Principles of Corporate Finance

Principles of Corporate Finance

Chapter 8. Introduction to risk, return, and the opportunity cost of capital

Ciclo Profissional 2^o Semestre / 2009

Gradua¸c˜ao em Ciˆencias Econˆomicas

Topics covered

1 Over a century of capital market history

2 Measuring portfolio risk

3 Calculating portfolio risk

4 How individual securities affect portfolio risk

5 Diversification and value additivity

The value of an investment of $1 in 1900

How an investment of $1 at the start of 1900 would have grown in nominal terms, assuming reinvestment of all dividends and interests payments

The value of an investment of $1 in 1900

How an investment of $1 at the start of 1900 would have grown in real terms, assuming reinvestment of all dividends and interests payments

Average rates of return from 1900 to 2006 in % per year

Definition

The risk premium is the extra return versus Treasury bills returns (riskfree rate of interest)

Risk premium of portfolio p = Return ofp−Return of Treasury bills

Using historical evidence to evaluate today’s cost of capital

Suppose there is an investment project that youknow belongs to the same risk class as Standard & Poor’s Composite Index

What rate should we use to discount this project’s forecasted cash flows?

We should use the currentlyexpected rate of return on the market index, i.e., the return rm investors would forgo (if they have correct expectations) by investing in the proposed project One way to estimater_m is to assume that the future will be like the past:

I Today’s investors expect to receive the same rate of return revealed by the averages of the past realization

I The Law of Large Numbers

Following data, we should set rm = 11.7%

Using historical evidence to evaluate today’s cost of capital

Recall that

r_m=r_f + risk premium of the Index The riskfree interest rate r_f is observable today

Therefore, a more sensible procedure to estimate the interest rate on the Index is to estimate the risk premium of the Index using average risk premium

I Mid-2006, the interest rate on Treasury bills is about 5%

I The average risk premium of the Index is 7.6%

Therefore

r = 0.05 + 0.076 = 12.6%

Using historical evidence to evaluate today’s cost of capital

The crucial assumption is that there is a stable risk premium on the Index, so that the expectedfuturerisk premium can be measured by the average past risk premium

Even with over 100 years of data, we can’t estimate the market risk premium exactly

We cannot be sure that investors today are demanding the same reward for risk that they were 50 or 100 years ago

All this leaves a plenty of room for argument about what the risk premiumreally is

Market portfolio

Definition

A market portfoliois a portfolio consisting of a weighted sum of every asset in the market, with weights in the proportions of the value (price) of each asset in the market

Denote by K the finite family of assets in the market Denote by P_0,k and Pe_1,k the today’s price and tomorrow’s (random) prices of each asset kand let

P0 = X

k∈K

P_0,k and Pe1= X

k∈K

Pe_1,k

Denote by α the proportionP /P

Market portfolio

Assume I want to invest the amount x >0

I will invest the proportion α_kx in assetk by buyingy ≡x/P₀ units of asset k

The rate of returnr_m of this portfolio is P

k∈KyPe_1,k−P

k∈KyP_0,k

x =

P

k∈KyPe_1,k−P

k∈KyP_0,k P

k∈KyP0,k

= X

k∈K

α_kPe_1,k−P_0,k P0,k

= X

k∈K

α_ker_k

Wherere_k is the random rate of return of asset kdefined by Pe_1,k−P_0,k

P

Market portfolio

In theory, the market portfolio would necessarily include every single possible available asset, including real estate, precious metals, stamp collections, jewelry, and anything with any worth, as the theoretical market being referred to what would be the world market

In practise, good proxies for the market are indexes

I S&P500 in the US

I DAX in Germany

I TSE100 in the UK

Average risk premia in the world

Nominal return of the market portfolio minus nominal return on bills, from 1900 to 2006

The equity premium puzzle

There is a high (historic) risk premium earned in the market This seems to imply that that investors are extremely risk-averse If that is true, investors should cut back their consumption when stock prices fall and wealth decreases

But evidence suggests that when stock prices fall, investors spend at nearly the same rate

This is difficult to reconcile with high risk aversion and a high market risk premium

Mehra, S. and Prescott, E.

The Equity Premium: A Puzzle.

Journal of Monetary Economics 15, 1985

The equity premium puzzle

A large number of explanations for the puzzle have been proposed The puzzle is a statistical illusion

Modifications to the assumed preferences of investors Market imperfections

The equity premium puzzle

The historical average of risk premium in U.S. is 7.6%

Survey of chief financial officers commonly suggest that they expect a market risk premium that is several percentage points below the historical average

Possible explanations:

I Growth of mutual funds has made it easier for individuals to diversify away part of their risk

I Pension funds and other financial institutions have found that they also could reduce their risk by investing part of their funds overseas

I If investors can eliminate more of their risk than in the past, they may become content with a lower return

The dividend yields

If a stock is paying dividends with a constant growth rate g, then we should have

r = DIV1

P0

+g

Dividend yields in U.S. have averaged about 4.4% since 1900 The annual growth in dividends has averaged about 5.6%

If the dividend growth was representative of what investors expected, then the expected market return over this period should

be DIV₁

P0

+g= 4.4 + 5.6 = 10.0%

or 6% above the risk-free interest rate

This figure is 1.6% lower than the realizedrisk premium

The dividend yields

Average dividend yields in the U.S.A. since 1900

The dividend yields

Dividend yields fluctuated quite sharply

There are only two possible reasons for the yield changes:

I some years investors were unusually optimistic or pessimistic aboutg

I the required returnrwas unusually high or low

Following several studies, Economists have concluded that very little of the variation is related to the rate of dividend growth If they are right, the level of yields ought to be telling us something about the return that investors require

Measuring portfolio risk

The stock market has been a profitable but extremely variable investment

Wide spread of returns from investment in common

stocks

Standard statistical measures

Consider a portfolio pand denote by erp the function erp : Ω−→[0,∞)

describing the possible values of the return of the portfoliop The set Ω describes possible “states of nature”

There exists a probability P^erp ∈Prob(Ω), called the law ofer_p and defined as follows

∀A⊂[0,∞), P^erp(A) =P{er_p ∈A}

IfA= [0.02,0.04], thenP^erp(A) is the probability that the return r_p is between 2% and 4%

Standard statistical measures

Assume that Ω is finite and let Imer_p be the range (or image) ofer_p, i.e., the set of possible values of er_p:

Imer_p ={r ∈[0,∞) : ∃ω∈Ω, er_p(ω) =r}

Definition

The expected value of er_p is the number denoted byE^P[er_p] or r_p defined by

E^P[erp] = X

ω∈Ω

P(ω)rep(ω)

= X

r∈Imrep

rP^erp(r)

Standard statistical measures

Assume that Ω is infinite and that the law of P^erp has a density h

erp in the sense that

∀06a < b, P^erp[a, b] = Z b

a

herp(r)dr.

Definition

The expected value of er_p is the number denoted byE^P[er_p] or r_p defined by

E^P[erp] = Z ∞

0

rherp(r)dr This expectation is also denoted by

∞

Variance and standard deviation

Definition

The varianceof the return erp is the expected squared deviation from the expected return, i.e.,

var(erp) =E^P

(rep−rp)²

=E^P h

erp−E^P[rep]2i Definition

The standard deviation (orrisk) of the return erp is the square root of the variance:

Standard deviation ofrep = q

var(erp)

When there is no ambiguity concerning the portfolio, the standard deviation is denoted by σ and the variance by σ²

Example

Consider the following game You start by investing $100 Then two coins are flipped

For each head that comes up you get back your starting balance plus 20%

For each tail that comes up you get back your starting balance less 10%

There are four equally likely outcomes:

Head + Head: You gain 40%

Head + tail: You gain 10%

Tail + head: You gain 10%

Example

The expected return is

Expected return = (0.25×40%) + (0.50×10%) + (0.25× −20%)

= +10%

Measuring expected returns and variance

To compute E^P[er_p] and var(er_p), we should

I identify the possible outcomes

I assign a probability to each outcome Where do the probabilities come from?

Most analysts start by observing the past

It is reasonable to consider that portfolios with histories of high variability also have the least predictable future performance

Measuring expected returns and variance

Consider a sample of observed returns {erm(t)}_16t6N whereN is the number of observations One may consider the following estimates

E^P[erm]≈ 1 N

N

X

t=1

erm(t)

var(er_m)≈ 1 N −1

N

X

t=1

er_m(t)−E^P[er_m]2

Measuring risk

The annual standard deviations and variances (in nominal terms) observed over the period 1900-2006

Measuring risk

The risk (standard deviation of annual returns) of markets around the world, 1900-2006

Measuring risk

Annualized standard deviation of the Dow Jones Industrial Average, 1900-2006

Measuring risk

Standard deviations for selected U.S. common stocks, July 2001-June 2006 (figures in percent per year)

During this period, the market portfolio’s standard deviation was about 16%

Measuring risk

Standard deviation for selected foreign stocks and market indexes, July 2001–June 2006 (figures in percent per year)

Some of the stocks are much more volatile than others

But individual stocks are for the most part more variable that the

Diversification reduces risk

The risk (standard deviation) of randomly selected portfolios containing different numbers of New York Stock Exchange stocks

Diversification reduces risk

Diversification reduces risk because prices of different stocks do not move exactly together

Different kinds of risk

Definition

The risk that potentially can be eliminated by diversification is called unique risk

Unique risk may be called unsystematic risk residual risk specific risk diversifiable risk

Unique risk stems from the fact that many of the perils that surround an individual company are peculiar to that company and perhaps its immediate competitors

Different kinds of risk

Definition

The risk that cannot be avoided, regardless of how much you diversy, is called market risk

Market risk may be called systematic risk undiversifiable risk

Market risk stems from the fact that there are economywide perils that threaten all businesses

Different kinds of risk

Calculating portfolio risk

Definition

Consider two portfolios with returns re₁ and er₂ respectively. The covariance of the two portfolios is defined as follows

cov(er₁,re₂) =E^P

(re₁−E^P[er₁])(er₂−E^P[er₂])

=E^P[(re₁−r₁)(er₂−r₂)]

When there is no ambiguity concerning the two portfolios, the covariance is denoted byσ₁₂

The covariance is symmetric, i.e.,σ₁₂=σ₂₁ Observe that

cov(er₁,re₂) =E^P[er₁er₂]−E^P[er₁]E^P[er₂] Moreover, we haveσ =σ², i.e.,

Calculating portfolio risk

Definition

Consider two portfolios with returns re₁ and er₂ respectively. The

correlation coefficientρ(er₁,er₂) of the two portfolios is defined as follows cov(re₁,er₂) =ρ(er₁,re₂)p

var(re₁)p var(er₂) When there is no ambiguity concerning the two portfolios, the correlation coefficient is denoted by ρ₁₂, in particular we have

σ₁₂=ρ₁₂σ₁σ₂

Calculating portfolio risk

From Cauchy-Schwartz, we have ρ₁₂∈[−1,+1]:

if the returns of the portfolios are positively and perfectly correlated then ρ₁₂= 1

if the returns of the portfolios are wholly unrelated thenρ₁₂= 0 if the returns of the portfolios are negatively and perfectly correlated then ρ₁₂=−1

If two stocks are perfectly negatively correlated then there is always a portfolio strategy that will completely eliminate risk

Calculating portfolio risk

Consider a portfoliop composed of two portfolios with returnsre1

and er₂

We let x_i ∈[0,1] the proportion holding of portfolioi, x1+x2= 1

in the sense that the returnerp of portfoliop is defined by re_p=x₁er₁+x₂er₂

The expected return r_p of portfoliop is given by

rp =x1r1+x2r2 or E^P[rep] =x1E^P[er1] +x2E^P[er2]

Calculating portfolio risk

The variance σ_p² of portfoliop is given by

var(re_p) =x²₁var(er₁) +x²₂var(er₂) + 2x₁x₂cov(er₁,re₂) or equivalently

σ²_p =x²₁σ₁²+x2σ₂²+ 2(x1x2ρ12σ1σ2)

Calculating portfolio risk

Consider the general case of a portfolio p composed of a finite family of stocks with returns

(er_i)_16i6N with proportions (x_i)_16i6N, i.e.,

N

X

i=1

xi= 1 and xi >0, ∀i We have

rep= X

16i6N

xieri

implying that

r_p= X

16i6N

x_ir_i and

σ_p= X X

x_ix_jσ_ij

Calculating portfolio risk

Calculating portfolio risk

Assume that the portfolio p is composed with the same proportion of each stock, i.e., xi = 1/N for each i

σ_p² = N 1

N 2"

PN i=1σ²_i

N

#

+ (N²−N) 1

N

2 P

i6=jσij

N²−N

= 1

N ×average variance +

1− 1 N

×average covariance Passing to the limit

N→∞lim σ²_p = average covariance we see that only the average covariance matters

Calculating portfolio risk

Consider the general case of a portfolio p composed of a finite family of stocks with returns

(eri)_16i6N with proportions (xi)_16i6N, i.e.,

re_p= X

16i6N

x_ier_i

The risk of the portfolio is measured by the variance σ_p = var(er_p) One may wonder what is the relative contribution of a specific stockj to the risk of portfoliop

How individual securities affect portfolio risk

Recall that the risk σ_p² of portfoliop is given by σ_p² =

N

X

i=1 N

X

k=1

xixkσik

Therefore, the contribution of stock j is

∂ σ²_p

∂xj

= 2 [x₁σ_j1+x₂σ_j2+. . .+x_Nσ_jN]

= 2

N

X

i=1

xicov(rej,rei)

= 2 cov er_j,

N

X

i=1

x_ire_i

!

= 2 cov(erj,erp)

How individual securities affect portfolio risk

The relative contribution of stockj to the standard deviationσp of the portfolio pis

∂σp/∂xj

σ_p = 1

2σ_p²

∂ σ_p²

∂x_j

= cov(erj,rep) var(er_p)

= σjp

σ_p²

How individual securities affect portfolio risk

Definition

Consider a portfolio pwith return erp and a stock iwith returneri. The beta of stock irelative to portfolio p is denotedβ_i(p) and defined by

βi(p) = σ_ip

σ_p² = cov(er_i,re_p) var(erp) Therefore, we have

cov(rei,erp) =βi(p) var(rep)

The beta of a stock

Recall that the market portfolio mis defined by the returnre_m given by

erm =

N

X

i=1

αieri where αi= V_i PN

k=1V_k and Vi is the value (or price) of equity i

Definition

The beta of stock irelative the market portfoliom is denoted byβ_i

An intuition

Imagine that a stock iis perfectly correlated with the market portfolio m, i.e., there exists a correlation coefficient α_im such that

eri−ri =αim(erm−rm) In that case, we have βi=αim

if the market rises 1% then the price of stockiwill rise by β%

if the market falls 2% then the price of stock iwill fall 2×β%

In general stocks are not perfectly correlated with market returns, each firm is subject to unique risk

Examples

Betas for selected U.S. common stocks, July 2001-June 2006

Example

Disney had a beta of 1.26

If the future resembles the past, this means thaton average when the market rises an extra 1%

Disney’s stock prices will rise by an extra 1.26%

When the market falls an extra 2%, Disney’s stock prices will fall an extra 2.52%

Foreign betas

Betas for foreign stocks, July 2001–June 2006

Beta is measured relative to the stock’s home market

Diversification and value additivity

Diversification reduces risk for investors, and therefore makes sense for them

Does it make sense for firms to diversify their activities? Is a diversified firm more attractive to investors than an undiversified one?

If the answer is yes, then it contradicts the additivity of the net present value since the value of a diversified package would be greater than the sum of the parts

If investors can diversify then firms don’t have to do diversify since investors will not pay any extrafor firms that diversify

I It is very easy for an investor to diversify: he can buy shares of mutual funds that hold diversified portfolios

I A firm cannot diversify so easily