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BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory
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BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

Dec 29, 2015

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Page 1: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

BM410: Investments

Theory 1: Risk and Return

The beginnings of portfolio theory

Page 2: 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

Page 3: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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!

Page 4: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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?

Page 5: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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

Page 6: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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%

Page 7: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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

Page 8: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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

Page 9: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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%

Page 10: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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%

Page 11: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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

Page 12: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

Questions

Any questions on returns and rates of returns?

Make sure you understand the type of return you are looking at!

Page 13: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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

Page 14: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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

Page 15: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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

Page 16: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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.

Page 17: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

rrNegativeNegative PositivePositive

Skewed Distribution: Large Negative Returns Possible

Median

Page 18: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

rrNegativeNegative PositivePositive

Skewed Distribution: Large Positive Returns Possible

Median

Page 19: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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

Page 20: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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

Page 21: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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:

Page 22: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

Questions

Any questions on scenario analysis and probabilities?

Page 23: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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?

Page 24: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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)

Page 25: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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?

Page 26: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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

Page 27: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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?

Page 28: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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

Page 29: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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

Page 30: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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 -

Page 31: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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 --

Page 32: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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

Page 33: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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

Page 34: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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

Page 35: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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

Page 36: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

E(r)E(r)

E(rE(rpp) = 15%) = 15%

rrff = 7% = 7%

22%22%00

PP

FF

Possible Combinations

Page 37: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

ppcc ==

SinceSince rfrf

yy

Variance on the Possible Combined Portfolios

= 0, then= 0, then

Page 38: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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

Page 39: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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)

Page 40: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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.

Page 41: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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

.

Page 42: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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?

Page 43: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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%

Page 44: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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?

Page 45: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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.

Page 46: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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%

Page 47: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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%

Page 48: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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

Page 49: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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

Page 50: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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?

Page 51: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

Answer

Portfolio standard deviation 20% = (y) x 25%

Y = 20/25 = 80.0%

Mean return = (.80 x 12%) + (.20 x 4%) = 10.4%

Page 52: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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?

Page 53: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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.

Page 54: BM410: Investments Theory 1: Risk and Return The beginnings of portfolio theory.

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?