bm410: investments theory 1: risk and return the beginnings of portfolio theory
TRANSCRIPT
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, dollar-weighted?
• 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
HPR P P DP
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 = $24
Beginning Price = 20
Dividend = 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?
1 2 3 4Assets(Beg.) 1.00 1.20 2.00 0.80HPR .10 .25 (.20) .25Total Assets: Before Net Flows 1.10 1.50 1.60 1.00Net Flows 0.10 0.50 (0.80) 0.00Ending Assets 1.20 2.00 .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 weightedDon’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 returnsExpectation of future payoff given a
specific set of assumptions.Key is how you determine those
assumptionsWAG (wild ask guess)Probability distributionsScenario analysisOther 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
rr
Symmetric distributionSymmetric distribution
Normal Distribution
s.d. s.d.
Remember: 68.3% of returns are +/- 1 S.D. 95.4% of returns are +/- 2 S.D. 99.7% of returns are +/- 3 S.D.
rrNegativeNegative PositivePositive
Skewed Distribution: Large Negative Returns Possible
Median
rrNegativeNegative PositivePositive
Skewed Distribution: Large Positive Returns Possible
Median
Measuring Mean: Scenario or Subjective Returns
E(r) = p(s) r(s)s
Subjective Returns
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
Standard deviation = [variance]1/2
Problem 5: Measuring Variance or Dispersion of Returns
Subjective or Scenario
Variance = s
p(s) [rs - E(r)]2
Var =[(.1)(-.05-.15)2+(.2)(.05- .15)2...+ .1(.35-.15)2]Var= .01199S.D.= [ .01199] 1/2 = .1095
Using Our Example:
Questions
Any questions on scenario analysis and probabilities?
Problem 6: Scenario Analysis
Original ScenarioScenario Scenario Probability HPRRecession 1 .25 +44%Normal 2 .50 +14%Boom 3 .25 -16% New ScenarioScenario Scenario Probability HPRRecession 1 .30 +44%Normal 2 .40 +14%Boom 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 (-16-14)2 = 4501/2 = 21.21%
New Std Dev= (.3 (44-14)2 + .4(14-14)2 + .3 (-16-14)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=0Later, 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 premiumThe 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
Geometric Standard Real
Series Mean (%) Deviation (%) Return (%)
Large Stock 10.0 20.2 6.5
Small Stock 13.7 32.9 10.1
Treasury Bond 05.5 09.5 2.1
Treasury Bills 03.7 03.2 0.4
Inflation 03.3 04.3 -
Annual Holding Period Risk Premiums and Real Returns (after inflation)
Real Risk
Series Return (%) Premium (%)
Large Stock 6.5 6.3
Small Stock 10.1 10.0
Treasury Bond 2.1 1.8
Treasury Bills 0.4 --
Inflation --
The Two Asset Case
Asset Allocation is the process of investing your funds in various asset classesIt is the most important investment
decision you will makeMake it wisely!
Now assume you only have 2 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
Allocating Capital Between Risky and Risk-Free Assets
rf = 7%rf = 7% rf = 0%rf = 0%
E(rp) = 15%E(rp) = 15% p = 22%p = 22%
y = % in py = % in p (1-y) = % in rf(1-y) = % in rf
Problem 7: Two Asset Portfolio
E(rc) = yE(rp) + (1 - y)rfE(rc) = yE(rp) + (1 - y)rf
rc = complete or combined portfoliorc = complete or combined portfolio
For example, y = .75For example, y = .75E(rc) = .75(.15) + .25(.07)E(rc) = .75(.15) + .25(.07)
= .13 or 13%= .13 or 13%
Expected Returns for Combinations
E(r)E(r)
E(rE(rpp) = 15%) = 15%
rrff = 7% = 7%
22%22%00
PP
FF
Possible Combinations
ppcc ==
SinceSince rfrf
yy
Variance on the Possible Combined Portfolios
= 0, then= 0, then
cc = .75(.22) = .165 or 16.5%= .75(.22) = .165 or 16.5%
If y = .75, thenIf y = .75, then
cc = 1(.22) = .22 or 22%= 1(.22) = .22 or 22%
If y = 1If y = 1
cc = 0(.22) = .00 or 0%= 0(.22) = .00 or 0%
If y = 0If y = 0
Combinations Without Leverage
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
c = (1.5) (.22) = .33 Note that we assume the T-bill is totally risk free (bear with me again)
E(r)E(r)
E(rE(rpp) = 15%) = 15%
rrff = 7% = 7%
= 22%= 22%00
PP
FF
PP
) S = 8/22) S = 8/22
E(rE(rpp) - ) - rrff = 8% = 8%
CAL: (Capital
AllocationLine)
Capital Allocation Line
Slope: Reward to variability ratio: ratio of risk premium to std. dev.
Risk premium
This graph is the risk return combination available by choosing different values of y. Note we have E(r) and variance on the axis.
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 historyYear Beginning stock price Dividend Yield2001 $100 $4 2002 110 $4 2003 90 $42004 95 $4An 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 time-weighted 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 FundE(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 % Slope = .3704 17 P 14 Client
Standard Deviation 18.9 27
7
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?