On risk and return Objective Learn the math of portfolio diversification Measure relative risk...
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Transcript of On risk and return Objective Learn the math of portfolio diversification Measure relative risk...
![Page 1: On risk and return Objective Learn the math of portfolio diversification Measure relative risk Estimate required return as a function of relative risk.](https://reader030.fdocuments.in/reader030/viewer/2022032704/56649d635503460f94a45867/html5/thumbnails/1.jpg)
On risk and return
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Objective
• Learn the math of portfolio diversification
• Measure relative risk
• Estimate required return as a function of relative risk
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Important observation
(portfolio) < (rA)wA + (rB) wB
In general, the standard deviation of the portfolio is less than the average of the individual standard
deviations
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The standard deviation of a portfolio return
Number of stocks in the portfolio
(portfolio)
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The standard deviation of expected return: A summary
• Standard deviation and variance measure the variability of the return
• Standard deviation is a measure of absolute risk
• The standard deviation of a portfolio is less than the weighted average of
individual standard deviations
• This is true because the returns of various securities are not perfectly correlated,
i.e. changes in returns are not perfectly synchronized.
• By adding individual securities to a portfolio, the overall standard deviation
of the portfolio is likely to decrease.
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More on the standard deviation of a portfolio return
• Bundling stocks & bonds into portfolios is called diversification
• Diversification is useful because it reduces risk
• The amount of risk (standard deviation) that can be eliminated is called diversifiable or non-systematic risk.
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More on the standard deviation of a portfolio return
Number of securities in the portfolio
(portfolio)
Non-systematic risk
Systematic (market) risk
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Remember
E(rA) = 6.6%
E(rB) = 5.2%
(rA) = 2.94%
(rB) = 0.979%
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Standard deviation
Expected return
(rA) = 2.94(rB) = 0.98
E(rB) = 5.2%
E(rA) = 6.6% A
B
(p) = 1.64
E(p) = 5.9
P
Portfolio P when (A,B) = 0.2
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Standard deviation %
Expected return %
(rA) = 2.94(rB) = 0.98
E(rB) = 5.2
E(rA) = 6.6 A
B
(p) = 1.64
E(p) = 5.9
All possible portfolio combinations of A and B when (A,B) = 0.2
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Variations
What if the returns of A and B were perfectly correlated?
(A,B) = 1
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Standard deviation %
Expected return %
(rA) = 2.94(rB) = 0.98
E(rB) = 5.2
E(rA) = 6.6 A
B
Portfolio P when (rA,B) = 1
(p) = 1.96
E(p) = 5.9P
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Standard deviation %
Expected return %
(rA) = 2.94(rB) = 0.98
E(rB) = 5.2
E(rA) = 6.6 A
B
(p) = 1.96
E(p) = 5.9
All possible portfolio combinations of A and B when (rA,B) = 1
P
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More variations
What if the returns of A and B were perfectly negatively correlated?
(rA,B) = - 1
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Standard deviation %
Expected return %
(rA) = 2.94(rB) = 0.98
E(rB) = 5.2
E(rA) = 6.6 A
B
All possible portfolio combinations of A and B when (rA,B) = -1
(p) = 1.45
E(p) = 5.9P
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Standard deviation
Expected return
A
B
All possible portfolio combinations of A and B, for all possible correlations between the return of A and B
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Reality check
There are thousands of securities in the market
Their returns are highly correlated, but not perfectly correlated
0 < < 0.8
There are benefits from diversification!
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Standard deviation
Expected return
All possible portfolio combinations in a world with n securities
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Question
Of all possible combinations, which portfolios would you rather hold?
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Answer
It is expected that you would want to hold the portfolios that have:
• the highest expected return for a given standard deviation, or
• the lowest standard deviation for a given level of expected return
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Standard deviation
Expected return
All possible portfolio combinations in a world with n securities
The efficient set
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Question
From the efficient set, which portfolios would you rather hold?
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Answer
It depends on your risk preference.
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Yet another reality check
Individuals can borrow and lend money fairly easily...
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Question(s)
How many individuals/families have a savings account/GIC?
How many individuals/families invest in the stock market directly, or through mutual funds, pension plans etc?
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Facts
Almost everyone holds (directly or indirectly) a combination of risky assets and risk-free investments.
Risky assets: Stocks, bonds, etc.
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A portfolio of risky assets and risk-free investments
Risky assets: A and B
Risk-free investment: T-bill
E(rA) = 6.6%
E(rB) = 5.2%
(rA) = 2.94%
(rB) = 0.979%
A,B = 0.2
E(rT) = 3%
(rT) = 0
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Portfolio P
Weights: A (50%) and B (50%)
(portfolio) = 1.64%
ER(p) = 5.9%
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Standard deviation %
Expected return %
(rA) = 2.94(rB) = 0.98
E(rB) = 5.2
E(rA) = 6.6 A
B
(p) = 1.64
E(p) = 5.9
Portfolio P when (A,B) = 0.2
P
E(rT) = 3
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Various combinations between P and T
Combination C1:
Invest $5,000 in T and $5,000 in P
E(C1) = (1/2)3% + (1/2)5.9% = 4.45%
(C1) = (1/2)1.64% = 0.82%
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Standard deviation %
Expected return %
(p) = 1.64
E(p) = 5.9
P
E(rT) = 3
(C1) =0.82%
E(C1) =4.45
C1
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Various combinations between P and T
Combination C2:
Invest $2,500 in T and $7,500 in P
E(C2) = (1/4)3% + (3/4)5.9% = 5.175%
(C2) = (3/4)1.64% = 1.23%
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Standard deviation %
Expected return %
(p) = 1.64
E(p) = 5.9
P
E(rT) = 3
C1
(C2) = 1.23
E(C2) =5.175
C2
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Various combinations between P and T
Combination C3:
Invest $7,500 in T and $2,500 in P
E(C3) = (3/4)3% + (1/4)5.9% = 3.725%
(C3) = (1/4)1.64% = 0.41%
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Standard deviation %
Expected return %
(p) = 1.64
E(p) = 5.9
P
E(rT) = 3
C1
C2
(C3) = 0.41
E(C3) = 3.725C3
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Important
Combinations among risky assets lie on a curved line
Combinations between risky assets and the risk-free investment lie on a straight line
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Question
How many possible combinations of risky assets and risk free investments are there?
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Standard deviation
Expected return
All possible portfolio combinations in a world with n securities and a risk-free investment
Risk-free return
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Standard deviation
Expected return
Risk-free return
Question: Of all possible portfolios in the world, which ones would you rather hold?
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Standard deviation
Expected return
Answer: Efficient portfolios only!
Risk-free return
The efficient set
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Important
Again, note that all portfolios from the efficient set have:
- The highest expected return for a given level of risk
- The lowest level of risk for a given level of expected return
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Standard deviation
Expected return
Risk-free return
The efficient set
Question: Of all the efficient portfolios, which ones would you hold?
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Answer
The choice is dictated by individual risk preferences
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Standard deviation
Expected return
Question: Of all possible portfolios of risky assets, which one(s) would you rather hold?
Risk-free return
The efficient set
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Standard deviation
Expected return
Answer: Of all possible combinations of risky assets, investor would want to hold only M
Risk-free return
The efficient set
M
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Question
Why only M?
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Answer
M is the only portfolio of risky assets that produces efficient portfolios when combined with the risk-free investment
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Important
All investors should buy the same portfolio of risky assets, regardless of their risk preference
Adjusting for risk:
In order to reflect individual risk preferences, each investor would combine M with the risk-free asset:
- More audacious investors would borrow money to buy more of M
- More prudent investors would park a fraction of their wealth in the risk-free investment
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Consequences
M is very important !
Out of respect, let’s call it The Optimal Portfolio.
Optimal portfolio aka Market portfolio
Due to its importance, M becomes the yardstick for risk in the marketplace
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More consequences
If M is the yardstick for risk, we should compare each risky security/portfolio to M
The result of the comparison would yield the relative risk of any given security
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Comparing risky securities to M
Comparison by regression:
Ri = + RM +e
i = the relative risk of security “i”
In other words, measures the contribution of each stock
to the volatility of the market portfolio
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Comparing risky securities to M
Convention: M = 1
i < 1, the security is less risky than the market
i > 1, the security is riskier than the market
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Risk and return: The climax
We want to find how to estimate the expected return that would compensate for bearing the aforementioned risk
Again, use M
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The Facts of life
On average, M earns a return above and beyond the risk free rate.
In other words, M earns a risk premium, which is the reward for bearing risk.
returnM = risk free rate + risk premiumM
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Risk and return: The climax
Using algebra, we can prove that:
(Ri - Rf)/ = (RM - Rf)/1
Interpretation:
The required risk premium per unit of relative risk is constant among all securities in this world
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Summary
Diversification reduces absolute risk
Some combinations of risky securities result in efficient portfolios
When there is a risk-free investment, only one efficient portfolio of risky assets is desirable: M
Investors combine M with the risk-free asset in different proportions
M is the yardstick for risk (CAPM)
The risk premium per unit of relative risk is constant across all securities (CAPM)