Risk and Return (Chs. 7 & 8)

 

RISK AVERSION

We assume investors are risk averse

Willing to bear risk only if they are compensated for it

Required return for risky asset = Ri = Rf + compensation for risk = Rf + RPi

Need higher returns from riskier assets: RPi is proportional to risk (twice the risk means twice the risk premium, not twice the required return)

Why do we assume this: capital market history (riskier assets have given higher returns, on average)

 

SYSTEMATIC AND UNSYSTEMATIC RISK

·        Each stock has some total risk to start with (variance of Ri)

·        Investors hold diversified portfolios

·        Some of the total risk is diversified away, some remains undiversified

·        Systematic risk is the risk that remains after diversification

·        Unsystematic risk comes from firm specific news: randomly positive and negative for different firms (uncorrelated).  Effects cancel out in a large portfolio.  This type of news/risk doesn’t affect a large portfolio

·        Systematic risk comes from economy-wide news.  This systematically moves most firms in the same direction.  Movements of different stocks are positively correlated; hence they don’t cancel out

·        Large portfolio moves up and down only in response to economy-wide news

 

SYSTEMATIC RISK OF A STOCK & BETA

·        Loosely, systematic risk of a stock is the risk that remains after diversification when you hold the stock as part of a diversified portfolio

·        CAPM: everyone holds market portfolio m as their stock portfolio

·         sm2 is the risk that remains in the market portfolio after diversification; this is the systematic risk investors actually bear

·         sm2 = systematic risk jointly contributed to the market portfolio by all stocks

·        Formally, systematic risk of stock 1 is the marginal risk the stock contributes to the market portfolio after diversification

·        i.e., it is stock 1’s marginal contribution to sm2, or the rate at which sm2 increases as you invest more in stock 1 (i.e. increase its portfolio weight)

·        turns out that  sm2 can be decomposed as:    sm2 = x1 s1m  + x2 s2m  + … + xN sNm

·        total contribution of stock 1 to  sm2 is x1 s1m

·        marginal contribution of stock 1 to  sm2 = d sm2/ dx1 =  s1m

·        So, in absolute terms, systematic risk of a stock is measured by  s1m

·        beta is a relative measure (or a % measure).  It measures systematic risk of a stock relative to the systematic risk of the market portfolio :  bi =  sim/ sm2

·        so beta of .8 means the systematic risk of the stock is 80% of the systematic risk of the market portfolio

 

CAPM

·        Basic equation is the Security Market Line equation (SML): 3 versions

o    Ri = Rf +bi [E(Rm) - Rf]

o    Ri = Rf + bi * RPm

o    RPi = bi * RPm

·        Basic message of this equation: required return depends only on systematic risk (which is captured by beta); total risk doesn’t matter

·        systematic risk versus total risk: wildcat oil driller is in a business with high total risk; would you expect systematic risk to be high?

·        When is systematic risk high? 

o    bi =  sim/ sm2 = ( si/ sm) * rim

o    Systematic risk is high or low depending on rim  => high when returns on an asset or project are positively correlated with market return

·        Back to the question: should systematic risk for wildcatting be high or low?  Returns  for wildcatting would be positively correlated with market return if you are more likely to strike oil when the market goes up.  Clearly there’s no relation => correlation is zero => systematic risk of wildcatting is zero (even though total risk is high)

·         Meaning of positive beta and negative beta

o    Remember that bi =  sim/ sm2 = ( si/ sm) * rim

o    positive beta stock is positively correlated with market portfolio; moves with the market

o    negative beta stock is negatively correlated with market portfolio; counter-cyclical stock; moves opposite to the market

·        negative beta stock will have E(R) < Rf.  so we have a risky asset offering a return less than riskless rate.  why would anyone hold this stock? 

o    you don’t hold stocks in isolation; you hold them as part of the market portfolio, m

o    positive beta means positive sim => stock makes a positive contribution to the risk you bear when you hold market portfolio (i.e. it increases the risk you bear)

o    negative beta means the stock makes a negative contribution to  sm2 (reduces the risk you bear)

o    When a stock increases the risk you bear, you demand a return > Rf.  When a stock reduces the risk you bear, you accept a return < Rf.

o    So you willingly hold this stock as part of the market portfolio, even though it has an E(R) < Rf

o    That’s what it means for a stock to have negative systematic risk; it’s a risk-reducing stock

 

BETA AS SENSITIVITY TO MARKET MOVEMENTS

A positive beta stock moves with the market.  The sign of beta tells us the direction in which it will move.  The magnitude of beta tells us the amount it will move, on average

e.g. Rf = 5%, RPm = 9%, and we have a stock whose beta is 1.2

First of all, E(Rm) = Rf + RPm = 14%

E(Ri) = Rf + bi * RPm = .05 + 1.2*.09 = 15.8%

This is the ex ante expected return, the return we expect from the stock at the beginning of the year (when all we know of the market return is that it is expected to be 14%)

Now suppose at the end of the year, the actual return on the market portfolio turns out to be 16%.  So the market gives a return 2% higher than expected.  This is the extra return on the market, or the unexpected return on the market

The stock is expected to move with the market, so if the market went up we expect the stock to go up along with it.  What return do we now expect from the stock, once we know the market return is 16%?

The ex post E(R) = ex ante E(R) + extra return from the stock

= ex ante E(R) + bi * (extra return on the stock)

                                    = .158 + 1.2*.02 = 18.2%

This is what it means that beta is the sensitivity of the stock to market movements.  Beta acts as a multiplier.  With a beta of 1.2, when the market moves 2%, the stock moves 2.4%.  The higher the beta, the more the stock will move

What we’re saying here is that initially we expected the stock to produce a return of 15.8%.  But when economy-wide news makes the market go up 2%, we expect the stock to go up 2.4% in response to this economy-wide news. 

Note that ex post E(R i) = ex ante E(Ri) + bi * (extra return on the stock)

= Rf + bi [E(Rm) - Rf] + bi * [Actual Rm - E(Rm)]

= Rf + bi [Actual Rm - Rf]

 

EVENT STUDIES:  STUDYING IF A CORPORATE ANNOUNCEMENT IS GOOD NEWS OR BAD NEWS

·        We want to look at the effect of a given type of corporate news (e.g. stock split or earnings increase).  Question is whether this type of event is good news or bad news for s/h: does it increase or decrease s/h wealth?

·        We want to know the average reaction to this type of firm-specific news

·        The actual return on a stock has three components:   

         Actual return = ex ante E(R) + effect of economy-wide news + effect of firm-specific news

     (Actual return deviates from expected return only due to news, and there are two types of news)

·        In an event study, we want to capture just the effect of firm-specific news.  So we need to subtract away the first two components from the actual return.  And the first two components together are just the ex post E(R)

·        We define Abnormal Return (AR) to be the difference between actual return and ex post E(Ri):

    ARi = Actual Ri - ex post E(R)

       Abnormal return captures the impact of firm-specific news only

·        If you compute the abnormal return for a stock on the day a stock split was announced, the AR will reflect the impact of that piece of firm-specific news plus any other firm-specific news announced the same day

        ·        So we take a large sample of firms all of which announced stock splits, and we compute the AR for the announcement day.  And we take the average AR for the sample.  Each firm will have some other firm-specific news that day (apart from the stock split).  But this other news will be randomly positive or negative for different firms, so it will average out to zero. This means that the average AR represents just the average effect of stocks splits.  We have managed to subtract away every thing else: the normal expected return, the effect of economy-wide news, and the effect of other firm-specific news.

 

ACTUALLY ESTIMATING REQUIRED RETURN

Ri = Rf + bi * RPm

To estimate required return, we use Rf  today and average historical RPm (with adjustments to taste)  

How we come up with Rf  today will be discussed further in Ch. 9.  For now we can think of it as the yield on 3-month T-bills

There’s no magic number for the average historical market risk premium.  Using different periods will give slightly different numbers:

1926-2000             9.1%

1900-2000             7.5%

There’s no consensus on which period should be used (key issue is: is it more representative if we include the Great Depression or exclude it)

There are also reasons why you might want to make adjustments to the average historical number.

So what number you use becomes ultimately a matter of taste or preference.  Something around 8 or 8.5% seems to make sense.  We’ll use 8.5% in group project.

 

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