investment science - part ii: single-period random cash flows

Investment SciencePart II: Single-Period Random Cash Flows

Dr. Xi CHEN

Department of Management Science and EngineeringInternational Business School

Beijing Foreign Studies University100089, Beijing, People’s Republic of China

Xi CHEN ([email protected]) Investment Science 1 / 81

Outline

1 Asset Return

2 Random Variables (self-learning)

3 Random Returns (self-learning)

4 Portfolio Mean and Variance

5 The Feasible Set

6 The Markowitz Model

7 The Two-Fund Theorem

8 Inclusion of a Risk-Free Asset

9 The One-Fund Theorem


Asset Return

Outline

1 Asset Return




5 The Feasible Set






Asset Return

An investment instrument that can be bought and sold is frequently calledan asset.

Suppose that you purchase an asset at time zero, and 1 year later yousell the asset. The total return on your investment is defined to be

total return =amount received

amount investedor R =

X1

X0.

For simplicity, the term return is used for total return.

The rate of return is

rate of return =amount received - amount invested

amount invested

r =X1 − X0

X0= R − 1.

The shorter expression return is also frequently used for the rate ofreturn.


Asset Return

Short Sales

Sometimes it is possible to sell an asset that you do not own through theprocess of short selling, or shorting, the asset.

Short selling is considered quite risky-even dangerous-by manyinvestors.

When short selling a stock, you are essentially duplicating the role ofthe issuing corporation. You sell the stock to raise immediate capital.If the stock pays dividends during the period that you have borrowedit, you too must pay that same dividend to the person from whomyou borrowed the stock.

We receive X0 initially and pay X1 later, so the outlay is −X0 and thefinal receipt is −X1, and hence

R =−X1

−X0⇒ −X1 = −X0R = −X0(1 + r).

Therefore, the return value R applies algebraically to both purchasesand short sales.


Asset Return

Example (A short sale)

Suppose I decide to short 100 shares of stock in company CBA. This stockis currently selling for $10 per share. I borrow 100 shares from my brokerand sell these in the stock market, receiving $1,000. At the end of 1 yearthe price of CBA has dropped to $9 per share. I buy back 100 shares for$900 and give these shares to my broker to repay the original loan.Because the stock price fell, this has been a favorable transaction for me. Imade a profit of $100.

Someone who purchased the stock at the beginning of the year and sold itat the end would have lost $100, with R = 0.9 or r = −0.1.

The rate of return is clearly negative as r = −0.1. Shorting converts anegative rate of return into a profit because the original investment is alsonegative. For my shorting activity on CBA my original outlay was -$1,000;hence my profit is −$1, 000× r = $100.


Asset Return

Portfolio Return

Suppose now that n different assets are available. We can form a masterasset, or portfolio, of these n assets.

Suppose that this is done by apportioning a total amount X0 among the nassets. We then select amounts X0i , i = 1, 2, . . . , n, such that∑n

i=1 X0i = X0, where X0i represents the amount invested in the ith asset.If we are allowed to sell an asset short, then some of the X0i ’s can benegative; otherwise we restrict the X0i ’s to be nonnegative.

The amounts invested can be expressed as fractions of X0, i.e.,

X0i = wiX0, i = 1, 2, . . . , n,n∑

i=1

wi = 1,

where wi is the weight or fraction of asset i in the portfolio.


Asset Return

Let Ri denote the total return of asset i . We have

R =1

X0

n∑i=1

RiwiX0 =n∑

i=1

wiRi ⇒ 1 + r =n∑

i=1

wi (1 + ri )

⇒ r =n∑

i=1

wi ri .

Formula (Portfolio return)

Both total return and the rate of return of a portfolio of assets are equalto the weighted sum of the corresponding individual asset returns, with theweight of an asset being its relative weight (in purchase cost) in theportfolio; that is,

R =n∑

i=1

wiRi , r =n∑

i=1

wi ri .


Asset Return


Random Variables (self-learning)

Outline

1 Asset Return




5 The Feasible Set






Random Returns (self-learning)

Outline

1 Asset Return




5 The Feasible Set






Portfolio Mean and Variance

Outline

1 Asset Return




5 The Feasible Set







Mean Return of a Portfolio

Suppose that there are n assets with (random) rates of return ri ’s andexpected values r i ’s, i = 1, 2, . . . , n.

Suppose that we form a portfolio of these n assets using the weights wi ,i = 1, 2, . . . , n. The rate of return of the portfolio in terms of the return ofthe individual returns is

r =n∑

i=1

wi ri .

We may take the expected values of both sides, and obtain

E(r) = r =n∑

i=1

wiE(ri ) =n∑

i=1

wi r i .

The expected rate of return of the portfolio is found by taking theweighted sum of the individual expected rates of return!



Variance of Portfolio Return

Denote the variance of the return of asset i by σ2i , the variance of thereturn of the portfolio by σ2, and the covariance of the return of asset iwith asset j by σij .

var(r) = E[(r − r)2

]= E

[

n∑i=1

wi (ri − r i )

] n∑j=1

wj(rj − r j)

= E

n∑i=1

n∑j=1

wiwj(ri − r i )(rj − r j)

=n∑

i=1

n∑j=1

wiwjσij .

Example (Two-asset portfolio)

Suppose that there are two assets with r1 = 0.12, r2 = 0.15, σ1 = 0.20,σ2 = 0.18 and σ12 = 0.01 (values typical for two stocks). A portfolio isformed with weights w1 = 0.25 and w2 = 0.75. The mean and thevariance of this portfolio are r = 0.1425 and σ2 = 0.024475.



Diversification

The variance of the return of a portfolio can be reduced by includingadditional assets in the portfolio, a process referred to as diversification.

Don’t put all your eggs in one basket!

Suppose that the rate of return of each of uncorrelated assets has mean mand variance σ2. Let wi = 1/n for each i . Then

r =1

n

n∑i=1

ri ⇒ E(r) = m, var(r) = var

(1

n

n∑i=1

ri

)=σ2

n.



Suppose again that each asset has a rate of return with mean m andvariance σ2, but now each return pair has a covariance ofcov(ri , rj) = 0.3σ2 for i 6= j . Then

var(r) = var

(1

n

n∑i=1

ri

)=

1

n2

∑i=j

σij +∑i 6=j

σij

=

1

n2

[nσ2 + 0.3(n2 − n)σ2

]=

0.7σ2

n+ 0.3σ2.

It is impossible to reduce the variance below 0.3σ2, no matter howlarge n is made!

In general, diversification may reduce the overall expected return whilereducing the variance. Blind diversification is not necessarily desirable.This is the motivation behind the general mean-variance approachdeveloped by Markowitz.



Diagram of a Portfolio

Consider two assets as indicated in the following figure.

Introduce a variable α. Define w1 = 1− α and w2 = α. As α varies,the new portfolios trace out a curve that includes assets 1 and 2 butits exact shape depends on σ12.

The solid portion of the curve corresponds to positive combinations oftwo assets; the dashed portion corresponds to shorting one of them.

The solid portion of the curve must lie within the shaded region.



Lemma (Portfolio diagram lemma)

The curve in an r − σ diagram defined by nonnegative mixtures of twoassets 1 and 2 lies within the triangular region defined by the two originalassets and the point on the vertical axis of height

A =r1σ2 + r2σ1σ1 + σ2

.

Proof.

The mean of the rate of return of the portfolio is r(α) = (1− α)r1 + αr2,which indicates that the mean value is between the original means. Thestandard deviation of the portfolio is

σ(r(α)) =√

(1− α)2σ21 + 2α(1− α)σ12 + α2σ22

=√

(1− α)2σ21 + 2ρα(1− α)σ1σ2 + α2σ22.



contd.

We know that ρ ∈ [−1, 1]. Using ρ = 1, we have the upper bound

σ∗(r(α)) = (1− α)σ1 + ασ2.

This implies that as α varies from 0 to 1, the portfolio point will trace outa straight line between the two points. Using ρ = −1, we likewise obtainthe lower bound

σ∗(r(α)) =√

(1− α)2σ21 − 2α(1− α)σ1σ2 + α2σ22

= |(1− α)σ1 − ασ2|.

It is nearly linear as well, except for the absolute-value sign. Then

α =σ1

σ1 + σ2⇒ A =

r1σ2 + r2σ1σ1 + σ2

.


The Feasible Set

Outline

1 Asset Return




5 The Feasible Set






The Feasible Set

Definition

Suppose there are n basic assets. The set of points corresponding toportfolios made by letting wi range over all possible combinations suchthat

∑ni=1 wi = 1 is called the feasible set or feasible region.

If there are at least three assets (not perfectly correlated and withdifferent means), the feasible set will be a solid two-dimensionalregion.


The Feasible Set

The feasible region is convex to the left, which means that given anytwo points in the region, the straight line connecting them does notcross the left boundary of the feasible set.

In general, the feasible legion defined with short selling allowed will containthe region defined without short selling, and the leftmost edges of thesetwo regions may partially coincide.


The Feasible Set

The Minimum-Variance Set and Efficient Frontier

Definition

The left boundary of a feasible set is called the minimum-variance set.The minimum-variance set has a characteristic bullet shape, and there is aspecial point on this set having minimum variance termed theminimum-variance point (MVP).


The Feasible Set

Definition

An investor who prefers the portfolio corresponding to the point withthe smallest standard deviation for the given mean is said to be riskaverse.

An investor who would select a point other than the one of minimumstandard deviation is said to be risk preferring, or risk seeking.

Most investors would select the portfolio of the largest mean for agiven level of standard deviation. This property of investors is termednonsatiation.

Only the upper part of the minimum-variance set will be of interest toinvestors who are risk averse and satisfy nonsatiation. This upperportion of the minimum-variance set is termed the efficient frontierof the feasible region.


The Markowitz Model

Outline

1 Asset Return




5 The Feasible Set






The Markowitz Model

Assume that there are n assets. Each has a mean rate of return r i , andcovariances are σij , i , j = 1, 2, . . . , n. A portfolio is defined by a set of nweights wi that sum to 1. We formulate the Markowitz model

min1

2

n∑i=1

n∑j=1

wiwjσij ,

s.t.n∑

i=1

wi r i = r ,n∑

i=1

wi = 1.

It explicitly addresses the trade-off between expected rate of returnand variance of the rate of return in a portfolio.

It can be solved either numerically or analytically.

It is used mainly when a risk-free asset and risky assets are available.


The Markowitz Model

Solution to the Markowitz Model

Using Lagrange multipliers λ and µ, the Lagrangian is

L(w, λ, µ) =1

2

n∑i=1

n∑j=1

wiwjσij − λ

(n∑

i=1

wi r i − r

)− µ

(n∑

i=1

wi − 1

).

Formula (Equations for efficient set)

The n portfolio weights wi for i = 1, 2, . . . , n and the two Lagrangemultipliers λ and µ for an efficient portfolio (with short selling allowed)having mean rate of return r satisfy

n∑j=1

σijwj − λr i − µ = 0, i = 1, 2, . . . , n,n∑

i=1

wi r i = r ,n∑

i=1

wi = 1.


The Markowitz Model

Example (Three uncorrelated assets)

Suppose there are three uncorrelated assets. Each has variance 1, and themean values are 1, 2, and 3, respectively.

w1 =4

3− r

2, w2 =

1

3, w3 =

r

2− 2

3, λ =

r

2− 1, µ =

7

3− r ,

minσ(r) = min√

w21 + w2

2 + w23 ⇒ r∗ = 2, σ∗ =

√3

3.


The Markowitz Model

Nonnegativity Constraints

We can prohibit short selling by restricting each wi to be nonnegative,which leads to

min1

2

n∑i=1

n∑j=1

wiwjσij ,

s.t.n∑

i=1

wi r i = r ,n∑

i=1

wi = 1, wi ≥ 0, i = 1, 2, . . . , n.

(1)

This problem is a quadratic program, since the objective is quadratic andthe constraints are linear equalities and inequalities.

When short selling is allowed, most, if not all, of the optimal wi ’shave nonzero values (either positive or negative), so essentially allassets are used.

When short selling is not allowed, typically many weights are equal tozero.


The Markowitz Model

Example (The three uncorrelated assets)

Consider again the assets of the previous example, but with shorting notallowed. Efficient points must solve problem (1) with the previousparameters.

In this case the problem cannot be reduced to a system of equations.Nevertheless, by considering combinations of pairs of assets, the efficientfrontier can be found.


The Markowitz Model

Example

Suppose the expected rates of return of five assets are

r = (0.16, 0.11, 0.08, 0.13,−0.02)′,

and their covariance matrix is0.56 0.11 0.09 0.08 0.130.11 0.32 0.20 0.14 0.110.09 0.20 0.22 0.13 0.120.08 0.14 0.13 0.48 0.110.13 0.11 0.12 0.11 0.24

.

The transaction costs for these assets are

ci (xi ) = 0.002xi + 0.002(1− e−xi ), i = 1, 2, . . . , 5.

Construct a portfolio such that it has an expected rate of return r = 10%with minimal risk.


The Two-Fund Theorem

Outline

1 Asset Return




5 The Feasible Set







Points in the minimum-variance set satisfy the system of n + 2 equations:

n∑j=1

σijwj − λr i − µ = 0, i = 1, 2, . . . , n,n∑

i=1

wi r i = r ,n∑

i=1

wi = 1.

Suppose that there are two known solutions w1, λ1, µ1 and w2, λ2, µ2,with expected rates of return r1 and r2. Then, the combination portfolio

αw1 + (1− α)w2

also represents a point in the minimum-variance set.

Theorem (The two-fund theorem)

Two efficient funds (portfolios) can be established so that any efficientportfolio can be duplicated, in terms of mean and variance, as acombination of these two. In other words, all investors seeking efficientportfolios need only invest in combinations of these two funds.



Example (A securities portfolio)

The information concerning the 1-year covariances and mean values of therates of return on five securities is shown in the top part of the followingtable. The mean values are expressed on a percentage basis, whereas thecovariances are expressed in units of (percent)2/100.

For example, the first security has an expected rate of return of15.1% = 0.151 and a variance of return of 0.023, which translates into astandard deviation of

√0.023 = 0.152 = 15.2% per year.

We shall find two funds in the minimum-variance set (note −0.27).



Example (contd.)

1 We set λ = 0 and µ = 1, and solve the system of equations

5∑j=1

σijv1j = 1, i = 1, 2, . . . , 5.

Next, we normalize the v1j ’s so that they sum to one, obtaining w1

j ’s,which defines the minimum-variance point.

2 We set µ = 0 and λ = 1, solve the system of equations for v2, andnormalize the v2

j ’s to obtain w2j ’s.


Inclusion of a Risk-Free Asset

Outline

1 Asset Return




5 The Feasible Set







A risk-free asset has a deterministic return and thus σ = 0.

Suppose that there is a risk-free asset with a rate of return rf , andconsider any other risky asset with rate of return r , having mean rand variance σ2. Then the covariance of these two returns is

E {(r − r)(rf − r f )} = 0.

Now suppose that these two assets are combined to form a portfoliousing a weight of α for the risk-free asset and 1− α for the riskyasset, with α ≤ 1. Define σf = 0. We have{

r(α) = αrf + (1− α)r ,

σ(α) = ασf + (1− α)σ.

Both r(α) and σ(α) vary linearly with α!



Suppose now that there are n risky assets with known mean rates ofreturn r i , and known covariances σij . In addition, there is a risk-freeasset with rate of return rf .

In forming the portfolios, we allow borrowing or lending of therisk-free asset, but only purchase of the risky asset.

What if borrowing of the risk-free asset is not allowed?


The One-Fund Theorem

Outline

1 Asset Return




5 The Feasible Set







When risk-free borrowing and lending are available, the efficient setconsists of a single straight line, which is the top of the triangular feasibleregion. This line is tangent to the original feasible set of risky assets. Theportfolio represented by the tangent point, F , can be thought of as a fundmade up of assets and sold as a unit.

Theorem (The one-fund theorem)

There is a single fund F of risky assets such that any efficient portfolio canbe constructed as a combination of the fund F and the risk-free asset.



Solution

Given a point in the feasible region, we draw a line between the risk-freeasset and that point, and denote the angle between that line and thehorizontal axis by θ. For any feasible (risky) portfolio p, we have

tan θ =rp − rfσp

.

The tangent portfolio is the feasible point that maximizes tan θ.

We assign weights w1,w2, . . . ,wn to the risky assets such that∑ni=1 wi = 1. There is zero weight on the risk-free asset in the tangent

fund. Then we have

tan θ =

∑ni=1 wi (r i − rf )√∑ni=1

∑nj=1 wiwjσij

.



In order to maximize tan θ, we set the derivative of tan θ with respect toeach wk equal to zero,

(rk − rf )

n∑i=1

n∑j=1

wiwjσij

=

[n∑

i=1

wi (r i − rf )

](n∑

i=1

wiσki

)

⇒n∑

i=1

σkiλwi = rk − rf , k = 1, 2, . . . , n,

where λ is an (unknown) constant. Let vi = λwi for each i , we have

n∑i=1

σkivi = rk − rf , k = 1, 2, . . . , n.

Solve these linear equations for the vi ’s and then normalize for the wi ’s,

wi =vi∑nj=1 vj

.



Example (Three uncorrelated assets)

We consider again the previous example, where the three risky assets wereuncorrelated and each had variance equal to 1. The three mean rates ofreturn were r1 = 1, r2 = 2, and r3 = 3. We assume in addition that thereis a risk-free asset with rate of return r f = 0.5. We then find that

w =

(1

9,

1

3,

5

9

)′.

Example (A larger portfolio)

Consider the previous example with five risky assets. Assume also thatthere is a risk-free asset with rf = 10%. We can easily find the specialfund F with the solution

v = (2.242,−0.427, 2.728,−0.786, 3.306)′,

w = (0.317,−0.060, 0.386,−0.111, 0.468)′.


Outline

10 Market Equilibrium

11 The Capital Market Line

12 The Pricing Model

13 The Security Market Line

14 Investment Implications (self-learning)

15 Performance Evaluation

16 The CAPM as a Factor Model

17 Project Choice


Market Equilibrium

Outline








17 Project Choice


Market Equilibrium

What will happen with the following assumptions?

Everyone is a mean-variance optimizer.

Everyone agrees on the probabilistic structure of assets; that is,everyone assigns to the returns of assets the same mean values, thesame variances, and the same covariances.

There is a unique risk-free rate of borrowing and lending that isavailable to all, and there are no transactions costs.

From the one-fund theorem we know that

Everyone will purchase a single fund of risky assets, and they may, inaddition, borrow or lend at the risk-free rate.

Since everyone uses the same means, variances, and covariances,everyone will use the same risky fund.

The mix of these two assets, the risky fund and the risk-free asset,will likely vary across individuals according to their individual tastesfor risk by adjusting α.


Market Equilibrium

Definition

The market portfolio is the summation of all assets. The weight of anasset in the market portfolio is equal to the proportion of that asset’s totalcapital value to the total market capital value. These weights are termedcapitalization weights.

In the situation where everyone follows the mean-variance methodologywith the same estimates of parameters, the efficient fund of risky assetswill be the market portfolio!


Market Equilibrium

How can the market portfolio be reached?

The return on an asset depends on both its initial price and its finalprice. The other investors solve the mean-variance portfolio problemusing their common estimates, and they place orders in the market toacquire their portfolios.

If the orders placed do not match what is available, the prices mustchange. The prices of assets under heavy demand will increase; theprices of assets under light demand will decrease. These price changesaffect the estimates of asset returns directly, and hence investors willrecalculate their optimal portfolios.

This process continues until demand exactly matches supply; that is,it continues until there is equilibrium.


The Capital Market Line

Outline








17 Project Choice



Definition

Given that the single efficient fund of risky assets is the market portfolio,the efficient set therefore consists of a single straight line, emanating fromthe risk-free point and passing through the market portfolio. This line,shown in the following figure, is called the capital market line.

r = rf +rM − rfσM

σ.

The slope of the capital market line is

K =rM − rfσM

,

and this value is frequently called the price of risk.



Example (The impatient investor)

Mr Smith is young and impatient. He notes that the risk-free rate is only6% and the market portfolio of risky assets has an expected return of 12%and a standard deviation of 15%. He figures that it would take about 60year for his $1,000 nest egg to increase to $1 million if it earned themarket rate of return. However, he wants that $1 million in 10 years.

Mr. Smith easily determines that he must attain an average rate of returnof about 100% per year to achieve his goal ($1,000×210 = $1,024,000).Correspondingly, his yearly standard deviation according to the capitalmarket line would be the value of σ satisfying

1.0 = 0.06 +0.12− 0.06

0.15σ ⇒ σ = 10.

This corresponds to σ = 1, 000%. So this young man is certainly notguaranteed success (even if he could borrow the amount required to movefar beyond the market on the capital market line).



Example (An oil venture)

Consider an oil drilling venture. The price of a share of this venture is$875. It is expected to yield the equivalent of $1,000 after 1 year, but dueto high uncertainty about how much oil is at the drilling site, the standarddeviation of the return is σ = 40%. Currently the risk-free rate is 10%.The expected rate of return on the market portfolio is 17%. and thestandard deviation of this rate is 12%.

Given the level of σ, r predicted by the capital market line is

r = 0.10 +0.17− 0.10

0.12× 0.40 = 33%.

But the actual expected rate of return is only r = 1, 000/875− 1 = 14%.Therefore the point representing the oil venture lies well below the capitalmarket line (This does not mean that the venture is necessarily a poorone, but it certainly does not, by itself, constitute an efficient portfolio).


The Pricing Model

Outline








17 Project Choice


The Pricing Model

Theorem (The capital asset pricing model (CAPM))

If the market portfolio M is efficient, the expected return r i of any asset isatisfies

r i − rf = βi (rM − rf ) =cov(ri , rM)

σ2M(rM − rf ).

Proof.

For any α consider the portfolio consisting of a portion α invested in asseti and a portion 1− α invested in the market portfolio M. Then we have

rα = αr i + (1− α)rM ,

σα =√α2σ2i + 2α(1− α)cov(ri , rM) + (1− α)2σ2M .

As α varies, these values trace out a curve in the r -σ diagram.


The Pricing Model

contd.

The tangency condition can be translated into the condition that the slopeof the curve is equal to the slope of the capital market line at M.

drαdα

∣∣∣∣α=0

= (r i − rM)|α=0 = r i − rM ,

dσαdα

∣∣∣∣α=0

=ασ2i + (1− 2α)cov(ri , rM) + (α− 1)σ2M

σα

∣∣∣∣α=0

=cov(ri , rM)− σ2M

σM.

We then use the relation

drαdσα

∣∣∣∣∣α=0

=drα/dα

dσα/dα

∣∣∣∣∣α=0

=(r i − rM)σM

cov(ri , rM)− σ2M.


The Pricing Model

contd.

This slope must equal the slope of the capital market line. Hence,

(r i − rM)σMcov(ri , rM)− σ2M

=rM − rfσM

.

We now just solve for r i obtaining the final result

r i = rf +

(rM − rfσ2M

)cov(ri , rM) = rf + βi (rM − rf ).

The proof is complete.


The Pricing Model

Definition

In CAPM, the value βi is referred to as the beta of an asset. The valuer i − rf is termed the expected excess rate of return of asset i ; it is theamount by which the rate of return is expected to exceed the risk-freerate. Likewise, rM − rf is the expected excess rate of return of the marketportfolio.

The CAPM says that the expected excess rate of return of an asset isproportional to the expected excess rate of return of the marketportfolio, and the proportionality factor is β.

Since β is a normalized version of the covariance of the asset with themarket portfolio, the CAPM states that the expected excess rate ofreturn of an asset is directly proportional to its covariance with themarket. It is this covariance that determines its expected excess rateof return.

What if β is zero or even negative?


The Pricing Model

Example (A simple calculation)

We illustrate how simple it is to use the CAPM formula to calculate anexpected rate of return.

Let the risk-free rate be rf = 8%. Suppose the rate of return of themarket has an expected value of 12% and a standard deviation of 15%.

Now consider an asset that has covariance 0.045 with the market. Thenwe find

β =cov(ri , rM)

σ2M=

0.045

0.152= 2.0.

The expected return of the asset is

r = 0.08 + 2.0× (0.12− 0.08) = 0.16 = 16%.


The Pricing Model

Betas of Common Stocks

Aggressive companies or highly leveraged companies are expected tohave high betas.

Conservative companies whose performance is unrelated to thegeneral market behavior are expected to have low betas.

Companies in the same business will have similar, but not identical,beta values.

Beta of a PortfolioA portfolio containing n assets with the weights w1,w2, . . . ,wn should have

r =n∑

i=1

wi ri ⇒ cov(r , rM) = cov

(n∑

i=1

wi ri , rM

)=

n∑i=1

wicov(ri , rM),

which immediately leads to

β =n∑

i=1

wiβi .


The Security Market Line

Outline








17 Project Choice



The CAPM formula can be expressed in graphical form by regarding theformula as linear relationship. This relationship is termed the securitymarket line.

It expresses the risk-reward structure of assets according to the CAPM,and emphasizes that the risk of an asset is a function of its covariancewith the market or, equivalently, a function of its beta.



Systematic Risk

Write the (random) rate of return of asset i as

ri = rf + βi (rM − rf ) + εi .

The CAPM formula tells us several things about εi .

1 Taking the expected value, the CAPM says that E(εi ) = 0.

2 Taking the correlation with rM and using the definition of βi , we find

cov(ri , rM) = cov(rf + βi (rM − rf ) + εi , rM)

= cov(rf , rM) + βicov(rM − rf , rM) + cov(εi , rM).

By the definition of β, cov(εi , rM) = 0, and we can therefore write

σ2i = β2i σ2M + var(εi ).

The first part, β2i σ2M is termed the systematic risk. The second part,

var(εi ), is termed the nonsystematic, idiosyncratic, or specific risk.



Diversification

Systematic risk is the risk associated with the market as a whole andcannot be reduced by diversification because every asset with nonzerobeta contains this risk.

Nonsystematic risk is uncorrelated with the market and can bereduced by diversification.


Investment Implications (self-learning)

Outline








17 Project Choice


Performance Evaluation

Outline








17 Project Choice



Example (ABC fund analysis)

The ABC mutual fund has the 10-year record of rates of return shown inthe column labeled ABC in the following table. We would like to evaluatethis fund’s performance in terms of mean-valiance portfolio theory and theCAPM. Is it a good fund that we could recommend? Can it serve as theone fund for a prudent mean-variance investor?



Example (contd.)

We begin our analysis by computing the three quantities: the average rateof return, the standard deviation of the rate as implied by the 10 samples,and the geometric mean rate of return. These quantities are estimatesbased on the available data.

r̂ =1

n

n∑i=1

ri ,

s2 =1

n − 1

n∑i=1

(ri − r̂

)2,

µ =

[n∏

i=1

(1 + ri )

]1/n− 1

= [(1 + r1)(1 + r2) · · · (1 + rn)]1/n − 1.



Example (contd.)

Next we obtain data on both the market portfolio and the risk-free rate ofreturn over the 10-year period. We use the Standard & Pool’s 500 stockaverage and the 1-year Treasury bill rate, respectively.

We also calculate an estimate of the covariance of the ABC fund with theS&P 500 by using the estimate

cov(r , rM) =1

n − 1

n∑i=1

(ri − r̂

) (rMi − r̂M

).

We then calculate beta from the standard formula,

β =cov(r , rM)

σ2M.

This gives us enough information to carry out an interesting analysis.



Example (contd.)

We write the formula

r̂ − rf = J + β(r̂M − rf

),

where J stands for the Jensen index. According to the CAPM, the valueof J should be zero when true expected returns are used. Therefore, Jmeasures, approximately, how much the performance of ABC has deviatedfrom the theoretical value of zero.



Example (contd.)

In order to measure the efficiency of ABC, we must see where it fallsrelative to the capital market line. Only portfolios on that line are efficient.We do this by writing the formula

r̂ − rf = Sσ,

where S stands for Sharpe index, which is the slope of the line drawnbetween the risk-free point and the ABC point on the r -σ diagram.

For ABC, S = 0.43577, which must be compared with the correspondingvalue for the market represented by the S&P 500. We find the value forthe S&P 500 is S = 0.46669.

We conclude that ABC may be worth holding in a portfolio. By itself it isnot quite efficient, so it would be necessary to supplement this fund withother assets or funds to achieve efficiency.


The CAPM as a Factor Model

Outline








17 Project Choice



Formula (Pricing form of the CAPM)

The price P of an asset with payoff Q is

P =Q

1 + rf + β(rM − rf ).

Proof.

Suppose that an asset is purchased at price P and later sold at price Q.The rate of return is then r = (Q − P)/P. Putting this in the CAPMformula, we have

Q − P

P= rf + β(rM − rf ).

Solving for P we obtain the result.

Here, rf + β(rM − rf ) can be regarded as the risk-adjusted interest ratein comparison with rf .



Example (The price is right)

Gavin Jones is good at math, but his friends tell him that he doesn’talways see the big picture.

Right now, Gavin is thinking about investing in a mutual fund. The fundinvests 10% at the risk-free rate of 7% and the remaining 90% in a widelydiversified portfolio that closely approximates the market portfolio, whichhas an expected rate of return equal to 15%. One share of the mutualfund represents $100 of assets in the fund.

Having just studied the CAPM, Gavin wants to know how much such ashare should cost. With Q and β, the price of a share of the fund is

P =Q

1 + rf + β(rM − rf )

=$114.20

1 + 0.07 + 0.90× (0.15− 0.07)= $100.



Example (The oil venture)

Consider again the possibility of investing in a share of a certain oil well.

The expected payoff is $1,000 and the standard deviation of return is arelatively high 40%. The beta of the asset is β = 0.6, which is relativelylow because, although the uncertainty in return due to oil prices iscorrelated with the market portfolio, the uncertainty associated withexploration is not. The risk-free rate is rf = 10%, and the expected returnon the market portfolio is rM = 17%. What is the value of this share ofthe oil venture, based on CAPM?

P =Q

1 + rf + β(rM − rf )=

$1000

1 + 0.10 + 0.60× (0.17− 0.10)= $876.

The venture may be quite risky in the traditional sense of having a highstandard deviation associated with its return. Nevertheless, it is fairlypriced because of the relatively low beta.



Formula (Certainty equivalent pricing formula)

The certainty equivalent of payoff Q of an asset is defined as the term inthe brackets of the following pricing formula,

P =1

1 + rf

[Q − cov(Q, rM)(rM − rf )

σ2M

].

Proof.

By the definition of beta and the pricing form of the CAPM, we have

β =cov[(Q/P − 1), rM ]

σ2M⇒ β =

cov(Q, rM)

Pσ2M,

1 =Q

P(1 + rf ) + cov(Q, rM)(rM − rf )/σ2M.

Solving for P, we obtain the result.



The certainty equivalent form shows clearly that the pricing formula islinear because both terms in the brackets depend linearly on Q. Therefore,

P1 =Q1

1 + rf + β1(rM − rf ),

P2 =Q2

1 + rf + β2(rM − rf ),

⇒ P1 + P2 =Q1 + Q2

1 + rf + β1+2(rM − rf ),

where β1+2 is the beta of a new asset, which is the sum of assets 1 and 2.

Example (Gavin tries again)

Gavin Jones decides to use the certainty equivalent form of the pricingequation to calculate the share price of the mutual fund. In this case henotes that cov(Q, rM) = 90σ2M , where Q is the value of the fund after 1year. Hence,

P =1

1 + rf


σ2M

]=

114.20− 90× 0.08

1 + 0.07= $100.


Project Choice

Outline








17 Project Choice


Project Choice

Two Criteria

The net present value (NPV) of a project based on the certaintyequivalent form of the CAPM is

NPV = −P +1

1 + rf


σ2M

].

The firm selects the group of projects that maximize NPV.

If investors base their investment decisions on a mean-variancecriterion, they want an individual firm to operate so as to push theefficient frontier, of the entire universe of assets, as far upward andleftward as possible.

Theorem (Harmony theorem)

If a firm does not maximize NPV, then the efficient frontier can beexpanded.


Project Choice

Proof.

Suppose firm i is planning to operate in a manner that leads to an NPV of∆, which does not maximize the NPV available. The initial cost of theproject is P0

i . Investors pay Pi = P0i + ∆; and plan to receive the reward

Qi , for which ri = (Qi − Pi )/Pi .

We assume that firm i has a very small weight in the market portfolio ofrisky assets and that projects have positive initial cost.

The current rate of return ri satisfies the CAPM relation

r i − rf = βi (rM − rf )⇔ 0 = −Pi +1

1 + rf

[Q i −

cov(Qi , rM)(rM − rf )

σ2M

].

Suppose now that the firm could operate to increase the present value byusing a project with cost P0′

i and reward Q ′i ; Investors pay ∆ to buy thecompany and pay the operating cost P0′

i .


Project Choice

contd.

The total P ′i = P0′i + ∆ satisfies

−P ′i +1

1 + rf

[Q′i −

cov(Q ′i , rM)(rM − rf )

σ2M

]> 0,

which, since P ′i > 0, implies that

r ′i − rf −cov(r ′i , rM)(rM − rf )

σ2M= r ′i − β′i (rM − rf )− rf > 0.

Now consider the portfolio with return rα = rM + αr ′i − αri , where α is theoriginal weight of the firm i in the market portfolio. We want to show thatthis portfolio lies above the old efficient frontier. For small α > 0, we have

tan θα =rα − rfσα

⇒ d tan θαdα

=1

σα

drαdα− (rα − rf )

σ2α

dσαdα

.


Project Choice

contd.

Usingdrαdα

∣∣∣∣α=0

= r ′i − r i ,dσαdα

∣∣∣∣α=0

=σMi ′ − σMi

σM,

r0 = rM and σ0 = σM , we find

d tan θαdα

∣∣∣∣α=0

=r ′i − r iσM

− (rM − rf )

σ2M

(σMi ′ − σMi )

σM

=1

σM

[r ′i − β′i (rM − rf )

]− 1

σM[r i − βi (rM − rf )]

=1

σM

[r ′i − β′i (rM − rf )− rf

]> 0.

Since α is small, this means that tan θα > tan θ0. Hence the efficientfrontier is larger than it was originally.

The proof is complete.


investment science - part ii: single-period random cash flows

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