Download BM410-08 Theory 1 - Risk and Return 20Sep05

BM410: Investments
Theory 1: Risk and Return
The beginnings of portfolio
theory
Objectives
•
•
•
A. Understand rates of return
B. Understand return using scenario,
probabilities, and other key statistics used to
describe your portfolio return
C. Understand risk and the implications of
using a risky and a risk-free asset in a
portfolio
Portfolio Theory
•
Portfolio Theory is an attempt to answer two
critical questions:
1. How do you build an optimal portfolio?
2. How do you price assets?
The next 4 class periods will be devoted to
answering those two questions!
A. Understand Rates of Return
• Portfolio Theory – the Basics
• Return: What it is?
• Accounting
• ROI, ROA, ROE, ROS?
• Market
• Monthly, expected, geometric, arithmetic, dollarweighted?
• Portfolio Return
• What is it? How do you measure it?
• Expected (or prospective) Return?
• What is it? How do you measure it?
Rates of Return:
Single Period
P D
P
HPR 
P
1
0
1
0
HPR = Holding Period Return
P1 = Ending price
P0 = Beginning price
D1 = Dividend during period one
Problem 1: Rates of Return:
Single Period Example
You paid $20 per share for Apple Computer stock at
the end of 1998. At the end of 1999, it increased to
$24. Assuming it distributed $1 in dividends, what
is your HPR for Apple?
Ending Price =
Beginning Price =
Dividend =
$24
20
1
HPR = ( 24 - 20 + 1 )/ ( 20) = 25%
Problem 2: Rates of Return:
Multiple Period Example (p. 154)
What is your geometric and arithmetic return for
the above assets for the four years?
Assets(Beg.)
HPR
Total Assets:
Before Net Flows
Net Flows
Ending Assets
1
1.00
.10
1.10
0.10
1.20
2
1.20
.25
1.50
0.50
2.00
3
4
2.00 0.80
(.20) .25
1.60 1.00
(0.80) 0.00
.80 1.00
Rates of Return:
Arithmetic and Geometric Averaging
Arithmetic
ra = (r1 + r2 + r3 + ... rn) / n
ra = (.10 + .25 - .20 + .25) / 4
= .10 or 10%
Geometric
rg = {[(1+r1) (1+r2) .... (1+rn)]} 1/n - 1
rg = {[(1.1) (1.25) (.8) (1.25)]} 1/4 - 1
= (1.5150) 1/4 -1 = .0829 = 8.29%
Dollar weighted
Don’t worry about it for now. Just know that it
is the IRR of an investment
Return Conventions
•
APR = annual percentage rate
Total interest paid / total amount borrowed
(periods in year) X (rate for period)
• EAR = effective annual rate (includes compounding)
( 1+ (annual %/periods year))Periods year - 1
Example: monthly return of 1%
APR = 1% x 12 = 12%
EAR = (1+ .12/12)12 - 1 =
EAR = 12.68%
Real vs. Nominal Rates
Fisher effect: Approximation
Nominal rate = real rate + inflation premium
(1+R) = (1+rr) * (1+ i) multiply out
R = rr + i + rr*i assuming rr*i is small
R = rr + i or R – I = rr
Example Nominal (R) = 6% and inflation (i) = 3%
rr = 6% - 3% or 3%
Fisher effect: Exact. This is the way it is done! Divide
both sides by (1 + i) to get:
rr = (1 + R)/(1 + i) –1
2.9% = (6%-3%) / (1.03) or (1.06/1.03) –1 = 2.9%
Problem 3:
Why Use the Exact Formula?

The approximation overstates the real return
• Return 5% and inflation 3%
• Approximation 5-3 = 2% real
• Exact (1+.05)/(1+.03) = 1.942%
• .01942/.02 -1 = Real return overstated by 2.9%
• Return 50% and inflation 30%
• Approximation 50-30 = 20% real
• Exact (1+.5)/(1+.3) = 15.385%
• .15385/.2 -1 = Real return overstated by 23.1%
• The higher the numbers, the more overstated the Fisher
approximation
• Calculate it correctly in all situations
Questions
Any questions on returns and rates of
returns?
Make sure you understand the type of
return you are looking at!
B. Key Statistics to Describe
your Portfolio Return
 Expected returns
Expectation of future payoff given a
specific set of assumptions.
Key is how you determine those
assumptions
WAG (wild ask guess)
Probability distributions
Scenario analysis
Other logical method
Scenario Analysis /
Probability Distributions
 Estimate the probability of an event occurring
and the likely outcome for each occurrence
during some specific period
 Characteristics of Probability Distributions
• 1. Mean: most likely value
• 2. Variance or standard deviation: volatility
• 3. Skewness: direction of the tails
 If a distribution is approximately normal, the
distribution is described by characteristics 1
and 2
Scenario Analysis –
Its use in class
•
Your financial analysis is based on your
assumptions for the economy, industry, and
company.
• What happens when you vary your assumptions
based on differing economic forecasts, industry
forecasts, and company ratios?
• What will be the outcome of your company
analysis under varying assumptions?
• Your analysis is really your forecast based
on your preferred scenario
Normal Distribution
Symmetric distribution
s.d.
Remember:
s.d.
68.3% of returns are +/- 1 S.D.
95.4% of returns are +/- 2 S.D.
99.7% of returns are +/- 3 S.D.
r
Skewed Distribution:
Large Negative Returns Possible
Median
Negative
r
Positive
Skewed Distribution:
Large Positive Returns Possible
Median
Negative
r
Positive
Measuring Mean:
Scenario or Subjective Returns

Subjective Returns
E(r) = S p(s) r(s)
s

p(s) = probability of a state occurring
 r(s) = return if that state occurs
Over the range from 1 to s states
Problem 4:
Subjective or Scenario Distributions
State
Prob. of State
Return in State
1
.10
-.05
2
.20
.05
3
.40
.15
4
.20
.25
5
.10
.35
 What is the expected return of this scenario?
• E(r) = (.1)(-.05) + (.2)(.05) + (.4)(.15) +
(.2)(.25) + (.1)(.35)
• E(r) = .15
Problem 5: Measuring Variance or
Dispersion of Returns
Subjective or Scenario
Variance = S p(s) [rs - E(r)]
2
s
Standard deviation = [variance]1/2
Using Our Example:
Var =[(.1)(-.05-.15)2+(.2)(.05- .15)2...+ .1(.35-.15)2]
Var= .01199
S.D.= [ .01199] 1/2 = .1095
Questions
 Any
questions on scenario analysis and
probabilities?
Problem 6: Scenario Analysis
Scenario
Recession
Normal
Boom
Scenario
Recession
Normal
Boom
Original Scenario
Scenario Probability HPR
1
.25
+44%
2
.50
+14%
3
.25
-16%
New Scenario
Scenario Probability HPR
1
.30
+44%
2
.40
+14%
3
.30
-16%
Calculate and compare the mean and standard
deviation of each scenario. What differences
have occurred?
Problem 6: Answer
Old E(r) = .25 x 44 + .5 x 14 + .25 x –16 = 14%
New E(r) = .3 x 44 + .4 x 14 + .3 x –16 = 14%
Old Std Dev= (.25 (44-14)2 + .5(14-14)2 + .25 (-1614)2 = 4501/2 = 21.21%
New Std Dev= (.3 (44-14)2 + .4(14-14)2 + .3 (-1614)2 = 5401/2 = 23.24%
The mean is unchanged, but the standard deviation
has increased (due to the greater probability of
extreme returns)
C.

Understand the implications of using
risky and risk-free assets
What is risk?
• Possibility of a loss?
• Possibility of not achieving a goal?
• Market-risk, i.e. business cycles, economic
conditions, inflation, interest rates, exchange rates,
etc.?
• Variability of returns?
• Uncertainty about future holding period returns?

What risk are we referring to?
Investment Risk
 What is investment risk?
 It is the risk of not achieving a specific HP return
 How is it measured?
 Historically, government securities were considered
risk-free, hence variance=0
 Later, analysts started using variance (standard
deviation) as a better measure of risk
Investment Risk (continued)
 Is Standard Deviation still the best measure?
Do you care about risk if it is in your favor,
i.e. if it adds positive return?
What about other measures, such as
downside variance, i.e. semi-standard
deviation?
Key Risk Concepts
 Risk
 Investment risk. The probability of not achieving
some specific return objective
 Risk-free rate
 The rate of return that can be obtained with
certainty
 Risk premium
 The difference between the expected holding period
return and the risk-free rate
 Risk aversion
 The reluctance to accept risk
The difference between
investing and gambling

Investors
• Are willing to take on risk because they
expect to earn a risk premium from
investing, a favorable risk-return tradeoff
 Gamblers
• Are willing to take on risk even without the
prospect of a risk premium, there is no
favorable risk-return tradeoff
Building a Portfolio: Annual Holding Period
Returns from 1926- 2004
Series
Large Stock
Small Stock
Treasury Bond
Treasury Bills
Geometric Standard
Real
Mean (%) Deviation (%) Return (%)
10.0
20.2
6.5
13.7
32.9
10.1
05.5
09.5
2.1
03.7
03.2
0.4
Inflation
03.3
04.3
-
Annual Holding Period Risk Premiums and Real
Returns (after inflation)
Series
Large Stock
Small Stock
Treasury Bond
Treasury Bills
Inflation
Real
Return (%)
6.5
10.1
2.1
0.4
--
Risk
Premium (%)
6.3
10.0
1.8
--
The Two Asset Case
 Asset Allocation is the process of investing
your funds in various asset classes
It is the most important investment decision
you will make
Make it wisely!
 Now assume you only have 2 assets
Allocating Capital Between
Risky and Risk-Free Assets

Lets split our investment funds between safe
and risky assets
• Risk free asset: proxy; T-bills.
• We assumes no risk for this asset class by
definition
• Risky asset: A portfolio of stocks similar to an
index fund

Issues
• Examine risk/ return tradeoff
• Demonstrate how different degrees of risk aversion
will affect allocations between risky and risk free
assets
Problem 7: Two Asset Portfolio
rf = 7%
srf = 0%
E(rp) = 15%
sp = 22%
y = % in p
(1-y) = % in rf
Expected Returns for Combinations
E(rc) = yE(rp) + (1 - y)rf
rc = complete or combined portfolio
For example, y = .75
E(rc) = .75(.15) + .25(.07)
= .13 or 13%
E(r)Possible
Combinations
E(rp) = 15%
rf = 7%
0
P
F
22%
s
Variance on the
Possible Combined Portfolios
Since
s r = 0, then
f
sc = y s p
Combinations Without Leverage
If y = .75, then
s
c
= .75(.22) = .165 or 16.5%
If y = 1
s
c
= 1(.22) = .22 or 22%
If y = 0
s c = 0(.22) = .00 or 0%
Using Leverage with
Capital Allocation Line
Borrow at the Risk-Free Rate and invest in stock
(while not really possible, lets assume we can
do it)
Using 50% Leverage
rc = (-.5) (.07) + (1.5) (.15) = .19
sc = (1.5) (.22) = .33 Note that we assume the Tbill is totally risk free (bear with me again)
E(r)
Capital Allocation Line
CAL:
(Capital
Allocation
Line)
This graph is the risk return combination
available by choosing different values of y. Note
P
we have E(r) and variance on the axis.
E(rp) = 15%
Risk premium
E(rp) - rf = 8%
rf = 7%
) S = 8/22
F
Slope: Reward to variability ratio:
ratio of risk premium to std. dev.
0
P = 22%
s
Risk Aversion and Allocation

Key concepts
• Greater levels of risk aversion lead to larger
proportions of the risk free rate
• Lower levels of risk aversion lead to larger
proportions of the portfolio of risky assets
• Willingness to accept high levels of risk for
high levels of returns would result in
leveraged combinations
 .
Problem 9: Portfolio Return
Stock price and dividend history
Year Beginning stock price Dividend Yield
2001
$100
$4
2002
110
$4
2003
90
$4
2004
95
$4
An investor buys three shares at the beginning of 2001,
buys another 2 at the beginning of 2002, sells 1
share at the beginning of 2003, and sells all 4
remaining at the beginning of 2004.
A. What are the arithmetic and geometric average timeweighted rates of return?
B. What is the dollar weighted rate of return?
Answer

Time weighted return
• 2001 (110-100+4)/100 =
14%
• 2002 (90-110+4)/110 =
- 14.6%
• 2003 (95-90+4)/90 =
10%
 Arithmetic mean return
(14-14.6+10)/3 = 3.13%
 Geometric mean return
(1+.14)*(1-.146)*(1+.1)]1/3 = 1.078.33 –1 = 2.3%
Problem 11: Risk Premiums

Using the historical risk premiums as your
guide from the chart earlier, what is your
estimate of the expected annual HPR on the
S&P500 stock portfolio if the current risk-free
interest rate is 5.0%. What does the risk
premium represent?
Answer
For the period of 1926- 2004 the large cap
stocks returned 10.0%, less t-bills of 3.7%
gives a risk premium of 6.3%.
•
•
•
•
If the current risk free rate is 5.0%, then
E(r) = Risk free rate + risk premium
E(r) = 5.0% + 6.3% = 11.3%
The risk premium represents the additional return
that is required to compensate you for the
additional risk you are taking on to invest in this
asset class.
Problem 12: Client Portfolios

You manage a risky portfolio with an expected return
of 12% and a standard deviation of 25%. The T-bill
rate is 4%. Your client chooses to invest 70% of a
portfolio in your fund and 30% in a T-bill money
market fund. What is the expected return and
standard deviation of your client’s portfolio?
• Clients Fund
E(r) (expected return) =.7 x 12% + .3 x 4% =
9.6%
σ (standard deviation) = .7 x .25 =
17.5%
Problem 13: Portfolio Allocations


Suppose your risky portfolio includes investments in
the following proportions. What are the investment
proportions in your clients portfolio
Stock A
27%
Stock B
33%
Stock C
40%
Investment proportions: T-bills = 30%
Stock A = .7 x 27% = 18.9%
Stock B = .7 x 33% = 23.1%
Stock A = .7 x 40% = 28.0%
Check: 30 + 18.9 + 23.1 + 28 = 100%
Problem 14: Reward to Variability
C. What is the reward-to-variability ratio (s) of your risky
portfolio and your clients portfolio?
• Reward to Variability (risk premium / standard deviation)
•
•
Fund = (12.0% – 4%) / 25 = .32
Client = (9.6% – 4%) / 17.5 = .32
Problem 15: The CAL Line
D. Draw the CAL of your portfolio. What is the slope of
the CAL?
Slope of the CAL line
%
17
14
Slope = .3704
P
Client
7
Standard Deviation
18.9
27
Problem 16:
Maximizing Standard Deviation
Suppose the client in Problem 12 prefers to invest in your
portfolio a proportion (y) that maximizes the expected
return on the overall portfolio subject to the constraint
that the overall portfolio’s standard deviation will not
exceed 20%. What is the investment proportion?
What is the expected return on the portfolio?
Answer
Portfolio standard deviation 20% = (y) x
25%
Y = 20/25 = 80.0%
Mean return = (.80 x 12%) + (.20 x 4%) =
10.4%
Problem 17:
Increasing Stock Volatility

What do you think would happen to the expected
return on stocks if investors perceived an increased
volatility of stocks due to some recent event, i.e.
Hurricane Katrina?
Answer

Assuming no change in risk aversion, investors
perceiving higher risk will demand a higher risk
premium to hold the same portfolio they held before.
If we assume the risk-free rate is unchanged, the
increase in the risk premium would require a higher
expected rate of return in the equity market.
Review of Objectives
• A.
Do you understand rates of return?
• B. Do you know how to calculate return using
scenario, probabilities, and other key statistics
used to describe your portfolio?
• C. Do you understand the implications of using
a risky and a risk-free asset in a portfolio?