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

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?