Chapter 6 investments efficient diversification

- The key determinant of portfolio risk is the extent to which the returns on the two assets tend to vary either in tandem or in opposition.. - Portfolio risk depends on the covarian

Chapter 6 Efficient Diversification

6.1 Diversification and Portfolio Risk

 Market risk

- The risk that has to do with general economic

conditions.

- The risk that remains even after diversification.

- Systematic risk or non-diversifiable risk.

 Firm-specific risk

- Diversifying into many more securities reduce

exposure to firm-specific factors.

- Unique risk, nonsystematic risk, or diversifiable risk

Figure 6.1 Portfolio risk as a function of the number of stocks in the portfolio

Figure 6.2 Portfolio risk decreases as

diversification increases.

6.2 Asset Allocation With Two Risky Assets

- Need to understand how the uncertainties of asset

returns interact when we form a risky portfolio

- The key determinant of portfolio risk is the extent to

which the returns on the two assets tend to vary

either in tandem or in opposition

- Portfolio risk depends on the covariance between

the returns of the assets in the portfolio

Asset Allocation With Two Risky Assets

= W1 + W2 W1 = Proportion of funds in Security 1 W2 = Proportion of funds in Security 2

= Expected return on Security 1

= Expected return on Security 2

Two-Security Portfolio: Return

in securities #

n

; r W )

r E(

n

1 i

i i

=

E(rp) = W1r1 + W2r2 W1 =

0.6 0.4 9.28%

11.97%

r1 r2

Combinations of risky assets

When Stock 1 has a return <

E[r1] it is likely that Stock 2 has

a return > E[r2] so that rp that contains stocks 1 and 2 remains close to E[rp]

What statistics measure the tendency for r1 to be below expected when r2 is above expected?

Covariance and Correlation

n = # securities

Portfolio Variance and Standard Deviation

Variance of a Two Stock Portfolio:

Covariance: A measure of the extent to which the returns tend to vary with each other, that is, to co-vary

(The covariance between any stock such as Stock 1 and itself is simply the variance of Stock 1.)

∑∑

= =

= Q

1 I

Q 1 J

J I J

in stocks of

number total

The Q

ly respective J

and I stock in

invested portfolio

total the

of Percentage W

I Stock of

returns the

of Covariance )

r , Cov(rI J =

) r , r ( Cov )

r , Cov(r

&

σ ) r , (r Cov then

J I

2 2

2 2 2

1 2

1

2 1

2 1

2

),(

σ

2 1

n

) r (r

) r

(r )

r ,

ns observatio of

# n

2 stock for

return expected

or average r

1 stock for

return expected

or average r

2 1

=

=

=

Covariance and correlation

The problem with covariance

- Covariance does not tell us the intensity of the comovement of the stock returns, only the direction.

and calculate the correlation coefficient which will tell us not only the direction but provides a scale to estimate the degree to which the stocks move together.

Measuring the correlation coefficient

Standardized covariance is called

the correlation coefficient or ρ

Correlation coefficient can range from values of -1 to +1.

Values of -1 indicate perfect negative correlation.

Values of +1 indicate perfect positive correlation.

Values of 0 indicate that the returns on assets are unrelated

2 1

2

1 12

σ σ

) r ,

Cov(r ρ

×

=

ρ and diversification in a 2 stock portfolio

ρ is always in the range inclusive.

What does ρ 12 = +1.0 imply?

What does The two are perfectly ρ 12 = -1.0 imply? positively correlated Means?

ρ and diversification in a 2 stock portfolio

What does -1 < ρ 12 < 1 imply?

If -1 < ρ 12 < 1, then

σp2 = W12σ12 + W22σ22 + 2W1W2 Cov(r1r2) And since Cov(r1r2) = ρ12σ 1 σ 2

There are some diversification benefits from combining stocks 1 and 2 into a portfolio.

The effects of correlation &

covariance on diversification

Asset A Asset B Portfolio AB

The effects of correlation &

covariance on diversification

Asset C Asset C Portfolio CD

Most of the diversifiable risk eliminated at 25 or so stocks

The power of diversification

Two-Risky Assets Portfolios

rp = W1r1 +W2r2

E(rp) = W1E(r1) + W2E(r2)

σ p2 = W12 σ 12 + W22 σ 22 + 2W1W2 Cov(r1r2)

= W12 σ 12 + W22 σ 22 + 2W1 σ 1W2 σ 2 ρ 12

Using scenario analysis with probabilities, the covariance can be calculated wit

h the following formula:

Linear Function

Not Linear Function

( , ) S ( ) ( ) ( )

Cov r r =∑p i r i −r r i −r 

Two-Security Portfolio Risk

0.36(0.15265) + 0.1115019 = variance of the portfolio 33.39%

Q 1

2

p [W W Cov(I, J)]

σ

ρ = 0

Summary: Portfolio Risk/Return Two Security Portfolio

Amount of risk reduction depends critically on correlations or covariances

Adding securities with correlations _ will result in risk reduction.

If risk is reduced by more than expected return, what happens to the return per u nit of risk (the Sharpe ratio)? < 1

Minimum Variance Combinations

-1< ρ < +1

- Cov(r1r2) W1 =

Minimum Variance Combinations -1< ρ < +1

2E(r2) = 14 = 20 Stk 2 Stk 1 E(r1) = 10 σ = 15 12 = 2

Cov(r1r2) =

ρ1,2σ1σ2

2 2

2

p (0.6733 ) (0.15 ) (0.3267 ) (0.2 ) 2 (0.6733) (0.3267) (0.2) (0.15) (0.2)

%

p = 0 01711 2 = 13 08

σ

Minimum Variance Combination with ρ = -.3

-.3

1

Cov(r1r2) = ρ1,2σ1σ2

Minimum Variance Combination with ρ = -.3

ρ12 = 2 E(rp) = 11.31%

1/2 2

2 2

2

p (0.6087 ) (0.15 ) (0.3913 ) (0.2 ) 2 (0.6087) (0.3913) (- 0.3) (0.15) (0.2) σ

p = 0 0102 1 2 = 10 09

σ

Expected

Return

The minimum-variance frontier of risky assets

Efficient Frontier is the best diversified set of investments with the highest returns

Global minimum variance portfolio

Efficient frontier

Individual assets

Minimum variance frontier

St Dev.

Found by forming portfolios of securities with the lowest

covariances at a given E(r) level.

Find the mean-variance efficient portfolios!

The EF and asset allocation

Efficient frontier

St Dev.

20% Stocks 80% Bonds

100% Stocks

EF including international &

alternative investments

80% Stocks 20% Bonds 60% Stocks

40% Bonds 40% Stocks 60% Bonds

100% Stocks

Expected

Return

Extending Concepts to All Securities

 Consider all possible combinations of securities, with all possible different weightings and keep track of combinations that provide more return for less risk or the least risk for a given level of return and graph the result

 The set of portfolios that provide the optimal trade-offs are

described as the efficient frontier

 The efficient frontier portfolios are dominant or the best diversified possible combinations All investors should want a portfolio on the efficient frontier

… Until we add the riskless asset

6.3 The Optimal Risky Portfolio With A Risk-Free Asse

t 6.4 Efficient Diversification With Many Risky Assets

Including Riskless Investments

 The optimal combination becomes linear

 A single combination of risky and riskless assets

will dominate

CAL (Global minimum variance) G

oThe optimal CAL is

called the Capital Market Line or CML

oThe CML dominates

the EF

6-35

Dominant CAL with a Risk-Free Investment (F)

 CAL(P) = Capital Market Line or CML dominates

other lines because it has the largest slope.

 Slope = (E(rp) - rf) / σp

(CML maximizes the slope or the return per unit of risk

or it equivalently maximizes the Sharpe ratio.)

 Regardless of risk preferences, some combinations

of P & F dominate.

Efficient Frontier

portfolio P and the risk free asset F, but they choose different proportions of each.

σP&F E(rP&F)

Practical Implications

 The analyst or planner should identify what they believe

will be the best performing well diversified portfolio, call it P

 This portfolio will serve as the starting point for all their clients

 The planner will then change the asset allocation between the risky portfolio and “near cash” investments according to risk tolerance of client

 The risky portfolio P may have to be adjusted for

individual clients for tax and liquidity concerns if relevant and for the client’s opinions

6.5 A Single Index Asset Market

Individual securities

 We have learned that investors should diversify

 What do we call the risk that cannot be diversified away, i.e., the risk that remains when the stock is put into a portfolio?

 How do we measure a stock’s systematic risk?

Systematic risk

Systematic risk

Systematic risk arises from events that effect the entire economy such as a change

in interest rates or GDP or a financial crisis such as occurred in 2007and 2008

If a well diversified portfolio has no unsystematic risk, then any risk that remains must be systematic

That is, the variation in returns of a well diversified portfolio must be due to chang

es in systematic factors

Δ interest rates,

Δ GDP,

Δ consumer spending, etc.

β

Systematic Factors

Single Factor Model

Ri = E(Ri) + ßiM + ei

Ri = Actual excess return = ri – rf E(Ri) = expected excess return Two sources of Uncertainty

M ßi ei

= some systematic factor or proxy; in this case M is unanticipated movement in a well diversified broad market index like the S&P500

= sensitivity of a securities’ particular return to the factor (cyclical stocks vs defensive stocks)

= unanticipated firm specific events, average out to 0.

Single Index Model Parameter Estimation

or Index Risk Premium

= A stock’s expected return beyond that induced by the market index  “Positive alpha is attractive.” ßi(rm - rf) = the component of excess return due to

movements in the market index

ei = firm specific component of excess return that is not

due to market movements αi

(ri − r f ) = α i + β i (rm − r f ) + ei