Transcript 4464-Chapter-04.ppt

HFT 4464
Chapter 4
Risk & Return
7/12/2016
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Chapter 4 Introduction
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This chapter will discuss the concept of risk and how
it is measured.
Furthermore, this chapter will discuss:
 Risk aversion
 Mean return
 Variance and standard deviation of return
 Systematic and unsystematic risk
 Capital asset pricing model (CAPM)
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What is Risk?
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Risk is the uncertainty that an outcome will vary
from our expectations.
 For an investment, it is the notion that cash
flows or percentage returns will be different
than our expectations.
 This includes the “upside” potential as well as
the “downside.”
 As the potential outcomes widen, so does the
risk.
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Declining Marginal Utility for Money
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“Who Wants to be a Millionaire” example
 Why do most contestants stop and not “take a
chance” for the $1 million question?
 Because the potential utility gain is not as much
as the utility lost from an incorrect answer (and
loss of a major portion of their winnings up to that
point)
4-4
Risk Aversion
and Risk Averse Behavior
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Does this mean we never take risks?
Risk aversion means that we must be
compensated adequately for bearing risk.
This applies to returns for individual investors
investing in stocks or bonds.
It also applies to companies deciding on investing
in new projects for their shareholders.
4-5
Returns and Distributions
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An example of a distribution is a potential distribution
of dividends.
Each potential dividend is an outcome.
Each outcome has a probability of occurrence
associated with it.
Expected return = average return
 Simple average vs. weighted average
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What is the Expected Dividend for the
Following Distribution?
Dividend
$10
$7
$5
$3
$1
Probability
10%
20%
40%
20%
10%
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Expected Dividend / Return Calculation
Dividend
Probability
Return
$10
10%
$1.00
$7
20%
$1.40
$5
40%
$2.00
$3
20%
$0.60
$1
10%
$0.10
Expected Return
$5.10
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Expected Value Calculation
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We have an expected outcome (mean) and a
number of outcomes around the mean.
 This is called a distribution.
 A normal “bell-shaped” curve has half the
outcomes to the right of the mean.
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What Can a Distribution Tell Us?
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In this example, the shape of the distribution tells us about
the risk of the investment.
Variance is a measure of risk.
 The variance examines the differences between each
outcome and the expected value.
 Variance is a positive number.
 In general form, variance is the sum of:
((Outcome 1 – expected value)2 x probability of outcome 1))
+ ((Outcome 2 – expected value)2 x probability of outcome 2))+…
((Outcome n – expected value)2 x probability of outcome n))
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Variance Calculation
Dividend
Expected
Return
Difference
Diff Sq
Prob
Result
10.00
5.10
4.90
24.01
0.10
2.40
7.00
5.10
1.90
3.61
0.20
0.72
5.00
5.10
(0.10)
0.01
0.40
0.00
3.00
5.10
(2.10)
4.41
0.20
0.88
1.00
5.10
(4.10)
16.81
0.10
1.68
Total Variance
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5.69
Standard Deviation
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Standard Deviation is the Sq Root of the Variance
$5.69 = $2.38  (Std Dev)
68.3% of outcomes within +/- 1 (Std Dev)
$2.71 thru $7.49
95.4% of outcomes within +/- 2 (Std Dev)
$0.34 thru $9.86
99.7% of outcomes within +/- 3 (Std Dev)
($2.04) thru $12.24
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Standard Deviation and Risk Aversion
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Standard deviation is an indication of the risk of the
investment.
Given that most people are risk averse, what can we
say about investments and their standard deviations?
If investment A and investment B have the same
expected return, but investment B has a higher
standard deviation, which investment would you
choose? Why?
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Coefficient of Variation
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Suppose two investments have different expected
returns and different standard deviations.
How do we know which one to choose?
Coefficient of variation = standard deviation /
expected return.
The risk averse investor will choose the lowest risk
for the greater return and thus, the lower ratio.
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Diversification
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The concept of a portfolio:
 A financial portfolio is a collection of two or more
assets.
 Why do investors hold more than one asset?
 “Don’t put all your eggs in one basket.”
 Investors hold more than one asset in order to be
diversified.
 Investors diversify to improve or hold returns
constant and reduce overall risk.
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Diversification (continued)
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If we have two assets that are very similar, they could
both increase or both decrease.
 What does risk aversion tell us about these
alternatives?
If we have two assets that are different, we can
maintain a return by holding them simultaneously—if
one decreases, the other increases.
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Correlation
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How do we know if the returns of two assets move in
the same direction (or not)?
Correlation coefficient
 Abbreviated  (lowercase Greek or “rho”)
 A statistical measure of the relationship between
two variables
What kind of relationship would you expect to find
between the amount of rainfall and umbrella sales?
 In this case, would be positive
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Correlation (continued)
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The correlation coefficient ranges from –1.0 to +1.0.
 If two assets have returns that move together in
perfect lockstep, we can say their returns have a
 (rho) of +1.0.
 If they move in exactly opposite directions, then
the  (rho) of their returns is –1.0.
Given what you know about portfolios, the ideal pair
of assets would have a  (rho) of _____.
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Correlation and Risk
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If the correlation coefficient of the returns of two
assets is +1.0, then the standard deviation (risk) of
the portfolio is simply the weighted average of the
standard deviations of the two assets.
 Thus, there is no risk reduction in this case.
 There would be no benefit from holding these
two assets in portfolio.
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Correlation and Risk (continued)
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On the other hand, what happens if we find two
assets with a correlation of –1.0?
 Risk would be completely eliminated.
 The standard deviation of the portfolio would be 0.
What is the likelihood of finding two assets with
perfect negative correlation?
 It is rare if not impossible.
However, all we have to find is two assets with
correlation of less than +1.0 to achieve some benefits
of risk reduction.
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Portfolio Standard Variation
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W2a  2a + W2b  2b + 2WaWb  ab  a  b
(0.66672 * 42) + (0.33332 * 82) + 2(0.6667)(0.333)(1.0)(4)(8)
7.11 + 7.11 + 14.22
= 5.333%
W = Proportion of investment of portfolio
  = Standard Deviation for investment
  = RHO = Correlation Coefficient
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See Page 73
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The Market Portfolio
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If we are very risk averse, can we hold an asset that
has no risk?
 Risk-free asset has a guaranteed return.
An example would be a security issued by the
U.S. Government, such as a treasury bill.
 If the return is guaranteed, what is the standard
deviation of the return for the risk-free asset?
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The Market Portfolio (continued)
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The market portfolio is a theoretical portfolio
comprising all assets in the appropriate proportion.
It is the most “efficient.”
 It provides the most return for a given level of risk.
Since the market portfolio is the best portfolio in
terms of risk and return, we must assume investors
will own it.
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The Capital Market Line
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Depending on risk preferences, investors can invest
part of their funds in the risk-free asset and part in the
market portfolio.
They could also increase their return (but also the
risk) by holding the market portfolio and borrowing
funds.
The line that extends from the risk-free asset through
the market portfolio is called the capital market line.
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Risk and Its Components
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Total Risk = Systematic Risk + Unsystematic Risk
Systematic risk relates to those factors that affect all
assets in the market.
Unsystematic risk relates to those factors that are
specific to a particular asset.
The market portfolio is so diversified that all
unsystematic risk is removed as assets are added to it.
Therefore, the only risk in the market portfolio is
systematic.
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How Can We Hold the Market Portfolio?
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Do we need to hold all the assets in the world to
obtain the benefits of the market portfolio?
Research indicates that if we have approximately 30
assets in a portfolio we will have obtained the
maximum benefit from diversification.
Investors hold a “proxy” for the market portfolio—a
mutual fund such as the S&P 500 index fund.
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What is the Relevant Risk of New Assets?
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We assume investors are diversified and hold a proxy
for the market portfolio.
Therefore, the only risk component relevant to them is
the systematic risk because the unsystematic risk of
an investment will be diversified away.
Total risk (standard deviation) includes both types of
risk.
Is there a measure of systematic risk?
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Beta
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Beta is the measure of an asset’s systematic risk
relative to the market portfolio.
Beta = xmx / m
It is found by multiplying the correlation coefficient of
any asset (asset x) and the market portfolio by the
standard deviation of asset x. This product is divided
by the standard deviation of the market portfolio.
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Beta (continued)
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Betas are compared to the overall market.
The market portfolio has a beta of 1.
If the stock of a company has a beta of 2, it is twice
as risky as the market.
Where can I find betas?
 Use linear regression
 Yahoo! Finance website
 Various brokerage firm websites
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Beta as a Predictive Tool
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The capital market line examines return versus total
risk.
The security market line (SML) measures return of a
security against beta.
The SML represents a minimum expected return
given the relevant risk of a security.
Expected Return = Rf + [(Rm – Rf) x ]
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The Capital Asset Pricing Model
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The SML is an equation for a straight line.
Beta is the slope of the line.
This is also known as the Capital Asset Pricing
Model (CAPM).
If a project generates a return higher than the
required rate of return as shown by the SML, value is
created and the project is accepted. If not, then
value is lost and the project should be rejected.
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Limitations of CAPM
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The CAPM cannot always predict the returns of
assets accurately and it has limitations.
 The market portfolio is a theoretical concept; no
consensus on which proxy for the market portfolio
is best.
 Betas are calculated based upon historical
returns and then used to predict future returns.
Despite the limitations, CAPM is useful in getting
investors to understand a fundamental relationship
between risk and return.
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Homework:
Problems 1,2,3,4,5 & 6
7/12/2016
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