Lecture_10-11 Risk and Return

8/17/2019 Lecture_10-11 Risk and Return

http://slidepdf.com/reader/full/lecture10-11-risk-and-return 1/38

Professor Sang Byung [email protected]

Risk and Expected Return



About the exam

• Mean (average): 79.5

• Standard deviation: 14.4

• Lower quartile (25% percentile): 68.5

• Median (50% percentile): 84

•Upper quartile (75% percentile): 90

0

5

10

15

20

25

40-45 45-50 50-55 55-60 60-65 65-70 70-75 75-80 80-85 85-90 90-95 95-100



Announcement

• No class this Thursday (2/25)

• Due to the Republican presidential primary debate

• Traffic congestion and parking disruptions are expected.

• The university recommends “alternative academic

experience” to the students.

• I will give you a reading assignment instead.

• Exam 2

• Tentatively March 31st

• Note that

• The date and material covered will be different from what Ioriginally planned and put in the syllabus.



Big picture

•So far, we have studied…• Time value of money

• Investment criteria

•Fixed income valuation

• Here we assumed …

• The future cash flows are already identified.

• The discount rate is given.



Big picture

•Now we study:• How we value a firm.

• How we value its risky projects.

• What’s new?

• Need to identify future cash flows.

• Need to find the appropriate discount rate.



Big picture

• Basic theory: the risk/return tradeoff.

• Capital Asset Pricing Model

• The dividend-discount model

• Using the valuation principle we studied.

• The discounted cash flow model (DCF) intro

• Identifying free cash flows from financial statements

• The DCF model with financing decisions

• Capital structure

• Weighted average cost of capital (WACC)

•

APV (Adjusted present value)



Reading assignment

• To identify free cash flows, we need some

accounting knowledge.

• Don’t worry – I know that this is not an accounting course.

• But students should know some basic accounting facts.

• Two helpful, short, and free guides to help you

brush up on your accounting skills:

•

Merrill Lynch Guide to Understanding Financial Reports• Merrill Lynch How to Read a Financial report

•

Both are uploaded on Blackboard



Uncertain cash flows

• So far, we’ve used this equation a lot:

=

1 +

• Is a guaranteed cash flow? Usually not…

• is risky, and its risk is measured by the discount rate

• Furthermore, we should write the equation as:

=

1 +

• where [] is the expected cash flow.



Example

• Consider an investment that pays off at time 1.

• The payoff is $60 with a probability of 25%, and

$100 with a probability of 75%.

• Assume that the appropriate discount rate is 12%.

• What is the value of the investment?

=

1 + =

60 × 0.25 + 100 × 0.75

1.12 =

90

1.12 = $80.36



Risk-adjusted discount rate

•

Up until now we’ve been vague about thediscount rate (“opportunity cost of capital”).

• How do we measure this?

• In case of assured (or riskless) cash flows, we can use adiscount rate derived from government bond prices.

• What if future cash flows are uncertain or risky?

• Short answer: It’s difficult.

• But we have some formal ways to think about it.

• In particular, we use CAPM to estimate the cost of

capital.



Risk Return Tradeoff

$0

$1

$10

$100

$1,000

$10,000

1 9 2 6

1 9 3 6

1 9 4 6

1 9 5 6

1 9 6 6

1 9 7 6

1 9 8 6

1 9 9 6

2 0 0 6

Stock Market

T-Bill




-60

-40

-20

0

20

40

60

80

1 9 2 7

1 9 3 7

1 9 4 7

1 9 5 7

1 9 6 7

1 9 7 7

1 9 8 7

1 9 9 7

2 0 0 7

Stock Market

T-Bill




• Why does the stock market outperform the T-bills?• Stocks are risker!

• Owning stocks makes the good times better, and the bad

times worse.

• When the economy is doing poorly and your job prospects

are worse, you get the double whammy of knowing your

investments have tanked in value.

• In equilibrium, stocks should provide higher expected

returns in order to induce investors to hold them.

• You earn higher returns when you take systemic

risk, not casino risk!



What is return? What is risk?

• To understand how we quantify these, let’s

consider the following example:

• Suppose you invest $10 in a stock today.

• A year later,

• It pays a dividend of $0.50.

• The stock price will be

• $11 with 50% probability

• $9 with 50% probability



Return

• Two scenarios

• In case the stock price after a year = $11:

Return () =$0.5 + $(11 − 10)

$10 = 5% + 10% = 15%


Return () =$0.5 + $(9 − 10)

$10 = 5% − 10% = −5%

• Since the two scenarios are with 50% chance,

• ( ) = 0.5(15%) + 0.5(−5%) = 5%

• In statistics, we call this the expectation of R (or E[R])



Risk

• Another stock with the same price/dividend but

• The price will be either $15 or $5 after a year. (50% vs 50%)

• Two scenarios


Return () =$0.5 + $(15 − 10)

$10 = 5% + 50% = 55%


Return =$0.5 + $ 5 − 10

$10 = 5% − 50% = −45%

• Since the two scenarios are with 50% chance,

• Expected return ( ) = 0.5(55%) + 0.5(−45%) = %



Risk

• Both stocks have 5% expected returns.

• However, which stock is risker?

• $11 or $9

• $15 or $5

• How can we capture this?

• In statistics, we say that the second stock has a higher

variance (or Var(R)).



Stat review - Expectation, Variance

• Let’s consider a return on a single stock .

• What is expected to be?

• What is the risk?

• We do not know the stock’s future return today!

• is a random variable!

Probability Value of R

0.1 0 (or 0%)

0.5 0.1 (or 10%)

0.4 0.2 (or 20%)



What do we expect to get?

• What is the expected return?

[] = 0.1(0) + 0.5(0.1) + 0.4(0.2) = 0.13 ( 13%)

• It is also called the mean return.

• Note that

• We’re taking an average of the different outcomes.

• But it is not a simple average!

• Weighing each outcome by the likelihood that it will occur.

Probability Value of R

0.1 0 (or 0%)0.5 0.1 (or 10%)

0.4 0.2 (or 20%)



What’s the risk?

•For risk, we use the notion of variance: Var = −

= 0.1 0 − 0.13 + 0.5 0.1 − 0.13 + 0.4 0.2 − 0.13

= 0.0041

• We take each observation, subtract the mean,

and square it.

• We square because this measures distances from the

mean.

• If we did not square, we’d end up with 0 by definition.



What’s the risk?

• However, by squaring, we’ve made this into the

wrong units.

• Returns are in % , but variance is in %

• To bring it back to % , we have to take the

square root:

σ = Var = . = . = . %

• We call this the standard deviation of R.



What’s the risk?

• Example – normal distribution



Not all risk is created equal

• Total risk

• the variance of an investment’s returns

• Firm-specific risk

• The risk that can be diversified away, only affects one firm

• Aka diversifiable risk, idiosyncratic risk, unique risk, or‘casino’ risk

• Systematic risk

• The risk that cannot be diversified away, affects every firm

• Aka non-diversifiable risk or market risk



Portfolio

• As investors, we don’t just choose individual assets.

• We can combine them to form portfolios.

• For simplicity, consider two stocks.

• How do the risk and return of the portfolio relate those of our

underlying assets?

•

Notation• = Weight in stock 1; = Weight in stock 2

• = Return on stock 1; = Return on stock 2

• = Return on portfolio



Portfolio return

• Note that

= +

• Example

• $100 investment put $20 in stock 1 and $80 in stock 2

• =

= 0.2 and =

= 0.8 ( + = 1)

• If returns are: = 12% and = 7%

=$20 1 + 0.12 + $80 1 + 0.07

$100 − 1

= 0.2 1 + 0.12 + 0.8 1 + 0.07 − 1

= 0.2 0.12 + 0.8 0.07 = 0.08 +



• In the example,

• The given and represent only one of a set ofpossible outcomes.

• Three scenarios, each with equal probability

• Given the and outcomes, we can use therelationship to compute portfolio return.

• For example, in the recession case:

0.2(−0.07) + 0.8(0.17) = 0.122

Portfolio return



Expected return

• Expected returns on two stocks?

= 13

−0.07 + 13

0.12 + 13

0.28 = 0.11

=1

3 0.17 +

1

3 0.07 +

1

3 −0.03 = 0.07

• Expected portfolio return

=1

3 0.122 +

1

3 0.08 +

1

3(0.032) = 0.078



Variance / Standard deviation

= =1

3 −0.18 +1

3 0.01 +1

3 0.17 = 0.143

= =1

3 0.10 +

1

3 0 +

1

3 −0.10 = 0.082

= =1

3 0.044 +

1

3 0.002 +

1

3 −0.046 = 0.037

− [] − [] − []



Let’s think about this

• The portfolio mean is 7.8%

• Which makes sense because it is between 11% and 7%.

• Also, it makes sense that it is close to 7%, as we have

80% in stock 2.

• The standard deviation is 3.7%

• This seems a bit strange.

• It is below both the standard deviation for and , which are 14.3% and 8.2%, respectively.

• I assure that there has been no mistake!

• Then, how should we interpret this result?



Diversification

• Two assets’ returns move in opposite directions!

•In a recession, holding stock 2 was very helpful – itbalanced the loss of stock 1.

• In a boom, holding stock 1 was very helpful – it balanced

the loss of stock 2.

• This is diversification!

• The risk (or variance) gets smaller when assets are held

together as a portfolio.



Stat review – Covariance

• How diversification works?

• To better understand this, we need a measure of how

much two asset returns change together.

• The covariance measures the extent to which

two assets move together:

, = − −

• Note: the variance is the covariance of an asset with

itself.



Example (revisited)

, = − −

=1

3 −0.18 0.10 +

1

3 0.01 0 +

1

3 0.17 −0.10

= −0.01167

• Here, covariance is negative

• Relative to their mean, stock 1 has good outcomes when

stock 2 has bad outcomes, and vice-versa.

− [] − [] − []

S C



Stat review - Correlation

•

The magnitude of the covariance is not easy tointerpret.

• What does the covariance of -0.01167 mean?

• Correlation

• To interpret covariance, it is useful to scale it by the

standard deviations.

= = ,

• We can show that −1 ≤ ≤ 1.

E l ( i i d)



Example (revisited)

•

,

= −0.01167

• = 0.14306

• = 0.08165

=

−0.01167

(0.14306)(0.08165) = −0.999

• Returns on stock 1 and 2 show an almost perfect

negative correlation.

− [] − [] − []

G i f di ifi ti



Gains from diversification

•

The takeaway from the previous example:• If two asset returns are (perfectly) negatively correlated,

we obtain gains from holding them together.

• The variance (or risk) of the portfolio is smaller than

those of the two original stocks.

• Is this the case even when the correlation is

positive?• Yes – as long as it is not equal to 1 (perfect positive

correlation).

A th l



Another example

• Suppose that there are two assets:

• = 10% and = 12%

• = 17% and = 25%

• = .

• Suppose that we invest 80% in asset 1 and 20%

in asset 2.

•

= 11.4%• = 11.7%

Note: here I use the above (shortcut) formulas. (You learn

them in FINA 4320 or some stat courses.)

= wE R + wE[R]

= +

+ 2(, )

Wh t i t iki b t thi ?



What is striking about this?

•

Note that asset 1 is less risky than asset 2• = 12% vs = 25%

•

Interestingly, you can make your portfolio evenless risky by adding a bit of asset 2!

• = 12% vs = 11.7%

• Furthermore, your portfolio has the higher expected

return! (10% vs 11.4%)

• Gains from diversification!

Di ifi bl i k t ti i k



Diversifiable risk vs systematic risk

1 11 21 31

σ

Number of Stocks

SYSTEMATIC RISKTOTAL RISK

UNIQUE RISK

Lecture_10-11 Risk and Return

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