Corporate finance chapter 012 porfolio selection and diversification

Chapter 12 Contents• 12.1 The process of personal portfolio selection • 12.2 The trade-off between expected return and risk • 12.3 Efficient diversification with many risky assets...

Chapter 12 Contents

• 12.1 The process of personal portfolio

selection

• 12.2 The trade-off between expected

return and risk

• 12.3 Efficient diversification with many

risky assets

• To understand the process of personal

portfolio selection in theory and practice

Security Prices

10 100 1000 10000 100000

Security Prices

100 1000 10000 100000

Probability of Future Price

Probabilistic Stock Price Changes Over Time

Probabilistic Bond Price Changes over Time

Mode =104

Mode =106

Median=104 Mean =104 Median=111 Mean = 113

Two Years Out

Mode = 122

Mode = 135

Median=

126 Mean = 128

Median=

165 Mean = 182

Mode =503

Mode =1,102

Median=650 Mean =739 Median=5,460 Mean =12,151

Value of Central Tendency Statistics for the LogNormal

mode The most probable price

median 50% of prices are equal or lower that this

mean The expected or average price

Deaths Per Thousand M & F

Combining the Riskless Asset and a Single Risky Asset

– The expected return of the portfolio is the

weighted average of the component returns

µ p = W 1* µ 1 + (1- W 1 ) * µ 2

Combining the Riskless Asset and a Single Risky Asset

– The volatility of the portfolio is not quite as

simple:

σ p = (( W 1* σ 1) 2 + 2 W 1* σ 1* W 2* σ 2 + ( W 2* σ 2) 2 ) 1/2

Combining the Riskless Asset and a Single Risky Asset

portfolio, namely that security 2 is riskless, so

Combining the Riskless Asset and a Single Risky Asset

A Portfolio of a Risky and a Riskless Security

Capital Market Line

Long risky and short risk-free

Long both risky and risk-free

100%

Risky

100%

Risk-less

Mutual Fund Average % Total Returns

14.81 30.40 15.87 14.15 16.53 16.96

To obtain a 20% Return

• You settle on a 20% return, and decide

not to pursue on the computational issue

= (0.20 - 0.05)/(0.15 - 0.05) = 150%

To obtain a 20% Return

• Assume that your manage a $50,000,000

portfolio

• A W1 of 1.5 or 150% means you invest

(go long) $75,000,000, and borrow (short)

$25,000,000 to finance the difference

• Borrowing at the risk-free rate is moot

Portfolio of Two Risky Assets

• Recall from statistics, that two random

variables, such as two security returns, may be combined to form a new random variable

• A reasonable assumption for returns on

different securities is the linear model:

1 with

2 2 1

r p

Equations for Two Shares

• The sum of the weights w1 and w2 being

1 is not necessary for the validity of the following equations, for portfolios it

happens to be true

• The expected return on the portfolio is

the sum of its weighted expectations

2 2

1

µ p = w + w

Equations for Two Shares

• Ideally, we would like to have a similar

result for risk

– Later we discover a measure of risk with this

property, but for standard deviation:

(wrong)

2 2

1

σ p = w + w

2 2

2 2 2

, 1 2

1 2

1

2 1

2 1

• There is a mnemonic that will help you

remember the volatility equations for two

or more securities

• To obtain the formula, move through

each cell in the table, multiplying it by

the row heading by the column heading, and summing

Variance with 2 Securities

W1*Sig1 W2*Sig2 W1*Sig1 1 Rho(1,2)

W2*Sig2 Rho(2,1) 1

2 , 1 2

1 2

1

2 2

2 2

2 1

2 1

σ p = w + w + w w

Variance with 3 Securities

W1*Sig1 W2*Sig2 W3*Sig3

3 , 2 3 2 3 2 3

, 1 3 1 3 1

2 , 1 2 1 2 1

2 3

2 3

2 2

2 2

2 1

2 1 2

2 2

2

ρ σ σ ρ

σ σ

ρ σ σ σ

σ σ

σ

w w w

w

w w w

+ +

=

Correlated Common Stock

• The next slide shows statistics of two

common stock with these statistics:

– mean return 1 = 0.15 – mean return 2 = 0.10 – standard deviation 1 = 0.20 – standard deviation 2 = 0.25 – correlation of returns = 0.90

– initial price 1 = $57.25 – Initial price 2 = $72.625

2-Shares: Is One "Better?"

Portfolio of Two Securities

Efficient

optimalMinimumVariance

Sub-Fragments of the Output

Table

Data For two securities

This data has been constructed

to produce the mean-varience paradox mu_1 15.00%

-0.30 1.30 0.2723 0.0850 -0.20 1.20 0.2646 0.0900 -0.10 1.10 0.2571 0.0950

0.10 0.90 0.2432 0.1050 0.20 0.80 0.2366 0.1100 0.30 0.70 0.2305 0.1150

Sample of the Excel Formulae

=SQRT(w_1^2*sig_1^2 + 2*w_1*w_2*sig_1*sig_2*rho + w_2^2*sig_2^2)

=w_1*mu_1 + w_2*mu_2

Formulae for Minimum

Variance Portfolio

* 1

2 2 2

1 2

, 1

2 1

2 1

2 , 1

2 1

* 2

2 2 2

1 2

, 1

2 1

2 1

2 , 1

2 2

* 1

1

2

2

w w

σ ρ

σ

σ σ

ρ σ

σ σ

σ ρ

σ

σ σ

ρ σ

Formulae for Tangent

tan

2

3 2

tan

1

2 2

2 tan

2 1 2 , 1 2

1

2 1 2

2 1 2 , 1 2

2 2 1

tan

1

1 2

25 0

* 10 0 25

0

* 20 0

* 90 0

* 05 0 10 0 20

0

* 05 0

25 0

* 20 0

* 90 0

* 05 0 25

0

* 10 0 1

−

=

=

+ +

− +

r r

r r

r

r w

f f

f f

f f

σ µ

σ σ ρ µ

µ σ

µ

σ σ ρ µ

σ µ

Example: What’s the Best

Return given a 10% SD?

1261

0 05

0 10

0 2409

0

05 0 2333

0

2409

0

90 0

* 25 0

* 2 0

* 3

5 3

8 2 25

.

0 3

5 20

.

0 3

8

2

2333

0

10

0 3

5 15

.

0 3

8

tan tan tan

2

2 2

2 2

tan

2 , 1 2 1

tan 2

tan 1

2 2

2 tan 2

2 1

2 tan 1

tan 1 tan

= +

−

= +

w w

w w

w w

σ σ

σ σ

µ

µ

µ µ

µ

Achieving the Target

Expected Return (2): Weights

• Assume that the investment criterion is to

1 05

0 2333

0

05 0 30

0

1

1

1 1

=

f tangent

f criterion

f tangent

criterion

r

r w

w r

w

µ µ

µ µ

Achieving the Target

Expected Return (2):Volatility

• Now determine the volatility associated

with this portfolio

• This is the volatility of the portfolio we

seek

3285

0 2409

0

* 3636

1

σ

Achieving the Target

Expected Return (2): Portfolio Weights