Risk & Return- Integrated Problem Solving
Table of Contents
Introduction ............................................................................................................ 3
Question no. a .......................................................................................................... 5
Answer to the Question no. a (1) .............................................................................. 5
Answer to the Question no. a (2) .............................................................................. 5
Question no. b ......................................................................................................... 6
Answer to the Question no. b ................................................................................... 6
Question no. c .......................................................................................................... 7
Answer to the Question no. c (1) .............................................................................. 8
Answer to the Question no. c (2) .............................................................................. 8
Answer to the Question no. c (3) .............................................................................. 9
Question no. d ....................................................................................................... 10
Answer to the Question no. d ................................................................................. 10
Question no. e ........................................................................................................ 11
Answer to the Question no. e ................................................................................. 11
Question no. f ........................................................................................................ 14
Answer to the Question no. f (1) ............................................................................. 14
Answer to the Question no. f (2) ............................................................................. 16
Question no. g ........................................................................................................ 18
Answer to the Question no. g (1) ............................................................................ 18
Answer to the Question no. g (2) ............................................................................ 19
Question no. h ....................................................................................................... 20
Answer to the Question no. h (1) ............................................................................ 20
Answer to the Question no. h (2) ............................................................................ 20
Question no. i ........................................................................................................ 21
Answer to the Question no. i (1) ............................................................................. 21
1
Answer to the Question no. i (2) ............................................................................. 22
Answer to the Question no. i (3) ............................................................................. 23
Question no. j ........................................................................................................ 27
Answer to the Question no. j (1) ............................................................................. 28
Answer to the Question no. j (2) ............................................................................. 29
Answer to the Question no. j (3) ............................................................................. 30
Answer to the Question no. j (4) ............................................................................. 30
Conclusion ............................................................................................................ 32
Reference .............................................................................................................. 33
2
Introduction
This chapter serves as a foundation for understanding the core principles of risk and return that are
central to Merrill Finch's investment analysis and decision-making. Merrill Finch, a prominent financial
services firm, faces the critical challenge of guiding clients towards investment portfolios that align
with their individual risk tolerances and financial objectives. This requires a deep understanding of risk
and return concepts, including expected return, risk measurement, portfolio diversification, and the
inherent trade-off between risk and reward. By exploring these concepts, this chapter will provide a
framework for evaluating investment alternatives, constructing optimal portfolios, and ultimately,
helping Merrill Finch clients achieve their financial goals.
1. Investment Risk
•
Defining Investment Risk: Investment risk refers to the uncertainty surrounding the actual
return that an investment will generate. It encompasses various forms, including market risk,
credit risk, liquidity risk, and operational risk.
o
Systematic Risk: Market risk, also known as systematic risk, is inherent in the overall
market and cannot be eliminated through diversification.
o
Unsystematic Risk: Company-specific risk, or unsystematic risk, can be mitigated
through diversification by investing in a portfolio of assets with low correlations.
•
The Impact of Risk: Understanding and managing investment risk is crucial for Merrill Finch.
High levels of risk can significantly impact investment performance, potentially leading to
substantial losses and jeopardizing client portfolios.
2. Measuring Investment Risk
•
Key Risk Metrics:
o
Standard Deviation: Measures the dispersion of possible outcomes around the
expected return.
o
Variance: The square of the standard deviation, providing a measure of the overall
dispersion of returns.
o
Coefficient of Variation (CV): A standardized measure of risk per unit of expected
return, useful for comparing investments with different expected returns.
o
Beta: A measure of a security's systematic risk relative to the market.
o
Value at Risk (VaR): Estimates the potential loss in value of an investment or portfolio
over a specific time horizon and at a given confidence level.
3
•
Application for Merrill Finch: Merrill Finch utilizes these risk metrics to assess the risk
profiles of individual securities, construct diversified portfolios, and manage risk within client
portfolios.
3. Expected Return
•
Defining Expected Return: Expected return represents the anticipated profit or loss on an
investment. It is calculated by considering the probability of different outcomes and their
associated returns.
•
Factors Influencing Expected Return: Factors that influence expected return include market
conditions, economic growth, industry trends, company performance, and interest rates.
•
Expected Return and Risk: Expected return and risk are inherently linked. Generally, higher
expected returns are associated with higher levels of risk.
4. Portfolio Diversification
•
The Benefits of Diversification: Portfolio diversification is a fundamental investment strategy
that aims to reduce overall portfolio risk. By combining assets with low correlations, investors
can reduce the impact of individual asset fluctuations on the overall portfolio.
•
Correlation and Diversification: The correlation between assets plays a crucial role in
portfolio diversification. Assets with low or negative correlations tend to offset each other's
movements, reducing overall portfolio volatility.
•
Constructing Diversified Portfolios: Merrill Finch utilizes diversification strategies by
investing in a variety of asset classes, sectors, and geographic regions to create well-diversified
portfolios for its clients.
5. Risk-Return Trade-off
•
The Fundamental Relationship: The risk-return trade-off is a central principle in finance. It
highlights the inherent relationship between risk and expected return – higher expected returns
typically require investors to accept higher levels of risk.
•
The Efficient Frontier: The efficient frontier represents a set of portfolios that offer the highest
possible expected return for a given level of risk, or the lowest possible risk for a given level of
expected return.
4
Question no. a
1. Why is the T-bill’s return independent of the state of the economy? Do T-bills
promise a completely risk-free return? Explain.
2. Why are High Tech’s returns expected to move with the economy, whereas
Collections Inc’s are expected to move counter to the economy?
Answer to the Question no. a (1)
1. Why is the T-bill’s return independent of the state of the economy? Do T-bills promise
a completely risk-free return? Explain.
Ans:
The T-bill return does not depend on the state of the economy because the Treasury must (and
will) redeem the bills at par regardless of the state of the economy. The T-bills are risk free in
the default risk sense because
Answer to the Question no. a (2)
2. Why are High Tech’s returns expected to move with the economy, whereas Collections
Inc’s are expected to move counter to the economy?
Ans:
The return will be realized in all possible economic states. However, this return is composed
of the real risk-free rate. Let’s say the rate of return is 1% and inflation premium is 2%. Since
there is uncertainty about inflation, it is unlikely that the realized real rate of return would equal
the expected 1%. For example, if inflation averaged 2.5% over the year, then the realized real
return would only be 3.0% - 2.5% = 0.5%, not the expected 1%. Thus, in terms of purchasing
power, T-bills are not riskless.
5
Question no. b
Calculate the expected rate of return on each alternative and fill in the blanks on
the row for 𝒓̂ in the previous table.
Answer to the Question no. b
Ans:
High Tech’s returns move with the economy, hence are positively correlated with the with
economy because the firm’s sales, or profits, will generally experience the same type of ups
and downs as the economy. If the economy is booming, High Tech will too. On the other hand,
Collections Inc. is considered by many investors to be a hedge against both bad times and high
inflation, so if the stock market crashes, investors in this stock should do relatively well. Stocks
such as Collections are thus negatively correlated with (move counter to) the economy. (Note:
In actuality, it is almost impossible to find stocks that are expected to move counter to the
economy.)
The expected rate of return, 𝑟̂ , is expressed as follows:
𝑁
∑ 𝑃𝑖 𝑟𝑖
𝑖=1
Here,
Pi = is the probability of occurrence of the ith state
ri = is the estimated rate of return for that state of the economy
N = is the number of states of the economy.
Here is the calculation for High Tech:
r̂ High Tech = 0.1(−29.5%) + 0.2(−9.5%) + 0.4(12.5%) + 0.2(27.5%) + 0.1(42.5%).
6
We use the same formula to calculate ’s for the other alternatives:
𝑟̂ T-bills = 0.1(3%) + 0.2(3%) + 0.4(3%) + 0.2(3%) + 0.1(3%).
= 3.0%.
𝑟̂ Collections Inc. = 0.1(24.5%) + 0.2(10.5%) + 0.4(−1%) + 0.2(−5%) + 0.1(−20%).
= 1.2%.
𝑟̂ U.S Rubber = 0.1(3.5%) + 0.2(−16.5%) + 0.4(0.5%) + 0.2(38.5%) + 0.1(23.5%).
= 7.3%.
𝑟̂ Market = 8.0%.
Question no. c
You should recognize that basing a decision solely on expected returns is only
appropriate for risk-neutral individuals. Because your client, like virtually
everyone, is risk averse, the riskiness of each alternative is an important aspect of
the decision. One possible measure of risk is the standard deviation of returns.
1. Calculate this value for each alternative, and fill in the blank on the row for
 in the table
2. What type of risk is measured by the standard deviation?
3. Draw a graph that shows roughly the shape of the probability distributions
for High Tech, U.S. Rubber, and T-bills
7
Answer to the Question no. c (1)
Calculate this value for each alternative, and fill in the blank on the row for  in the table
Ans:
Standard Deviation for High Tech:
σ = formula
σ High Tech = [(-29.5 – 9.9)2(0.1) + (-9.5 – 9.9)2(0.2) + (12.5 – 9.9)2(0.4) + (27.5 – 9.9)2(0.2) +
(42.5 – 9.9)2(0.1)]1/2
= 20.0%.
Standard deviations for the other alternatives:
σT-bills = 0.0%.
σCollections = 11.2%.
σU.S. Rubber = 18.8%.
σM = 15.2%
Answer to the Question no. c (2)
What type of risk is measured by the standard deviation?
Ans:
The standard deviation is a measure of a security’s (or a portfolio’s) stand-alone risk. The larger
the standard deviation, the higher the probability that actual realized returns will fall far below
the expected return, and that losses rather than profits will be incurred.
8
Answer to the Question no. c (3)
Draw a graph that shows roughly the shape of the probability distributions for High
Tech, U.S. Rubber, and T-bills
Probability
T-bills
U.S Rubber
High Tech
3
7.3
9.9
Rate of Return
9
Question no. d
Suppose you suddenly remembered that the coefficient of variation (CV) is
generally regarded as being a better measure of stand-alone risk than the standard
deviation when the alternatives being considered have widely differing expected
returns. Calculate the missing CVs and fill in the blanks on the row for CV in the
table. Does the CV produce the same risk rankings as the standard deviation?
Explain.
Answer to the Question no. d
The coefficient of variation (CV) is a standardized measure of dispersion about the expected
value; it shows the amount of risk per unit of return.

CV = 𝑟̂
CVT-bills =
0.0%
5.5%
= 0.0.
CVHigh Tech =
20.0%
12.4%
= 1.6.
CCollections =
13.2%
1.0%
= 13.2.
18.8%
CVU.S. Rubber =
9.8%
= 1.9.
CVM
=
15.2%/
10.5%
= 1.4.
10
When we measure risk per unit of return, Collections, with its low expected return, becomes
the riskiest stock. The CV is a better measure of an asset’s stand-alone risk than  because CV
considers both the expected value and the dispersion of a distribution—a security with a low
expected return and a low standard deviation could have a higher chance of a loss than one
with a high  but a high 𝑟̂ .
Question no. e
Someone mentioned that you might also want to calculate the Sharpe ratio as a
measure of standalone risk. Calculate the missing ratios and fill in the blanks on
the row for the Sharpe ratio in the table. Briefly explain what the Sharpe ratio
actually measures.
Answer to the Question no. e
Expected Rate of
Return
Standard
Deviation
Risk Free Rate
High Tech
9.9%
11.2
3%
U.S Rubber
7.3%
18.8
3%
Market Portfolio
8.0%
15.2
3%
The Sharpe ratio measures the risk-adjusted return of an investment. It is calculated as:
Sharpe Ratio =
𝐄𝐱𝐩𝐞𝐜𝐭𝐞𝐝 𝐫𝐞𝐭𝐮𝐫𝐧−𝐑𝐢𝐬𝐤 𝐟𝐫𝐞𝐞 𝐫𝐞𝐭𝐮𝐫𝐧
𝐒𝐭𝐚𝐧𝐝𝐚𝐫𝐝 𝐝𝐞𝐯𝐢𝐚𝐭𝐢𝐨𝐧
11
High-tech:
Sharpe Ratio =
9.9%−3%
11.2%
= .6161
U.S Rubber:
Sharpe Ratio =
7.3%−3%
18.8%
= .2287
Market Portfolio:
Sharpe Ratio =
8.0%−3%
15.2%
= .3289
Securities
Sharpe Ratio
High Tech
.6161
U.S Rubber
.2287
Market
Portfolio
.3289
Collections
-.16
Two-Stock
Portfolio
.54
12
•
High Tech
High Tech offers the highest risk-adjusted return among the individual investments, which
indicates it provides significant returns compared to its risk. It will make it appealing to the
customers.
•
U.S. Rubber
U.S Rubber has a much lower sharp ratio that indicates that its return is not well compared
to its risk and it may only be appealing for very specific diversification needs.
•
Market Portfolio
The market portfolio provides average risk-adjusted performance.
•
Collections Inc.
Collections Inc. has a negative Sharpe ratio, meaning its return is below the risk-free rate.
•
Two-Stock Portfolio
The portfolio of High Tech and Collections has a solid Sharpe ratio, although slightly lower
than High Tech alone which shows the benefit of diversification.
Conclusion:
High tech gives the investors highest return for taking the risk, U.S Rubber has comparatively
lower reward for taking risk, the Market portfolio shows average performance, Collections
underperforms with returns below the risk-free rate, and the Two-Stock Portfolio balances risk
and return well.
13
Question no. f
Suppose you created a two-stock portfolio by investing $50,000 in High Tech and
$50,000 in Collections Inc.
1. Calculate the expected return (r⁄p), the standard deviation (sp), the coefficient
of variation (CVP),
and the Sharpe ratio for this portfolio and fill in the appropriate blanks in the
table.
2. How does the riskiness of this two-stock portfolio compare with the riskiness of
the individual stocks if they were held in isolation?
Answer to the Question no. f (1)
Calculate the expected return (r⁄p), the standard deviation (sp), the coefficient of
variation (CVP).
Ans:
•
Expected Return (rp):
The two-stock portfolio allocates 50% to High Tech and 50% to collections. The portfolio
return in each state is:
State
Recession
probability
.1
High Tech
-29.5%
Collections
24.5%
Below
Average
Average
.2
-9.5
10.5%
.4
12.5%
-1.0%
Above
Average
Boom
.2
27.5%
-5.0%
.1
42.5%
-20%
Portfolio Return
0.5⋅ (−29.5%) +0.5⋅24.5%
= −2.5%
0.5⋅ (−9.5%) +0.5⋅10.5%
= 0.5%
0.5⋅12.5%+0.5⋅ (−1.0%)
=5.75%
0.5⋅27.5%+0.5⋅ (−5.0%)
=11.25%
0.5⋅42.5%+0.5⋅ (−20.0%)
=11.25%
14
•
Expected return
rp= ∑Pi⋅rp,i
rp= (0.1⋅−2.5) +(0.2⋅0.5) +(0.4⋅5.75) +(0.2⋅11.25) +(0.1⋅11.25)
= 5.8%
Standard Deviation:
State
rp
rpi-rp
(rpi-rp)2
Recession
-2.5%
-8.3
0.006889
.1
0.0006889
Below
.5%
-5.3%
0.002809
.2
0.0005618
Average
5.75%
.05%
0.000000025
.4
0.00000001
Above
11.25%
5.45%
0.002970
.2
0.000594
11.25%
5.45%
0.002970
.1
0.002970
Probability Pi⋅(rp,i−rp)2
average
average
Boom
S =√∑ Pi⋅(rp,i−rp)2
=√ (0.0006889+0.0005618+0.00000001+0.000594+0.002970)
= 4.6%
• Coefficient of Variation (CVP):
CVP=SP/RP
= 4.6% / 5.8%
=.79
• Sharpe Ratio:
𝑅𝑃 − 𝑅𝐹
𝑆𝑝
=
5.8%−3%
4.6%
= .61
15
Updated table:
Expected Return
5.8%
Std. Deviation
4.6%
Coefficient of Variation
.79
Sharpe Ratio
.61
Answer to the Question no. f (2)
2. How does the riskiness of this two-stock portfolio compare with the riskiness of the
individual stocks if they were held in isolation?
Ans:
Comparison of Portfolio vs. Individual Stocks
Risk (Standard Deviation):
High Tech:
11.2%
Collections:
18.8%
Portfolio:
4.6%
➢ The portfolio has significantly lower risk due to diversification benefits, as High Tech
and Collections have negative correlations.
16
Coefficient of Variation (CV):
High Tech:
2.6
Collections:
1.9
Portfolio:
0.79
➢ The portfolio provides better returns relative to its risk.
Sharpe Ratio:
High Tech:
20.16
Collections:
0.54
Portfolio:
0.61
➢ While High Tech has the highest Sharpe ratio, the portfolio achieves a balanced tradeoff between risk and return.
Conclusion
The advantages of diversification are shown by the two-stock portfolio's far lower risk when
compared to owning each stock separately. But when it comes to return per unit of risk, High
Tech continues to be the best performer.
17
Question no. g
Suppose an investor starts with a portfolio consisting of one randomly selected
stock.
1. What would happen to the riskiness and to the expected return of the portfolio
as more randomly selected stocks were added to the portfolio?
2. What is the implication for investors? Draw a graph of the two portfolios to
illustrate your answer.
Answer to the Question no. g (1)
1. What would happen to the riskiness and to the expected return of the portfolio as more
randomly selected stocks were added to the portfolio?
Ans:
As more stocks are being added to the portfolio, the unsystematic risk, which arises from firmspecific factors, reduces because the negative impacts from one stock may be offset by positive
impacts from another, and so the risk is diversifiable and reduces with increasing stocks.
However, systematic risk, which is market-wide and cannot be diversified away, remains
constant regardless of the number of stocks added. As a result, the total portfolio risk, (as
measured by standard deviation) which is a sum of systematic and unsystematic risk,
experiences an overall decline with increasing stocks in the portfolio. Moreover, the portfolio's
expected return will stabilize as more stocks are added, converging toward the market’s average
expected return. This is because the weights of individual assets are diluted due to the
diversification, reducing the portfolio’s dependence on any one stock’s performance and
forming an average closer to the market’s average rate of return.
18
Answer to the Question no. g (2)
2. What is the implication for investors? Draw a graph of the two portfolios to illustrate
your answer.
Ans:
Investors should hold a portfolio consisting of a diverse range of stocks as opposed to a
portfolio consisting of only the same kind of stocks because diversification is a critical strategy
for reducing risk without sacrificing expected returns. Investors can lower unsystematic risk
significantly while maintaining expected returns by holding a sufficiently diversified portfolio,
leaving only the systematic risk, which cannot be diversified away. Additionally, selecting
negatively correlated stocks (like High Tech and Collections in this case) further enhances
diversification benefits.
Portfolio consisting of a single stock Portfolio consisting of more than one stock
19
Question no. h
1. Should the effects of a portfolio impact the way investors think about the
riskiness of individual stocks?
2. If you decided to hold a one-stock portfolio (and consequently were exposed to
more risk than diversified investors), could you expect to be compensated for all
of your risk; that is, could you earn a risk premium on the part of your risk that
you could have eliminated by diversifying?
Answer to the Question no. h (1)
Should the effects of a portfolio impact the way investors think about the riskiness of
individual stocks?
Ans:
Yes, the risk of an individual stock should be assessed based on its contribution to the overall
systematic risk of a well-diversified portfolio, which is usually measured by beta (β), rather
than its stand-alone risk. This is because, as investors add more stocks to their portfolios, the
overall unsystematic risk declines. Therefore, the risk an investor should consider for
individual stocks is their beta rather than total risk (standard deviation).
Answer to the Question no. h (2)
If you decided to hold a one-stock portfolio (and consequently were exposed to more risk
than diversified investors), could you expect to be compensated for all of your risk; that
is, could you earn a risk premium on the part of your risk that you could have eliminated
by diversifying?
Ans:
No. Investors are not compensated for unsystematic risk because it can be eliminated through
diversification. The expected risk premium for holding a stock depends only on its systematic
risk, as measured by its beta (β) relative to the market. Hence, a one-stock portfolio exposes
the investor to unnecessary risk without providing additional compensation.
20
Question no. i
The expected rates of return and the beta coefficients of the alternatives supplied
by an independent analyst are as follows:
Security
Return, 𝒓̂
Risk (Beta)
High Tech
9.9%
1.31
Market
8.0
1.00
U.S. Rubber
7.3
0.88
T-bills
3.0
0.00
Collections
1.2
(0.50)
1. What is a beta coefficient, and how are betas used in risk analysis?
2. Do the expected returns appear to be related to each alternative’s market risk?
3. Is it possible to choose among the alternatives on the basis of the information
developed thus far? Use the data given at the start of the problem to construct a
graph that shows how the T-bill’s, High Tech’s, and the market’s beta coefficients
are calculated. Then discuss what betas measure and how they are used in risk
analysis.
Answer to the Question no. i (1)
1. What is a beta coefficient, and how are betas used in risk analysis?
Ans:
Beta Coefficient: It is a metric that shows the extent to which a given stock’s returns move up
and down with the stock market. Beta measures market risk (Brigham & Houston, 2019).
Use of Beta in Risk Analysis: The volatility of returns in relation to the market is measured
by the beta (β) of an investment security, such as a stock. It is a crucial component of the Capital
Asset Pricing Model (CAPM) and is utilized as a risk indicator. Both risk and projected returns
are higher for a corporation with a higher beta (CFI Team, n.d.).
21
The beta coefficient can be interpreted as follows:
•
β =1 exactly as volatile as the market
•
β >1 more volatile than the market
•
β <1>0 less volatile than the market
•
β =0 uncorrelated to the market
•
β <0 negatively correlated to the market (CFI Team, n.d.)
So, Beta accomplishes two main goals. They are-
1) Assessing Market Risk: The amount of risk an investment adds to a diversified
portfolio is indicated by its beta. Low-beta stocks are safer and yield steady returns,
whereas high-beta stocks are riskier but have the potential for larger profits.
2) Calculating Required Return: The needed return on an investment is calculated by
the Capital Asset Pricing Model (CAPM) using beta. This aids investors in determining
if the risk of an investment is worthwhile.
Answer to the Question no. i (2)
2. Do the expected returns appear to be related to each alternative’s market risk?
Ans:
The relation between each alternative’s expected returns and market risk are given belowHigher Beta = Higher Expected Return:
1) High Tech (beta = 1.31) has the highest return (9.9%), consistent with its higher risk.
2) The Market (beta = 1.00) has a return of 8.0%, aligning with average market risk.
Lower Beta = Lower Expected Return:
1)
U.S. Rubber (beta = 0.88) has a lower return (7.3%) than the market return.
2)
T-bills (beta = 0.00) have a return equal to the risk-free rate (3%).
22
Negative Beta:
1)
Collections (beta = -0.50) has the lowest return (1.2%), as its negative beta reflects an
inverse relationship with the market. Investors use such assets for hedging rather than
growth.
Since higher-beta securities require larger returns to offset their risk, the relationship between
beta and expected return is usually accurate. Other factors, such as industry conditions and
company performance, may cause actual returns to differ.
Answer to the Question no. i (3)
3. Is it possible to choose among the alternatives on the basis of the information developed
thus far? Use the data given at the start of the problem to construct a graph that shows
how the T-bill’s, High Tech’s, and the market’s beta coefficients are calculated. Then
discuss what betas measure and how they are used in risk analysis.
Ans:
If we consider T-bills return as risk free return as the beta of T-bills is 0.00 and consider the
market portfolio return as the market return, then we can know the required rates of returns for
various alternatives and choose among them.
The Security Market Line (SML) is derived using the Capital Asset Pricing Model (CAPM):
𝑟𝑖 = 𝑟𝑓 +β𝑖 ×(𝑟𝑚 −𝑟𝑓 )
𝑟𝑓 = 3.0% (T-bill return, the risk- free rate)
𝑟𝑚 = 8.0% (Market Return)
(𝑟𝑚 −𝑟𝑓 )= 5% (Market Risk Premium)
23
Required returns for each security areSecurity
Expected Return (𝒓̂) Risk (Beta)
Required Return (𝒓𝒊 )
High Tech
9.9%
1.31
3.0+1.31×5.0= 9.55%
Market
8.0
1.00
3.0+1.00×5.0= 8.0%
U.S. Rubber
7.3
0.88
3.0+0.88×5.0= 7.40%
T-bills
3.0
0.00
3.0+0.00×5.0= 3.0%
Collections
1.2
-0.50
3.0+(-0.50)×5.0= 0.50%
High Tech & Collections: The expected rate of returns of High Tech and Collections
consecutively (9.9% & 1.2%) exceed the required returns (9.55% & 0.50%). So, we can say
that these two securities can be considered attractive to investors because it’s expected to
provide more value than what is needed for the investment’s risk (Brigham & Ehrhardt, 2024).
U.S. Rubber: The expected rate of return (7.3%) is lower than the required rate of return
(7.4%). So, the investment may not be deemed worthwhile because it does not compensate for
its level of risk. However, it may still appeal to moderately risk-averse investors, as its beta, β
(0.88) is lower than the market average (Brigham & Ehrhardt, 2024).
Market & T-bills: The expected rate of returns of Market and T-bills consecutively (8% &
3%) are equal to the required rate of returns (8% & 3%). The market provides exactly the return
predicted by CAPM. It serves as a benchmark for other securities but doesn’t offer additional
value above its risk. T-bills are risk-free investments (β=0) and provide a return that exactly
matches their required return. They are ideal for highly risk-averse or short-term investors
(Brigham & Ehrhardt, 2024).
24
Construction of Graph:
We can calculate beta using the CAPM formula.
𝑟𝑖 = 𝑟𝑓 +β𝑖 ×(𝑟𝑚 −𝑟𝑓 )
So, β𝑖 =
𝑟𝑖 −𝑟𝑓
𝑟𝑚 −𝑟𝑓
So, Beta of High Tech =
Beta of Market =
Beta of T-bills =
(9.55−3)
(8−3)
(8−3)
(8−3)
(3−3)
(8−3)
= 1.31
= 1.00
= 0.00
But to construct a graph we need a data set where we can find the returns on individual
securities in different states of economy or periods and returns on the market in different
states of economy or periods.
From the data given at the start of the problem we can find our required information,
State of
Return on the
Security-1
Security-2
Security-3
Economy
Market
High-Tech
T-bills
Market
Portfolio
Recession
-19.50%
-29.50%
3%
-19.50%
Below Average
-5.50%
-9.50%
3%
-5.50%
Average
7.50%
12.50%
3%
7.50%
Above Average
22.50%
27.50%
3%
22.50%
Boom
35.50%
42.50%
3%
35.50%
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Now using this data set, we can calculate the beta coefficients in excel using its slope
function.
Example is, Beta of High Tech: = SLOPE (High Tech data set, Market data set)
In the graph, returns on securities will be on y-axis and returns on market will be on x-axis.
Return on Securities (%)
50.00%
High Tech=beta=slope=1.31
40.00%
Market=beta=slope=1.00
30.00%
20.00%
10.00%
0.00%
-30.00% -20.00% -10.00% 0.00%
T-bills=beta=slope=0.00
10.00%
20.00%
30.00%
40.00%
High Tech
-10.00%
T-bills
-20.00%
Return on Market (%)
Market
-30.00%
-40.00%
Figure: Beta Coefficients in Securities Return vs Market Return Graph
Analysis of the Graph
1. T-bills (Risk-Free Rate): As expected, T-bills have a beta of 0 and offer the lowest
return of 3%, representing the risk-free rate.
2. Market Portfolio: The market portfolio has a beta of 1 and an expected return of 8%.
This aligns with the CAPM, where the market return is proportional to its beta.
3. High Tech: High Tech's beta is 1.31, showing it has 31% more risk compared to the
market. Its expected return of 9.9% reflects this additional risk premium.
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The relative volatility of a certain stock (high tech, for example) compared to an average stock
(market) is measured by beta coefficients. The beta of the typical stock is 1.0. The majority of
stocks have betas between 0.5 and 1.5. Although they can be negative in theory, betas are
typically positive in practice. The slope of the "characteristic" line—the regression line that
illustrates the correlation between a particular stock and the overall stock market—is used to
compute betas. In risk analysis, beta is mainly used to assess the amount of risk an investment
adds to a diversified portfolio.
Question no. j
The yield curve is currently flat; that is, long-term Treasury bonds also have a
3.0% yield. Consequently, Merrill Finch assumes that the risk-free rate is 3.0%.
1)
Write out the security market line (SML) equation; use it to calculate the
required rate of return on each alternative and graph the relationship
between the expected and required rates of return.
2)
How do the expected rates of return compare with the required rates of
return?
3)
Does the fact that Collections has an expected return that is less than the Tbill rate make any sense? Explain.
4)
What would be the market risk and the required return of a 50-50 portfolio
of High Tech and Collections? Of High Tech and U.S. Rubber?
27
Answer to the Question no. j (1)
1. Write out the security market line (SML) equation; use it to calculate the required rate
of return on each alternative and graph the relationship between the expected and
required rates of return.
Ans:
The Security Market Line (SML) equation is:
𝑟𝑖 = 𝑟𝑓 +β𝑖 ×(𝑟𝑚 −𝑟𝑓 )
𝑟𝑖 = Required Rate of Return
𝑟𝑓 = 3.0% (The Risk- Free Rate)
𝑟𝑚 = 8.0% (Market Return)
(𝑟𝑚 −𝑟𝑓 ) = 5% (Market Risk Premium)
Required returns for each security and relation between expected return and required return
areSecurity
Expected
Return (𝒓̂)
Risk (Beta)
High Tech
9.9%
1.31
Market
8.0
1.00
U.S. Rubber
7.3
0.88
T-bills
3.0
0.00
Collections
1.2
-0.50
Required
Conditions
Return (𝒓𝒊 )
𝑟𝑖 =𝑟𝑓 +β𝑖 ×(𝑟𝑚 −𝑟𝑓 )
Undervalued:
3.0+1.31×5.0
𝑟̂ > 𝑟𝑖
= 9.55%
Fairly valued:
3.0+1.00×5.0
= 8.0%
𝑟̂ = 𝑟𝑖
Overvalued:
3.0+0.88×5.0
= 7.40%
𝑟̂ < 𝑟𝑖
Fairly valued:
3.0+0.00×5.0
𝑟̂ = 𝑟𝑖
= 3.0%
3.0+(-0.50)×5.0 Undervalued:
𝑟̂ > 𝑟𝑖
= 0.50%
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Relationship Line Graph between the Expected Return & Required Return:
Using the data set from the box above we can make a graph where expected return and
Required Return & Expected
Return (%)
required return will be on y-axis and beta will be on x-axis.
-1
12
High Tech
10
U.S. Rubber
8
Market
6
4
Collections
T-bills
2
0
-0.5
0
0.5
Required Return
Expected Return
Linear (Required Return)
1
1.5
Risk (Beta)
Figure: Relationship Between Expected & Required Return
Answer to the Question no. j (2)
2. How do the expected rates of return compare with the required rates of return?
Ans:
(Note: The X-axis extends to the left of zero, giving the plot an odd appearance. Since our
stock has a negative beta, the necessary return is lower than the risk-free rate.)
High Tech and Collections plot are above the SML, U.S. Rubber plots are below the SML, and
the T-bills and market portfolio plot are on the SML. Consequently, the market portfolio and
T-bills offer a reasonable return; High Tech and Collections are attractive investments since
their projected returns exceed their needed returns; and U.S. Rubber has an expected return
below its needed return making it less attractive for investors.
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Answer to the Question no. j (3)
3. Does the fact that Collections has an expected return that is less than the T-bill rate
make any sense? Explain.
Ans:
Collections is a stock worth considering. It has a negative beta a negative beta is when an asset
moves in the opposite direction of the stock market (Liberto, 2024). Because of its negative
beta, which indicates negative market risk, adding it to a portfolio of conventional equities will
help lower the portfolio's total risk. Its required rate of return is therefore less than the risk-free
rate or T-bills rate. In essence, this makes Collections a good investment for logical, wellrounded
investors.
Insurance plans are comparable to negative-beta stocks. For instance, while fire insurance
policies pay out in emergency situations, like when a fire disrupts operations, businesses
continue to purchase them even though their expected returns are frequently negative due to
commissions and insurance company profits. In a similar vein, life insurance provides financial
stability in the event that an income source stops.
Answer to the Question no. j (4)
4. What would be the market risk and the required return of a 50-50 portfolio of High
Tech and Collections? Of High Tech and U.S. Rubber?
Ans:
The beta of a portfolio is the weighted average of the betas of the stocks in the portfolio.
High Tech & Collections:
Market risk or beta of a 50-50 portfolio of High Tech and Collections is𝛽𝑝 = ∑𝑁
𝑖=1 𝑤𝑖 × 𝛽𝑖
Weight of High Tech, 𝑤1=50%=0.5, Beta of High Tech, 𝛽1=1.31
Weight of Collections, 𝑤2 =50%=0.5, Beta of Collections, 𝛽2= -0.50
So, Beta of this portfolio is, 𝜷𝒑 = [(0.5× 1.31)+{0.5× (−0.50)}] = 0.405
30
And, to calculate the required return for portfolio𝑟𝑓 = 3.0% (The Risk- Free Rate)
𝑟𝑚 = 8.0% (Market Return)
(𝑟𝑚 −𝑟𝑓 ) = 5% (Market Risk Premium)
𝑟𝑝 = 𝑟𝑓 +β𝑝 ×(𝑟𝑚 −𝑟𝑓 )
𝒓𝒑 = 3%+0.405(5%) = 5.025%
High Tech & U.S. Rubber:
Market risk or beta of a 50-50 portfolio of High Tech and U.S. Rubber is𝛽𝑝 = ∑𝑁
𝑖=1 𝑤𝑖 × 𝛽𝑖
Weight of High Tech, 𝑤1=50%=0.5, Beta of High Tech, 𝛽1=1.31
Weight of Collections, 𝑤2 =50%=0.5, Beta of Collections, 𝛽2= 0.88
So, Beta of this portfolio is, 𝜷𝒑 = [(0.5× 1.31)+(0.5× 0.88)] = 1.095
And, to calculate the required return for portfolio𝑟𝑓 = 3.0% (The Risk- Free Rate)
𝑟𝑚 = 8.0% (Market Return)
(𝑟𝑚 −𝑟𝑓 ) = 5% (Market Risk Premium)
𝑟𝑝 = 𝑟𝑓 +β𝑝 ×(𝑟𝑚 −𝑟𝑓 )
𝒓𝒑 = 3%+1.095(5%) = 8.475%
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Conclusion
This integrated case's analysis of financial options highlights how crucial it is to weigh risk and
return when choosing an investment. Investments like T-bills offer a steady, stable return
regardless of the state of the economy, but they don't have the same potential for large returns
as riskier assets like U.S. Rubber and High Tech do. This highlights how important it is to
match investments to the client's objectives and risk tolerance. Combining assets that are
negatively correlated can dramatically reduce risk while still yielding a decent return, as
demonstrated by the two-stock portfolio (High Tech and Collections). This demonstrates that
diversification is an essential tactic for lowering independent risk. High Tech and U.S. Rubber
are two examples of how the expected and necessary rates of return highlight the necessity of
assessing assets both independently and in accordance with market conditions. Additionally,
Investors must assess the projected return as well as the risks involved (coefficient of variation
and standard deviation). Such decisions are supported by a thorough grasp of market risk when
instruments such as the Security Market Line (SML) are used.
Lastly, Investors should keep a diverse portfolio that is suited to their risk tolerance to
maximize returns while lowering risk. Making wise financial decisions in line with both shortterm and long-term goals can be facilitated by having a solid understanding of the link between
expected and required returns as well as the advantages of diversification.
32
Reference
Brigham, E. F., & Ehrhardt, M. C. (2024). Financial management: Theory & practice (17e
ed.). Cengage.
Brigham, E. F., & Houston, J. F. (2019). Fundamentals of financial management (15e ed.).
Cengage.
CFI Team. (n.d.). Beta. Corporate Finance Institute. Retrieved January 15, 2025, from
https://corporatefinanceinstitute.com/resources/valuation/what-is-beta-guide/
Liberto, D. (2024, June 6). What Beta Means When Considering a Stock’s Risk. Investopedia.
https://www.investopedia.com/investing/beta-know-risk/
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