Risk and rate of return in finance

1. RISK
&
RATES OF RETURN

2. NO RISK NO GAIN!
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Investors like returns & dislike risk. Therefore, people invest in risky
assets only if they expect to receive higher returns. Bonds offer
relatively low returns, but with relatively little risk. Stocks offer the
chance of higher returns, but stocks are generally riskier than bonds.
Deviation from/Difference between actual and expected return. The
chance that some unfavorable event will occur. If you engage in
skydiving, you are taking a chance with your life—skydiving is risky.

3. CLASSIFICATION OF RISK
Risk
Portfolio Risk
Diversifiable Risk/Unsystematic
Risk
Market Risk/Systematic Risk
Stand Alone Risk
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4. STAND ALONE RISK
Stand Alone Risk:
The risk an investor would face if he or she held only one asset. No investment
should be undertaken unless the expected rate of return is high enough to
compensate for the perceived risk.
To illustrate stand-alone risk, suppose an investor buys $100,000 of short-term
Treasury bills with an expected return of 5%. In this case, the investment’s
return, 5%, can be estimated quite precisely; and the investment is defined as
being essentially risk-free.
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5. STAND ALONE RISK (CONT’D)
Expected Rates of Return: The rate of return expected to be realized from an
investment; the weighted average of the probability distribution of possible
results.
𝑬𝒙𝒑𝒆𝒄𝒕𝒆𝒅 𝑹𝒆𝒕𝒖𝒓𝒏 𝑬(𝑹𝒊) = σ𝒊=𝟏
𝒏
PiRi
Measuring Stand Alone Risk:
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The standard deviation is a measure of
how far the actual return is likely to
deviate from the expected return.
The coefficient of variation shows the risk per
unit of return, and it provides a more
meaningful risk measure when the expected
returns on two alternatives are not the same.

6. PORTFOLIO RISK
Portfolio: Combination of assets/stocks.
Expected Portfolio Returns, r^p: the expected return on a portfolio is a weighted
average of expected returns on the stocks in the portfolio, with the weights being
the percentage of the total portfolio invested in each asset :
Measuring Portfolio Risk:
The portfolio’s risk, 𝜎p, is not the weighted average of the individual stocks’
standard deviations. The portfolio’s risk is generally smaller than the average of
the stocks because diversification lowers the portfolio’s risk.
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7. PORTFOLIO RISK (CONT’D)
Correlation: The tendency of two variables to move together.
Correlation Coefficient, r: A measure of the degree of relationship between
two variables.
Reduction of portfolio risk through diversification:
•Diversification is completely useless for reducing risk if the stocks in the
portfolio are perfectly positively correlated(r=+1).
•Diversification is quite meaningful/significant if two assets are perfectly
negatively correlated. (r=-1).
•Diversification to some extend will become meaningful (r=0).
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8. PORTFOLIO RISK (CONT’D)
Diversifiable Risk: That part of a security’s risk associated with random
events; can be eliminated by proper diversification. This risk is also known as
company-specific, or unsystematic, risk.
Market Risk: The risk that remains in a portfolio after diversification has
eliminated all company-specific risks. This risk is also known as non-
diversifiable or systematic or beta risk.
Risk Aversion: Risk-averse investors dislike risk and require higher rates of
return as an inducement to buy riskier securities.
Risk Premium (RP): The difference between the expected rate of return on a
given risky asset and that on a less risky asset
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9. MEASUREMENT OF MARKET RISK
Beta Coefficient, 𝜷: Beta is a statistical measure of the volatility of a stock versus the
overall market. It's generally used as both a measure of systematic risk and a
performance measure. The market is described as having a beta of 1. The beta for a
stock describes how much the stock's price moves compared to the market.
Beta =
𝑪𝒐𝒗𝒂𝒓𝒊𝒂𝒏𝒄𝒆
𝑽𝒂𝒓𝒊𝒂𝒏𝒄𝒆
where:
Covariance=Measure of a stock’s return relative to that of the market
Variance=Measure of how the market moves relative to its mean
•A beta above 1 means a stock is more volatile than the overall market.
•A beta below 1 means a stock is less volatile than the overall market.
•If the beta is below 1, the stock has lower volatility than the market. The price
of Treasury bills (T-bills) has a beta lower than 1 because T-bills don't move in relation to
the overall market.
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10. CAPITAL ASSET PRICING MODEL(CAPM)
CAPM: A model based on the proposition that any stock’s required rate of
return is equal to the risk-free rate of return plus a risk premium that
reflects only the risk remaining after diversification.
Market Risk Premium: The additional return over the risk-free rate needed
to compensate investors for assuming an average amount of risk.
Market Risk Premium= Rm- Rf
The market risk premium, RPM, shows the premium that investors require
for bearing the risk of an average stock. Let us assume that at the current
time, Treasury bonds yield Rf is 6% and an average share of stock has a
required rate of return of Rm is 11%. Therefore, the market risk premium is
5%, calculated as follows:
Market Risk Premium= Rm- Rf =11%-6% =5%
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11. SECURITY MARKET LINE (SML)
SML/ Characteristic Line: The line on a graph that shows the relationship between
risk as measured by beta and the required rates of return on individual securities. So
SML is characteristic line/ graphical representation of market risk & return at a given
time.
Required rate of return on stock= Risk free rate of return + 𝛃 𝐱 𝐌𝐚𝐫𝐤𝐞𝐭 𝐑𝐢𝐬𝐤 𝐏𝐫𝐞𝐦𝐢𝐮𝐦
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12. DECISION MAKING THROUGH CAPM
From CAPM, we can take 03 types of decision:
Whether the share is 1) Overpriced 2) Under Priced or 3) Fairly Priced
Rp=
𝑷𝒕− 𝑷𝒕−𝟏 +𝑫𝑷𝑺
𝑷𝒕−𝟏
x 100
If my E(Rp)< actual return, expectation was understated
If E(Rp)> actual return, expectation was overstated
If E(Rp)= actual return, stock is fairly priced.
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13. LIMITATION OF CAPM
▪ Market as a single risk factor (Doesn’t consider macroeconomic factors like
interest, inflation, etc.)
▪ 𝛃 is calculated from past return which doesn’t necessarily indicate future
riskiness correctly.
▪ Betas do not remain stable over time.
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14. CAPITAL MARKET LINE (CML)
• The capital market line (CML) represents portfolios that optimally combine
risk and return. It is a theoretical concept that represents all the portfolios
that optimally combine the risk-free rate of return and the market portfolio of
risky assets.
•Under the capital asset pricing model (CAPM), all investors will choose a
position on the capital market line, in equilibrium, by borrowing or lending at
the risk-free rate, since this maximizes return for a given level of risk.
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15. CAPITAL MARKET LINE (CML) CONT’D
• The intercept point of CML and efficient frontier
would result in the most efficient portfolio called the
tangency portfolio.
• Calculating the capital market line is done as follows:
Rp= Rf +
𝑹𝒎−𝑹𝒇
𝝈m x 𝝈p
Rp=portfolio return
Rf= risk-free rate
Rm=market return
σm=standard deviation of market returns
σp=standard deviation of portfolio returns
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16. DIFFERENCE BETWEEN CML & SML
Issues CML SML
Risk Total Risk Only systematic/ Market
Risk
Risk measured
by
Standard Deviation Beta coefficient which helps
to find securities' risk
contribution for the
portfolio.
Portfolio Efficient portfolio
(Considers Risk Return
Trade Off)
All portfolio
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17. MATHEMATICAL PROBLEMS
Problem 01:
An individual has $35,000 invested in a stock with a beta of 0.8 and another $40,000
invested in a stock with a beta of 1.4. If these are the only two investments in her
portfolio, what is her portfolio’s beta?
Problem 02:
A stock has a required return of 11%, the risk-free rate is 7%, and the market risk
premium is 4%.
a. What is the stock’s beta?
b. If the market risk premium increased to 6%, what would happen to the stock’s
required rate of return? Assume that the risk-free rate and the beta remain
unchanged.
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18. MATHEMATICAL PROBLEMS
Problem 03:
Stocks X and Y have the following probability distributions of expected future returns:
a. Calculate the expected rate of return, r^y, for Stock Y (r^x = 12%).
b. Calculate the standard deviation of expected returns, 𝜎x, for Stock X (𝜎y =20.35%).
Now calculate the coefficient of variation for Stock Y. Is it possible that most investors
will regard Stock Y as being less risky than Stock X? Explain.
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Probability X Y
0.1 (10%) (35%)
0.2 2 0
0.4 12 20
0.2 20 25
0.1 38 45

19. MATHEMATICAL PROBLEMS
Problem 04:
Suppose you are the money manager of a $4 million investment fund. The
fund consists of four stocks with the following investments and betas:
If the market’s required rate of return is 14% and the risk-free rate is 6%,
what is the fund’s required rate of return?
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Stock Investment Beta
A $ 400,000 1.50
B 600,000 (0.50)
C 1,000,000 1.25
D 2,000,000 0.75

20. MATHEMATICAL PROBLEMS
Problem 05:
Given the following information, determine the beta coefficient for Stock J
that is consistent with equilibrium: Rj= 12.5%; Rf= 4.5%; Rm= 10.5%.
Problem 06:
Bradford Manufacturing Company has a beta of 1.45, while Farley
Industries has a beta of 0.85. The required return on an index fund that
holds the entire stock market is 12.0%. The risk-free rate of interest is 5%.
By how much does Bradford’s required return exceed Farley’s required
return?
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21. MATHEMATICAL PROBLEMS
Problem 07:
Suppose you held a diversified portfolio consisting of a $7,500 investment in each
of 20 different common stocks. The portfolio’s beta is 1.12. Now suppose you
decided to sell one of the stocks in your portfolio with a beta of 1.0 for $7,500 and
use the proceeds to buy another stock with a beta of 1.75. What would your
portfolio’s new beta be?
Problem 08:
You have been managing a $5 million portfolio that has a beta of 1.25 and a
required rate of return of 12%. The current risk-free rate is 5.25%. Assume that
you receive another $500,000. If you invest the money in a stock with a beta of
0.75, what will be the required return on your $5.5 million portfolio?
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22. SOLUTIONS
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Solution 01: Solution 02:
Hints:
Answer: 1.12
Hints:
Calculate Rm-Rf &
Answer: a) 1 b) 13%
Solution 03:
Hints: 𝑬(𝑹𝒊) = σ𝒊=𝟏
𝒏
PiRi
Answer: a) 14% b) 12.19% , CVy=1.45
CVx=1.015 Select Stock X
Solution 04:
Hints: Portfolio Beta & SML equation
Answer: 12.1%

23. SOLUTIONS
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Solution 05: Solution 06:
Hints: SML Equation
Answer: 1.33
Hints: SML Equation
Answer: 4.2%
Solution 07:
Hints: First Calculate the weight & then
calculate beta without old stock. Lastly again
calculate portfolio beta with new stock beta
Answer: 1.16
Solution 08:
Hints:
S1: Calculate new Portfolio Beta
S2: Find out Rm from SML equation with old
return
S3: Calculate new return using the new beta
Answer: 11.75%

24. Thank you!
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