Transcript document

MEASURING INVESTMENT RETURNS AND RISKS
COMPARING RETURN INFORMATION
Holding period return
Pt
Rt
CFt
R1
=
= Price of asset at time t
= % return from time t-1 to t
= cash flow from time t-1 to t - e.g. dividend
P1  P0  CF1 P1  P0 CF1


P0
P0
P0
= (%capital gain) + (%dividend yield)
QUESTION: How can you measure the return you expect
from an asset?
The best guess at the future return i.e., what one should
expect is the mean return:

n
R   Ri / n
i 1
where Ri is an observation of the variable (return) and

R is the arithmetic sample mean of the variable (return)
EXAMPLE OF ARITHMETIC MEAN
P0
P1
= 100 - price now
= 200 - price at time 1 period from now
R1
= (200 - 100)/100
= 1.00 or 100%
Suppose P2 = 100 - period 2 price
R2
= ((100 - 200)/200
= -.5 or -50%

Arithmetic mean return
R
= (R1 + R2 )/2 = .25 or 25%
QUESTION: Is an average return of 25% a good return?
Is it accurate?
ANSWER: Not for past returns. Instead use Geometric
return (also called compound return) especially when returns
are volatile. Geometric mean ≤ arithmetic mean.
n
G
G
Ri

(1  Ri ) 1/ n  1
= i
1
= is the geometric mean return
= the return in period i
= the product operator
The geometric mean using the previous data is:
G = (1 + 1.0)1/2 x (1 + (-.5))1/2 - 1
= [(2)(.5)]1/2 - 1
=0
Question: Who knows what this symbol means?
Answer: This is the Chinese ideograph for Risk (or crisis).
It is a combination of the ideograph for Danger (first symbol) and the
ideograph for Opportunity (second one). Why are they together?
Who likes taking risks? –playing card games – poker?
Who wants to make millions of dollars?
RISK AVERSE INVESTORS REQUIRE MORE
RETURN TO HOLD ASSETS WITH MORE RISK
QUESTION: What is risk?
ANSWER: The likelihood that you will not receive
what you expect - i.e. the mean risk free
you always get what you expect
Variance and Standard Deviation - ivolatility.com
s2
sample variance = 
sample standard deviation = s
for s = 10% and
-20%
-10%
0
10%
n

 ( Ri  R ) 2 /n
i 1

R = 10%
20%
30%
40%
-3s
-2s
-1s
Mean
+1 s
+2s
+3s



For the normal distribution, one, two and three standard deviations about the
mean delineate where observations of a variable should fall 68, 95 and 99
percent of the time, respectively.
COEFFICIENT OF VARIATION (CV)
CV
= (Std. Dev. of return)/(mean return)
= risk/ arithmetic expected return
says, for each percent of mean return, how much volatility
must you bear.
PROBLEM: Suppose asset 1 has a standard deviation of
.08 and a mean return of .06 while asset 2 has a
standard deviation of .05 and a mean return of .03.
Which is best? (Asset 1)
CV1
CV2
= .08/.06 = 1.33 still has more total risk
= .05/.03 = 1.67
1
2
These two distributions have the same mean but
1’s variance is smaller than 2’s.
If these represent stock returns, a risk averse investor should choose stock
1.
1
2
When two stock return distributions have different means and variances, a risk
averse investor choosing between them needs a method that compares mean return
relative to risk, such as coefficient of variation or the capital asset pricing model.
Ibbotson Sinquifield Data – 1926-2009
Stocks
Large Cos.
Small Cos.
Geometric
Mean
10.4
12.7
Arithmetic
Mean
12.4
17.5
Standard
Deviation
20.3
33.1
Bonds
Long Corp.
Long Treas.
Med. Treas.
Tbills
5.9
5.4
5.4
3.7
6.2
5.8
5.5
3.8
8.6
9.3
5.7
3.1
QUESTION: Compare the Risk/Return tradeoff of Bonds
to stocks and T Bills to Bonds. -anything
unusual here?
ANSWER:
Bonds have relatively low returns but large
variation due to unexpected inflation
SMALL STOCKS
mean=18%
standard deviation=35%
-87% -52% -17%
-48
-28
-8
18% 53%
12
32
88% 123%
52
72
LARGE STOCKS
mean=12%
standard deviation=20%
LONGTERM U.S.
GOVERNMENT BONDS
mean=5%
standard deviation=9%
-22
-13
-4
5
14
23
32
U.S. TREASURY BILLS
mean=4%
standard deviation=3%
-5 -2 1
4 7
10 13
There are considerable differences among return distributions for these common asset types.
Expected Return and Variance Using Probability
Distributions
Probabilities are weights attached to scenarios or
observation classes where i indexes scenarios.
Expected return = E(R1) =  Probabilityi x Return1i
Return Variance = s2(R1)
=  Probabilityi x [Return1i - E(R1)]2
Sample mean and variance assumes that each observation
has equal probability which is acceptable if the sample
covers a full economic cycle.
Return standard deviation = s(R1) = [s2(R1)]1/2
How to Get Probabilities
You can get some probabilities from past data. If stocks
earned 10% in 30 years over the last 100 years, then
the probability of earning 10% is .30.
But this approach uses past data which may not reflect
expected future events.
Another approach is to use data from websites like
intrade.com that offer bets on events for prices that
reflect the probability that the event will occur.
Price = probability1(payoff1) + probability2(payoff2)
= probability1(1) + probability2(0)
Price = probability1
EXPECTED RETURN, VARIANCE, AND
COVARIANCE OF A PORTFOLIO USING JOINT
PROBABILITY DISTRIBUTION
Assume portfolio consisting of stock 1 and stock 2:
Expected portfolio return =
E(Rp) = Weight1 x E(R1) + Weight2 x E(R2)
Portfolio Variance = sp2
=Weight12 x s2(R1) + Weight22 x s2(R2)
+ 2 x Weight1 x Weight2 x s(R1, R2)
where s(R1, R2) = covariance of stock 1 and
stock 2 returns.
NO COST TO DIVERSIFY
Diversifiable risk can be eliminated easily so - no
compensation. Only undiversified risk should receive
compensation - Covij risk
where Covariance = s(R1, R2)
  Probabilityi x [Return1i - E(R1)][Return2i - E(R2)]
In a portfolio, there is a covariance for each asset pairing –
the many covariances account for most of a portfolio’s
variance.
All else equal, covariance is large when the data points fall
along the regression line instead of away from it because,
on the line, the deviations from the means of each variable
are equal – the products are squares - larger than otherwise.
Illustrate surprising probabilities with student birthdays. Question:
What is the probability that two students in class have the same
birthday? Individual pairings have small probability but there are
many pairings.
It’s usually best to diversify, except in this case.
Stock 1
Return


 
Stock 1
Return














Stock 1
Return

Negative
Correlation


 

 






 



Stock 2 Return


Stock 2 Return
Positive
Correlation
Stock 2 Return
Zero
Correlation
Portfolio
Risk
Diversifiable Risk
Nondiversifiable Risk
Number of securities in the portfolio
Diversifiable risk drops as more securities
are added to a portfolio.
Example: Bill Gates started with $100 billion in Microsoft
- now sells $50 million each month and buys other stocks.
QUESTION: Consider two stocks with the following
return distributions. Find the variance for a portfolio with
40% invested in stock 1 and 60% invested in stock 2.
Obama
Liberal Conservative Moderate
Probability
.50
.10
.40
stock 1 return.10
.15
.20
stock 2 return.25
.10
.05
Portfolio Return Distribution
Example: Probability Distribution Results
E(R1) = .50(.10) + .40(.20) + .10(.15) = .145
E(R2) = .50(.25) + .40(.05) + .10(.10) = .155
s2(R1) = .50(.10 - .145)2 + .40(.20 - .145)2
+ .10(.15 - .145)2 = .0022375
s2(R2) = .50(.25 - .155)2 + .40(.05 - .155)2
+ .10(.10 - .155)2 = .009235
s(R1, R2) = .50(.10 - .145)(.25 - .155) + .40(.20 - .145)
(.05 - .155) + .10(.15 - .145)(.10 - .155) = -.004476
correlation = s(R1, R2)/s(R1) s(R2)
= -.004475/[(.0473022)(.0960989)] = -.98
E(Portfolio return) = .4(.145) + .6(.155) = .151
Portfolio Variance = (.4)2 (.0022375) + (.6)2 (.009235)
+ 2(.4)(.6)(-.004475) = .0015346
QUESTION: Suppose Obama chose Chris Dodd as
Treasury Secretary instead of New York Fed President
Timothy Geithner. How would you restructure the
portfolio? - more of stock 2 and less of stock 1.
Illustrate correlation and optimal portfolio weights using
www.wolframalpha.com (click “Examples”, then Money
and Finance” then under “compare several stocks” put
in up to 4 stock ticker symbols)
Explain the optimal portfolio return and volatility.
For more complex portfolio optimization statistics and
to input more than 4 stocks go to Macroaxis.com,
register for free and create a portfolio – then try the
“management” option and try “optimize” or “suggest”.
CORRELATION AND HEDGING - FIRE INSURANCE
Correlation - in general - hedging takes advantage of
negative correlation but less than perfect
correlation can be used to reduce risk.
EXAMPLE
Suppose you own a $200,000 house.
QUESTION: If the probability of fire = .001 and the cost
of a fire insurance policy is $700, what is your
expected return on the insurance policy alone. Is it
risky? Is insurance a good investment?
Probability
Fire
.001
Outcome
% Return
200,000-700 199,300/700=
28471%
No fire
.999
-700
-700 / 700 = -100%
Expected return and variance of policy on its own (like
Walmart buys life insurance policies on its employees).
E(R)
= .001(28471) + .999(-100) = -71.43%
s2 = .001( 284.71 - (-0.7143))2 + .999( -1 - (-0.7143))2
= 81.46 + .0815 = 81.54 = 8154%
Expected return on house without insurance
= [.999 (0) + .001 (-1.00)] = -.001 = -.1%
Variance of return on house without insurance
= [.999 (0 - (-.001))2 + .001 (-1.00 - (-.001))2]
= .0001
Expected return with insurance
= [ .001 (-700/200,700) + .999 (- 700/200,700)]
= 1.00 (- 700/200,700) = -.0035
= -.35%
Variance of return with insurance
= [.001 [(- 700/200,700) - (-.0035)]2
+ .999 [(-700/200,700) - (-.0035)]2
= 1.00 (0) = 0%
Insurance doesn't look like a good investment but return
variance is zero with insurance, hence insurance is valuable
for hedging or risk reduction purposes.
Question: Many stores offer insurance or service contracts
on items such as DVD players to cover costs after
warranties run out. Why is this coverage typically an even
poorer investment than home or automobile coverage?
Beta is a Standardized Covariance
Beta is the slope of a regression line of an asset’s returns
on the market portfolio’s return.
Beta1 =
Cov1,m
s
2
m
=
Corr1ms 1s m
s m2
where m signifies market and 1 signifies stock 1.
All asset betas are measured against the market so all are
being compared with the same gauge.
If B1 = 1 and B2 =.5, stock 1 is twice as risky as stock 2
CAPM - Capital Asset Pricing Model
Beta is used as a measure of risk in a theoretical return
equation called CAPM.
E(Ri) = Rf + Bi[E(Rm) - Rf]
where E(Ri) = expected return on stock i
Rf = the risk-free rate of return
Bi = stock i’s beta
E(Rm) = expected return on the market portfolio
Illustrate diversification with
stockcharts.com – carpets
Holding a diversified portfolio is best unless you have
better estimates of payoff probabilities for a stock than
the market.
Annual return pairs for the S&P 500 and Homestake Mining's stock
H
o
m
e
s
t
a
k
e
'
s
Year
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1
0.8
0.6
Slope is 0.54
0.4
0.2
R
0
e
t
-0.2
u
r
n -0.4
-0.1
0
0.1
0.2
0.3
0.4
S&P 500 Return
The correlation between Homestake and the S&P 500 is 0.18 and its beta is 0.54
S&P Homestake
0.23
0.01
0.06
-0.26
0.32
0.1
0.18
0.09
0.05
0.39
0.17
-0.27
0.31
0.55
-0.03
-0.09
0.3
-0.16
0.08
-0.25
0.1
0.83
0.01
-0.19
Year
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
Annual return pairs for the S&P 500 and gasoline
0.8
G
a
s
o
l
i
n
e
R
e
t
u
r
n
0.7
0.6
0.5
Slope is -2.11
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.1
0
0.1
0.2
0.3
0.4
S&P 500 Return
Gasoline's correlation with the S&P 500 is -0.47 and its beta is -2.11.
S&P
0.23
0.06
0.32
0.18
0.05
0.17
0.31
-0.03
0.3
0.08
0.1
0.01
Gas
0.08
-0.1
0.09
-0.45
0.19
-0.04
-0.08
0.73
-0.33
-0.07
-0.29
0.2
Annual return pairs for the S&P 500 and Gold
Year
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
0.4
G
o
l
d
0.2
R
e
t
u
r
n
Slope is zero
0
-0.2
-0.1
0
0.1
0.2
0.3
0.4
S&P 500 Return
The correlation between gold and the S&P 500 and its beta is approximately zero.
S&P
0.23
0.06
0.32
0.18
0.05
0.17
0.31
-0.03
0.3
0.08
0.1
0.01
Gold
-0.1
-0.16
0
0.25
0.2
-0.17
0
-0.05
-0.05
-0.06
0.12
0.02
High Beta
Stock
Return
Market
Low Beta
During this time period the market rises, falls, and then rises again. A
beta stock varies more (less) than the market.
high (low)
Positive Beta
Stock
Return
Negative Beta
Positive and negative beta stock returns
move opposite one another.
Illustrate beta and return distributions using
www.wolframalpha.com (click “Examples”, then Money
and Finance” then under Stock Data put in a company)