variance. We can
determine beta by the following formula:
𝑅 −𝑅
β = 𝑅 𝑖 −𝑅𝑓
𝑚
𝑓
17. What kind of risk is measured by beta (β)?
Answer: Beta (β) is a tool of measuring risk.
Mainly non-diversifiable market risk is measured by beta. It is a risk determining index that
measures change in income rate of a security compared to change in market rate.
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18. Why does value of an investment fluctuate?
Answer: Return from an investment is called profit or interest.
Value of an investment depends on its return or interest rate. Interest rate in the market rises
and falls for different reasons. And value of an investment fluctuates due to this fluctuation
in interest rate.
Item Wise Board Creative Questions & Answers:
Standard Deviation Related Math
1. Following are the rate of return of last four years of Asa Ltd. and Shawpno Ltd.:
1st
12%
15%
Project Year
Asa Ltd.
Shawpno Ltd.
2nd
15%
10%
3rd
4th
13%
10%
5%
20%
(Dhaka Board-2023)
a) Calculate standard deviation of the rate of return of Asa Ltd.
b) In which of the company an investor should invest considering the level of risk? Analyze.
Answer:
a) Calculating standard deviation of Asa Ltd.:
Here,
Average rate of return, 𝑅̅𝐴
=
=
∑ 𝑅𝑖
𝑛
12 + 15 + 13 + 10
4
50
= 4
= 12.50%
We know,
∑(𝑅𝑖 −𝑅̅𝐴 )2
Standard deviation, σA = √
𝑛−1
(12−12.50)2 +(15−12.50)2 +(13−12.50)2 +(10−12.50)2
=√
4−1
0.25 + 6.25 + 0.25 + 6.25
=√
3
13
=√3
= √4.33
= 2.08%
Therefore, standard deviation of the rate of return of Asa Ltd. is 2.08%.
b) We get from a,
Standard deviation of the rate of return of Asa Ltd. is 2.08%.
Determining standard deviation of Shawpno Ltd.:
Here,
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∑𝑅
Average rate of return, 𝑅̅𝑆
= 𝑛𝑖
=
15 + 10 + 5 + 20
4
50
= 4
= 12.50%
We know,
Standard deviation, σS = √
∑(𝑅𝑖 −𝑅̅𝑆 )2
𝑛−1
(15−12.50)2 +(10−12.50)2 +(5−12.50)2 +(20−12.50)2
=√
4−1
6.25 + 6.25 + 56.25 + 56.25
=√
3
125
=√ 3
= √41.67
= 6.46%
Here, the average rate of return of both Asa Ltd. and Shawpno Ltd. are is 12.50% and 12.50%
respectively, that is, equal. But Asa Ltd.’s standard deviation or risk is 2.08%, which is less than
Shawpno Ltd.’s standard deviation of 6.46%. Therefore, an investor should invest in Asa
Ltd. considering the level of risk.
2. Rate of returns of two projects for the last four years are given below:
Year
Rate of return of Jamuna Project
Rate of return of Padma Project
2015
20%
5%
2016
2017
2018
6%
15%
23%
15%
20%
24%
(Barishal Board-2023)
a) Determine standard deviation of Jamuna Project.
b) Which of the two projects is more acceptable? Show logic with mathematical analysis.
Answer:
a) Determining the standard deviation of Jamuna Project:
Here,
∑𝑅
Average rate of return, 𝑅̅𝐽
= 𝑛𝑖
=
20 + 6 + 15 + 23
4
64
= 4
= 16%
We know,
2
∑(𝑅𝑖 −𝑅̅𝐽 )
Standard deviation, σJ = √
𝑛−1
(20−16)2 +(6−16)2 +(15−16)2 +(23−16)2
=√
4−1
Chapter-9: Risk and Rate of Return
2/15
16 + 100 + 1 + 49
=√
3
166
=√ 3
= √55.33
= 7.44%
Therefore, the standard deviation of Jamuna Project is 7.44%.
b) We get from a,
Average rate of return of Jamuna Project, 𝑅̅𝐽 = 16%
Standard deviation of Jamuna Project, σJ
= 7.44%
Determining the standard deviation of Padma Project:
Here,
∑𝑅
Average rate of return, 𝑅̅𝑃
= 𝑛𝑖
=
5 + 15 + 20 + 24
4
64
= 4
= 16%
We know,
Standard deviation, σP = √
∑(𝑅𝑖 −𝑅̅𝑃 )2
𝑛−1
(5−16)2 +(15−16)2 +(20−16)2 +(24−16)2
=√
4−1
121 + 1 + 16 + 64
=√
3
202
=√ 3
= √67.33
= 8.21%
Here, the average rate of return of both Jamuna and Padma projects are 16% and 16%
respectively, that is, equal. But the standard deviation or risk of Jamuna project is 7.44%,
which is less than the standard deviation of Padma project of 8.21%. In this consideration,
Jamuna project is more acceptable.
3. Sadia Afrin wished to invest in share market. Share collected Star Foods and Moon Lamp’s
last 4 years’ profit related information. The information of Star Foods and Moon Lamp are as
follows:
Year
Star Foods
Moon Lamp
1
16%
26%
2
18%
08%
3
20%
30%
4
22%
12%
(Board-2022)
Chapter-9: Risk and Rate of Return
2/16
a) Calculate standard deviation of Star Foods of the stem.
b) Which project’s risk between Star Foods and Moon Lamp is less? Explain mathematically.
Answer:
a) Calculating standard deviation of Star Foods:
Here,
̅ 𝑆𝐹 = ∑ 𝑅𝑖𝑆𝐹
Average rate of return of Star Foods, R
𝑛
16+18+ 20+22
=
4
76
=
4
= 19%
We know,
∑(𝑅𝑖𝑆𝐹 − 𝑅̅𝑆𝐹 )2
Standard deviation of Star Foods, σSF = √
=√
𝑛−1
(16 −19)2 + (18 −19)2 + (20 −19)2 + (22 −19)2
4 −1
9+1+1+9
=√
3
20
=√3
= √6.67
= 2.58%
Therefore, standard deviation of Star Foods is 2.58%.
b) To determine which project’s risk between Star Foods and Moon Lamp is less, we need to
compare standard deviation of both organizations.
Determining standard deviation of Moon Lamp:
Here,
̅ 𝑀𝐿 =
Average rate of return of Moon Lamp, R
=
∑ 𝑅𝑖𝑀𝐿
𝑛
26+8+ 30+12
4
76
= 4
= 19%
We know,
Standard deviation of Moon Lamp, σML = √
=√
∑(𝑅𝑖𝑀𝐿 − 𝑅̅𝑀𝐿 )2
𝑛−1
(26 −19)2 + (8 −19)2 + (30 −19)2 + (12 −19)2
4 −1
49+121+121+49
=√
3
340
=√ 3
= √113.33
Chapter-9: Risk and Rate of Return
2/17
= 10.65%
It is noticeable that standard deviation of both Star Foods and Moon Lamp are 19%, that is,
equal. But standard deviation or risk of Star Foods is 2.58% (from a), which is less than
standard deviation or risk of Moon Lamp 10.65%. Therefore, Star Foods project’s risk is less
between the two.
4. Mr. Karim has interest to invest in capital market. For investment, he is considering some
information of the two following securities:
Probability
Expected rate of return
Security-A
Security-B
0.40
30%
25%
0.20
20%
35%
0.40
18%
32%
(Dhaka Board’17)
a) Determine standard deviation of security-A.
b) “Security-B is more risky” - Analyze this statement mathematically.
Answer:
a) Determining standard deviation of security-A:
Expected rate of return of Security-A,
̅A
R
= ∑𝑛𝑖=1(𝑅𝑖 × 𝑃𝑖 )
= (0.30 × 0.40) + (0.20 × 0.20) + (0.18 × 0.40)
= 0.12 + 0.04 + 0.072
= 0.232
= 23.20%
∴ Standard deviation, σA = √∑(𝑅𝑖 − 𝑅̅)2 × 𝑃𝑖
= √(0.30 − 0.232)2 × 0.40 + (0.20 − 0.232)2 × 0.20 + (0.18 − 0.232)2 × 0.40
= √0.0018496 + 0.0002048 + 0.0010816
= √0.003136
= 0.056
= 5.60%
Therefore, standard deviation of security-A is 5.60%.
Answer: 5.60%.
b) Determining risk of security-B:
Expected rate of return of security-B,
̅B
R
= ∑𝑛𝑖=1(𝑅𝑖 × 𝑃𝑖 )
= (0.25 × 0.40) + (0.35 × 0.20) + (0.32 × 0.40)
= 0.10 + 0.07 + 0.128
= 0.298
= 29.80%
∴ Standard deviation,
σB = √∑(𝑅𝑖 − 𝑅̅)2 × 𝑃𝑖
= √(0.25 − 0.298)2 × 0.40 + (0.35 − 0.298)2 × 0.20 + (0.32 − 0.298)2 × 0.40
Chapter-9: Risk and Rate of Return
2/18
= √0.0009216 + 0.0005408 + 0.0001936
= √0.001656
= 0.040694
= 4.07%
After calculation, it is found that standard deviation of security-B is 4.07% which is less than
standard deviation of security-A, 5.60%. In other words, security-B is less risky than securityA. Hence, “Security-B is more risky”- this statement is not appropriate.
5. The rate of return of Sajeeb food Limited and Aziz Food Limited of last 3 years are as follows:Year
Rate of return
Sajeeb Food Limited
12%
11%
14%
2012
2013
2014
Rate of return
Aziz Food Limited
15%
-03%
18%
(Rajshahi Board’16)
a) Calculate standard deviation of income for Sajeeb Food Limited.
b) Considering risk of the two companies, which company should be preferred for investment?
Answer:
a) Determining standard deviation of Sajeeb Food Limited:
Here,
Rate of income in three years (R) = 12%, 11% and 14%.
∑𝑅
Average rate of return, (𝑅̅) =
=
𝑛
12+11+14
3
37
= 3
= 12.33%
We know,
Standard deviation, 𝜎 = √
=√
∑(𝑅𝑖 −𝑅̅)2
𝑛−1
(12−12.33)2 +(11−12.33)2 +(14−12.33)2
3−1
=√
0.1089+1.7689+2.7889
=√
4.6667
2
2
= √2.3333
= 1.5275 or 1.53%
So, standard deviation of Sajeeb Food Limited is 1.53%.
Answer: 1.53%.
b) Determining standard deviation of Aziz Food Limited:
Here,
Chapter-9: Risk and Rate of Return
2/19
Rate of income in three years (R) = 15%, -03% and 18%.
∑𝑅
Average rate of return, (𝑅̅) = 𝑛
=
=
=
15+(−03)+18
3
15−03+18
3
30
3
= 10%
We know,
Standard deviation, 𝜎 = √
=√
=√
∑(𝑅𝑖 −𝑅̅)2
𝑛−1
(15−10)2 +(−03−10)2 +(18−10)2
3−1
25+169+64
2
258
=√ 2
= √129
= 11.36%
So, standard deviation of Aziz Food Limited is 11.36% that is greater than Sajeeb Food
Limited’s (1.53%). That means investing in Sajeeb Food Limited is less risky. Hence, Sajeeb
Food Limited should be preferred for investment.
6. The rate of return of Bengal Foods Ltd. and Modern Foods Ltd. for the last 3 years are:Year
Rate of Return of
Bengal Foods Ltd.
12%
11%
14%
Rate of Return of
Modern Foods Ltd.
2012
15%
2013
-03%
2014
18%
(Sylhet Board’16)
a) Calculate standard deviation of returns for Bengal Foods Ltd.
b) In which company’s investment will be justified considering the level of risk of these two
companies?
Answer:
a) Determining standard deviation of Bengal Foods Ltd.:
Here,
Rate of income in three years (R) = 12%, 11% and 14%.
∑𝑅
Average rate of return, (𝑅̅) = 𝑛
=
12+11+14
3
37
= 3
= 12.33%
Chapter-9: Risk and Rate of Return
2/20
We know,
Standard deviation, 𝜎 = √
=√
∑(𝑅𝑖 −𝑅̅)2
𝑛−1
(12−12.33)2 +(11−12.33)2 +(14−12.33)2
3−1
=√
0.1111+1.7778+2.7778
=√
4.6667
2
2
= √2.3333
= 1.53%
So, standard deviation of Bengal Foods Ltd. is 1.53%.
Answer: 1.53%.
b) Determining standard deviation of Modern Foods Ltd.:
Here,
Rate of income in three years (R) = 15%, -03% and 18%.
∑𝑅
Average rate of return, (𝑅̅) = 𝑛
=
15−03+18
3
30
= 3
= 10%
We know,
Standard deviation, 𝜎 = √
=√
=√
∑(𝑅𝑖 −𝑅̅)2
𝑛−1
(15−10)2 +(−03−10)2 +(18−10)2
3−1
25+169+64
2
258
=√ 2
= √129
= 11.36%
So, standard deviation of Modern Foods Ltd. is 11.36% that is greater than that of Bengal
Foods Ltd. (1.53%). That means investment in Bengal Foods Ltd. is less risky. Hence,
investing in this company will be logical.
Average/Expected Rate of Return and Standard Deviation Related Math
7. Mr. Abdul Alim is an undergraduate student. He is an investor of capital market. He is
concerned about which one of the two securities to include in his portfolio due to market
instability. The rate of returns of the two securities are as follows:
Chapter-9: Risk and Rate of Return
2/21
Rate of return
Rakib Foods Ltd.
Zakir Foods Ltd.
2019
22%
20%
2020
15%
22%
2021
-10%
7%
2022
14%
-8%
(Dinajpur Board-2023)
a) Determine the average rate of return of Rakib foods Ltd.
b) Which security should be included by Mr. Abdul Alim in his portfolio? Give your opinion
with reasons.
Answer:
a) Determining the average rate of return of Rakib foods Ltd.:
We know,
Probability
∑𝑅
Average rate of return, 𝑅̅𝑅
= 𝑛𝑖
=
22 + 15 –10 + 14
4
41
= 4
= 10.25%
So, the average rate of return of Rakib foods Ltd. is 10.25%.
b) To determine which security should be included by Mr. Abdul Alim in his portfolio, we can
compare the standard deviations or risks of both the securities.
Determining the standard deviation of Rakib foods Ltd.:
We get from a,
Average rate of return, 𝑅̅𝑅
= 10.25%
We know,
∑(𝑅𝑖 −𝑅̅𝑅 )2
Standard deviation, σR = √
𝑛−1
(22−10.25)2 +(15−10.25)2 +(−10−10.25)2 +(14−10.25)2
=√
4−1
138.0625 + 22.5625 + 410.0625 + 14.0625
=√
3
584.75
=√
3
= √194.92
= 13.96%
Determining the standard deviation of Zakir foods Ltd.:
Here,
Average rate of return, 𝑅̅𝑍
=
=
∑ 𝑅𝑖
𝑛
20 + 22 + 7−8
4
41
= 4
Chapter-9: Risk and Rate of Return
2/22
= 10.25%
We know,
∑(𝑅𝑖 −𝑅̅𝑍 )2
Standard deviation, σZ = √
𝑛−1
(20−10.25)2 +(22−10.25)2 +(7−10.25)2 +(−8−10.25)2
=√
4−1
95.0625 + 138.0625 + 10.5625 + 333.0625
=√
3
576.75
=√
3
= √192.25
= 13.87%
Here, though the average rate of return of both Rakib Foods Ltd. and Zakir Foods Ltd. are
10.25% and 10.25% respectively, that is, equal, the standard deviation or risk of Zakir Foods
Ltd. is 13.87%, which is less than the standard deviation of Rakib Foods
Ltd. 13.96%. Therefore, Zakir Foods Ltd.’s security should be included by Mr. Abdul Alim
in his portfolio.
8. Mr. Hasan is considering to invest in one of two projects. Based on two similar projects
carried out in the recent past, the required information are as follows:
Rate of return
Economic
Probability
condition
Project-X
Project-Y
Good
.20
25%
30%
Normal
.50
10%
8%
Recession
.30
3%
3%
Expected return of Project-X is 10.9%. (Mymensingh Board-2023)
a) Determine expected rate of return of Project-Y.
b) Which project is more suitable for Mr. Hasan? Comment with logic.
Answer:
a) Determining the expected rate of return of Project-Y:
Here,
Expected rate of return, 𝑅̅𝑌
= ∑(𝑅𝑖 × 𝑃𝑖 )
= (30 × 0.20) + (8 × 0.50) + (3 × 0.30)
= 6 + 4 + 0.90
= 10.90%
Hence, the expected rate of return of project-Y is 10.90%.
b) Determining the standard deviation of project-X:
Given,
Expected return, 𝑅̅𝑋 = 10.90%
We know,
2
Standard deviation, σX = √∑(𝑅𝑖 − ̅̅
𝑅̅̅
𝑋 ) × 𝑃𝑖
Chapter-9: Risk and Rate of Return
2/23
=
√(25 – 10.90)2 × 0.20 + (10 − 10.90)2 × 0.50 + (3 − 10.90)2 × 0.30
= √39.762 + 0.405 + 18.723
= √58.89
= 7.67%
Determining the standard deviation of project-Y:
We get from a,
Expected return, 𝑅̅𝑌 = 10.90%
⸫ Standard deviation, σY = √∑(𝑅𝑖 − ̅𝑅̅̅𝑌̅)2 × 𝑃𝑖
= √(30 – 10.90)2 × 0.20 + (8 − 10.90)2 × 0.50 + (3 − 10.90)2 × 0.30
= √72.962 + 4.205 + 18.723
= √95.89
= 9.79%
Here, though the average rate of return of both project-X and project-Y are 10.90% and
10.90% respectively, that is, equal, the standard deviation or risk of project-X is 7.67%, which
is less than project-Y standard deviation of 9.79%. Therefore, project-X is more suitable for
Mr. Hasan.
9. Mizan has collected the following information about stock-A and stock-B:
Economic Condition
Recession
Normal
Good
Rate of return
Stock-A
Stock-B
15%
20%
20%
30%
60%
40%
(Chattogram, Barishal & Cumilla Board’18)
Probability
20%
50%
30%
a) Calculate average rate of return of stock-A.
b) Which stock is more suitable for investment? Give your opinion with mathematical analysis.
Answer:
a) Calculating average rate of return of stock-A:
Given,
R1 = 15%
P1 = 20% or 0.20
R2 = 20%
P2 = 50% or 0.50
R3 = 60%
P3 = 30% or 0.30
We know,
̅ A = ∑𝑛𝑖=1(𝑅𝑖 × 𝑃𝑖 )
Average rate of return, R
= 𝑅1 𝑃1 + 𝑅2 𝑃2 + 𝑅3 𝑃3
= (15% × 0.20) + (20% × 0.50) + (60% × 0.30)
= 3% + 10% + 18%
= 31%
Answer: 31%.
Chapter-9: Risk and Rate of Return
2/24
b) To know which stock is more suitable for investment, standard deviation of both stocks is to
be determined.
Determining standard deviation of stock-A:
Economic
̅
̅ )𝟐
̅ ) 𝟐 × 𝑷𝒊
Ri
Pi
𝑹𝒊 − 𝑹
(𝑹𝒊 − 𝑹
(𝑹𝒊 − 𝑹
condition
Recession
15
˗16
256
0.20
51.2
Normal
20
˗11
121
0.50
60.5
Good
60
29
841
0.30
252.3
𝟐
̅ ) × 𝑷𝒊
∑(𝑹𝒊 − 𝑹
= 𝟑𝟔𝟒
We know,
Standard deviation, σA = √∑(𝑅𝑖 − 𝑅̅)2 × 𝑃𝑖
= √364
= 19.08%
Determining standard deviation of stock-B:
Given,
R1 = 20%
P1 = 20% or 0.20
R2 = 30%
P2 = 50% or 0.50
R3 = 40%
P3 = 30% or 0.30
We know,
̅B
Average rate of return, R
= ∑𝑛𝑖=1(𝑅𝑖 × 𝑃𝑖 )
= (20% × 0.20) + (30% × 0.50) + (40% × 0.30)
= 4% + 15% + 12%
= 31%
Economic
̅
̅ )𝟐
̅ ) 𝟐 × 𝑷𝒊
Ri
Pi
𝑹𝒊 − 𝑹
(𝑹𝒊 − 𝑹
(𝑹𝒊 − 𝑹
condition
Recession
20
˗11
121
0.20
24.2
Normal
30
˗1
1
0.50
0.5
Good
40
9
81
0.30
24.3
𝟐
̅ ) × 𝑷𝒊
∑(𝑹𝒊 − 𝑹
= 𝟒𝟗
∴ Standard deviation, σB
= √∑(𝑅𝑖 − 𝑅̅)2 × 𝑃𝑖
= √49
= 7%
Although average rate of return of both the stocks is equal (31%), standard deviation of stockA (19.08%) is more than that of stock-B (7%). We know, smaller value of standard deviation
indicates less risk. So, stock-B is comparatively less risky. Therefore, stock-B is more suitable
for investment for Mr. Mizan.
10. After retirement, Mr. Raihan got Tk. 25 lakh from his company. He is eager to invest Tk. 10
lakh in capital market. To take investment decision, he is evaluating previous expenses of two
alternative securities ‘X’ and ‘Y’. Expected rate of return of the last 3 years of security ‘X’ and
‘Y’ are given below:
Chapter-9: Risk and Rate of Return
Year
2010
2011
2012
2/25
Expected rate of return (Security-Y)
13%
-5%
22%
(Chattogram Board’17)
a) Compute expected return of the two securities mentioned in the stem.
b) In the light of standard deviation, which investment will be the better one for Mr. Raihan and
why? Give your opinion considering risk.
Answer:
a) Determining expected return of security-X:
Expected return,
̅𝑋
R
Expected rate of return (Security-X)
8%
6%
13%
=
=
=
∑ 𝑅𝑖𝑋
𝑛
8% + 6% + 13%
3
27%
3
= 9%
Determining expected return of security-Y:
̅𝑌
R
=
=
=
∑ 𝑅𝑖𝑌
𝑛
13%+(− 5%) + 22%
3
30%
3
= 10%
So, expected return of security-X and security-Y are 9% and 10% consecutively.
Answer: 9% and 10%.
b) Calculating standard deviation of security-X:
Standard deviation, σx
∑(𝑅𝑖− 𝑅̅)2
=√
𝑛−1
(8% −9%)2 + (6% −9%)2 + (13% −9%)2
=√
3 −1
1%+9%+16%
=√
2
26%
=√ 2
= √13%
= 3.61%
Calculating standard deviation of security-Y:
Standard deviation, σy
∑(𝑅𝑖− 𝑅̅)2
=√
=√
𝑛−1
(13% −10%)2 + (−5% −10%)2 + (22% −10%)2
3 −1
Chapter-9: Risk and Rate of Return
2/26
9%+225%+144%
=√
=√
2
378%
2
= √189%
= 13.75%
Here, standard deviation of security-Y is more than that of security-X. That is to say, higher
risk is involved in security-Y. So, investment in security-X will be the better one for Mr.
Raihan.
11. Mr. Khokon got 50 lac taka from his company after retirement. He is interested to invest 20
lac taka from it in capital market. To make the investment decision, he is considering historical
cost of security A and B. Rate of return of this two securities of the last 3 years are as bellow:
Year
Rate of return
(Security A)
8%
6%
13%
2012
2013
2014
Rate of return
(Security B)
13%
-5%
22%
(Barisal Board’16)
a) Calculate expected rate of return of the two securities mentioned in the stem.
b) Which security should Mr. Khokon in invest and why? Give your opinion considering risk.
Answer:
a) Expected rate of return of Security A:
Here,
Rate of income in 3 years (R) = 8%, 6% and 13%
n
= 3 years
We know,
∑𝑅
Average rate of return,𝑅̅𝐴 = 𝑛
=
8+6+13
3
27
= 3
= 9%
Expected rate of return of Security B:
Here,
Rate of income in 3 years (R) = 13%, -5% and 22%
n
= 3 years
We know,
Average rate of return,𝑅̅𝐴 =
=
∑𝑅
𝑛
13+(−5)+22
3
30
= 3
Chapter-9: Risk and Rate of Return
2/27
= 10%
Answer: 9% and 10%
b) We know,
Standard deviation, 𝜎 = √
∑(𝑅𝑖 −𝑅̅)2
𝑛−1
So, standard deviation of security A, 𝜎𝐴 = √
(8−9)2 +(6−9)2 +(13−9)2
=√
3−1
1+9+16
2
26
= √2
= √13
= 3.6056% or 3.61%
=√
Standard deviation of security B, 𝜎𝐵
(13−10)2 +(−5−10)2 +(22−10)2
3−1
= √189
= 13.7477% or 13.75%
As standard deviation of security A is less than that of security B, security A is less risky.
Therefore, Mr. Khokon had better invest in security A.
Average/Expected Rate of Return and Coefficient of Variation Related Math:
Method-1
12. Mr. Zakir is interested in investing in financial market. He wants to invest in any one of the
following two securities:
year
Income rate
Security-A
20%
18%
14%
2016
2017
2018
Income rate
Security-B
15%
12%
20%
(Cumilla Board-2021)
(a)
(b)
Determine expected rate of return from the stem.
In the stem, which security do you think is reasonable for Mr. Zakir to invest?
Answer:
(a)
Security A:
Expected rate of return
(R ) =  R = 20 + 18 + 14 = 52 = 17.33%
A
i
n
Security B:
3
3
Chapter-9: Risk and Rate of Return
Expected rate of return
(R ) =  R = 15 + 12 + 20 = 47 = 15.67%
B
2/28
i
n
3
3
Therefore, expected rate of return of both security A and B is 17.33% and 15.67%
respectably.
(b)
Determining coefficient of variance of security A:
We get,
Expected rate of return of A is 17.33%. (from a)
 (R − R )
Standard deviation ( ) =
2
i
n −1
A
=
(20 − 17.33)2 + (18 − 17.33)2 + (14 − 17.33)2
3 −1
=
7.1111 + 0.4444 + 11.1111
2
=
18.6667
2
= 9.3333
= 3.0550or 3.06
We know,
Coefficient of variance (CV A )=   100
R
3.06
 100
17.33
= 17.63%
=
Determining coefficient of variance of security B:
We get from (a)
Expected rate of return of B is 15.67%.
 (R − R )
Standard deviation ( ) =
2
i
n −1
B
=
(15 − 15.67 )2 + (12 − 15.67 )2 + (20 − 15.67 )2
3 −1
=
0.4444 + 13.4444 + 18.7778
2
=
32.6667
2
= 16.3333
= 4.04%
Chapter-9: Risk and Rate of Return
2/29
We know,
Coefficient of variance (CV B )=   100
R
4.04
 100
15.67
= 25.80%
=
Therefore Mr Jakir should invest on security A. Because coefficient of variance, that is,
amount of risk of security A is 17.63% which is lower than coefficient of variance of stock B
(25.80%).
13. Mr. Shahin has accumulated Taka 30,00,000, during his service life. He is eager to invest
Taka 20,00,000 in the capital market. To make the investment decision, he studied of previous
expenses of two securities X and Y. the rate of return of the two securities for four years are given
below:
Year
2017
2018
2019
2020
Rate of return (Security-X)
8%
6%
13%
18%
Rate of Return (Security-Y)
13%
-5%
22%
15%
(Sylhet Board-2021)
(a)
(b)
(a)
Calculate expected rate of return of the two securities.
Which security is the best for Mr. Shahin to invest and why? Express your opinion, in
consideration of risk or co-efficient of variance.
Answer:
Determining expected rate of return of the two securities:
Security X:
Expected rate of return
(R ) =  R = 8 + 6 + 13 + 18 = 45 = 11.25%
X
i
n
Security Y:
Expected rate of return
3
4
(R ) = nR = 13 + (−5)4+ 22 + 15 = 454 = 11.25%
i
Y
Therefore, expected rate of return of both X security and Y security is 11.25.
(b)
Determining coefficient of variance of Security X:
( ) = 11.25%
Expected rate of return RX
 (R − R )
Standard deviation ( ) =
2
i
X
n −1
Chapter-9: Risk and Rate of Return
=
2/30
(8 − 11.25)2 + (6 − 11.25)2 + (13 − 11.25)2 + (18 − 11.25)2
4 −1
=
10.5625 + 27.5625 + 3.0625 + 45.5625
3
=
86.75
3
= 28.92
= 5.38%
So, coefficient of variance (CV X ) =   100
R
5.38
 100
11.25
= 47.82%
=
Determining coefficient of variance of Security Y:
We get from (a)
( )
Expected rate of return RY =11.25%
 (R − R)
Standard deviation ( ) =
2
i
n −1
Y
=
(13 − 11.25)2 + (− 5 − 11.25)2 + (22 − 11.25)2 + (15 − 11.25)2
4 −1
=
3.0625 + 264.0625 + 115.5625 + 14.0625
3
=
396.75
3
= 132.25
= 11.5%
We know,
Coefficient of variance (CVY ) =   100
R
11.5
 100
11.25
= 102.22%
=
Therefore, Mr. Shahin should invest in security X. Because, coefficient of variance, that is,
amount of risk of security X is 47.82% which is lower than coefficient of variance of Security
Y (102.22%).
Chapter-9: Risk and Rate of Return
2/31
14. Mr. Sajib is thinking about investing in any one of the two companies. Information regarding
income of the last 5 years of the two companies are as follows: Year
2012
2013
2014
2015
2016
(a)
(b)
(a)
Company-A
15%
16%
20%
22%
30%
Company-B
12%
20%
30%
40%
35%
(Rajshahi Board-2019)
Determine average rate of return of the two companies.
Comment on which company Mr. Sajib should invest in after calculating standard deviation
and coefficient of variance.
Answer:
Determining average rate of return of the two companies:
Company A
()
 Ri 15 + 16 + 20 + 22 + 30 103
=
=
= 20.6%
n
5
5
()
 Ri 12 + 20 + 30 + 40 + 35 137
=
=
= 27.4%
n
5
5
Average rate of return, R =
Company B
Average rate of return, R =
(b)
Standard deviation of Company A:

A
=
(
 Ri − R
)2
n −1
(15 − 20.6)2 + (16 − 20.6)2 + (20 − 20.6)2 + (22 − 20.6)2 + (30 − 20.6)2
=
5 −1
31 .36 + 21 .16 + 0.36 + 1.96 + 88 .36
=
4
143 .2
=
4
= 35 .8
= 5.98 %
Standard deviation of Company B:

B
=
(
 Ri − R
n −1
)2
Chapter-9: Risk and Rate of Return
=
2/32
(12 − 27.4)2 + (20 − 27.4)2 + (30 − 27.4)2 + (40 − 27.4)2 + (35 − 27.4)2
5 −1
=
237.16 + 54.76 + 6.76 + 158.76 + 57.76
4
=
515.2
4
= 128.8
= 11.35%
Coefficient of variance of Company A:

Coefficient of variance CV= 100
R
5.98
=
 100
20.6
= 29.03%
Coefficient of variance of Company B:

Coefficient of variance CV= 100
R
11.35
=
 100
27.4
= 41.42%
As coefficient of variance of Company A is lower than that of Company B, the investment
should be made in Company A.
15. Suma is willing to invest in securities. She has collected information of two securities from
the market. The rate of return of the last 3 years is as follows :
Year
1
2
3
(a)
(b)
(a)
Security A (rate of return)
10%
15%
5%
Security B (rate of return)
12%
-8%
20%
(Dhaka Board-2019, Set-1)
Calculate average rate of return of Security A and Security B.
Compute co-efficient of variation of the two securities. Which security is suitable for
investment and why?
Answer:
Average rate of return of Security A:
( )
Average rate of return, R A =
Average rate of return of Security B:
 Ri 10 + 15 + 5 30
=
=
= 10%
n
3
3
Chapter-9: Risk and Rate of Return
2/33
( )
Average rate of return, R B =
(b)
 Ri 12 + (− 8) + 20 24
=
=
= 8%
n
3
3
Standard deviation of Security A:
A =
(
 Ri − R A
n −1
) = (10 − 10) + (15 − 10) + (5 − 10)
2
2
2
2
3 −1
=
0 + 25 + 25
2
= 25
=5
Coefficient of variance of security A:
CV =

5
 100 =  100 = 50%
R
10
Standard deviation of security B:
B =
(
 Ri − RB
n −1
) = (12 − 8) + (− 8 − 8) + (20 − 8)
2
=
2
2
2
3 −1
16 + 256 + 144
2
= 208
= 14.42%
Coefficient of variance of security B:

14.42
CV =  100 =
 100 = 180.28%
8
R
Security A is suitable for investment. Because coefficient of variance of A is lower than
that of Security B.
Average/Expected Rate of Return and Coefficient of Variation Related Math:
Method-2
16. Probability and rate of return of two companies are as follows:
Probability
0.20
Rate of return of Moni Company
0.15
Rate of return of Roni Company
0.12
0.40
0.40
0.14
0.18
0.20
0.10
(Rajshahi Board-2022)
a) Calculate average rate of return of the two companies.
b) Which company is better in consideration of coefficient of variation? Evaluate.
Answer:
a) Calculate average rate of return Moni Company:
We know,
Chapter-9: Risk and Rate of Return
2/34
Average rate of return, 𝑅̅
= ∑(𝑅𝑖 × 𝑃𝑖 )
= (.15×0.20) + (.14×0.40) + (.18×0.40)
= 0.03 + 0.056 + 0.072
= 0.158
= 15.8%
Calculate average rate of return of Roni Company:
We know,
Average rate of return, 𝑅̅
= ∑(𝑅𝑖 × 𝑃𝑖 )
= (.12×0.20) + (.20×0.40) + (.10×0.40)
= 0.024 + 0.08 + 0.04
= 0.144
= 14.4%
Therefore, average rate of return of Moni Company and Roni Company are 15.8% and 14.4%
respectively.
b) Determining coefficient of variation of Moni Company:
We get from a,
Average rate of return, 𝑅̅ = 15.8% or 0.158
We know,
Standard deviation, σ = √∑(𝑅𝑖 − 𝑅̅)2 × 𝑃𝑖
= √(0.15 − 0.158)2 × 0.20 + (0.14 − 0.158)2 × 0.40 + (0.18 − 0.158)2 × 0.40
= √0.0000128 + 0.0001296 + 0.0001936
= √0.000336
= 0.0183303
= 1.83%
We know,
σ
Coefficient of variation (CV) = 𝑅̅ × 100
1.83
= 15.8 × 100
= 1.1158 × 100
= 11.58%
Determining coefficient of variation of Roni Company:
We get from a,
Average rate of return, 𝑅̅ = 14.4% or 0.144
We know,
Standard deviation, σ = √∑(𝑅𝑖 − 𝑅̅)2 × 𝑃𝑖
= √(0.12 − 0.144)2 × 0.20 + (0.20 − 0.144)2 × 0.40 + (0.10 − 0.144)2 × 0.40
= √0.0001152 + 0.0012544 + 0.0007744
= √0.002144
= 0.0463033
= 4.63%
Chapter-9: Risk and Rate of Return
2/35
σ
⸫ Coefficient of variation (CV) = 𝑅̅ × 100
4.63
= 14.4 × 100
= 0.3215 × 100
= 32.15%
Here, we can see that coefficient of variation or risk of Moni Company is 11.58%, which is
less than that of Roni Company 32.15%. Therefore, Moni Company is better between the two.
17. Mr. Hasan has decided to invest in two stocks of a company considering its financial
condition. Probability and rate of return of the two stocks are given below:
Economic
condition
Recession
Normal
Booming
Probability
25%
50%
30%
Rate of return
Stock-P
15%
20%
60%
Stock-Q
25%
18%
28%
(Cumilla Board-2022)
a) Determine average rate of return of stock-P and stock-Q.
b) In which stock should Mr. Hasan invest and why? Evaluate in the light of the stem.
Answer:
a) Determining average rate of return of stock-P:
We know,
Average rate of return, 𝑅̅𝑃
= ∑(𝑅𝑖 × 𝑃𝑖 )
= (15×0.25) + (20×0.50) + (60×0.30)
= 3.75 + 10 + 18
= 31.75%
Determining average rate of return of stock-Q:
We know,
Average rate of return, 𝑅̅𝑄
= ∑(𝑅𝑖 × 𝑃𝑖 )
= (25×0.25) + (18×0.50) + (28×0.30)
= 6.25 + 9 + 8.40
= 23.65%
Therefore, average rate of return of stock-P and stock-Q are 31.75% and 23.65% respectively.
b) To determine in which stock Mr. Hasan should invest, we need to compare coefficient of
variation of both stocks.
Determining coefficient of variation of stock-P:
From a we get,
Average rate of return, 𝑅̅𝑃 = 31.75%
We know,
Standard deviation, σP = √∑(𝑅𝑖 − 𝑅̅)2 × 𝑃𝑖
= √(15 − 31.75)2 × 0.25 + (20 − 31.75)2 × 0.50 + (60 − 31.75)2 × 0.30
= √70.140625 + 69.0312 + 239.41875
Chapter-9: Risk and Rate of Return
2/36
= √ 378.590625
= 19.46%
⸫ Coefficient of variation, CVP
𝜎
= 𝑃 × 100
𝑅𝑃
19.46
= 31.75 × 100
= 61.29%
Determining coefficient of variation of stock-Q:
From a we get,
Average rate of return, 𝑅̅𝑄 = 23.65%
We know,
Standard deviation, σQ = √∑(𝑅𝑖 − 𝑅̅)2 × 𝑃𝑖
= √(25 − 23.65)2 × 0.25 + (18 − 23.65)2 × 0.50 + (28 − 23.65)2 × 0.30
= √0.455625 + 15.96125 + 5.67675
= √ 22.093625
= 4.70%
⸫ Coefficient of variation, CVP
𝜎
= 𝑄 × 100
𝑅𝑄
4.70
= 23.65 × 100
= 19.87%
Here, coefficient of variation or risk of stock-Q is 19.87%, which is less than that of stock-P
61.29%. Hence, Mr. Hasan should invest in stock-Q.
18. Mr. Monir is the new manager of Alfa Ltd. He collects some information for himself. The
information are as follows:
Project ‘A’ probable return
Project ‘B’ probable return
Probability distribution
10%
9%
0.20
15%
12%
0.30
18%
20%
0.05
25%
30%
0.40
20%
15%
0.05
(Dhaka Board-2021)
(a) Calculate expected rate of return of project ‘B’ of Alpha Ltd. of Mr. Monir.
(b) Which of Mr. Munir’s projects is more risky in the light of the stem based on standard
deviation and co-efficient of variances? Analyze it.
Answer:
(a) Determining expected rate of return:
We know,
( )
Expected rate of return RB =  Ri  Pi
= (9  0.20) + (12  0.30) + (20  0.05) + (30  0.40) + (15  0.05)
= 1.8 + 3.6 + 1 + 12 + 0.75
= 19.15%
Therefore expected rate of return of project ‘B’ of Alpha Ltd. is 19.15%.
Chapter-9: Risk and Rate of Return
(b) Project A:
Here,
2/37
( )
Expected rate of return RA =  Ri  Pi
= (10 0.20)+ (15 0.30)+ (18 0.05)+ (25 0.40)+ (20 0.05)
= 2 + 4.5 + 0.9 + 10 + 1
= 18.4%
 Standard Deviation ( A ) =
=
 (R − R )  P
2
i
i
(10 − 18.4)2  0.20 + (15 − 18.4)2  0.30 + (18 − 18.4)2  0.05 + (25 − 18.4)2  0.40 + (20 − 18.4)2  0.05
= 14.112 + 3.468 + 0.008 + 17.424 + 0.128

= 35.14%
= 5.93%
Coefficient of variance (CVA ) =  A 100
RA
5.93
 100
18.4
= 32.22%
=
Project-B
We get from (a)
Expected rate of return ( RB ) = 19.15%
 Standard Deviation
=
( B ) =  (Ri − R)  Pi
2
(9 − 19.15)2  0.20 + (12 − 19.15)2  0.30 + (20 − 19.15)2  0.05 + (30 − 19.15)2  0.40 + (15 − 19.15)2  0.05
= 20.6045 + 15.33675 + 0.036125 + 47.089 + 0.861125
= 83.9275
= 9.16%
 Coefficient of variance
(CV B ) =  B  100
RB
9.16
 100
19.15
= 47.83%
=
Therefore, Mr. Monir’s project B is more risky. Because, standard deviation and coefficient
of variance of project B is higher than that of project A.
19. Mr. Jane Alom is interested to invest in project. He will take decision to invest in a project
out of two projects. Information regarding this tow projects are as follows:
Chapter-9: Risk and Rate of Return
2/38
Cash inflows
Project – A
Project – B
35,000
40,000
20,000
15,000
15,000
15,000
20,000
20,000
Probabilities
Project – A
Project – B
0.40
0.50
0.30
0.15
0.20
0.20
0.10
0.15
(Jashore Board-2021)
(a)
(b)
(a)
Calculate expected cash flows of the mentioned projects.
Investment in which project will be beneficial to Mr. Jane Alom? Give opinion with
mathematical explanation.
Answer:
Determining expected cash flow of project A:
( )
Expected cash flow RA =  Ri  Pi
= (35,000  0.40 ) + (20,000  0.30 ) + (15,000  0.20 ) + (20,000  0.10 )
= 14,000 + 6,000 + 3,000 + 2,000
= 25,000
Determining expected cash flow of project B:
(R ) =  R  P
i
i
Expected cash flow B
= (40,000  0.50 ) + (15,000  0.15) + (15,000  0.20 ) + (20,000  0.15)
= 20,000 + 2,250 + 3,000 + 3,000
= 28,250
(b)
Determining coefficient of variance of project A:
We get from (a)
( )
Expected cash flow RA = 25,000
 Standard deviation
=
( A ) =  (Ri − R)  Pi
2
(35,000 − 25,000)2  0.40 + (20,000 − 25,000)2  0.30 + (15,000 − 25,000)2  0.20 + (20,000 − 25,000)2  0.10
= 4,00,00,000 + 75,00,000 + 2,00,00,000 + 25,00,000
= 7,00,00,000
= 8,366.60
 Coefficient of variance (CV A ) =
A
 100
RA
8,366.60
 100
25,000
= 33.47%
=
Determining coefficient of variance of project B:
We get from (a)
Chapter-9: Risk and Rate of Return
2/39
( )
Expected cash flow RB = 28,250
 Standard Deviation ( B ) =
=
 (R − R )  P
2
i
i
(40,000 − 28,250)2  0.50 + (15,000 − 28,250)2  0.15 + (15,000 − 28,250)2  0.20 + (20,000 − 28,250)2  0.15
= 6,90,31,250 + 2,63,34,375 + 3,51,12,500 + 1,02,09,375
= 14,06,87,500
= 11,861.18
 Coefficient of variance
(CV B ) =  B  100
RB
11,861.18
 100
28,250
= 41.99 %
=
Therefore, investment in project A will be beneficial to Mr. Jane Alom. Because the coefficient
of variance or amount of risk of project A is lower than that of project B.
20. Mr. Anis invested in stocks of Jamuna Company. Beta of that company is 0.70 Risk free rate
of return and market return of the company is 6% and 10% respectively. As the expected rate of
return for Jamuna Company is not satisfactory, Anis is now eager to invest in the following two
securities. Data relating to these securities are as follows:
Probability
Rate of return
Security X
25%
28%
26%
.20
.50
.30
(a)
(b)
(a)
Security Y
30%
25%
27%
(Chattogram Board-2021)
Calculate expected rate of return of the securities of Jamuna Company.
Investment in which security is safer to Mr. Anis considering risk? Analyze.
Answer:
Determining expected rate of return of Security-X:
Here,
( )
Expected rate of return RX =  Ri  Pi
= (25  0.20) + (28  0.50) + (26  0.30)
= 5 + 14 + 7.80
= 26.80%
Determining expected rate of return of Security-Y:
Here,
Expected rate of return
(R ) =  R  P
Y
i
i
Chapter-9: Risk and Rate of Return
2/40
= (30  0.20 ) + (25  0.50 ) + (27  0.30 )
= 6 + 12.50 + 8.10
= 26.60%
Therefore, expected rate of return of both security X and Y is 26.80% and 26.60%
respectively.
(b)
Determining coefficient of variance of Security-X:
We get from (a)
( )
Standard Deviation ( ) =  (R − R )  P
Expected rate of return of security-X, R X = 26.80%
2
X
i
i
(25 − 26.80)2  0.20 + (28 − 26.80)2  0.50 + (26 − 26.80)2  0.30
=
= 0.648 + 0.72 + 0.192
= 1.56
= 1.25%
Coefficient of variance (CV X ) =   100
R
1.25
 100
26.80
= 4.66%
Determining coefficient of variance of Security-X:
We get from (a)
=
( )
Expected rate of return of security-Y, RY = 26.60%
( ) =  (R − R)  P
Standard Deviation  Y
=
2
i
i
(30 − 26.60)2  0.20 + (25 − 26.60)2  0.50 + (27 − 26.60)2  0.30
= 2.312 + 1.28 + 0.048
= 3.68
= 1.91%
Coefficient of variance (CVY )=   100
R
1.91
 100
26.60
= 7.18%
=
Therefore, Mr. Anis should invest in security-X. Because, coefficient of variance, that is,
amount of risk of security X is 4.66 % which is lower than coefficient of variance of
security-Y (7.18%).
Chapter-9: Risk and Rate of Return
2/41
21. Mr. Shipon is planning for investment either Chanachur project or Chips project. Relevant
data are given below.
Economic
condition
Boom
Probability
Rate of return
Chanchur project
Chips project
18%
22%
0.30
Normal
0.50
14%
17%
Recession
0.20
10%
09%
(a)
(b)
(Chattogram Board-2019, Set-1)
Calculate expected rate of return of Chanachur project.
For investment decision, which project is more logical for Mr. Shipon? Analyze your
comment.
(a)
Answer:
Expected rate of return of Chanachur project:
We know,
()
Expected rate of return R =  RiPi
= (18  0.3) + (14  0.5) + (10  0.2)
= 5.4 + 7 + 2
= 14.4%
Expected rate of return of Chanachur project is 14.4%.
(b)
To determine which project is more logical for Mr. Shipon to invest, we need to determine
coefficient of variance of both projects.
Chanchur project:
 (R − R)  Pi
Standard deviation ( ) =
2
=
(18 − 14.4)2  0.3 + (14 − 14.4)2  0.5 + (10 − 14.4)2  0.2
= 3.89 + 0.08 + 3.87
= 7.84
= 2.8%

Coefficient of variance CV= 100
R
2 .8
=
 100
14.4
= 19.44%
Chips project:
()
Expected rate of return R =  RiPi
Chapter-9: Risk and Rate of Return
2/42
= (22  0.3) + (17  0.5) + (9  0.2)
= 6.6 + 8.5 + 1.8
= 16.9%
Standard deviation ( ) =
 (R − R)  Pi
2
=
(22 − 16.9)2  0.3 + (17 − 16.9)2  0.5 + (9 − 16.9)2  0.2
= 7.8 + 0.005 + 12.48
= 20.285
= 4.50%

Co efficient of variance CV= 100
R
4.5
=
 100
16.9
= 26.65%
Here, coefficient of variance of chanachur project is lower than that of chips project. Hence,
chanachur project is less risky. So investing in chanachur project is more logical for Mr.
Shipon.
22. Probability distribution and rate of return of Keya and Asa Company are as follows:Provability
Keya Company’s income
Asa Company’s income
0.30
0.15
0.15
0.40
0.16
0.25
0.30
0.17
0.05
(Cumilla Board’16)
a) Calculate average rate of return of Keya and Asa Company.
b) In consideration of coefficient of variance, which of the two companies is better and why?
Give logic to support your answer.
Answer:
a) We know,
Average rate of return, 𝑅̅ = ∑ 𝑅𝑖 × 𝑃𝑖
Here,
R = Rate of return
P = Possibility
Average rate of return of Keya Company, 𝑅̅ = ∑ 𝑅𝑖 × 𝑃𝑖
= (0.15×0.30) + (0.16×0.40) + (0.17×0.30)
= 0.16 or 16%
Average rate of return of Asa Company, 𝑅̅ = ∑ 𝑅𝑖 × 𝑃𝑖
= (0.15×0.30) + (0.25×0.40) + (0.05×0.30)
= 0.16 or 16%
Answer: 16% and 16%
Chapter-9: Risk and Rate of Return
2/43
b) We know,
Standard deviation, 𝜎 = √∑ 𝑃𝑖 (𝑅𝑖 − 𝑅̅)2
Here,
𝑅𝑖 = 𝑅𝑎𝑡𝑒 𝑜𝑓 𝑟𝑒𝑡𝑢𝑟𝑛 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑦𝑒𝑎𝑟
𝑅̅ = 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑟𝑒𝑡𝑢𝑟𝑛
𝑃 = 𝑃𝑜𝑠𝑠𝑖𝑏𝑖𝑙𝑖𝑡𝑦
Standard deviation of Keya Company,
𝜎 = √(0.15 − 0.16)2 × 0.30 + (0.16 − 0.16)2 × 0.40 + (0.17 − 0.16)2 × 0.30
= √0.00003 + 0 + −0.00003
= √0.00006
= 0.007746
= 0.7746%
Standard deviation of Asa Company,
𝜎 = √(0.15 − 0.16)2 × 0.30 + (0.25 − 0.16)2 × 0.40 + (0.05 − 0.16)2 × 0.30
= √0.00003 + 0.00324 + 0.00363
= √0.0069
= 0.083066
= 8.3066%
𝜎
Keya Company’s coefficient of variance, CV = 𝑅 × 100
=
0.007746
0.16
× 100
= 4.84%
𝜎
Asa Company’s coefficient of variance, CV = 𝑅 × 100
=
0.083066
0.16
× 100
= 51.92%
In the stem, Keya Company is comparatively better than Asa Company. Though both the
companies have same rate of return, Asa Company’s coefficient of variance is far greater.
That means Asa Company is more risky than Keya Company. So, Keya Company is better
between the two.
Average/Expected Rate of Return and Coefficient of Variation Related Math:
Method-3
23. Mr. Rasel is an investor. He has identified two securities for investment. The securities are as
follows:
Rate of Return
Year
Security-A
Security -B
2019
30%
25%
2020
20%
35%
Chapter-9: Risk and Rate of Return
2/44
2021
15%
45%
Value of beta of Moon Company is 1.8. Rate of return of treasury bill is 9%. Market rate of
return is 18%. (Jashore Board-2022)
a) Determine expected rate of return of Moon Company.
b) Which security will be better for Mr. Rasel to invest? Analyze mathematically.
Answer:
a) Determining expected rate of return of Moon Company:
Here,
Risk free rate of return, Rf
= 9%
Market rate of return, Rm
= 18%
Unavoidable risk, β
= 1.8
We know,
̅
Expected rate of return, R
= 𝑅𝑓 + (𝑅𝑚 − 𝑅𝑓 ) × 𝛽
= 9% + (18% − 9%) × 1.8
= 9% + (9% × 1.8)
= 9% + 16.2%
= 25.20%
Therefore, expected rate of return of Moon Company is 25.20%.
b) To determine which security will be better for Mr. Rasel to invest, we need to compare
coefficient of variation or risk of the two securities.
Determining coefficient of variation of security-A:
Here,
∑ 𝑅𝑖𝐴
̅𝐴 =
Average rate of return of security-A, R
=
=
𝑛
30% + 20% + 15%
3
65%
3
= 21.67%
∑(𝑅𝑖𝐴 − 𝑅̅𝐴 )2
Standard deviation of security-A, σA = √
=√
𝑛−1
(30% −21.67%)2 + (20% −21.67%)2 + (15% −21.67%)2
3 −1
69.3889+2.7889+44.4889
=√
2
116.6667
=√
2
= √58.3334
= 7.64%
We know,
Coefficient of variation, CVA
𝜎
= 𝐴 × 100
𝑅𝐴
7.64
= 21.67 × 100
Chapter-9: Risk and Rate of Return
2/45
= 35.26%
Determining coefficient of variation of security-B:
Here,
̅ 𝐵 = ∑ 𝑅𝑖𝐵
Average rate of return of security-B, R
𝑛
=
25% +35% + 45%
=
105%
3
3
= 35%
∑(𝑅𝑖𝐵 − 𝑅̅𝐵 )2
Standard deviation of security-B, σB = √
=√
=√
𝑛−1
(25% −35%)2 + (35% −35%)2 + (45% −35%)2
3 −1
100+0+100
2
200
=√ 2
= √100
= 10%
𝜎
⸫ Coefficient of variation of security-B, CVB = 𝑅𝐵 × 100
𝐵
10
= 35 × 100
= 28.57%
It is noticeable that coefficient of variation or risk of security-B is 28.57%, which is lower
than that of security-A 35.26%. Therefore, security-B will be better for Mr. Rasel to invest.
24. Quantum Securities is an investor in the capital market. The firm’s portfolio manager has
identified the following two securities for investment and presented the following information
regarding risk and rate of return:
Security
Expected rate of return
Standard deviation
Beta (β)
A
12%
8%
2
B
18%
10%
3
Risk free rate of return is 6% and market rate of return is 14%. (Barishal Board-2022)
a) Determine required rate of return of the two securities mentioned in the stem.
b) What should be the investment decision of Quantum Securities? Explain.
Answer:
a) Determining required rate of return of security-A:
Here,
Risk free rate of return, Rf
= 6%
Market rate of return, Rm
= 14%
Unavoidable risk, β
=2
We know,
Chapter-9: Risk and Rate of Return
2/46
Required rate of return, 𝑅̅𝑖
= 𝑅𝑓 + (𝑅𝑚 − 𝑅𝑓 ) × 𝛽
= 6% + (14% − 6%) × 2
= 6% + (8% × 2)
= 6% + 16%
= 22%
Determining required rate of return of security-B:
We know,
Required rate of return, 𝑅̅𝑖
= 𝑅𝑓 + (𝑅𝑚 − 𝑅𝑓 ) × 𝛽
= 6% + (14% − 6%) × 3
= 6% + (8% × 3)
= 6% + 24%
= 30%
Therefore, required rate of return of security-A and security-B are 22% and 30% respectively.
b) To determine what the investment decision of Quantum Securities should be, we need to
compare coefficient of variation or risk of the two securities.
Determining coefficient of variation of security-A:
Given,
̅𝐴
Expected rate of return, R
= 12%
Standard deviation, σA = 8%
We know,
𝜎
= 𝑅𝐴 × 100
Coefficient of variation, CVA
𝐴
=
8
12
× 100
= 66.67%
Determining coefficient of variation of security-B:
Given,
̅𝐵
Expected rate of return, R
= 18%
Standard deviation, σB = 10%
𝜎
⸫ Coefficient of variation, CVB = 𝑅𝐵 × 100
𝐵
10
= 18 × 100
= 55.56%
It is noticeable that coefficient of variation or risk of security-B is 55.56%, which is lower
than coefficient of variation or risk of security-A 66.67%. Therefore, Quantum Securities
should invest in security-B.
25. Mijan Chemical Company wants to invest in any one of the following two stocks. Economic
condition and probability distribution are given bellow:
Economic
condition
Bullish
Probability distribution
0.40
Rate of return
Stock-A
Stock-B
12%
16%
Chapter-9: Risk and Rate of Return
2/47
Bearish
0.30
10%
-5%
Normal
0.30
11%
14%
Expected rate of return of stock-A and B are 11.1% and 9.1% respectively. On the other hand,
beta of Finix Company is 1.8%. Market rate of return is 15% and rate of return of treasury
bill is 6%. (Dinajpur Board-2022)
a) Determine expected rate of return of Finix Company.
b) In which stock should Mijan Chemical Company invest? Give your opinion with
mathematical logic.
Answer:
a) Determining expected rate of return of Finix Company:
Here,
Risk free rate of return, Rf
= 6%
Market rate of return, Rm
= 15%
Unavoidable risk, β
= 1.8
We know,
Expected rate of return, 𝑅̅𝑖
= 𝑅𝑓 + (𝑅𝑚 − 𝑅𝑓 ) × 𝛽
= 6% + (15% − 6%) × 1.8
= 6% + (9% × 1.8)
= 6% + 16.20%
= 22.20%
Therefore, expected rate of return of Finix Company is 22.20%.
b) To determine which stock Mijan Chemical Company should invest in, we need to compare
coefficient of variation or risk of both stocks.
Determining coefficient of variation of stock-A:
Given,
Expected rate of return of stock-A, 𝑅̅𝐴 = 11.1%
⸫ Standard deviation, σA = √∑(𝑅𝑖 − 𝑅̅)2 × 𝑃𝑖
= √(12 − 11.1)2 × 0.40 + (10 − 11.1)2 × 0.30 + (11 − 11.1)2 × 0.30
= √0.324 + 0.363 + 0.003
= √ 0.69
= 0.83%
We know,
σ
Coefficient of variation (CVA) = 𝑅̅𝐴 × 100
𝐴
0.83
= 11.1 × 100
= 7.48%
Determining coefficient of variation of stock-B:
Given,
Expected rate of return of stock-B, 𝑅̅𝐵 = 9.1%
⸫ Standard deviation, σB = √∑(𝑅𝑖 − 𝑅̅)2 × 𝑃𝑖
Chapter-9: Risk and Rate of Return
2/48
= √(16 − 9.1)2 × 0.40 + (−5 − 9.1)2 × 0.30 + (14 − 9.1)2 × 0.30
= √19.044 + 59.643 + 7.203
= √ 85.89
= 9.27%
We know,
σ
Coefficient of variation (CVB) = 𝑅̅𝐵 × 100
𝐵
9.27
= 9.1 × 100
= 101.87%
Here, it is noticeable that coefficient of variation or risk of stock-A is 7.48%, which is lower
than coefficient of variation or risk of stock-B (101.87%). Therefore, Mijan Chemical
Company should invest in stock-A.
26. Mr. Nazrul, a retired government officer, is keen to invest his pension money in the stock
market. He is reviewing the last four years earnings of the following two companies:
Year
Rate of return (%)
Alif ltd.
15
10
10
5
10
-
2017
2018
2019
2020
Average return
Standard deviation
Mim ltd.
20
0
10
10
10
8.16
On the other hand, his younger brother Kamrul has invested in shares of Lam Company Ltd.
which market risk value is 1.2. Market rate of return and risk free rate of return are 12% and
5% respectively. (Barishal Board-2021)
(a) Determine expected rate of return of Kamrul.
(b) “Investing in shares of Alif Ltd. is less risky than Mim Ltd.” - Analyze the statement.
Answer:
(a) Here,
( )
Risk free rate of return, R f = 5%
Market rate of return, (Rm ) = 12%
Market risk, ( ) = 1.2
We know,
(
)
Expected rate of return, (Ri ) = R f + Rm − R f  
= 5% + (12%-5%)  1.2
= 5% + 8.40%
= 13.40%
Chapter-9: Risk and Rate of Return
2/49
Therefore, expected rate of return of Kamrul is 13.40%.
(b) Determining coefficient of variance of Alif Ltd.:
()
Average rate of return R = 10% (given)
(
 Ri − R
Standard Deviation:  =
n −1
=
)
2
(15 − 10 )2 + (10 − 10 )2 + (10 − 10)2 + (5 − 10)2
4 −1
=
25 + 0 + 0 + 25
3
=
50
3
= 16.67
= 4.08%
Coefficient of variance CV=   100
R
4.08%
 100
10
= 40.80%
=
Determining coefficient of variance of Mim Ltd.:
Average rate of return
(R) = 10 %
Standard Deviation ( ) = 8.16%
Coefficient of variance CV=   100
R
8.16%
 100
10
= 81.60%
It is noticeable that, coefficient of variance or risk of Alif Ltd is 40.80% which is lower than
coefficient of variance or risk of Mim Ltd. (81.60%). That means risk of share of Alif Ltd. is
less. So, it can be said that Investing in Alif Ltd.’s share is less risky than investing Mim Ltd.
27. Mr. Shihab is analyzing last 4 years’ income of the following two companies for investment:
=
Year
2015
1016
2017
2018
Padma Ltd.
Meghna
Ltd.
10%
6%
14%
10%
16%
20%
20%
24%
Average
return
15%
15%
Deviation
48.10%
?
Chapter-9: Risk and Rate of Return
2/50
On the other hand, Mr. Miraj, a friend of Mr. Shihab, has bought Surma Ltd.’s share market
risk of which is 1.40. Mention that market rate of return is 14% and risk-free rate of return
is 5%. (Cumilla Board-2019)
(a)
(b)
What is the expected rate of return of Mr. Miraj? Calculate.
Justify the investment decision of Mr. Shihab in the stem.
(a)
Answer:
Here,
Rf = 5%
Rm = 14%
 = 1.4
We know,
Expected rate of return = Rf + (Rm − Rf )
= 5 + (14-5)  1.4
=17.6%
(b)
()
Here, average rate of return of Meghna Ltd R = 15%
(
 R−R
So, standard deviation of Meghna Ltd.,  =
n −1
=
)
2
(6 − 15)2 + (10 − 15)2 + (20 − 15)2 + (24 − 15)2
4 −1
=
81 + 25 + 25 + 81
3
=
212
3
= 70.67
= 8.41%
So, coefficient of variance of Meghna Ltd, CV=

R
100
8.41
 100
15
= 56.07%
=
Coefficient of variance of Padma Ltd, CV=

R
100
48.10
 100
15
= 320.67%
As coefficient of variance of Padma Ltd is higher than coefficient of variance of Meghna
Ltd., Mr. Shihab should invest in Meghna Ltd. Because Meghna Ltd. is less risky.
=
Chapter-9: Risk and Rate of Return
2/51
28. Mr. Fayez Ahmed is interested to invest in financial market. He knows about portfolio
investment. Now he wants to invest in any of the two securities:
Year
Rate of return
Security-A
Security-B
2014
30%
20%
2015
21%
15%
2016
16%
32%
Beta of Simon Company is 1.70. Market rate of return is 15% and rate of return of treasury bill is
6%. (Sylhet Board’17)
a) Determine expected rate of return of Simon Company in the stem.
b) Which security is logical for Fayez Ahmed to invest according to the stem? Analyze.
Answer:
a) Determining expected rate of return of Simon Company:
Here,
Risk free rate of return, Rf
= 6%
Market rate of return, Rm
= 15%
Unavoidable risk, β
= 1.70
We know,
̅
Expected rate of return, R
= 𝑅𝑓 + (𝑅𝑚 − 𝑅𝑓 ) × 𝛽
= 6% + (15% − 6%) × 1.70
= 6% + (9% × 1.70)
= 6% + 15.30%
= 21.30%
Therefore, expected rate of return of Simon Company is 21.30%.
Answer: 21.30%.
b) To know investing in which security will be logical for Fayez Ahmed, coefficient of variance
of both the securities is to be determined.
Determining coefficient of variance of security-A:
̅𝐴 = ∑ 𝑅𝑖𝐴
Expected rate of return of security-A, R
𝑛
=
30% + 21% + 16%
=
67%
3
3
= 22.33%
∑(𝑅𝑖𝐴 − 𝑅̅𝐴 )2
Standard deviation of security-A, σA = √
=√
𝑛−1
(30% −22.33%)2 + (21% −22.33%)2 + (16% −22.33%)2
3 −1
100.6667
=√
2
= √50.33335
Chapter-9: Risk and Rate of Return
2/52
= 7.095%
Now
𝜎
Coefficient of variance of security-A, CVA = 𝑅𝐴 × 100
𝐴
7.095
= 22.33 × 100
= 31.77%
Determining coefficient of variance of security-B:
̅ 𝐵 = ∑ 𝑅𝑖𝐵
Expected rate of return of security-B, R
𝑛
=
20% + 15% + 32%
=
67%
3
3
= 22.33%
∑(𝑅𝑖𝐵 − 𝑅̅𝐵 )2
Standard deviation of security-B, σB = √
=√
𝑛−1
(20% −22.33%)2 + (15% −22.33%)2 + (32% −22.33%)2
3 −1
152.66667
=√
2
= √76.33335
= 8.74%
𝜎
Now, coefficient of variance of security-B, CVB = 𝑅𝐵 × 100
𝐵
8.74
= 22.33 × 100
= 39.14%
Therefore, security-B (39.14%) is more risky than security-A (31.77%). So, I think that it will
be logical for Mr. Fayez Ahmed to invest in security-A.
29. Mr. Musharraf is a general businessman. He has never invested in stock market before. Now
he has Tk. 2,00,000 in hand. He wants to invest this money. He knows that you are studying in
Finance and is asking for your advice. After market research, you have selected 3 securities which
are suitable for investment. Rate of return and standard deviation of the securities are as follows:
Security Expected return Standard deviation
A
12%
3.10%
B
12.5%
3.20%
C
13%
3.50%
(Dinajpur Board’17)
a) Determine coefficient of variance (CV) of the securities mentioned in the stem.
b) According to the information in the stem, which security will you suggest Mr. Musharraf to
invest and why?
Answer:
a) Determining coefficient of variance (CV) of the securities mentioned in the stem:
We know,
Chapter-9: Risk and Rate of Return
Coefficient of variance, CV
2/53
𝜎
= 𝑅̅ × 100
In case of security-A,
Standard deviation, σ = 3.10%
̅
Expected return, R
=12%
So, coefficient of variance of security-A, CVA
3.10
= 12
× 100
̅̅̅̅
= 25.83%
In case of security-B,
Standard deviation, σ = 3.20%
̅
Expected return, R
=12.50%
So, coefficient of variance of security-B, CVB
3.20
= ̅̅̅̅
12.50
× 100
= 25.60%
In case of security-C,
Standard deviation, σ = 3.50%
̅
Expected return, R
=13%
So, coefficient of variance of security-C, CVC
3.50
= 13 × 100
= 26.92%
Hence, coefficient of variance of security-A, B, and C are 25.83%, 25.60% and 26.92
respectively.
b) According to the information in the stem,
Coefficient of variance of security-A = 25.83%
Coefficient of variance of security-B = 25.60%
Coefficient of variance of security-C = 26.92%
Therefore, in consideration of risk, B < A < C.
Here, coefficient of variance of security-B is the least. So, its risk is the lowest and its
acceptability is also high. Hence, I shall recommend Mr. Musharraf to invest in security-B.
30. Mr. Azad and Mr. Jahangir are two friends who are interested to invest in stock market. Mr.
Azad is considering two securities for investment where he will take the decision based on relative
risk. Information about the securities are as follows:
Probability
Rate of return
Stock-X
Stock-Y
0.30
12%
19%
0.30
16%
14%
0.20
18%
9%
0.20
19%
11%
On the other hand, Mr. Jahangir has decided to invest in a security of which beta is 1.5. Note that
market rate of return is 12% and risk free return is 5%. (Barishal Board’17)
a) Calculate required rate of return of Mr. Jahangir's security.
b) In consideration of risk, which security should Mr. Azad invest in? Analyze.
Chapter-9: Risk and Rate of Return
2/54
Answer:
a) Calculating required rate of return of Mr. Jahangir's security:
We know,
Expected rate of return, Ri = 𝑅𝑓 + (𝑅𝑚 − 𝑅𝑓 ) × 𝛽
Here,
Risk free rate of return, Rf
Market rate of return, Rm
Beta, β
∴ Expected rate of return, Ri
= 5%
= 12%
= 1.50
= 5 + (12 − 5) × 1.50
= 5 + (7 × 1.50)
= 5 + 10.50
= 15.50%
Therefore, required rate of return of Mr. Jahangir's security is 15.50%.
Answer: 15.50%.
b) To know which security Mr. Azad should invest in consideration of risk, coefficient of
variance of both the securities is to be determined.
We know,
Coefficient of variance, CV
𝜎
= ̅ × 100
𝑅
Calculating coefficient of variance of stock-X:
Here,
̅ = ∑𝑛𝑖=1(𝑅𝑖 × 𝑃𝑖 )
Expected return, R
= (12 × 0.30) + (16 × 0.30) + (18 × .20) + (19 × .20)
= 15.80%
∴ Standard deviation, σ = √∑(𝑅𝑖 − 𝑅̅)2 × 𝑃𝑖
= √(12 − 15.8)2 × 0.30 + (16 − 15.8)2 × 0.30 + (18 − 15.8)2 × 0.20 + (19 − 15.8)2 × 0.20
= √4.332 + 0.012 + 0.968 + 2.048
= √7.36
= 2.71%
∴ Coefficient of variance, CV
2.71
= 15.80 × 100
= 17.15%
Calculating coefficient of variance of stock-Y:
Here,
̅ = ∑𝑛𝑖=1(𝑅𝑖 × 𝑃𝑖 )
Expected return, R
= (19 × 0.30) + (14 × 0.30) + (9 × .20) + (11 × .20)
= 13.90%
∴ Standard deviation, σ = √∑(𝑅𝑖 − 𝑅̅)2 × 𝑃𝑖
= √(19 − 13.9)2 × 0.30 + (14 − 13.9)2 × 0.30 + (9 − 13.9)2 × 0.20 + (11 − 13.9)2 × 0.20
= √7.803 + 0.003 + 4.802 + 1.682
= √14.29
Chapter-9: Risk and Rate of Return
2/55
= 3.78%
3.78
∴ Coefficient of variance, CV
= 13.90 × 100
= 27.20%
Here, coefficient of variance of stock-X is 17.15% and that of stock-Y is 27.20%. That is,
stock-Y is more risky. Hence, Mr. Azad should invest in stock-X in consideration of risk.
Standard Deviation and Coefficient of Variation Related Math
Method-1
31. Mr. Akhtar is interested to invest Tk. 1,00,000 in a profitable security. He has the following
two securities. He wants to select the security which has less risk.
Rate of return
Year
Security -Ka
Security-Kha
2020
16%
19%
2021
145
-7%
2022
15%
19%
(Jashore Board-2023)
a) Calculate standard deviation of Security-Kha based on stem.
b) "Security -Ka is more risky" – Comment on the statement based on mathematical logic.
Answer:
a) Determining the standard deviation of Security-Kha:
Here,
∑𝑅
Average rate of return, 𝑅̅𝐾ℎ𝑎
= 𝑛𝑖
=
19 − 7 + 19
3
31
= 3
= 10.33%
We know,
∑(𝑅𝑖 −𝑅̅𝐾ℎ𝑎 )2
Standard deviation, σKha = √
𝑛−1
(19−10.33)2 +(−7−10.33)2 +(19−10.33)2
=√
3−1
75.1689 + 300.3289 + 75.1689
=√
2
450.6667
=√
2
= √225.33335
= 15.01%
Therefore, the standard deviation of Security-Kha is 15.01%
b) Determining the coefficient of variation of Security-Kha:
We get from a,
Average rate of return of Security-Kha, 𝑅̅𝐾ℎ𝑎 = 10.33%
Chapter-9: Risk and Rate of Return
2/56
Standard deviation of Security-Kha, σKha
= 15.01%
We know,
𝜎
Coefficient of variation, CVKha = 𝑅𝐾ℎ𝑎 × 100
𝐾ℎ𝑎
15.01
= 10.33 × 100
= 145.30%
Determining the coefficient of variation of Security-Ka:
Here,
Average rate of return, 𝑅̅𝐾𝑎
=
=
∑ 𝑅𝑖
𝑛
16 + 14 + 15
3
45
= 3
= 15%
We know,
Standard deviation, σKa = √
∑(𝑅𝑖 −𝑅̅𝐾𝑎 )2
𝑛−1
(16−15)2 +(14−15)2 +(15−15)2
=√
3−1
1+1+0
=√
2
2
= √2
= √1
= 1%
⸫ Coefficient of variation, CVKa
𝜎
= 𝑅𝐾𝑎 × 100
𝐾𝑎
1
= 15 × 100
= 6.67%
Here, coefficient of variation or risk margin of Security-Ka is 6.67%. On the other hand,
coefficient of variation of Security-Kha is 145.30%. Hence, the statement "Security -Ka is
more risky" is incorrect.
32. Two friends, Shaon and Shovon, trades share regularly. Shaon invested in X Company’s share
and Shovon in Y Company’s share equally. Returns from the two shares for the last three years
are mentioned below:
Year
1
2
3
X Company’s income
17%
20%
16%
Y Company’s income
15%
12%
18%
(Cumilla Board-2023)
a) Determine the amount of Shaon’s risk.
b) Whose investment is more risky in light of the information given in the stem? Analyze.
Chapter-9: Risk and Rate of Return
2/57
Answer:
a) Determining the standard deviation or risk of Shaon’s investment in X Company’s share:
Here,
∑𝑅
Average rate of return, 𝑅̅𝑋
= 𝑛𝑖
17 + 20 + 16
=
3
53
= 3
= 17.67%
We know,
∑(𝑅𝑖 −𝑅̅𝑋 )2
Standard deviation, σX = √
𝑛−1
(17−17.67)2 +(20−17.67)2 +(16−17.67)2
=√
3−1
0.4489 + 5.4289 + 2.7889
=√
2
8.6667
=√
2
= √4.33335
= 2.08%
Therefore, the standard deviation or risk of Shaon’s investment in X Company’s share is
2.08%.
b) Determining the coefficient of variation of X Company’s share invested by Shaon:
We get from a, X Company’s
Average rate of return, 𝑅̅𝑋
= 17.67%
Standard deviation, σX
= 2.08%
We know,
𝜎
Coefficient of variation, CVX = 𝑅𝑋 × 100
𝑋
2.08
= 17.67 × 100
= 11.77%
Determining the coefficient of variation of Y Company’s share invested by Shovon:
Here,
Average rate of return, 𝑅̅𝑌
=
=
∑ 𝑅𝑖
𝑛
15 + 12 + 18
3
45
= 3
= 15%
We know,
∑(𝑅𝑖 −𝑅̅𝑌 )2
Standard deviation, σY = √
𝑛−1
Chapter-9: Risk and Rate of Return
2/58
(15−15)2 +(12−15)2 +(18−15)2
=√
3−1
0+9+9
=√
2
18
=√
2
= √9
= 3%
𝜎
⸫ Coefficient of variation, CVY = 𝑅𝑌 × 100
𝑌
3
= 15 × 100
= 20%
So, Shovon’s investment is more risky. Because the coefficient of variation of Y Company’s
share invested by Shovon is 20%, which is more than the coefficient of variation of 11.77%
of Y Company’s share invested by Shaon.
33. Mr. Azim is interested to invest in the better of the following two securities. The rates of
return of the two securities are given below:
Year
Security-M
Security-N
1st
- 10%
19%
2nd
5%
21%
3rd
4th
20%
35%
15%
9%
(Mymensingh Board-2023)
a) Determine standard deviation of security-M.
b) Which security will be profitable for Mr. Azim to invest in? Analyze.
Answer:
a) Determining the standard deviation of security-M:
Here,
∑𝑅
Average rate of return, 𝑅̅𝑀
= 𝑛𝑖
=
−10 + 5 + 20 + 35
4
50
= 4
= 12.50%
We know,
∑(𝑅𝑖 −𝑅̅𝑀 )2
Standard deviation, σM = √
𝑛−1
(−10−12.50)2 +(5−12.50)2 +(20−12.50)2 +(35−12.50)2
=√
4−1
506.25 + 56.25 + 56.25 + 506.25
=√
3
1,125
=√ 3
Chapter-9: Risk and Rate of Return
2/59
= √375
= 19.36%
Therefore the standard deviation of security-M is 19.36%.
b) Determining the coefficient of variation of security-M:
We get from a,
Average rate of return, 𝑅̅𝑀
= 12.50%
Standard deviation, σM
= 19.36%
We know,
𝜎
Coefficient of variation, CVM = 𝑅𝑀 × 100
𝑀
=
19.36
12.50
× 100
= 154.88%
Determining the coefficient of variation of security-N:
Here,
Average rate of return, 𝑅̅𝑁
=
=
∑ 𝑅𝑖
𝑛
19 + 21 + 15 + 9
4
64
= 4
= 16%
⸫ Standard deviation, σN
∑(𝑅𝑖 −𝑅̅𝑁 )2
=√
𝑛−1
(19−16)2 +(21−16)2 +(15−16)2 +(9−16)2
=√
4−1
9 + 25 + 1 + 49
=√
3
84
=√3
= √28
= 5.29%
We know,
𝜎
Coefficient of variation, CVN = 𝑅𝑁 × 100
𝑁
5.29
= 16 × 100
= 33.06%
Here, the coefficient of variation or risk of security-N is 33.06%, which is lower than securityM’s coefficient of variation of 154.88%. In this case, we can say that security-N will be
profitable for Mr. Azim to invest.
34. The rate of return of two projects for the last 3 years is as follows:
Year
Project-A
Project-B
2018
8%
10%
2019
9%
-2%
Chapter-9: Risk and Rate of Return
2/60
2020
12%
16%
(Rajshahi Board-2022)
a) Calculate standard deviation of project-A?
b) Which project should be selected to invest in consideration of risk? Analyze.
Answer:
a) Calculating standard deviation of project-A:
Here,
∑𝑅
Average rate of return, (𝑅̅𝐴 ) = 𝑛 𝑖
8+9+12
=
29
3
=
3
= 9.67%
We know,
∑(𝑅𝑖 −𝑅̅𝐴 )2
Standard deviation, σA = √
𝑛−1
(8−9.67)2 +(9−9.67)2 +(12−9.67)2
=√
3−1
2.7889+0.4489+5.4289
=√
2
8.6667
=√
2
= √4.3334
= 2.08%
Therefore, standard deviation of project-A is 2.08%.
b) Determining coefficient of variation (CV) of project-A:
We get from a,
Expected rate of return, 𝑅̅𝐴
= 9.67%
Standard deviation, σA = 2.08%
We know,
σ
Coefficient of variation (CVA) = 𝑅̅𝐴 × 100
𝐴
2.08
= 9.67 × 100
= 0.2151 × 100
= 21.51%
Determining coefficient of variation (CV) of project-B:
Here,
∑𝑅
Average rate of return, (𝑅̅𝐵 ) = 𝑛 𝑖
=
10+(−2)+16
3
24
= 3
= 8%
We know,
∑(𝑅𝑖 −𝑅̅𝐵 )2
Standard deviation, σB = √
𝑛−1
Chapter-9: Risk and Rate of Return
2/61
(10−8)2 +(−2−8)2 +(16−8)2
=√
3−1
4+100+64
=√
2
168
=√
2
= √84
= 9.17%
σ
⸫ Coefficient of variation (CVB) = 𝑅̅𝐵 × 100
𝐵
=
9.17
8
× 100
= 1.1463 × 100
= 114.63%
Here, we can see that coefficient of variation or risk of project-A is 21.51% which is less than
coefficient of variation of project-B 114.63%. Therefore, project-A should be selected to
invest in consideration of risk.
35. Mithun is an investor. He wants to invest some of his money in share market. So, he collected
last four years’ profit earning information of Bata shoes and Coca-Cola from share market. The
information are given bellow:
Year
Rate of return of Bata shoes
Rate of return of Coca-Cola
1
15%
25%
2
17%
7%
3
19%
32%
4
21%
10%
(Chattogram Board-2022)
a) Determine standard deviation of Bata shoes.
b) In which security should Mithun invest? Give your opinion.
Answer:
a) Determining standard deviation of Bata shoes:
Here,
̅ 𝐵 = ∑ 𝑅𝑖𝐵
Average rate of return of Bata shoes, R
𝑛
15+ 17 + 19+21
=
4
72
= 4
= 18%
We know,
∑(𝑅𝑖𝐵 − 𝑅̅𝐵 )2
Standard deviation of Bata shoes, σB = √
=√
𝑛−1
(15 −18)2 + (17 −18)2 + (19 −18)2 + (21 −18)2
4 −1
Chapter-9: Risk and Rate of Return
2/62
9+1+1+9
=√
3
20
=√3
= √6.6667
= 2.58%
So, Standard deviation of Bata shoes is 2.58%
b) To find out which security Mithun should invest in, we need to compare coefficient of
variation of both organizations.
Determining coefficient of variation of Bata shoes:
We get from a,
̅𝐵
Average rate of return of Bata shoes, R
= 18%
Standard deviation of Bata shoes, σB
= 2.58%
We know,
𝜎
Coefficient of variation, CVB = 𝐵 × 100
𝑅𝐵
2.58
= 18 × 100
= 14.33%
Determining coefficient of variation of Coca-Cola:
Here,
̅ 𝐶 = ∑ 𝑅𝑖𝐶
Average rate of return of Coca-Cola, R
𝑛
25+7+ 32+10
=
4
74
= 4
= 18.50%
We know,
Standard deviation of Coca-Cola, σC = √
=√
∑(𝑅𝑖𝐶 − 𝑅̅𝐶 )2
𝑛−1
(25 −18.50)2 + (7 −18.50)2 + (32 −18.50)2 + (10 −18.50)2
4 −1
42.25+132.25+182.25+72.25
=√
3
429
=√
3
= √143
= 11.96%
𝜎
⸫ Coefficient of variation, CVC = 𝑅𝐶 × 100
𝐶
11.96
= 18.50 × 100
= 64.65%
Chapter-9: Risk and Rate of Return
2/63
We can see that coefficient of variation of Bata shoes is 14.33% and that of Coca-Cola is
64.65%. Here, coefficient of variation or risk of Bata shoes is comparatively less. Therefore,
Mithun should invest in Bata shoes.
36. The particulars of last 4 years’ earning of Gobindaganj Ltd. is given below:
Year
1
2
3
4
Project-A
12
14
-2
16
Project-B
10
24
12
-6
(Sylhet Board2022)
a) Calculate standard deviation of project-A.
b) Which project of the stem is safer to invest? Analyze mathematically.
Answer:
a) Calculating standard deviation of project-A:
Here,
̅𝐴 = ∑ 𝑅𝑖𝐴
Average rate of return of project-A, R
=
𝑛
12+14+(−2)+16
4
40
= 4
= 10%
We know,
∑(𝑅𝑖𝐴 − 𝑅̅𝐴 )2
Standard deviation of project-A, σA = √
=√
𝑛−1
(12 −10)2 + (14 −10)2 + (−2 −10)2 + (16 −10)2
4 −1
4+16+144+36
=√
3
200
=√ 3
= √66.67
= 8.17%
Therefore, standard deviation of project-A is 8.17%.
b) To determine which project of the stem is safer to invest, we need to compare coefficient of
variation of both projects.
Determining coefficient of variation of project-A:
We get from a,
Average rate of return, 𝑅̅𝐴
= 10%
Standard deviation, σA = 8.17%
We know,
σ
Coefficient of variation (CVA) = 𝑅̅𝐴 × 100
𝐴
8.17
= 10 × 100
Chapter-9: Risk and Rate of Return
2/64
= 0.2514 × 100
= 81.70%
Determining coefficient of variation of project-B:
Here,
̅ 𝐵 = ∑ 𝑅𝑖𝐵
Average rate of return of project-B, R
=
𝑛
10+24+12+(−6)
4
40
= 4
= 10%
We know,
Standard deviation of project-B, σB = √
=√
∑(𝑅𝑖𝐵 − 𝑅̅𝐵 )2
𝑛−1
(10 −10)2 + (24 −10)2 + (12 −10)2 + (−6 −10)2
4 −1
0+196+4+256
=√
3
456
=√ 3
= √152
= 12.33%
σ
⸫ Coefficient of variation (CVB) = 𝑅̅𝐵 × 100
𝐵
=
12.33
10
× 100
= 1.2330 × 100
= 123.30%
It is noticeable that standard deviation or risk of project-A is 81.17%, which is less than that
of project-B 123.30%. Therefore, project-A of the stem is safer to invest.
37. The rate of return of last 3 years of Akash Food Ltd. And Shishir Food Ltd. Is as follows:
Project/year
Akash Food Ltd
Shishir Food Ltd
2014
12%
15%
2015
11%
-03%
2016
14%
18%
(Dhaka Board-2021)
(a) Determine standard deviation of Akash Food Ltd.
(b) Considering the level of risk, investment is reasonable in which company? Evaluate it.
Answer:
(a) Determining standard deviation of Akash Food Ltd.
The average rate of return of Akash Food Ltd:
()
Average rate of return R =  Ri = 12 + 11 + 14 = 37 = 12.33%
n
3
3
Chapter-9: Risk and Rate of Return
2/65
Standard deviation of Akash Food Ltd:
(
 Ri − R
=
n −1
)2 = (12 − 12.33)2 + (11 − 12.33)2 + (14 − 12.33)2
3 −1
=
0.1089 + 1.7689 + 2.7889
2
=
4.6667
2
= 2.33335
= 1.53%
Standard deviation of Akash Food Ltd is 1.53%.
(b) Determining coefficient of variance of Akash Foods Ltd foods:
We get from (a)
()
Average rate of return, R = 12.33%
Standard deviation, ( ) = 1.53%
Coefficient of variance CV=  100
R
1.53
 100
12.33
= 12.41%
Determining coefficient of variance of Shishir Foods Ltd:
Average rate of return of Shishir Foods Ltd:
=
()
Average rate of return R =  R i = 15 − 03 + 18 = 30 = 10%
n
3
3
Standard deviation of Shishir Foods Ltd:
 =
(
 Ri − R
n −1
) = (15 − 10) + (− 03 − 10) + (18 − 10)
2
2
2
2
3 −1
=
25 + 169 + 64
2
=
258
2
= 129
= 11.36%
Coefficient of variance, CV=   100
R
11.36
 100
10
= 113.57%
=
Therefore, investing in Akash Foods Ltd. will be appropriate. Because, coefficient of variance
that mean amount of risk of Akash foods Ltd is 12.41% which is lower than coefficient of
variance of Shishir Foods Ltd. (113.57%).
Chapter-9: Risk and Rate of Return
2/66
38. Mr. Simul Ahmed is interested in investing financial market. He has collected information of
two companies for last 3 years:
Rate of Return
Coca Cola Foods Ltd.
Alom Foods Ltd.
16%
13%
-4%
11%
19%
14%
Year
2018
2019
2020
(Jashore Board-2021)
(a)
(b)
Calculate standard deviation of Coca Cola Foods Ltd.
Based on stem, which company is more risky? Give opinion with mathematical explanation.
(a)
Answer:
Determining standard deviation of Coca cola’s Food Ltd.
The average rate of return of Coca cola’s Food Ltd:
()
Average rate of return R =  Ri = 16 − 4 + 19 = 31 = 10.33%
n
3
3
Standard deviation of Coca cola’s Food Ltd:
(
 Ri − R
=
n −1
) = (16 − 10.33) + (− 4 − 10.33) + (19 − 10.33)
2
2
2
2
3 −1
=
32.11 + 205.44 + 75.11
2
=
312.67
2
= 156.33
= 12.50%
Standard deviation of Coca cola’s Food Ltd is 12.50%.
(b)
Determining coefficient of variance of Coca cola Foods Ltd foods:
We get from (a)
()
Average rate of return, R = 10.33%
Standard Deviation ( ) = 12.50%
Coefficient of variance CV=   100
R
12.50
 100
10.33
= 121.01%
Determining coefficient of variance of Alam Foods Ltd:
Average rate of return of Alam Foods Ltd:
=
()
Average rate of return R =  R i = 13 + 11 + 14 = 38 = 12.67%
n
3
3
Chapter-9: Risk and Rate of Return
2/67
Standard deviation of Alam Foods Ltd:
 =
(
 Ri − R
n −1
) = (13 − 12.67) + (11 − 12.67) + (14 − 12.67)
2
2
2
2
3 −1
=
0.1089 + 2.7889 + 1.7689
2
=
4.6667
2
= 2.33335
= 1.53%
Coefficient of variance CV=   100
R
1.53
 100
12.67
= 12.08%
=
Therefore Coca cola foods Ltd is high risky. Because coefficient of variance, that means,
amount of risk of Coca cola foods Ltd is 121.01% which is higher than coefficient of
variance of Alam’s Foods Ltd. (12.08%).
39. Rate of returns of Shitolakkha Ltd. And Kornafuli Ltd. For the last three years are as follows:
Shitolakkha: 12%, 10%, 14%. Kornafuli: 20% 8%, 11%.
Mr. Riaj is not agreed to take more risk, so he makes decision for investing same amount of
investment in both the companies. (Chattogram Board-2021)
(a)
Calculate standard deviation of Shitolakkha Ltd.
(b)
Evaluate the investment decision of Mr. Riaj on the basis of co-Efficient of variance.
(a)
Answer:
Determining standard deviation Shitolakkha Ltd.
( ) = nRi = 12 + 103 + 14 = 363 = 12%
Average rate of return R
 (R − R )
Standard deviation ( ) =
2
i
n −1
(12 − 12 ) + (10 − 12 ) + (14 − 12 )
2
=
2
3 −1
2
Chapter-9: Risk and Rate of Return
2/68
=
0+4+4
2
=
8
2
= 4
=2
Therefore, standard deviation Shitolakkha Ltd. is 2%.
(b)
Determining coefficient of variance of Shitolakka Ltd.:
We get from (a)
( )
Standard Deviation ( ) = 2%
Coefficient of variance (CV ) =   100
Average rate of return R = 12%
R
2
 100
12
= 16.67%
Determining coefficient of variance of Kornofuli Ltd.:
=
( ) =  Ri = 20 + 8 + 11 = 39 = 13%
Average rate of return R
n
3
 (R − R)
Standard deviation ( ) =
3
2
i
=
n −1
(20 − 13)2 + (8 − 13)2 + (11 − 13)2
3 −1
=
49 + 25 + 4
2
=
78
2
= 39
= 6.24
( )
Coefficient of variance CV =   100
R
6.24
 100
13
= 48%
=
Chapter-9: Risk and Rate of Return
2/69
Therefore, Mr Jakir should invest Shitlokka Ltd. Because coefficient of variance, that is,
amount of risk of shitlokka Ltd. is 16.67% which is lower than coefficient of variance of
Karnofuli Ltd. (48%).
40. Mrs Hasnahena is a new investor. She is keen to invest in the stock market. From the stock
market, she collected information about the last 5 years’ profit earnings of Lisha Foods and Nisha
Garments. The collected data is as follows:
Year
Rate of earning profit
Lisha foods
Nisha Garments
15%
25%
17%
15%
19%
32%
21%
10%
23%
18%
1
2
3
4
5
(a)
(b)
(a)
(Mymensingh Board-2021)
Determine standard deviation of Lisha Foods based on the information given in the stem.
Judge Mrs Hasnahena's investment potentiality in the light of the stem.
Answer:
Determining standard deviation of Lisha Foods.
Average rate of return of Lisha Foods:
()
Average rate of return R =  Ri = 15 + 17 + 19 + 21 + 23 = 95 = 19%
n
5
5
Standard deviation of Lisha Foods:
=
(
 Ri − R
n −1
) = (15 − 19) + (17 − 19) + (19 − 19) + (21 − 19) + (23 − 19)
2
2
2
2
5 −1
=
16 + 4 + 0 + 4 + 16
4
=
40
4
= 10
= 3.16%
Standard deviation of Lisha Foods is 3.16%.
(b)
Determining coefficient of variance of Lisha foods:
We get from (a)
()
Average rate of return of Lisha Foods, R = 19%
Standard Deviation Lisha Foods, ( ) = 3.16%
2
2
Chapter-9: Risk and Rate of Return
2/70
Coefficient of variance CV=   100
R
3.16
 100
19
= 16.63%
=
Determining coefficient of variance of Nisha Garments:
The average rate of return of Nisha Garments:
()
Average rate of return R =  R i = 25 + 15 + 32 + 10 + 18 = 100 = 20%
n
5
5
Standard deviation of Nisha Garments:
 =
(
 Ri − R
n −1
) = (25 − 20) + (15 − 20) + (32 − 20) + (10 − 20) + (18 − 20)
2
2
2
2
2
2
5 −1
=
25 + 25 + 144 + 100 + 4
4
=
298
4
= 74.5
= 8.63%
Coefficient of variance CV=   100
R
8.63
 100
20
= 43.15%
=
Therefore Mrs. Hasnahena should invest on Lisha Foods project. Because coefficient of variance,
that is, amount of risk of this project is 16.63% which is lower than coefficient of variance of
Nisha Garments (43.15%).
41. Tamanna is new investor. She is eager to invest in share market. She collected Nira Foods’
and Lima Garments’ profit related information of the last 4 years. The collected information are
as follows:
Year
1
2
3
4
Nira Foods
15%
17%
19%
21%
(a)
(b)
Determine standard deviation of Nira Foods.
Investing in which security will be safe for Tamanna?
(a)
Answer:
Determining standard deviation of Nira Foods
Lima Garments
25%
07%
32%
10%
(Dinajpur Board-2019)
Chapter-9: Risk and Rate of Return
2/71
 R 15 + 17 + 19 + 21 72
=
=
= 18%
n
4
4
()
Average rate of return, R =
Standard deviation of Nira foods
(
 R−R
=
n −1
) = (15 − 18) + (17 − 18) + (19 − 18) + (21 − 18)
2
2
2
2
2
4 −1
=
9 +1+1+ 9
3
=
20
3
= 6.667
= 2.58%
(b)
Average rate of return of Lima Garments
 R 25 + 7 + 32 + 10 74
=
=
= 18.5%
n
4
4
()
Average rate of return, R =
Standard deviation of Lima Garments
(
 R−R
=
n −1
) = (25 − 18.5) + (7 − 18.5) + (32 − 18.5) + (10 − 18.5)
2
2
4 −1
=
42.25 + 132.25 + 182.25 + 72.25
3
=
429
3
= 143
= 11.96%
Coefficient of variance of Lima garments:
Here,
 = 11.96%
R = 18.5%

100
R
11.96
=
 100
18.5
= 64.65%
Coefficient of variance CV=
Coefficient of variance of Nira Foods:
 = 2.58%
R = 18%
2
2
2
Chapter-9: Risk and Rate of Return
2/72

100
R
2.58
=
100
18
= 14.33%
Co efficient of variance CV=
As coefficient of variance of Nira Foods is lower than that of Lima garments, Nira Foods is
less risky. So investing on securities of Nira foods will be safe for Tamanna.
42. Mr. Abir has some money to invest. He is considering to banks M and N. The last 5 years
information of both banks are shown below:
Rate of Return (%)
Year
M
12
18
20
17
13
1
2
3
4
5
N
10
18
24
16
12
(a) Calculate standard deviation of M bank.
(b) Which bank is more risky to Mr. Abir and why?
Answer:
(a) Determining standard deviation of M bank:
 R 12 + 18 + 20 + 17 + 13 80
=
=
= 16%
n
5
5
()
Average rate of return, R =
Standard deviation of M Bank:
(
 R−R
=
n −1
) = (12 − 16) + (18 − 16) + (20 − 16) + (17 − 16) + (13 − 16)
2
2
2
5 −1
=
16 + 4 + 16 + 1 + 9
4
=
46
4
= 11.5
= 3.39%
(b)
2
Co-efficient of variance of M Bank:
 = 3.39%
R = 16%
Co-efficient of variance CV=

R
100
2
2
Chapter-9: Risk and Rate of Return
2/73
3.39
 100
16
= 21.19%
=
Co-efficient of variance of N Bank:
Average rate of return of N Bank:
 R 10 + 18 + 24 + 16 + 12 80
=
=
= 16%
n
5
5
()
Average rate of return, R =
Standard deviation of N Bank:
(
 R−R
=
n −1
) = (10 − 16) + (18 − 16) + (24 − 16) + (16 − 16) + (12 − 16)
2
2
2
2
2
2
5 −1
=
36 + 4 + 64 + 0 + 16
4
=
120
4
= 30
= 5.48%
So, co-efficient of variance CV=

R
100
5.48
 100
16
= 34.23%
=
As co-efficient of variance of N Bank is more than that of M Bank, N Bank is more risky
to Mr. Abir.
Standard Deviation and Coefficient of Variation Related Math
Method-2
43. The rate of return and probability of two projects are as follows:
Probability
.10
.20
.30
.40
Project-1
-10%
12%
20%
22%
a) Calculate the standard deviation of project-1.
b) Analyze which project is less risky by coefficient of variation.
Answer:
a) Determining the standard deviation of project-1:
Project-2
12%
10%
15%
28%
(Rajshahi Board-2023)
Chapter-9: Risk and Rate of Return
Here,
Average rate of return, 𝑅̅1
2/74
= ∑(𝑅𝑖 × 𝑃𝑖 )
= (-10 × .10) + (12 × .20) + (20 × .30) + (22 × .40)
= 1 + 2.40 + 6 + 8.8
= 16.20%
We know,
Standard deviation, σ1 = √∑(𝑅𝑖 − ̅̅̅
𝑅1 )2 × 𝑃𝑖
= √(−10 – 16.20)2 × 0.10 + (12 − 16.20)2 × 0.20 + (20 − 16.20)2 × 0.30 + (22 − 16.20)2 × 0.40
= √68.644 + 3.528 + 4.332 + 13.456
= √89.96
= 9.48%
So the standard deviation of project-1 is 9.48%.
b) Determining the coefficient of variation of project-1:
We get from a,
Average rate of return of project-1, 𝑅̅1 = 16.20%
Standard deviation of project-1, σ1
= 9.48%
We know,
𝜎
Coefficient of variation, CV1 = 1 × 100
=
𝑅1
9.48
16.20
× 100
= 58.52%
Determining the coefficient of variation of project-2:
Here,
Average rate of return, 𝑅̅2
= ∑(𝑅𝑖 × 𝑃𝑖 )
= (12 × .10) + (10 × .20) + (15 × .30) + (28 × .40)
= 1.20 + 2 + 4.50 + 11.20
= 18.90%
We know,
Standard deviation, σ2 = √∑(𝑅𝑖 − ̅̅̅
𝑅2 )2 × 𝑃𝑖
= √(12 – 18.90)2 × 0.10 + (10 − 18.90)2 × 0.20 + (15 − 18.90)2 × 0.30 + (28 − 18.90)2 × 0.40
= √4.761 + 15.842 + 4.563 + 33.124
= √58.29
= 7.63%
⸫ Coefficient of variation, CV2
𝜎
= 𝑅2 × 100
2
7.63
= 18.90 × 100
= 40.37%
Here, the coefficient of variation or risk of project-2 is 40.37%, which is less than the
coefficient of variation of project-1 of 58.52%. Hence, we can conclude that project-2 is less
risky in consideration of coefficient of variation.
Chapter-9: Risk and Rate of Return
2/75
44. Mr. Redwan is considering the information of the following two securities for investment.
Expected return
Probability (%)
Security-A
15%
10%
9%
0.20
0.10
0.20
Security-B
12.5%
17.5%
16.0%
(Cumilla Board-2023)
a) Determine standard deviation of Security-A.
b) ‘Security-B is more risky’ – Explain the statement mathematically.
Answer:
a) Determining the standard deviation of Security-A:
Here,
Average rate of return, 𝑅̅𝐴
= ∑(𝑅𝑖 × 𝑃𝑖 )
= (15 × 0.20) + (10 × 0.10) + (9 × 0.20)
= 3 + 1 + 1.8
= 5.80%
We know,
2
̅̅̅
Standard deviation, σA = √∑(𝑅𝑖 − 𝑅
𝐴 ) × 𝑃𝑖
= √(15 – 5.80)2 × 0.20 + (10 − 5.80)2 × 0.10 + (9 − 5.80)2 × 0.20
= √16.928 + 1.764 + 2.048
= √20.74
= 4.55%
So the standard deviation of Security-A is 4.55%.
b) Determining coefficient of variation of Security-A:
We get from a,
Average rate of return of Security-A, 𝑅̅𝐴
= 5.80%
Standard deviation of Security-A, σA
= 4.55%
We know,
𝜎
Coefficient of variation, CVA = 𝑅𝐴 × 100
𝐴
4.55
= 5.80 × 100
= 78.45%
Determining coefficient of variation of Security-B:
Here,
Average rate of return, 𝑅̅𝐵
= ∑(𝑅𝑖 × 𝑃𝑖 )
= (12.5 × 0.20) + (17.5 × 0.10) + (16.0 × 0.20)
= 2.50 + 1.75 + 3.20
= 7.45%
⸫ Standard deviation, σB = √∑(𝑅𝑖 − ̅̅̅̅
𝑅𝐵 )2 × 𝑃𝑖
Chapter-9: Risk and Rate of Return
2/76
=
√(12.5 – 7.45)2 × 0.20 + (17.5 − 7.45)2 × 0.10 + (16.0 − 7.45)2 × 0.20
= √5.1005 + 10.10025 + 14.6205
= √29.82125
= 5.46%
We know,
𝜎
Coefficient of variation, CVB = 𝐵 × 100
𝑅𝐵
5.46
= 7.45 × 100
= 73.29%
Here, coefficient of variance or risk of Security-B is 73.29%, which is less than the risk of
Security-A of 78.45%. Hence, we can say that the statement ‘Security-B is more risky’ is
incorrect.
45. Mrs. Arpa is a newly retired employee. She is eager to invest in share market. She has
collected information of stock-A and stock-B after analyzing the market. The information is given
below:
Rate of return
Economic condition
Probability
A
B
Normal
.20
10%
14%
Good
.30
15%
18%
Excellent
.50
25%
20%
Expected rate of
19%
18.20%
return
Standard deviation
6.25%
(Chattogram Board-2023)
a) Calculate the standard deviation of stock-B in the light of the stem.
b) Which one is less risky stock to invest for Mrs. Arpa? Give your opinion with mathematical
analysis.
Answer:
a) Calculating the standard deviation of stock-B:
Here,
Expected return of stock-B, 𝑅̅𝐵 = 18.20%
We know,
Standard deviation, σB = √∑(𝑅𝑖 − ̅̅̅̅
𝑅𝐵 )2 × 𝑃𝑖
=
√(14 – 18.20)2 × 0.20 + (18 − 18.20)2 × 0.30 + (20 − 18.20)2 × 0.50
= √3.528 + 0.012 + 1.62
= √5.16
= 2.27%
Chapter-9: Risk and Rate of Return
2/77
Therefore, the standard deviation of stock-B is 2.27%.
b) Determining the coefficient of variation of stock-A:
Given,
Expected rate of return of stock-A, 𝑅̅𝐴 = 19%
Standard deviation of project-A, σA = 6.25%
We know,
𝜎
Coefficient of variation, CVA = 𝑅𝐴 × 100
𝐴
6.25
= 19 × 100
= 32.89%
Determining the coefficient of variation of stock-B:
Given,
Expected rate of return of stock-B, 𝑅̅𝐴 = 18.20%
We get from a,
Standard deviation of project-B, σA
= 2.27%
⸫ Coefficient of variation, CVA
𝜎
= 𝑅𝐴 × 100
𝐴
2.27
= 18.20 × 100
= 12.47%
Here, the coefficient of variation of stock-B is 12.47%, which is lower than the coefficient of
variation of stock-A of 32.89%. That is, stock-B is less risky for Mrs. Arpa to invest.
46. Mr. Babla is a security analyst. An investor asked for Mr. Babla’s opinion regarding which
of the two securities will be suitable for him to invest.
Expected rate of return (%)
Security-A
Security-B
0.20
-15%
20%
0.50
20%
30%
0.30
60%
40%
Analyzing the above information, Mr. Babla advised the investor to invest in Security B.
(Barishal Board-2023)
a) Determine standard deviation of Security-A.
b) Evaluate rationale of the advice Mr. Babla gave the investor in the stem.
Answer:
a) Determining the standard deviation of Security-A:
Here,
Average rate of return, 𝑅̅𝐴
= ∑(𝑅𝑖 × 𝑃𝑖 )
= (–15 ×0.20) + (20 ×0.50) + (60 ×0.30)
= - 3+10+18
= 25%
We know,
Probability
Chapter-9: Risk and Rate of Return
2/78
2
̅̅̅
Standard deviation, σA = √∑(𝑅𝑖 − 𝑅
𝐴 ) × 𝑃𝑖
=
√(−15 – 25)2 × 0.20 + (20 − 25)2 × 0.50 + (60 − 25)2 × 0.30
= √320 + 12.50 + 367.50
= √700
= 26.46%
So the standard deviation of Security-A is 26.46%.
b) Determining the coefficient of variation of Security-A:
We get from a,
Average rate of return, 𝑅̅𝐴
= 25%
Standard deviation, σA
= 26.46%
We know,
𝜎
Coefficient of variation, CVA = 𝑅𝐴 × 100
𝐴
=
26.46
25
× 100
= 105.84%
Determining the coefficient of variation of Security-B:
Here,
Average rate of return, 𝑅̅𝐵
= ∑(𝑅𝑖 × 𝑃𝑖 )
= (20 ×0.20) + (30 ×0.50) + (40 ×0.30)
= 4+15+12
= 31%
We know,
2
̅̅̅̅
Standard deviation, σB = √∑(𝑅𝑖 − 𝑅
𝐵 ) × 𝑃𝑖
=
√(20 – 31)2 × 0.20 + (30 − 31)2 × 0.50 + (40 − 31)2 × 0.30
= √24.20 + 0.50 + 24.30
= √49
= 7%
⸫ Coefficient of variation, CVB
𝜎
= 𝑅𝐵 × 100
𝐵
7
= 31 × 100
= 22.58%
Here, the coefficient of variation of Security-B is 22.58%, which is less than the coefficient
of variation of Security-A of 105.84%. Hence, Mr. Babla’s advice was rational.
47. Mr. Abdul Kadir is interested to invest in capital market. For this, he is considering the
following information of two securities:
Chapter-9: Risk and Rate of Return
2/79
Rate of returns
Probability
Security-A
10%
13%
15%
12%
.15
.30
.25
.30
Security-B
12%
15%
20%
14%
(Dinajpur Board-2023)
a) Calculate the standard deviation of security-A?
b) Which security is more logical to invest for Mr. Kadir? Give your opinion.
Answer:
a) Determining the standard deviation of security-A:
Here,
Average rate of return, 𝑅̅𝐴
= ∑(𝑅𝑖 × 𝑃𝑖 )
= (10 × 0.15) + (13 × 0.30) + (15 × 0.25) + (12 × 0.30)
= 1.50 + 3.90 + 3.75 + 3.60
= 12.75%
We know,
2
̅̅̅
Standard deviation, σA = √∑(𝑅𝑖 − 𝑅
𝐴 ) × 𝑃𝑖
=
√(10 – 12.75)2 × 0.15 + (13 − 12.75)2 × 0.30 + (15 − 12.75)2 × 0.25 + (12 − 12.75)2 × 0.30
= √1.134375 + 0.01875 + 1.265625 + 0.16875
= √2.5875
= 1.61%
So, the standard deviation of security-A is 1.61%.
b) Determining the coefficient of variation of security-A:
We get from a,
Average rate of return, 𝑅̅𝐴
= 12.75%
Standard deviation, σA
= 1.61%
We know,
𝜎
Coefficient of variation, CVA = 𝑅𝐴 × 100
𝐴
1.61
= 12.75 × 100
= 12.63%
Determining the coefficient of variation of security-B:
Here,
Average rate of return, 𝑅̅𝐵
= ∑(𝑅𝑖 × 𝑃𝑖 )
= (12 × 0.15) + (15 × 0.30) + (20 × 0.25) + (14 × 0.30)
= 1.80 + 4.50 + 5 + 4.20
= 15.50%
We know,
Standard deviation, σB = √∑(𝑅𝑖 − ̅̅̅̅
𝑅𝐵 )2 × 𝑃𝑖
Chapter-9: Risk and Rate of Return
2/80
=
√(12 – 15.50)2 × 0.15 + (15 − 15.50)2 × 0.30 + (20 − 15.50)2 × 0.25 + (14 − 15.50)2 × 0.30
= √1.8375 + 0.075 + 5.0625 + 0.675
= √7.65
= 2.77%
We know,
𝜎
Coefficient of variation, CVB = 𝐵 × 100
𝑅𝐵
2.77
= 15.50 × 100
= 17.87%
Here, the coefficient of variation or risk of security-A is 12.63%, which is less than securityB’s coefficient of variation of 17.87%. In this case, security-A will be more logical for Mr.
Kadir to invest.
48. Mrs. Irin wants to invest in a stock. She likes to take more risk. She has collected the following
information regarding stock 'X' and stock 'Y' after observing the market:
Rate of profit
Economic condition
Probability
Stock-X
Stock-Y
Recession
25%
5%
-2%
Normal
45%
11%
9%
Boom
30%
14%
19%
(Jashore Board-2023)
a) Calculate standard deviation and co-efficient of variance of stock-Y mentioned in stem.
b) In which stock will Mrs. Irin invest? Give opinion based on mathematical explanation.
Answer:
a) Determining standard deviation and co-efficient of variance of stock-Y:
Here,
Average rate of return, 𝑅̅𝑌
= ∑(𝑅𝑖 × 𝑃𝑖 )
= (-2 × 0.25) + (9 × 0.45) + (19 × 0.30)
= - 0.50+ 4.05+ 5.70
= 9.25%
We know,
Standard deviation, σY = √∑(𝑅𝑖 − ̅𝑅̅̅𝑌̅)2 × 𝑃𝑖
= √(−2 – 9.25)2 × 0.25 + (9 − 9.25)2 × 0.45 + (19 − 9.25)2 × 0.30
= √31.640625 + 0.028125 + 28.51875
= √60.1875
= 7.76%
We know,
𝜎
Coefficient of variation, CVY = 𝑅𝑌 × 100
𝑌
7.76
= 9.25 × 100
Chapter-9: Risk and Rate of Return
2/81
= 83.89%
So, standard deviation and co-efficient of variance of stock-Y are 9.25% and 83.89%
respectively.
b) Determining standard deviation of stock-X:
Here,
Average rate of return, 𝑅̅𝑋
= ∑(𝑅𝑖 × 𝑃𝑖 )
= (5 × 0.25) + (11 × 0.45) + (14 × 0.30)
= 1.25 + 4.95 + 4.20
= 10.40%
We know,
2
̅̅̅̅
Standard deviation, σX = √∑(𝑅𝑖 − 𝑅
𝑋 ) × 𝑃𝑖
=
√(5 – 10.40)2 × 0.25 + (11 − 10.40)2 × 0.45 + (14 − 10.40)2 × 0.30
= √7.29 + 0.162 + 3.888
= √11.34
= 3.37%
⸫ Coefficient of variation, CVX
𝜎
= 𝑅𝑋 × 100
𝑋
3.37
= 10.40 × 100
= 32.40%
Here, coefficient of variation or risk margin of stock-X is 83.89%. On the other hand,
coefficient of variation of stock-Y is 32.40%. Hence, Mrs. Irin will invest in stock-X.
49. Mr. Iqbal is interested to invest in the stock market along with his job. He has chosen two
stocks named E and F for investment considering different economic conditions. The rate of
return and probability of the two stocks are as follows:
Rate of return (%)
Economic
Probability
conditions
E
F
Promising
0.30
25
20
Normal
0.40
35
30
Disappointing
0.10
15
10
Expected rate of return of the two stocks is 23% and 19% respectively. (Dhaka Board-2022)
a) Determine standard deviation of stock F according to the stem.
b) Which stock between the two will Mr. Iqbal purchase according to the stem? Analyze in
consideration of risk.
Answer:
a) Determine standard deviation of stock F:
Given,
Expected rate of return, 𝑅̅ = 19%
We know,
Standard deviation, σ = √∑(𝑅𝑖 − 𝑅̅)2 × 𝑃𝑖
Chapter-9: Risk and Rate of Return
2/82
= √(20 − 19)2 × 0.30 + (30 − 19)2 × 0.40 + (10 − 19)2 × 0.10
= √0.3 + 48.4 + 8.1
= √56.8
= 7.54%
Therefore, standard deviation of stock F is 7.54%.
b) To determine which stock between the two Mr. Iqbal will purchase in consideration of risk,
we need to compare coefficient of variation of the two stocks.
Determining coefficient of variation of stock F:
From a we get,
Expected rate of return, 𝑅̅
= 19%
Standard deviation, σ
= 7.54%
We know,
σ
Coefficient of variation (CV) = 𝑅̅ × 100
7.54
= 19 × 100
= 0.3968 × 100
= 39.68%
Determining coefficient of variation of stock E:
Given,
Expected rate of return, 𝑅̅ = 23%
We know,
Standard deviation, σ = √∑(𝑅𝑖 − 𝑅̅)2 × 𝑃𝑖
= √(25 − 23)2 × 0.30 + (35 − 23)2 × 0.40 + (15 − 23)2 × 0.10
= √1.2 + 57.6 + 6.4
= √65.2
= 8.07%
⸫ Coefficient of variation (CV)
σ
= 𝑅̅ × 100
8.07
= 23 × 100
= 0.3509 × 100
= 35.09%
Here, we can see that coefficient of variation of stock E is 35.09%, which is less than
coefficient of variation of stock F (39.68%). Therefore, Mr. Iqbal will purchase stock E.
50. Mr. Sourav is interested to invest in any one of the two companies named P and Q. In case of
investment, he specially considers market risk. P and Q company’s last 3 years’ return and market
return are given below:
Year
P Company (%)
Q Company (%)
Market rate of return (%)
2016
10
8
10
2017
20
15
15
2018
8
6
7
Chapter-9: Risk and Rate of Return
2/83
(Dhaka Board-2022)
a) Determine co-variation of P Company on the basis of the above information?
b) Which company will be reasonable to invest for Mr. Sourav? Give your opinion.
Answer:
a) Determine co-variation of P Company:
∑𝑅
Average rate of return, (𝑅̅𝑃 ) = 𝑖
=
𝑛
10+20+8
3
38
= 3
= 12.67%
10+15+7
Average market rate of return, (𝑅̅𝑀 ) =
3
32
= 3
= 10.67%
⸫ Co-variation of P Company,
∑(𝑅𝑃 −𝑅̅𝑃 ) ∑(𝑅𝑀 −𝑅̅𝑀 )
Cov (RP, RM) =
=
=
=
𝑛−1
(10−12.67)(10−10.67)+(20−12.67)(15−10.67)+(8−12.67)(7−10.67)
3−1
1.7889+31.7389+17.1389
2
50.6667
2
= 25.33%
Therefore, co-variation of P Company is 25.33%.
b) To determine which company will be reasonable to invest for Mr. Sourav, we need to compare
co-variation of the two companies.
Determine co-variation of Q Company:
∑𝑅
Average rate of return, (𝑅̅𝑄 ) = 𝑛 𝑖
=
8+15+6
3
29
= 3
= 9.67%
From a we get,
Average market rate of return, (𝑅̅𝑀 ) = 10.67%
⸫ Co-variation of Q Company,
Cov (RQ, RM) =
=
=
=
∑(𝑅𝑄 −𝑅̅𝑄 ) ∑(𝑅𝑀 −𝑅̅𝑀 )
𝑛−1
(8−9.67)(10−10.67)+(15−9.67)(15−10.67)+(6−9.67)(7−10.67)
3−1
1.1189+23.0789+13.4689
2
37.6667
2
= 18.83%
Chapter-9: Risk and Rate of Return
2/84
Here, co-variation or risk of Q Company is 18.83%, which is less than that of P Company
25.33% (from a). Hence, Q Company will be reasonable to invest for Mr. Sourav.
51. Mr. Aslam is thinking of investing in a project. Information of two projects are as follows:
Rate of Return
Economic Condition
Probability
Project-P
Project-Q
Good
0.40
12%
10%
Recession
0.25
4%
6%
Normal
0.35
9%
8%
(Jashore Board-2022)
a) Determine standard deviation of project-P.
b) In which project should Mr. Aslam invest? Analyze.
Answer:
a) Determining standard deviation of project-P:
Here,
Average rate of return, 𝑅̅𝑃
= ∑(𝑅𝑖 × 𝑃𝑖 )
= (12×0.40) + (4×0.25) + (9×0.35)
= 4.8 + 1 + 3.15
= 8.95%
We know,
Standard deviation, σP = √∑(𝑅𝑖 − 𝑅̅)2 × 𝑃𝑖
= √(12 − 8.95)2 × 0.40 + (4 − 8.95)2 × 0.25 + (9 − 8.95)2 × 0.35
= √3.721 + 6.125625 + 0.000875
= √9.8475
= 3.14%
Therefore, standard deviation of project-P is 3.14%.
b) To determine which project Mr. Aslam should invest in, we need to compare coefficient of
variation or risk of both projects.
We get from a,
Average rate of return, 𝑅̅𝑃
= 8.95%
Standard deviation, σP = 3.14%
We know,
Coefficient of variation, CVP
𝜎
= 𝑅𝑃 × 100
𝑃
3.14
= 8.95 × 100
= 35.08%
Determining standard deviation of project-Q:
Here,
Average rate of return, 𝑅̅𝑄
= ∑(𝑅𝑖 × 𝑃𝑖 )
= (10×0.40) + (6×0.25) + (8×0.35)
= 4 + 1.50 + 2.80
Chapter-9: Risk and Rate of Return
2/85
= 8.30%
We know,
Standard deviation, σQ = √∑(𝑅𝑖 − 𝑅̅)2 × 𝑃𝑖
= √(10 − 8.30)2 × 0.40 + (6 − 8.30)2 × 0.25 + (8 − 8.30)2 × 0.35
= √1.156 + 1.3225 + 0.0315
= √ 2.51
= 1.58%
⸫ Coefficient of variation, CVQ
𝜎
= 𝑅𝑄 × 100
𝑄
=
1.58
8.30
× 100
= 19.04%
It is noticeable that coefficient of variation or risk of project-Q is 19.04%, which is lower than
that of project-P 35.08%. Therefore, Mr. Aslam should invest in project-Q.
52. Rate of return and probability of two securities are given below:
Probability
.25
.40
Security-A
.10
.20
Security -B
.30
.10
.35
.15
.20
(Sylhet
Board-
2022)
a) How much is the standard deviation of security–A?
b) Which security of the stem would be more acceptable to invest? Give mathematical
explanation.
Answer:
a) Determining standard deviation of security–A:
Here,
Average rate of return, 𝑅̅𝐴
= ∑(𝑅𝑖 × 𝑃𝑖 )
= (0.10×0.25) + (0.20×0.40) + (0.15×0.35)
= 0.025 + 0.08 + 0.0525
= 0.1575
= 15.75%
We know,
Standard deviation, σA = √∑(𝑅𝑖 − 𝑅̅)2 × 𝑃𝑖
= √(0.10 − 0.1575)2 × 0.25 + (0.20 − 0.1575)2 × 0.40 + (0.15 − 0.1575)2 × 0.35
= √0.000827 + 0.000723 + 0.000020
= √ 0.001569
= 0.039607
= 3.96%
Therefore, standard deviation of security-A is 3.96%.
Chapter-9: Risk and Rate of Return
2/86
b) To determine which security of the stem would be more acceptable to invest, we need to
compare coefficient of variation of both securities.
Determining coefficient of variation of security-A:
We get from a,
Average rate of return, 𝑅̅𝐴
= 15.75%
Standard deviation, σA = 3.96%
We know,
σ
Coefficient of variation (CVA) = ̅𝐴 × 100
=
𝑅𝐴
3.96
15.75
× 100
= 0.2514 × 100
= 25.14%
Determining coefficient of variation of security-B:
Here,
Average rate of return, 𝑅̅𝐴
= ∑(𝑅𝑖 × 𝑃𝑖 )
= (0.30×0.25) + (0.10×0.40) + (0.20×0.35)
= 0.075 + 0.04 + 0.07
= 0.185
= 18.50%
We know,
Standard deviation, σA = √∑(𝑅𝑖 − 𝑅̅)2 × 𝑃𝑖
= √(0.30 − 0.185)2 × 0.25 + (0.10 − 0.185)2 × 0.40 + (0.20 − 0.185)2 × 0.35
= √0.003306 + 0.00289 + 0.000079
= √ 0.006275
= 0.079215
= 7.92%
σ
⸫ Coefficient of variation (CVB) = 𝑅̅𝐵 × 100
𝐵
7.92
= 18.50 × 100
= 0.1620 × 100
= 42.81%
We can see that coefficient of variation of security-A is 25.14%, which is lower than
coefficient of variation of security-B is 42.81%. Therefore, security-A would be more
acceptable to invest.
53. Expected return and probability of two projects are given below:
Probability (Pi)
0.25
0.30
0.45
Expected rate of return (%)
Project-A
Project-B
-15
15
10
20
25
25
Chapter-9: Risk and Rate of Return
2/87
(Barishal Board-2022)
a) Determine standard deviation of project-A.
b) Investing in which project will be rational in consideration of comparative risk level of the
two projects?
Answer:
a) Determine standard deviation of project-A:
Here,
Average rate of return, 𝑅̅𝐴
= ∑(𝑅𝑖 × 𝑃𝑖 )
= (-15×0.25) + (10×0.30) + (25×0.45)
= −3.75 + 3 + 11.25
= 10.50%
We know,
Standard deviation, σA = √∑(𝑅𝑖 − 𝑅̅)2 × 𝑃𝑖
= √(−15 − 10.50)2 × 0.25 + (10 − 10.50)2 × 0.30 + (25 − 10.50)2 × 0.45
= √162.5625 + 0.075 + 94.612500
= √ 257.25
= 16.04%
Therefore, standard deviation of project-A is 16.04%.
b) To determine investing in which project will be rational in consideration of comparative risk
level of the two projects, we need to compare coefficient of variation of both projects.
Determining coefficient of variation of project-A:
We get from a,
Average rate of return, 𝑅̅𝐴
= 10.50%
Standard deviation, σA = 16.04%
We know,
σ
Coefficient of variation (CVA) = 𝑅̅𝐴 × 100
𝐴
16.04
= 10.50 × 100
= 152.76%
Determining coefficient of variation of project-B:
Here,
Average rate of return, 𝑅̅𝐵
= ∑(𝑅𝑖 × 𝑃𝑖 )
= (15×0.25) + (20×0.30) + (25×0.45)
= 3.75 + 6 + 11.25
= 21%
We know,
Standard deviation, σB = √∑(𝑅𝑖 − 𝑅̅)2 × 𝑃𝑖
= √(15 − 21)2 × 0.25 + (20 − 21)2 × 0.30 + (25 − 21)2 × 0.45
= √9 + 0.30 + 7.20
= √ 16.50
Chapter-9: Risk and Rate of Return
2/88
= 4.06%
σ
⸫ Coefficient of variation (CVB) = 𝑅̅𝐵 × 100
𝐵
4.06
= 21 × 100
= 19.33%
It is noticeable that coefficient of variation or risk of project-B is 19.33%, which is lower than
that of project-A 152.76%. Therefore, investing in project-B will be rational in consideration
of comparative risk level of the two projects.
54. Mr. Porosh, the Financial Manager of Srabonti Ltd., has collected the following information
of stock-A and stock-B from market.
Economic
condition
Good
Normal
Bad
Probability
distribution
20%
50%
30%
Rate of return
Stock-A
Stock-B
25%
20%
30%
40%
15%
10%
(Dinajpur Board-2022)
a) Determine standard deviation of stock-A of the stem.
b) Investing in which stock of the stem will be more logical? Give your opinion determining
coefficient of variation (CV).
Answer:
a) Determining standard deviation of stock-A:
Here,
Average rate of return, 𝑅̅𝐴
= ∑(𝑅𝑖 × 𝑃𝑖 )
= (25×0.20) + (30×0.50) + (15×0.30)
= 5 + 15 + 4.50
= 24.50%
We know,
Standard deviation, σA = √∑(𝑅𝑖 − 𝑅̅)2 × 𝑃𝑖
= √(25 − 24.50)2 × 0.20 + (30 − 24.50)2 × 0.50 + (15 − 24.50)2 × 0.30
= √0.05 + 15.125 + 27.075
= √ 42.25
= 6.50%
Therefore, standard deviation of stock-A is 6.50%.
b) To determine investing in which stock of the stem will be more logical, we need to compare
coefficient of variation or risk of both stocks.
Determining coefficient of variation of stock-A:
We get from a,
Average rate of return, 𝑅̅𝐴
= 24.50%
Standard deviation, σA = 6.50%
Chapter-9: Risk and Rate of Return
2/89
We know,
σ
Coefficient of variation (CVA) = 𝑅̅𝐴 × 100
𝐴
=
6.50
24.50
× 100
= 0.2653
= 26.53%
Determining coefficient of variation of stock-B:
Here,
Average rate of return, 𝑅̅𝐵
= ∑(𝑅𝑖 × 𝑃𝑖 )
= (20×0.20) + (40×0.50) + (10×0.30)
= 4 + 20 + 3
= 27%
We know,
Standard deviation, σB = √∑(𝑅𝑖 − 𝑅̅)2 × 𝑃𝑖
= √(20 − 27)2 × 0.20 + (40 − 27)2 × 0.50 + (10 − 27)2 × 0.30
= √9.8 + 84.5 + 86.7
= √ 181
= 13.45%
We know,
σ
Coefficient of variation (CVB) = 𝑅̅𝐵 × 100
𝐵
=
13.45
27
× 100
= 0.4981
= 49.81%
Here, it is noticeable that coefficient of variation or risk of stock-A is 26.53%, which is lower
than coefficient of variation or risk of stock-B (49.81%). Therefore, investing in stock-A will
be more logical.
55. Mr. Abdus Salam is considering the following two securities for investment:
Probability
0.20
0.10
0.20
Expected return
Security-X
15%
10%
9%
a) Determine standard deviation of security-X?
b) “Security-Y is more risky” – Explain mathematically.
Answer:
a) Determining standard deviation of security-X:
Here,
Security-Y
12.5%
17.5%
16%
(Mymensingh Board-2022)
Chapter-9: Risk and Rate of Return
Average rate of return, 𝑅̅𝑋
2/90
= ∑(𝑅𝑖 × 𝑃𝑖 )
= (15×0.20) + (10×0.10) + (9×0.20)
= 3 + 1 + 1.80
= 5.80%
We know,
Standard deviation, σX = √∑(𝑅𝑖 − 𝑅̅)2 × 𝑃𝑖
= √(15 − 5.80)2 × 0.20 + (10 − 5.80)2 × 0.10 + (9 − 5.80)2 × 0.20
= √16.928 + 1.764 + 2.048
= √20.74
= 4.55%
Therefore, standard deviation of security-X is 4.55%.
b) To justify the statement in the stem, we need to compare coefficient of variation or risk of
both securities.
Determining coefficient of variation of security-X:
We get from a,
Average rate of return, 𝑅̅𝑋
= 5.80%
Standard deviation, σX = 4.55%
We know,
σ
Coefficient of variation (CVX) = 𝑅̅𝑋 × 100
𝑋
=
4.55
5.80
× 100
= 0.2653
= 78.45%
Determining coefficient of variation of security-Y:
Here,
Average rate of return, 𝑅̅𝑌
= ∑(𝑅𝑖 × 𝑃𝑖 )
= (12.5×0.20) + (17.5×0.10) + (16×0.20)
= 2.50 + 1.75 + 3.20
= 7.45%
We know,
Standard deviation, σY = √∑(𝑅𝑖 − 𝑅̅)2 × 𝑃𝑖
= √(12.5 − 7.45)2 × 0.20 + (17.5 − 7.45)2 × 0.10 + (16 − 7.45)2 × 0.20
= √5.1005 + 10.10025 + 14.6205
= √ 29.82125
= 5.46%
σ
Coefficient of variation (CVY) = 𝑅̅𝑌 × 100
𝑌
5.46
= 7.45 × 100
= 0.2653
= 73.29%
Chapter-9: Risk and Rate of Return
2/91
Here, it is noticeable that coefficient of variation or risk of security-X is 78.45%, which is
more than that of security-Y (73.29%). Therefore, the statement that “Security-Y is more
risky” is not right.
56. Mr. Zahir is a financial manager. He will decide to invest in any one of the two selected
projects. The information of the projects are as follows: Cash inflow
Project-X
Project-Y
40,000
35,000
15,000
20,000
15,000
15,000
20,000
20,000
Probability
Project-X
.50
.15
.20
.15
(a)
(b)
Project-Y
.40
.30
.20
.10
(Jashore Board-2019, Set-1)
Calculate standard deviation of project-X according to the stem.
Which project is profitable for Mr. Zahir? Explain.
(a)
Answer:
Average net cashflow of project X
()
Expected rate of return R =  RiPi
= (40,000  0.50 ) + (15,000  0.15) + (15,000  0.2) + (20,000  0.15)
= 20,000 + 2,250 + 3,000 + 3,000
= 28,250
Standard deviation of project X:
Standard deviation ( ) =
=
 (R − R)  Pi
2
(40,000 − 28,250)2  0.5 + (15,000 − 28,250)2  0.15 + (15,000 − 28,250)2  0.2 + (20,000 − 28,250)2  0.15
= 6,90,31,250 + 2,63,34,375 + 3,51,12,500 + 1,02,09,375
= 14,06,87,500
= 11,861.18taka
Here, standard deviation of project X is 11,861.18 taka
(b)
Average net cash flow of project Y:
()
Expected rate of return R =  RiPi
= (35,000  0.4) + (20,000  0.3) + (15,000  0.2) + (20,000  0.1)
= 14,000 + 6,000 + 3,000 + 2,000
= 25,000
Standard deviation of project Y:
Standard deviation ( ) =
 (R − R)  Pi
2
Chapter-9: Risk and Rate of Return
=
2/92
(35,000 − 25,000)2  0.4 + (20,000 − 25,000)2  0.3 + (15,000 − 25,000)2  0.2 + (20,000 − 25,000)2  0.1
= 4,00,00,000 + 75,00,000 + 2,00,00,000 + 25,00,000
= 7,00,00,000
= 8,366.6
Coefficient of variance of Project Y:

100
R
8,366.6
=
100
25,000
= 33.47%
Coefficient of variance of Project X:
Co efficient of variance CV=

100
R
11,861.18
=
100
28,250
= 41.99%
Co efficient of variance CV=
Here standard deviation of project Y is lower than that of project X, so investing on project
Y is profitable for Mr. Jahir.
57. Mr. Zafor is interested to invest Tk. 10 lac into a profitable project. He has two alternative
project. The data of the two projects area follows:
Expected Return
Project - A
20,000
12,000
20,000
Financial Condition
Good
Normal
Bad
(a)
(b)
(a)
Project - B
16,000
10,000
15,000
Probability
50%
30%
20%
(Dhaka Board-2019, Set-3)
Calculate standard deviation of project-A according to the stem.
Which project is better between the two for Mr. Zafor? Analyze it in the light of coefficient of variance.
Answer:
Average net cash flow of project A:
()
Expected rate of return R =  RiPi
= (20,000  0.50 ) + (12,000  0.30 ) + (12,000  0.20 )
= 10,000 + 3,600 + 4,000
= 17,600
Standard deviation of project A:
Chapter-9: Risk and Rate of Return
 (R − R)  Pi
Standard deviation ( ) =
=
2/93
2
(20,000 − 17,600)2  0.5 + (12,000 − 17,600)2  0.30 + (20,000 − 17,600)2  0.20
= 28,80,000 + 94,08,000 + 11,52,000
= 1,34,40,000
= 3,666.06taka
Here, standard deviation of project A is 3,666.06 taka
(b)
Co-efficient of variance of Project A:

100
R
3,666.06
=
 100
17,600
= 20.83%
Co-efficient of variance of Project B:
Average net cash flow of project B:
Co-efficient of variance CV=
()
Expected rate of return R =  RiPi
= (16,000  0.50 ) + (10,000  0.30 ) + (15,000  0.20 )
= 8,000 + 3,000 + 3,000
= 14,000
Standard deviation of project B:
Standard deviation ( ) =
=
 (R − R)  Pi
2
(16,000 − 14,000)2  0.50 + (10,000 − 14,000)2  0.30 + (15,000 − 14,000)2  0.20
= 20,00,000 + 48,00,000 + 2,00,000
= 70,00,000
= 2,645.75

100
R
2,645.75
=
 100
14,000
= 18.90%
Co-efficient of variance CV=
Here standard deviation of project B is lower than that of project A, so project B is better
between the two for Mr. Zafor.
Chapter-9: Risk and Rate of Return
2/94
Standard Deviation and Coefficient of Variation Related Math
Method-3
58. Mr. Belal is interested to invest his savings of Tk. 10 lakh in a profitable project. He has the
following two projects in hand. He wants to pick the less risky project between the two.
Expected return
Probability
Project-X Project-Y
Booming
20,000
16,000
50%
Normal
12,000
10,000
30%
Downturn
20,000
15,000
20%
(Rajshahi Board’17)
a) In the light of the aforementioned information, determine standard deviation of project-X.
b) Which project do you think will be better for Mr. Belal to accept? Analyze in the light of
coefficient of variance of the two projects.
Answer:
a) Determining standard deviation of project-X:
Here,
Expected return = Ri
Possibility = Pi
̅ X = ∑𝑛𝑖=1(𝑅𝑖 × 𝑃𝑖 )
So, expected return, R
= (20,000 × 0.50) + (12,000 × 0.30) + (20,000 × 0.20)
= 10,000 + 3,600 + 4,000
= 17,600
̅
̅ )𝟐
̅ ) 𝟐 × 𝑷𝒊
Economic
Ri
Pi
𝑹𝒊 − 𝑹
(𝑹𝒊 − 𝑹
(𝑹𝒊 − 𝑹
condition
Booming
20,000
2,400
57,60,000
0.50
28,80,000
Normal
12,000
˗5,600
3,13,60,000 0.30
94,08,000
Downturn
20,000
2,400
57,60,000
0.20
11,52,000
𝟐
̅ ) × 𝑷𝒊 = 𝟏, 𝟑𝟒, 𝟒𝟎, 𝟎𝟎𝟎
∑(𝑹𝒊 − 𝑹
Economic condition
∴ Standard deviation, σX
= √∑(𝑅𝑖 − 𝑅̅)2 × 𝑃𝑖
= √1,34,40,000
= 3,666.0606
Answer: 3,666.0606.
b) To determine which project is better, we need to determine coefficient of variance of both
projects.
Determining coefficient of variance of project-Y:
̅Y
Expected return, R
= ∑𝑛𝑖=1(𝑅𝑖 × 𝑃𝑖 )
= (16,000 × 0.50) + (10,000 × 0.30) + (15,000 × 0.20)
= 8,000 + 3,000 + 3,000
= 14,000
Chapter-9: Risk and Rate of Return
Economic
condition
Booming
Normal
Downturn
2/95
Ri
̅
𝑹𝒊 − 𝑹
̅ )𝟐
(𝑹𝒊 − 𝑹
16,000
10,000
15,000
2,000
˗4,000
1,000
40,00,000 0.50
20,00,000
1,60,00,000 0.30
48,00,000
10,00,000 0.20
2,00,000
𝟐
̅ ) × 𝑷𝒊 = 𝟕𝟎, 𝟎𝟎, 𝟎𝟎𝟎
∑(𝑹𝒊 − 𝑹
∴ Standard deviation, σY
Pi
̅ ) 𝟐 × 𝑷𝒊
(𝑹𝒊 − 𝑹
= √∑(𝑅𝑖 − 𝑅̅)2 × 𝑃𝑖
= √70,00,000
= 2,645.7513
So, coefficient of variance, CVY
𝜎
= 𝑅̅ × 100
=
2,645.7513
14,000
× 100
= 18.9%
Determining coefficient of variance of project-X:
Coefficient of variance, CVX
𝜎
= ̅ × 100
=
𝑅
3,666.0606
17,600
× 100
= 20.83%
Hence, coefficient of variance (CV) of project-Y is less than that of project-X. That is, projectY is less risky. So, it will be better for Mr. Belal to accept project-Y.
59. Suppose KNP Collage has opportunity to invest in two projects. They are project M and N.
Other information about the two projects are:
Project M
Cash
Provability
inflow
distribution
60,000
.60
10,000
.10
10,000
.20
20,000
.10
Project N
Cash
Provability
inflow
distribution
50,000
.50
20,000
.30
10,000
.10
20,000
.10
(Dinajpur Board’16)
a) Determine standard deviation of project M as per information given in the stem.
b) Which project between M and N is more risky? Give your comment.
Answer:
a) Standard deviation of Project M:
(1)
(2)
(3) = 1×2 (4) = 1 −
(5) = 4×4
(6) =5×2
𝑅̅𝑀
Cash
Provability
Pi ×Ri
(𝑅𝑖 − 𝑅̅𝑀 )2
𝑅𝑖 − 𝑅̅𝑀
𝑃𝑖 (𝑅𝑖 − 𝑅̅𝑀 )2
Inflow
(Ri)
Distribution
(Pi)
Chapter-9: Risk and Rate of Return
60,000
10,000
10,000
20,000
.60
.10
.20
.10
2/96
36,000
1,000
2,000
2,000
19,000
-31,000
-31,000
-21,000
36,10,00,000
96,10,00,000
96,10,00,000
44,10,00,000
∑ 𝑃𝑖 (𝑅𝑖 − 𝑅̅𝑀 )2
21,66,00,000
9,61,00,000
19,22,00,000
4,41,00,000
54,90,00,000
So, expected rate of return, 𝑅̅𝑀 = ∑ 𝑃𝑖 𝑅𝑖
= 36,000+1,000+2,000+2,000
= 41,000
So, standard deviation, 𝜎𝑀 = √𝑃𝑖 (𝑅𝑖 − 𝑅̅𝑀 )2
= √54,90,00,000
= 23,430.75 or 23,431
Answer: 23,431.
b) Standard deviation of Project N:
Cash
Provability
Inflow
Distribution(Pi)
(Ri)
(1)
(2)
50,000
20,000
10,000
20,000
.50
.30
.10
.10
Pi ×Ri
𝑅𝑖 − 𝑅̅𝑁
(3) =
1×2
25,000
6,000
1,000
2,000
(4)
= 1 − 𝑅̅𝑁
16,000
-14,000
-24,000
-14,000
(𝑅𝑖
− 𝑅̅𝑁 )2
(5) = 4×4
(6) =5×2
25,60,00,000
19,60,00,000
57,60,00,000
19,60,00,000
12,80,00,000
5,88,00,000
5,76,00,000
1,96,00,000
26,40,00,000
∑ 𝑃𝑖 (𝑅𝑖 − 𝑅̅𝑁 )2
So, expected rate of return, 𝑅̅𝑁 = ∑ 𝑃𝑖 𝑅𝑖
= 25,000+6,000+1,000+2,000
= 34,000
So, standard deviation, 𝜎𝑁 = √𝑃𝑖 (𝑅𝑖 − 𝑅̅𝑁 )2
= √26,40,00,000
= 16,248.077 or 16,248
Coefficient of variance (CVN) of Project N:
CVN
𝜎
= ̅𝑁
𝑅𝑁
16,248
= 34,000
= 0.4778
= 47.78%
Coefficient of variance (CVM) of Project M:
CVM
𝜎
= 𝑅̅𝑀
𝑀
23,431
= 41,000
𝑃𝑖 (𝑅𝑖 − 𝑅̅𝑁 )2
Chapter-9: Risk and Rate of Return
2/97
= 0.5715
= 57.15%
Here, Coefficient of variance of Project M is 57.15% and Project N is 47.78%. That means
Project N is less risky than Project M. Therefore, between Project M and N in the stem, Project
M is more risky.
Portfolio Return Related Math
60. Padma Ltd. invested Tk. 50,000 in the stock of 'M' Co. and Tk. 30,000 in the stock of 'N' Co.
The Finance Manager of Padma Ltd. expects the following rate of return:
Rate of Return
State of economy
Probability
‘M’ Co.
‘N’ Co.
Recession
0.25
12%
10%
Normal
0.50
14%
12%
Boom
0.25
20%
16%
(Dhaka Board-2023)
a) Calculate expected rate of return of company 'M' and 'N'.
b) Analyze the rationality of investing in the portfolio if the expected rate of return of Padma
Ltd. is 12%.
Answer:
a) Determining expected rate of return of Company ‘M’:
We know,
Expected rate of return, 𝑅̅𝑀
= ∑(𝑅𝑖 × 𝑃𝑖 )
= (12 × 0.25) + (14 × 0.50) + (20 × 0.25)
= 3+7+5
= 15%
Determining expected rate of return of Company ‘N’:
We know,
Expected rate of return, 𝑅̅𝑁
= ∑(𝑅𝑖 × 𝑃𝑖 )
= (10 × 0.25) + (12 × 0.50) + (16 × 0.25)
= 2.50 + 6 + 4
= 12.50%
Therefore, expected rate of return of Company ‘M’ and ‘N’ are 15% and 12.50% respectively.
b) Determining Padma Ltd.’s portfolio rate of return:
Here,
50,000
Weight of the investment in Company ‘M’, WM = 80,000 = 0.63
30,000
Weight of the investment in Company ‘N’, WN = 80,000 = 0.37
We get from a,
̅𝑀
Expected return from Company ‘M’, R
̅𝑁
Expected return from Company ‘N’, R
We know,
= 15%
= 12.50%
Chapter-9: Risk and Rate of Return
2/98
̅ 𝑀 + 𝑊𝑁 R
̅𝑁
= 𝑊𝑀 R
= (0.63 ×15%) + (0.37 ×12.50%)
= 9.45% + 4.63%
= 14.08%
It is noticeable that Padma Ltd.’s portfolio return rate is 14.08%, which is more than the
company’s expected rate of return of 12%. Hence, investing in the portfolio was rational for
the company.
Portfolio return, 𝑅P
61. Mr. Parash Sams invested Tk. 50,000 in the share of ‘X’ Company and Tk. 20,000 in the
share of ‘Y’ Company. Rate of return of the two shares in different economic conditions are as
follows:
Rate of return
Economic condition
Probability
X Co.
Y Co.
Recession
0.30
13%
10%
Normal
0.40
12%
11%
Booming
0.30
16%
15%
(Chattogram Board-2022)
a) Determine expected rate of return of ‘X’ and ‘Y’ Company.
b) Analyze the rationale of portfolio investment if expected rate of return of Mr. Parash Sams is
12%.
Answer:
a) Determining expected rate of return of ‘X’ Company:
We know,
Expected rate of return, 𝑅̅𝑋
= ∑(𝑅𝑖 × 𝑃𝑖 )
= (13×0.30) + (12×0.40) + (16×0.30)
= 3.90 + 4.80 + 4.80
= 13.50%
Determining expected rate of return of ‘Y’ Company:
We know,
Expected rate of return, 𝑅̅𝑌
= ∑(𝑅𝑖 × 𝑃𝑖 )
= (10×0.30) + (11×0.40) + (15×0.30)
= 3 + 4.40 + 4.50
= 11.90%
Therefore, expected rate of return of ‘X’ and ‘Y’ Company are 13.50% and 11.90%
respectively.
b) Here,
̅𝑋
Expected return from ‘X’ Company, R
= 13.50% (from a)
50,000
Weight of investment in ‘X’ Company, WX
= 70,000 = 0.71
̅𝑌
Expected return from ‘Y’ Company, R
= 11.90% (from a)
Weight of investment in ‘Y’ Company, WY
= 70,000 = 0.29
20,000
Chapter-9: Risk and Rate of Return
2/99
We know,
Portfolio return, 𝑅P
̅ 𝑋 + 𝑊𝑌 R
̅𝑌
= 𝑊𝑋 R
= (0.71 × 13.50% ) + (0.29 × 11.90%)
= 9.59% + 3.45%
= 13.04%
Here, Mr. Parash Sams’s portfolio rate of return is 13.04%, which is higher than his
expected rate of return 12%. Therefore, his portfolio investment was rational.
62. Karnaphuli Limited invested 3,00,000 taka in X Company’s share and 2,00,000 taka in Y
Company’s share. Rates of return of the two shares are as follows in different economic
conditions:
Rate of return
X Company
Y Company
Recession
0.30
13%
10%
Normal
0.40
12%
11%
Booming
0.30
16%
15%
(Mymensingh Board-2022)
a) Determine expected rate of return of X Company and Y Company.
b) Analyze the rationale of portfolio investment if Karnaphuli Limited’s expected return is 12%.
Answer:
a) Determine expected rate of return of X Company:
We know,
Expected rate of return, 𝑅̅𝑋
= ∑(𝑅𝑖 × 𝑃𝑖 )
= (13×0.30) + (12×0.40) + (16×0.30)
= 3.90 + 4.80 + 4.80
= 13.50%
Determine expected rate of return of Y Company:
We know,
Expected rate of return, 𝑅̅𝑌
= ∑(𝑅𝑖 × 𝑃𝑖 )
= (10×0.30) + (11×0.40) + (15×0.30)
= 3 + 4.40 + 4.50
= 11.90%
Therefore, expected rate of return of X Company and Y Company are 13.50% and 11.90%
respectively.
b) Here,
̅ 𝑋 = 13.50% (from a)
Expected return of X Company, R
Economic condition
Probability
Weight of investment in X Company, WX
̅𝑌
Expected return of Y Company, R
= 11.90% (from a)
Weight of investment in Y Company, WY
We know,
3,00,000
= 5,00,000 = 0.60
2,00,000
= 5,00,000 = 0.40
Chapter-9: Risk and Rate of Return
2/100
̅ 𝑋 + 𝑊𝑌 R
̅𝑌
= 𝑊𝑋 R
= (0.60 × 13.50% ) + (0.40 × 11.90%)
= 8.10% + 4.76%
= 12.86%
Here, Karnaphuli Limited’s portfolio rate of return is 12.86%, which is higher than its
expected rate of return 12%. Therefore, Karnaphuli Limited’s portfolio investment was
rational.
Portfolio return, 𝑅P
63. ‘Sonali Ltd. invested Taka 3,00,000 in the share of X Company and Taka 2,00,000 in the
share of Y Company. The rate of return of the two shares in different economic conditions are
given below:
Economic
conditions
Recession
Normal
Boom
Probability
Rate of return
X company
Y company
13%
10%
12%
11%
16%
15%
.30
.40
.30
(Sylhet Board-2021)
(a)
(b)
(a)
Calculate expected rate of return of X and Y Company.
Analyze the rationality of portfolio investment of ‘Sonali Ltd.’ if the expected rate of return
is 12%
Answer:
Determining expected rate of return of Company-X:
Here,
Expected rate of return
(R ) =  R  P
X
i
i
= (13  0.30 ) + (12  0.40 ) + (16  0.30 )
= 3.9 + 4.8 + 4.8
= 13.5%
Determining expected rate of return of Company-Y:
Here,
( )
Expected rate of return RY =  Ri  Pi
= (10  0.30) + (11  0.40) + (15  0.30)
= 3 + 4.4 + 4.5
= 11.9%
Therefore, expected rate of return of company X and Y is 13.5% and 11.9% respectively.
(b)
Here,
( )
Expected rate of return of Company-X, R X = 13.5% ( from a)
Chapter-9: Risk and Rate of Return
2/101
Investment ratio in Company-X, (WX ) =
Again,
3,00,000
= 0.60
5,00,000
( )
Expected rate of return of Company-Y, RY = 11.9% ( from a)
Investment ratio in Company-Y, (WY ) =
We know,
Portfolio rate of return,
(
) (
R P = W A  R A + WB  R B
2,00,000
= 0.40
5,00,000
)
= (0.60  13.5) + (0.40  11.9 )
= 8.10 + 4.76
= 12.86%
Therefore, portfolio investment of Sonali bank is appropriate. Because portfolio rate of
return of Sonali bank is 12.86 which is higher than expected rate of return.
64. Karnafuli Limited invested Tk. 30,000 in X Company’s share and Tk. 20,000 in Y Company’s
share. In different economic conditions, rate of returns of the two shares are as follows:
Economic condition
Probabilities
Recession
Normal
Boom
0.30
0.40
0.30
Rate of return
X Co.
Y Co.
13%
10%
12%
11%
16%
15%
(Barishal Board-2019)
(a)
(b)
(a)
Calculate expected rate of return of X and Y Company.
If expected rate of return of Karnafuli Limited is 12%, evaluate logicality of portfolio
investment.
Answer:
Determining expected return of X Co.:
( )
Expected rate of return R X =  RiPi
= (13  0.30) + (12  0.40) + (16  0.30)
= 3.9  4.8  4.8
= 13.5%
Determining expected return of Y Co.:
( )
Expected rate of return R Y =  RiPi
Chapter-9: Risk and Rate of Return
2/102
= (10  0.30) + (11 0.40) + (15  0.30)
= 3  4.4  4.5
= 11.9%
Here, expected rate of return both X and Y Company is 13.5% and 11.9% respectively.
(b)
Here,
WX =
30,000
= 0.6
50,000
WY =
20,000
= 0.4
5,000
R X =13.5%
RY =11.9%
Portfolio rate of return (Rp) = WiRi
= Wx  Rx + Wy  Ry
= (0.6  13.5) + (0.4 11.9)
= 12.86%
Here portfolio rate of return is 12.86% which is higher than the company’s expected rate of
return 12%. So the portfolio investment is logical.
65. Mr. Sakib got Tk. 1,00,00,000 from company after his retirement. From it, he has invested
Tk. 50,00,000 in two projects. He has invested 60% of total investment in project-A and 40% in
project-B. Rate of return and probability of the two projects under different circumstances are
given below:
Economic Condition
Probability
Return
Project-A
Project-B
Recession
0.20
20%
-12%
Normal
0.60
15%
10%
Booming
0.20
30%
25%
Expected rate of return of Mr. Sakib is 14%. (Dhaka Board’17)
a) Calculate expected rate of return of project-A and project-B.
b) According to the stem, calculate portfolio return and justify rationale of portfolio investment.
Answer:
a) Calculating expected rate of return of project-A and project-B:
Expected rate of return of project-A,
̅A
R
= ∑𝑛𝑖=1(𝑅𝑖 × 𝑃𝑖 )
= 𝑅1 𝑃1 + 𝑅2 𝑃2 + 𝑅3 𝑃3
= (0.20 × 0.20) + (0.15 × 0.60) + (0.30 × 0.20)
= 0.04 + 0.09 + 0.06
= 0.19
Chapter-9: Risk and Rate of Return
2/103
= 19%
Expected rate of return of project-B,
̅B
R
= ∑𝑛𝑖=1(𝑅𝑖 × 𝑃𝑖 )
= (−0.12 × 0.20) + (0.10 × 0.60) + (0.25 × 0.20)
= −0.024 + 0.06 + 0.05
= 0.086
= 8.60%
Answer: 19% and 8.60%.
b) Calculating portfolio return according to the stem:
Given,
WA = 0.60
WB = 0.40
We know,
̅P
Portfolio return, R
= ∑𝑛𝑖=1(𝑊𝑖 × 𝑅𝑖 )
= (0.60 × 0.19) + (0.40 × 0.086)
= 0.114 + 0.0344
= 0.1484
= 14.84%
When an investor invests in more than one company or asset to minimize his risk, it is called
portfolio. In the stem, Mr. Sakib got Tk. 1,00,00,000 from company after his retirement. From
it, he has invested TK. 50,00,000 in two projects. That is, he has followed the principle of
portfolio investment. As a result, he will be able to compensate the loss of one investment
from the profit of other investment. Because, there is a little probability of having loss or
profit in both projects at the same time. Besides, portfolio return (14.84%) is also more than
expected rate of return of Mr. Sakib (14%). For this reason, Mr. Sakib’s portfolio investment
is rational.
66. Miss Maria has achieved degree from Department of Business Administration, Dhaka
University. She is eager to make investment decisions analyzing risk-return properly. She is
considering information regarding the following securities for investment:
Probability
Expected return
Security-A Security-B
0.30
0.10
-0.05
0.40
0.12
0.20
0.30
0.14
0.02
Expected return
0.12
0.0710
Standard deviation
0.0155
0.1088
Miss Maria will invest 40% on security-A and 60% on security-B. (Cumilla Board’17)
a) Determine Miss Maris’s portfolio return.
b) “Miss Maria is considering security-B to be more risky” – Evaluate rationality of the
statement.
Chapter-9: Risk and Rate of Return
2/104
Answer:
a) Determining Miss Maris’s portfolio return:
Given,
̅𝐴 = 12%
Expected return from security-A, R
̅ 𝐵 = 7.10%
Expected return from security-B, R
Weight of security-A, WA
= 40%
Weight of security-B, WB
= 60%
̅P
̅𝐴 + 𝑊𝐵 R
̅𝐵
So, portfolio return, R
= 𝑊𝐴 R
= (0.40 × 12%) + (0.60 × 7.10%)
= 4.80% + 4.26%
= 9.06%
Therefore, portfolio return of Miss Maris’s investment is 9.06%.
Answer: 9.06%.
b) To know which security is more risky, coefficient of variance of both the securities is to be
determined.
Determining coefficient of variance of security-A:
Standard deviation of security-A, σA = 1.55%
̅𝐴
Expected return of security-A, R
=12%
So, coefficient of variance of security-A, CV
𝜎
= 𝑅̅ × 100
=
1.55%
12%
× 100
= 12.92%
Determining coefficient of variance of security-B:
Standard deviation of security-A, σB = 10.88%
̅𝐵
Expected return of security-A, R
=7.10%
So, coefficient of variance of security-B, CV
𝜎
= ̅ × 100
𝑅
10.88%
= 7.10% × 100
= 153.24%
Therefore, coefficient of variance of security-B is more than that of security-A. Hence,
security-B is more risky. That is why Miss Maria has regarded security-B as more risky. So,
it can be said that her statement given in the stem is right.
Portfolio Return and Portfolio Risk Related Math
67. Mr. Harun is interested to invest in securities through portfolio analysis. He wants to utilize
the investable fund as follows:
Securities
Portfolio-1
X
Y
Invested
funds
60%
40%
Expected rate
of return
10%
20%
Standard
deviation
5%
10%
Corelation
0-45
Chapter-9: Risk and Rate of Return
Portfolio-2
2/105
Y
Treasury bill
30%
70%
20%
30%
10%
0
0%
(Sylhet Board-2023)
a) Determine the rate of return of portfolio-1.
b) Which portfolio is reasonable for Mr. Harun to invest? Give your opinion.
Answer:
a) Determining the rate of return of portfolio-1:
Here,
Weight of security X, WX
= 60% or 0.60
Weight of security Y, WY
= 40% or 0.40
̅
Expected return of security X, R𝑋
= 10%
̅𝑌
Expected return of security Y, R
= 20%
We know,
̅ 𝑌 + 𝑊𝑋 R
̅𝑌
Portfolio return, 𝑅P
= 𝑊𝑋 R
= (0.60 ×10%) + (0.40 ×20%)
= 6% + 8%
= 14%
So, the rate of return on portfolio-1 is 14%.
b) To determining which portfolio is reasonable for Mr. Harun to invest, we need to compare
the risk of the portfolios.
Determining the amount of risk of portfolio-1:
Here,
Weight of security X, WX
= 60% or 0.60
Weight of security Y, WY
= 40% or 0.40
Standard deviation of security X, σX = 5%
Standard deviation of security Y, σY = 10%
Co-relation, CORXY
= 0.45
We know, portfolio standard deviation or risk,
𝜎𝑃
2
= √𝑊𝑋 2 × 𝜎𝑋 2 + 𝑊𝑌 × 𝜎𝑌 2 + 2 × 𝑊𝑋 × 𝑊𝑌 × 𝜎𝑋 × 𝜎𝑌 × 𝐶𝑂𝑅𝑋𝑌
= √(0.60)2 × (5)2 + (0.40)2 × (10)2 + 2 × 0.60 × 0.40 × 5 × 10 × 0.45
= √9 + 16 + 10.8
= √35.80
= 5.98%
Determining the amount of risk of portfolio-1:
Here,
Weight of security Y, WY
= 30% or 0.30
Weight of treasury bill, WT
= 70% or 0.70
Standard deviation of security X, σX = 10%
Standard deviation treasury bill, σT
= 0%
Co-relation, CORYT
=0
Chapter-9: Risk and Rate of Return
2/106
We know, portfolio risk,
𝜎𝑃
2
= √𝑊𝑌 2 × 𝜎𝑌 2 + 𝑊𝑇 × 𝜎𝑇 2 + 2 × 𝑊𝑌 × 𝑊𝑇 × 𝜎𝑌 × 𝜎𝑇 × 𝐶𝑂𝑅𝑌𝑇
= √(0.30)2 × (10)2 + (0.70)2 × (0)2 + 2 × 0.30 × 0.70 × 10 × 0 × 0
= √9 + 0 + 0
= √9
= 3%
Here, the risk of portfolio-2 is 3%, which is less than the risk of portfolio-1 of
5.98%. Therefore, portfolio-2 will be reasonable for Mr. Harun to invest.
68. Mr. Ariz is the finance manager of an institution. Beta of the institution is 1.5, the rate of
market return is 16% and risk-free rate of return is 8%. Mr. Ariz personally invests in finance
market. He is interested to invest in multiple securities to reduce risk. Information of his invested
securities are as follow:
Security-A
Security-B
Expected rate of return
14%
16%
Standard Deviation
4%
4.35%
Weight
40%
60%
Correlation of securities
-0.85
(Chattogram Board-2023)
a) Calculate the required rate of return of the institution mentioned in the stem.
b) Evaluate Mr. Ariz’s investment decision, whether it is helpful to reduce risk or not.
Answer:
a) Determining the required rate of return of the institution:
Here,
Risk-free rate of return, Rf
= 8%
Market rate of return, Rm
= 16%
Beta, β
= 1.5
We know,
̅
Expected rate of return, R
= 𝑅𝑓 + (𝑅𝑚 − 𝑅𝑓 ) × 𝛽
= 8% + (16% − 8%) × 1.5
= 8% + (8% × 1.5)
= 8% + 12%
= 20%
⸫ The required rate of return of the institution is 20%.
b) Determining portfolio risk:
Here,
Weight of security-A, WA = 40% or 0.40
Weight of security-B, WB = 60% or 0.60
Standard deviation of security-A, σA = 4%
Standard deviation of security-B, σB = 4.35%
Chapter-9: Risk and Rate of Return
2/107
Correlation, CORAB = -0.85
We know, portfolio standard deviation or risk,
𝜎𝑃
2
= √𝑊𝐴 2 × 𝜎𝐴 2 + 𝑊𝐵 × 𝜎𝐵 2 + 2 × 𝑊𝐴 × 𝑊𝐵 × 𝜎𝐴 × 𝜎𝐵 × 𝐶𝑂𝑅𝐴𝐵
=
√(0.40)2 × (4)2 + (0.60)2 × (4.35)2 + 2 × 0.40 × 0.60 × 4 × 4.35 × (−0.85)
= √2.56 + 6.8121 – 7.0992
= √2.2729
= 1.51%
Here, standard deviation or risk of security-A and security-B are 4% and 4.35%
respectively. On the other hand, the portfolio risk is 1.51%. Hence, Mr. Ariz’s investment
decision will be helpful to reduce risk.
69. The information of share of two companies are as follows:
Company
Expected rate of return
Standard deviation
Alexa
40%
15%
Marcel
45%
17%
The covariance of the two shares is 0.90. The value of portfolio risk will be 68.03% if the
investment in both companies is equal. (Rajshahi Board-2023)
a) What will be the return of portfolio according to the stem?
b) "If the ratio of investment is 70: 30, it will be possible to reduce the portfolio risk." – Evaluate
the appropriateness of the statement.
Answer:
a) Determining portfolio rate of return:
Here,
Weight of the investment in Alexa Company, WA
= 50% or 0.50
Weight of the investment in Marcel Company, WM = 50% or 0.50
̅𝐴
Expected return from Alexa Company, R
= 40%
̅ 𝑀 = 45%
Expected return from Marcel Company, R
We know,
̅𝐴 + 𝑊𝑀 R
̅𝑀
Portfolio return, 𝑅P
= 𝑊𝐴 R
= (0.50 × 40%) + (0.50 × 45%)
= 20% + 22.50%
= 42.50%
Hence, portfolio rate of return is 42.50%.
b) Determining portfolio risk:
Here,
70
Weight of the investment in Alexa Company, WA = 100 = 0.70
30
Weight of the investment in Marcel Company, WM = 100 = 0.30
Standard deviation of Alexa Company, σA
= 15%
Chapter-9: Risk and Rate of Return
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Standard deviation of Marcel Company, σM
Co-variance, COVAM = 0.90
We know, portfolio standard deviation or risk,
= 17%
2
𝜎𝑃 = √𝑊𝐴 2 × 𝜎𝐴 2 + 𝑊𝑀 × 𝜎𝑀 2 + 2 × 𝑊𝐴 × 𝑊𝑀 × 𝐶𝑂𝑉𝐴𝑀
= √(0.70)2 × (15)2 + (0.30)2 × (17)2 + (2 × 0.70 × 0.30 × 0.90)
= √110.25 + 26.01 + 0.378
= √136.638
= 11.69%
It is noticeable that if money is invested in the two companies at 70: 30 ratio, portfolio risk
will be reduced from 68.03% to 11.69%. Therefore, the statement "If the ratio of investment
is 70: 30, it will be possible to reduce the portfolio risk." – is correct.
70. Mr. Sumon is interested to invest in capital market. Necessary information are as follows:
Security Investment Rate of return
Standard deviation
Co-relation
X
40,000
12%
8%
0.80
Y
60,000
10%
6%
(Jashore Board-2022)
a) Determine portfolio rate of return of Mr. Sumon.
b) Determine portfolio standard deviation and evaluate the rationale of portfolio investment.
Answer:
a) Determining portfolio rate of return of Mr. Sumon:
Here,
̅𝑋
Expected return from security-X, R
= 12%
̅𝑌
Expected return from security-Y, R
= 10%
4,0,000
Weight of investment in security-X, WX = 100,000 = 0.40
60,000
Weight of investment in security-Y, WY = 100,000 = 0.60
We know,
Portfolio return, 𝑅P
̅ 𝑋 + 𝑊𝑌 R
̅𝑌
= 𝑊𝑋 R
= (0.40 × 12%) + (0.60 × 10%)
= 4.80% + 6%
= 10.80%
Therefore, portfolio rate of return of Mr. Sumon is 10.80%.
b) Determining portfolio standard deviation:
Here,
Standard deviation of security-X, σX
= 8%
Standard deviation of security-Y, σY
= 6%
Weight of investment in security-X, WX = 0.40
Weight of investment in security-Y, WY = 0.60
Co-relation, CORXY = 0.80
Chapter-9: Risk and Rate of Return
2/109
We know, portfolio standard deviation,
2
𝜎𝑃 = √𝑊𝑋 2 × 𝜎𝑋 2 + 𝑊𝑌 × 𝜎𝑌 2 + 2 × 𝑊𝑋 × 𝑊𝑌 × 𝜎𝑋 × 𝜎𝑌 × 𝐶𝑂𝑅𝑋𝑌
= √(0.40)2 × (8)2 + (0.60)2 × (6)2 + 2 × 0.40 × 0.60 × 8 × 6 × 0.80
= √10.24 + 12.96 + 18.432
= √41.632
= 6.45%
It is noticeable that portfolio rate of return of Mr. Sumon is 10.80%, which is higher than his
portfolio standard deviation 6.45%. Therefore, Mr. Sumon’s portfolio investment is rational.
71. Mr. Zaman has received Tk. 10 lac as retirement benefit. He has decided to invest this money
in two types of shares. He will invest 60% of his money in share ‘X’ and rest of the money is
share ‘Y’ The rate for return and standard deviation of these share are as follows:
Share X
Share Y
Expected return
14%
12%
Standard deviation
15%
8%
Co-efficient of correlation
0.80
(Rajshahi Board–2021)
(a) Calculate the portfolio rate of return of Mr. Zaman.
(b) If the value of co-efficient of correlation is from + 1 to -1 of these shares, what will be the
effect on risk of these shares? Explain your comment.
Answer:
(a) Determining the portfolio rate of return of Mr. Zaman:
Here,
Weight of share X, (W X )
( )
= 60% or 0.60
Expected rate of return of X, R X
= 14%
Weight of share Y, (WY )
= 40% or 0.40
( )
Expected rate of return of Y, RY
We know,
(
= 12%
) (
Portfolio rate of return, (RP ) = W X  R X + WY  RY
)
= (0.60  14% ) + (0.40  12% )
= 8.40% + 4.80%
= 13.20%
Therefore, portfolio rate of return of Mr. Zaman is 13.20%.
(b) Here,
Weight of share X, (W X )
Weight of share Y, (WY )
= 60% or 0.60
= 40% or 0.40
Standard deviation of share X, ( X ) = 15%
Chapter-9: Risk and Rate of Return
2/110
Standard deviation of share Y, ( Y ) = 8%
When correlation is +1, then portfolio risk,
 P = WX 2 X 2 + WY 2 X 2 + 2  WX  WY   X   Y  rXY
=
(0.60)2  (15)2 + (0.40)2  (8)2 + 2  0.60  0.40  15  8  1
= 81 + 10.24 + 57.60
= 148.84
= 12.20%
When correlation is -1, then portfolio risk,
 P = WX 2 X 2 + WY 2 X 2 + 2  WX  WY   X   Y  rXY
=
(0.60)2  (15)2 + (0.40)2  (8)2 + 2  0.60  0.40  15  8  (− 1)
= 81 + 10.24 − 57.60
= 33.64
= 5.80%
It is noticeable that when correlation between the two shares is +1, then portfolio risk is 12.20%.
And when correlation between the two shares is -1, then portfolio risk is reduced to 5.80%. That
means risk is decreased.
72. Mr. Riaz formed a portfolio of two securities last year where his portfolio risk was 15%.
Currently he is interested in building a new portfolio with the following two securities:
Security-A
Security- B
E (rA ) = 7.5%
E (rb ) = 10%
 A = 10%
 A = 40%
 B = 20%
 B = 60%
Correlation of A and B security is 0.30
(Barishal Board-2021)
(a)
(b)
(a)
Determine Mr. Riaz's new portfolio rate of return.
Will the formation of Mr. Riaz's new portfolio be less risky for him? Give opinion based
on mathematical analysis.
Answer:
Determining Mr. Riaz's new portfolio rate of return:
Here,
Expected rate of return of security-A, E(RA ) = 7.5%
Expected rate of return of security-B, E (RB ) = 10%
Investment ratio in security-A, (WA ) = 40% or 0.40
Investment ratio in share-B, (WB ) = 60% or 0.60
We know,
Chapter-9: Risk and Rate of Return
2/111
Portfolio rate of return, RP = WA  E(RA )+ WB  E (RB )
= (0.40  7.5) + (0.60  10 )
= 3+6
= 9%
Therefore, Mr. Riaz's new portfolio rate of return is 9%.
(b)
Determining portfolio risk:
Here,
Standard deviation of security-A, ( A ) = 10%
Standard deviation of security-B, ( B ) = 20%
Investment ratio in security-A, (WA ) = 40% or 0.40
Investment ratio in security-B, (WB ) = 60% or 0.60
Correlation between security A and B, (RAB ) = 0.30
We know,
Portfolio risk,  P = WA  A + WB  B + 2  WA  WB   A   B  COR AB
2
=
2
2
2
(0.40)2  (10)2 + (0.60)2  (20)2 + 2  0.40  0.60  10  20  0.30
= 16 + 144 + 28.80
= 188.80
= 13.74%
Therefore, the formation of Mr. Riaz's new portfolio will be less risky for him. Because the
risk of new portfolio is 13.74 which is lower than the old portfolio risk.
73. Mr. M is desirous to make investment decision by analyzing the return and risk of shares. He
has considered the date of the following two shares for the purpose of investment.
Average return
Standard deviation
Share A
11%
9.50%
Share B
12%
11%
Mr. M will invest 45% in share A and 55% in share B of his total investment. (Chattogram
Board-2021)
(a)
(b)
Find out portfolio return of Mr. M.
Show the comparative analysis between portfolio risk and risks of two shares.
Answer:
(a)
Here,
Investment ratio in share-A, (WA ) = 45% or 0.45
Investment ratio in share-B, (WB ) = 55% or 0.55
Chapter-9: Risk and Rate of Return
2/112
Rate of return of share-A, (R A ) = 11%
Rate of return of share-B, (RB ) = 12%
We know,
(
) (
Portfolio rate of return, RP = WA  RA + WB  RB
)
= (0.45  11) + (0.55  12 )
= 4.95% + 6.60%
= 11.55%
Therefore, portfolio rate of return of Mr. M is 11.55%.
(b)
Determining coefficient of variance of share-A:
Coefficient of variance (CV )=  A
A
RA
 100
9.50
 100
11
= 86.36%
Determining coefficient of variance of share-B:
Coefficient of variance (CV ) =  B
=
B
RB
 100
11
 100
12
= 91.67%
=
Determining portfolio risk of Mr. M:
Here,
Investment ratio in share-A, (WA ) = 45% or 0.45
Investment ratio in share-B, (WB ) = 55% or 0.55
Standard deviation of share-A, ( A ) = 9.50%
Standard deviation of share-B, ( B ) = 11%
We know,
Portfolio risk,  P = WA  A + WB  B
2
=
2
2
2
(0.45)2  (9.50)2 + (0.55)2  (11)2
= 18.275626 + 36.6025
= 54.878125
= 7.41%
It is noticeable that coefficient of variance or risk of share-A is 86.36% and coefficient of
variance or risk of share-B is 91.67%. On the other hand, portfolio risk is 7.41%. That is, if
Mr. M invests according to this portfolio, his risk will reduce.
Chapter-9: Risk and Rate of Return
2/113
74. Miss Borsha has invested in two securities in equal proportion. Other information are as
follows:
Economic condition
Probabilities
Recession
Normal
Boom
Average return
Standard Deviation
Return (%)
M
8%
12%
16%
11.60%
3.32%
0.40
0.30
0.30
-----------
N
2%
20%
30%
15.80%
11.91%
Co-relation of security M and N is 0.95. (Cumilla Board-2019)
(a)
(b)
Determine portfolio return of Miss Borsha.
Evaluate rationality of Miss Borsha’s portfolio investment in the stem.
(a)
Answer:
Here,
WM = 0.5
WN = 0.5
RM = 11.6%
RN = 15.8%
We know,
Portfolio rate of return (Rp) = WiRi
= (0.5  11.6) + (0.5  15.8)
= 5.8 + 7.9
= 13.7%
Therefore, Miss Borsha’s portfolio return is 13.7%.
(b)
We know,
Portfolio risk (Rp) = WM  M + WN  N + 2.WM .WN . M  N rMN
2
=
2
2
2
(0.5)2  (3.32)2 + (0.5)2  (11.91)2 + 2(0.5)(0.5)(3.32)(11.92)(0.95)
= 2.76 + 35.46 + 18.78
= 57
= 7.55%
As portfolio risk is lower than the risk of N securities, investment on this portfolio is
appropriate. But portfolio risk is higher than the risk of M securities. So, investment on M
securities is better than investment on the portfolio.
75. Mr. Nayan has an investment of Tk. 40,000. He will decide to invest this money as follows:
Company
A
B
C
D
Chapter-9: Risk and Rate of Return
2/114
Investment(Taka)
Expected rate of returns
4,000
8%
8,000
12%
12,000
15%
16,000
18%
(Cumilla Board-2021)
(a)
(b)
(a)
Determine the rate of portfolio return.
Determine standard deviation of the portfolio.
Answer:
Here,
4,000
Weight of Company-A, (W A ) =
= 0.10
40,000
Weight of Company-B, (WB ) =
8,000
= 0.20
40,000
Weight of Company-C, (WC ) =
12,000
= 0.30
40,000
Weight of Company-D, (WD ) =
16,000
= 0.40
40,000
( )
Expected rate of return of Company-B, (R ) = 12%
Expected rate of return of Company-C, (R ) = 15%
Expected rate of return of Company-D, (R ) = 18%
Expected rate of return of Company-A, R A = 8%
B
C
D
We know,
(
) (
) (
) (
Portfolio rate of return, RP = WA  RA + WB  RB + WC  RC + WD  RD
)
= (0.10  8) + (0.20  12) + (0.30  15) + (0.40  18)
= 0.80 + 2.40 + 4.50 + 7.20
= 14.90%
Therefore, portfolio rate of return is 14.90%.
(b)
Determining standard deviation of the portfolio:
Let,
Standard deviation of Company A, B, C, D is 3%, 3.5%, 4%, 4.5% respectably.
We get from a
WA = 0.10 ; WB = 0.20 ; WC = 0.30 ; WD = 0.40
We know,
Standard deviation of portfolio,
Chapter-9: Risk and Rate of Return
2/115
 P = W A 2 A 2 + WB 2 B 2 + WC 2 C 2 + WD 2 D 2
=
(0.10)2  (3)2 + (0.20)2  (3.5)2 + (0.30)2  (4)2 + (0.40)2  (4.5)
= 0.09 + 0.49 + 1.44 + 3.24
= 5.26
= 2.29%
Therefore, standard deviation of the portfolio is 2.29%.
(Note: As no company’s standard deviation is given in the stem, the math is solved
assuming standard deviation of each company.)
Other Questions
76. Mr. Amin wants to invest in any one of three securities A, B and C. But he emphasizes to
avoidable risk to invest. The details of all three securities are as follows:
Security-A: Rate of return for 2018, 2019 and 2020 are 6%, 12% and 8% respectively and market
rate of return is 10%, 18% and 9% respectively.
Security-B: The variance between market rate of return and that security is 13.5% and the
standard deviation is 12%.
Security-C: Expected rate of return is 20%, while risk-free rate of return is 5% and market rate
of return is 10%. (Sylhet Board-2023)
a) Calculate the co-variance of security-A according to the stem.
b) Which one is to be selected for Mr. Amin of the three securities? Give your opinion.
Answer:
a) Determining the co-variance of security-A:
Here,
∑𝑅
Average rate of return, 𝑅̅𝐴
= 𝑛𝑖
=
6 + 12 + 8
3
26
= 3
= 8.67%
10+18+9
Average market rate of return, 𝑅̅𝑀 =
3
37
= 3
= 12.33%
⸫ Co-variance of security-A,
Cov (RA, RM) =
=
=
=
∑(𝑅𝐴 −𝑅̅𝐴 ) ∑(𝑅𝑀 −𝑅̅𝑀 )
𝑛−1
(6−8.67)(10−12.33)+(12−8.67)(18−12.33)+(8−8.67)(9−12.33)
3−1
6.2211 + 18.8811 + 2.2311
2
27.3333
2
Chapter-9: Risk and Rate of Return
2/116
= 13.67%
Therefore, co-variation of security-A is 13.67%.
b) To determine which of the three securities is to be selected for Mr. Amin, we can compare
their unavoidable risk.
Determining the unavoidable risk (Beta) of Security-A:
We get from a,
Co-variance Cov (RA, RM)
= 13.67%
̅
Average market rate of return, 𝑅𝑀
= 12.33%
⸫ Standard deviation of market return, σM = √
∑(𝑅𝑀 −𝑅̅𝑀 )2
=√
𝑛−1
(10−12.33)2 +(18−12.33)2 +(9−12.33)2
3−1
=√
5.4289 + 32.1489 + 11.0889
=√
48.6667
2
2
= √24.33335
= 4.93%
We know,
Beta, βA
=
Cov (𝑅𝐴 ,𝑅𝑀 )
σ𝑀 2
13.67
= (4.93)2
13.67
= 24.3049
= 0.56
Determining the unavoidable risk (Beta) of Security-B:
Given,
Co-variance Cov (RB, RM)
= 13.5%
Standard deviation, σM
= 12%
We know,
Beta, βB
=
Cov (𝑅𝐵 ,𝑅𝑀 )
σ𝑀 2
13.5
= (12)2
13.5
= 144
= 0.09
Determining the unavoidable risk (Beta) of Security-C:
Given,
Expected rate of return, RC
= 20%
Risk-free rate of return, Rf
= 5% and
Market rate of return, RM
= 10%
We know,
Chapter-9: Risk and Rate of Return
Beta, βB
2/117
𝑅 −𝑅
= 𝑅 𝐴 − 𝑅𝑓
𝑀
𝑓
20−5
= 10−5
=
15
5
=3
Here, the unavoidable risk (beta) of security-A, B and C is 0.56, 0.09 and 3 respectively. That
is, the unavoidable risk of security-B is relatively low. In this case, Mr. Amin should invest
in security-B.
77. Mr. Ali is an eminent garments businessman. His garment’s products are of international
quality. On the other hand, Mr. Siddique invests in consumer goods, real estate and different
renowned companies’ share besides his garments business. Last year, there was a blockade on
Bangladeshi garment products by the USA. As a result, the garments sector faced a huge financial
crisis. Mr. Ali lost his capital and became helpless. But though Mr. Siddique could not make
profit in his business like the previous years, he did not face financial loss. (Jashore Board’16)
a) According to the stem, explain the causes of loss in Mr. Ali’s business.
b) Explain the relationship between portfolio theory and risk according to Mr. Siddique’s
business.
Answer:
a) According to the stem, the cause of loss in Mr. Ali’s business is lack of diversity.
Not to invest in only one security or sector, rather to invest in many different securities or
sectors is called diversification. Though risk cannot be totally ignored through diversification,
it helps to minimize risk to some extent.
In the stem, Mr. Ali is a renowned garments businessman. He produces international quality
products in his garments. Last year, there was a blockade by the USA. As a result, the
garments sector faced a huge financial crisis. Mr. Ali lost his capital and became helpless.
The reason why Mr. Ali lost his capital was the business risk due to blockade imposed on
USA market that was out of Mr. Ali’s control. If he were conscious from the beginning, he
would bring diversification in investment and invest in other sector besides garment sector.
As a result, if he faced loss from garment sector, he would be able to recover it from other
sectors and sustain in the market.
So we conclude that the main reason of Mr. Ali’s failure is lack diversification.
b) Besides increasing number of securities, amount of avoidable risk is reduced in portfolio.
Portfolio theory is such a procedure by which investors try to maximize income against a
certain risk. Normally there are two or more securities in portfolio and in most cases, they are
of different characteristics. As a result all the securities are not equally affected if there is any
change in the market.
In the stem, Mr. Siddique invests in consumer goods, real estate and different renowned
companies share besides his garments business. Last year, there was a blockade on
Bangladeshi garment products by the USA. As a result, the garments sector faced a huge
financial crisis. As a result, though many garment businessmen like Mr. Ali lost their capital
Chapter-9: Risk and Rate of Return
2/118
and became helpless, Mr. Siddique could continue his business without any interrupt. It is
only the reason that Mr. Siddique followed the principle of portfolio diversity in his
investment.
Normally the more the number of securities in a portfolio is, the less the risk. In fact, there is
a negative relationship between number of portfolio securities and amount of risk. Increases
in number of securities decreases amount of risk. And this has been reflected in Mr.
Siddique’s business.