Download Market Risk

Introduction to Risk and
Return
Where does the discount
rate come from?
FIN 351: lecture 6
Today’s learning objective



Introduction to risk
•
•
•
•
•
How to measure investment performance
Rates of Return
73 Years of Capital Market History
Measuring risk and risk premium
Risk & Portfolio Diversification
Two types of risk
•
How to measure systematic risk
CAPM
How to measure the
performance of your investment

Suppose you buy one share of IBM at
$74 this year and sell it at the expected
price of $102. IBM pays a dividend of
$1.25 for your investment
• What profit do you expect to make for your
•
investment?
What profit do you expect to make for one
dollar investment?
Solution


Profit in total =102-74+1.25=$29.25
Profit per one dollar=29.25/74=0.395 or
39.5%
Rates of Return
Percetage Return or expected rate of return = Profit
Cost
Percetage Return or expected rate of return =
Capital Gain + Dividend
Initial Share Price
Percentage Return = 28 +741.25
= .395 or 39.5%
Rates of Return
Dividend Yield =
Dividend
Initial Share Price
Capital Gain Yield =
Capital Gain
Initial Share Price
Percentage return = Dividend yield  capital gain yield
Rates of Return
1.25
Dividend Yield =
74
 .017 or 1.7%
28
Capital Gain Yield =
74
 .378 or 37.8%
Rates of Return
Nominal vs. Real
1 + nominal rate
1 + real rate = 1 + inflation rate
Suppose that the
inflation rate is1.6%
1
+
.395
1 + real rate = 1 + .016  1.373
real rate  37.3%
Market Indexes
Dow Jones Industrial Average (The Dow)
Value of a portfolio holding one share in each of 30 large
industrial firms.
Standard & Poor’s Composite Index (The S&P 500)
Value of a portfolio holding shares in 500 firms. Holdings are
proportional to the number of shares in the issues.
The performance of $0.1
investment
1000
10
Common Stocks
Long T-Bonds
T-Bills
0.1
30
9
1
40
9
1
50
9
1
60
9
1
70
9
1
80
9
1
90 998
9
1 1
Volatility of portfolios
60
40
Volatility 20
0
-20
Common Stocks
Long T-Bonds
T-Bills
-40
-60 26
30
35
40
45
Year
50
55
60
65
70
75
80
85
90
95
Why are stock returns so high?

To invest in stocks, investors require a risk
premium with respect to relative risk-free
security such as government securities.
•
•
•

The expected return on a risky security is equal to the riskfree rate plus a risk premium
Expected return =risk-free rate + risk premium
Risk premium =expected return –risk-free rate
Example
•
•
23.3% (1981 on market portfolio)=14%+9.3%
14.1% (1999 on market portfolio)=4.8%+9.3%
How to Measure Risk


We can use the variance or the standard
deviation of the expected rate of return to
measure risk.
Variance or standard deviation measure
weighted average of squared deviation of
each observation from the mean.
Some formula

Suppose that there are N states, then the expected
rate of return (mean) is
N
r  E (r )   pi ri
i 1

The variance of the rate of return is
N
2
  Var (r )   pi * (ri  r ) 2
i 1

The standard deviation
1/ 2
N

2
  Var (r )    pi (ri  r ) 
 i 1

Example of risk

Stock A has the following returns
depending on the state of the economy
next year as follows:
State of economy
Probability of the state
Return rate
Good
0.6
20%
Average
0.3
10%
0.1
-5%
Bad
Measure risk (continue)




First, calculate the mean return or the expected rate
of return. Here N=3 (three states)
Expected rate of return is r-bar=
p1*r1+p2*r2+p3*r3=0.6*0.2+0.3*0.1+0.1*(-0.05)
=14.5%
The variance of return is
p1*(r1- r-bar)2+p2*(r2- r-bar)2+p3*(r3-r-bar)2
=0.003325
The standard deviation is 0.0577=5.7%
Two types of risks
Unique Risk - Risk factors affecting only that
firm. Also called “firm-level risk.”
Market Risk - Economy-wide sources of risk that
affect the overall stock market. Also called
“systematic risk.”
Can we reduce risk?


Yes, we can reduce risk by
diversification: that is, we invest our
money in different assets or form a
portfolio of different assets.
Can we understand intuitively why
diversification can reduce risk?
Portfolio weights


Let W be the total money invested in a
portfolio, a set of assets.
Let xi be the proportion of total wealth
invested in asset i. Then xi is called portfolio
weight for asset i. The sum of portfolio
weights for all the assets in the portfolio is 1,
that is,
N
 xi  1
i 1
Example



You invest $400 of your $1000 in IBM at
a price of $74 per share and the other in
Dell at a price of $28.
What is the portfolio weight for IBM and
Dell respectively?
Are you sure that you are right?
Solution



xIBM=400/1000=0.4
xDell=600/1000=0.6
xIBM+xDell=1
Some formula for portfolios

The return of a portfolio is the weighted average of
returns of the stocks in the portfolio. That is,
N
R   xi ri
i 1

The expected return of a portfolio is the weighted
average of expected returns of the stocks in the
portfolio. That is,
N
R   xi ri
i 1
Risk and Diversification
(example)

John puts his money half in stock A and
half in stock B, as shown in the following.
rA  10%, rB  15%,  A  20%
 B  50%,  AB  0.9, x A  0.5, xB  0.5

What is the mean and variance of the
return of John’s portfolio?
My solution

The mean of the return of a portfolio is the weighted
average of the returns of the stocks in the portfolio.
Thus the mean of the return of John’s portfolio is
x A * rA  xB * rB  0.5 * 0.1  0.5 * 0.15  12.5%

The variance of the return of the portfolio is
portfolio variance
x 2A2A  xB2  2B  2 AB x A xB  A B
 0.52 * 0.22  0.52 * 0.52  2 * (0.9) * 0.5 * 0.5 * 0.2 * 0.5  0.0275
 P  0.02751 / 2  16.6%
Portfolio standard deviation
Risk and Diversification
Unique
risk
Market risk
0
5
10
Number of Securities
15
Measuring Market Risk

Market Portfolio
• It is a portfolio of all assets in the economy.
In
practice a broad stock market index, such as the
S&P 500 is used to represent the market portfolio.
The market return is denoted by Rm

Beta (β)
• Sensitivity of a stock’s return to the return on the
•
market portfolio,
Mathematically,
Cov(ri , Rm )
i 
Var ( Rm )
An intuitive example for Beta
Turbo Charged Seafood has the following
% returns on its stock, relative to the
listed changes in the % return on the
market portfolio. The beta of Turbo
Charged Seafood can be derived from
this information.
Measuring Market Risk
(example, continue)
Month Market Return % Turbo Return %
1
+ 1
+ 0.8
2
+ 1
+ 1.8
3
+ 1
- 0.2
4
-1
- 1.8
5
-1
+ 0.2
6
-1
- 0.8
Measuring Market Risk
(continue)



When the market was up 1%, Turbo
average % change was +0.8%
When the market was down 1%, Turbo
average % change was -0.8%
The average change of 1.6 % (-0.8 to
0.8) divided by the 2% (-1.0 to 1.0)
change in the market produces a beta of
0.8. β=1.6/2=0.8
Another example

Suppose we have following information:
Market
Stock A
bad
-8%
-10%
-6%
good
32%
38%
24%
State
Stock B
a. What is the beta for each stock?
b. What is the expected return for each stock if each scenario is
equally likely?
c. What is the expected return for each stock if the probability
for good economy is 20%?
Solution
a.
b.
0.38  (0.1) 0.48

 1.2
0.32  (0.08) 0.40
0.24  (0.06) 0.30
B 

 0.75
0.32  (0.08) 0.40
A 
rA  0.5 * 0.38  0.5 * (0.1)  0.14
rB  0.5 * 0.24  0.5 * (0.06)  0.09
c.
rA  0.2 * 0.38  0.8 * (0.1)  0.004
rB  0.2 * 0.24  0.8 * (0.06)  0
Portfolio Betas


Diversification reduces unique risk, but not market
risk.
The beta of a portfolio will be an weighted average of
the betas of the securities in the portfolio.
n
 p   xii
i 1

What is the beta of the market portfolio?

What is the beta of the risk-free security?
Example

Suppose you have a portfolio of IBM and
Dell with a beta of 1.2 and 2.2,
respectively. If you put 50% of your
money in IBM, and the other in Dell,
what is the beta of your portfolio
Beta of your portfolio =0.5*1.2 +0.5*2.2=1.7
Market risk and risk premium

Risk premium for bearing market risk
• The difference between the expected return
•
•
required by investors and the risk-free asset.
Example, the expected return on IBM is 10%,
the risk-free rate is 5%, and the risk premium
is 10% -5%=5%
If a security ( an individual security or a
portfolio) has market or systematic risk, riskaverse investors will require a risk premium.
CAPM (Capital Asset Pricing
Model)

The risk premium on each security is
proportional to the market risk premium
and the beta of the security.
• That is,
ri  r f  i ( Rm  r f )
ri  r f  risk premium for sec urity i
Rm  r f  risk premium for the market portfolio
Security market line
The graphic representation of CAPM in
the expected return and Beta plane
Security Market Line
16
14
Expected Return (%) .

12
10
Rm
8
6
4
rf
2
0
0
0.2
0.4
0.6
Beta
0.8
1
1.2