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Risk and Return
Primer
Expectations
Expected value (μ) is weighted sum of possible
outcomes
 E(X) = μ = p1X1 + p2X2 + …. psXs

 E(X) – Expected value of X
 Xi – Outcome of X in state i
 pi – Probability of state i
 s – Number of possible states
 Probabilities have to sum to 1

p1 + p2 + …..+ ps = 1
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Horse Race

There are three horse racing in the Finance Derby.
Your horse is “Love of NPV”. If your horse has a
30% chance of coming in first, and a 40% chance of
coming in second. How much do you expect your
horse to win?
 1st
pays $1,500
 2nd pays $750
 3rd pays $250
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Horse Race

There are three horse racing in the Finance Derby.
Your horse is “Love of NPV”. If your horse has a
30% chance of coming in first, and a 40% chance of
coming in second. How much do you expect your
horse to win?
 1st


pays $1,500, 2nd pays $750, 3rd pays $250
Chance of coming in 3rd: 1-0.3-0.4 = 0.3
0.3*1,500 + 0.4*750 + 0.3*250 = $825
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What is risk?
Uncertainty
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Measuring Risk
 There
is no universally agreed-upon
measure
 However,
variance and standard deviation are both
widely accepted measures of total risk
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Statistics Review: Variance

Variance (σ2) measures the dispersion of
possible outcomes around μ
Standard deviation (σ) is the square root of
variance
 Higher variance (std dev), implies a higher
dispersion of possible outcomes

 More
uncertainty
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Different Variances
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Variance Calculation

Variance = σ2 =
Σpi * (Xi – μ)2: Use this one
 Alternative
formulas you may have seen
 σ2 = Σ(Xi – μ)2 / N
 σ2 = Σ(Xi – μ)2 / (N-1)
 All give similar answers with large samples
 BUT each give very different answers with small
samples

Ex. s=3
σ2 = p1 * (X1 – μ)2 + p2 * (X2 – μ)2 + p3 * (X3 – μ)2
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Risk Example
Economy is “Good” with 20% probability
DJIA will return 20%
 Economy is “Fair” with 30% probability DJIA
will return 5%
 Economy is “Bad” with 50% probability DJIA
will return -9%

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Calculations
Expected Return =
Variance =
Standard Deviation =
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Calculations
Expected Return = p1X1 + p2X2 + p3X3
= 0.2*0.20+0.3*0.05+0.5*(-0.09) = 0.01
Variance =
Standard Deviation =
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Calculations
Expected Return = 0.01
Variance = p1(X1- μX)2+p2(X2-μX)2+p3(X3-μX)2
=0.2*(0.20-0.01)2 +
0.3*(0.05-0.01)2 +
0.5*(-0.09-0.01)2
= 0.0127 =127 (%)2
Standard Deviation =
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Calculations
Expected Return = 0.01
Variance = 0.0127 =127 (%)2
Standard Deviation = √ σ2
√0.0127 = 0.113 = 11.3%
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Historical Data

In practice we do not know all of the possible
states of the world, so we use historical data to
form expectations
 Idea:
Look at what has happened in the past and
we can calculate the mean and variance

What is each states probability of occurring?
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Risk Example 2



1996
1997
1998
1999
2000
20%
15%
-5%
5%
10%
Sample Mean
= 0.2*0.20+0.2*0.15+0.2*(-0.05)+0.2*0.05+0.2*0.10 = 0.09 = 9%
Sample Variance =
= 0.2*(0.20-0.09)2 + 0.2*(0.15-0.09)2 + 0.2*(-0.05-0.09)2 +
0.2*(0.05-0.09)2 + 0.2*(0.10-0.09)2 = 74%2
Standard Deviation =
√0.0074 = 0.086 = 8.6%
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Risk
A risky asset is one in which the rate of return
is uncertain.
 Risk is measured by ________________

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Risk
A risky asset is one in which the rate of return
in uncertain.
 Risk is measured by standard deviation.
 higher σ → more uncertainty

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General Securities

T-bills are a very safe investment

No default risk, short maturity
 Risk free asset
Stocks are much riskier
 Bond’s riskiness is between T-bills and Stocks

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Why Do We Demand a Higher Return

Investors seem to dislike risk (ex. insurance)
 Risk

Averse
If the expected return on T-Bills (risk-free), is
10%, and the expected return for Ford is 10%,
which would you buy?
 The
10% offered by T-Bills is guaranteed while
this is not the case for Ford
 A guaranteed 10% dominates a possible 10%
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Return Breakdown

A risky asset’s return has two components:
 Risk
free rate + Risk premium
Risk free rate: The return one can earn from
investing in T-Bills
 Risk Premium: The return over and above the
risk free rate
Compensation for bearing risk

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Average Risk Premiums (1926-2005)
Small company stocks :
17.4% – 3.8% = 13.6%
 Large company stocks :
12.3% – 3.8% = 8.5%
 Long-term corporate bonds :
6.2% – 3.8% = 2.4%

 The
more risk the larger the risk premium
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The Risk-Return Tradeoff
18%
Small-Company Stocks
Annual Return Average
16%
14%
Large-Company Stocks
12%
10%
8%
6%
T-Bonds
4%
T-Bills
2%
0%
5%
10%
15%
20%
25%
30%
35%
Annual Return Standard Deviation
Highest Risk & Return: Small Cap Stocks, Large Cap
Stocks, L.T. Corp bonds, L.T. Gov Bonds, U.S. T-Bills
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Quick Quiz
Which of the investments discussed has had
the highest average return and risk premium?
 Which of the investments discussed has had
the highest standard deviation?

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Why we care?
This is the very basics of investing
 General knowledge that “finance” people
possess

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