1 Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation Fina2802: Investments and Portfolio...

1Fin 2802, Spring 10 - TangChapter 6: Asset Allocation

Fina2802: Investments and Portfolio Analysis

Spring, 2010Dragon Tang

Fina2802: Investments and Portfolio Analysis

Spring, 2010Dragon Tang

Lecture 9

Capital Allocation

February 9, 2010

Readings: Chapter 6

Practice Problem Sets: 1-5,12-14,26-28


Asset AllocationAsset Allocation

Objectives:

• Characterize the risk and return of portfolios containing risky and risk-free assets.

1. Evaluate the performance of a passive strategy.

3

Portfolio SelectionPortfolio Selection

1. Asset allocation

2. Security selection

Fin 2802, Spring 08 - TangChapter 6: Asset Allocation


4

Asset AllocationAsset Allocation

• John Bogle: “Asset allocation accounts for 94% of the differences in pension fund performance”

• Identify investment opportunities (risk-return combinations)

• Choose the optimal combination according to investor’s risk attitude




Portfolios: Basic Asset AllocationPortfolios: Basic Asset Allocation

The complete portfolio is composed of:

• The risk-free asset: Risk can be reduced by allocating more to the risk-free asset

• The risky portfolio: Composition of risky portfolio does not change (market portfolio)

This is called Two-Fund Separation Theorem.

The proportions depend on your risk aversion.

6

Risk-free Investment Risk-free Investment

0.00

0.20

0.40

0.60

0.80

1.00

-5% 0% 5% 10% 15%



7

Stock Returns Are Uncertain Stock Returns Are Uncertain

Example: Risky investment with ten possible rates of return

0.00

0.20

0.40

0.60

0.80

1.00

-40% -20% 0% 20% 40%




Risk and Risk PremiumRisk and Risk Premium

• Risk-free rate: determined by demand/supply and intermediaries (such as Fed)

• Risk premium (=Risky return –Risk-free return)Example

The expected return on the S&P500 is 9%The return on a 1-month T-bill is 3%The risk premium is 6% (9%-3%)

• Risk aversionE(rP) – rf = ½ A σp

2



Complete Portfolio Expected ReturnComplete Portfolio Expected Return

Example: Let the expected return on the risky portfolio, E(rP), be 15%, the return on the risk-free asset, rf, be 7%. What is the return on the complete portfolio if all of the funds are invested in the risk-free asset? What is the risk premium?

7%0

What is the return on the portfolio if all of the funds are invested in the risky portfolio?

15%8%


Complete Portfolio Expected ReturnComplete Portfolio Expected Return

Example: Let the expected return on the risky portfolio, E(rP), be 15%, the return on the risk-free asset, rf, be 7%. What is the return on the complete portfolio if 50% of the funds are invested in the risky portfolio and 50% in the risk-free asset? What is the risk premium?

0.5*15%+0.5*7%=11%

4%


Complete Portfolio Risk PremiumComplete Portfolio Risk Premium

fffPfC rryrrEyrrE )1(

assetrisky on the premiumrisk

portfolio complete on the premiumrisk

assetrisky in the invested funds offraction

fP

fC

rrE

rrE

y

In general:Equal to 0


Portfolio Standard DeviationPortfolio Standard Deviation

0 if

121 22222

f

fff

rPC

rPPrrPC

y

yyyy

where

c - standard deviation of the complete portfolioP - standard deviation of the risky portfoliorf - standard deviation of the risk-free rate y - weight of the complete portfolio invested in the risky asset


Portfolio Standard DeviationPortfolio Standard Deviation

Example: Let the standard deviation on the risky portfolio, P, be 22%. What is the standard deviation of the complete portfolio if 50% of the funds are invested in the risky portfolio and 50% in the risk-free asset?

22%*0.5=11%


Capital Allocation LineCapital Allocation Line

We know that given a risky asset (p) and a risk-free asset, the expected return and standard deviation of any complete portfolio (c) satisfy the following relationship:

c portfolio completeevery for )()(

))(()(

p

fp

c

fc

pc

fpfc

rrErrE

y

rrEyrrE

Where y is the fraction of the portfolio invested in the risky asset


Capital Allocation LineCapital Allocation Line

Risk Tolerance and Asset Allocation: •More risk averse - closer to point F

•Less risk averse - closer to P


Slope of the CALSlope of the CAL

S

E r rP f

P

S is the increase in expected return per unit of additional standard deviation

S is the reward-to-variability ratio or Sharpe Ratio


Example: Let the expected return on the risky portfolio, E(rP),

be 15%, the return on the risk-free asset, rf, be 7% and the

standard deviation on the risky portfolio, P, be 22%. What is

the slope of the CAL for the complete portfolio?

S = (15%-7%)/22% = 8/22

Slope of the CALSlope of the CAL

18

Historical Risk-Return Trade-offHistorical Risk-Return Trade-off

Risk Real Sharp

Asset Class Prem.(%) Retn(%) Ratio

Sm Stk 13.9 14.6 0.35

Lg Stk 9.3 8.9 0.45

LT Gov 1.9 2.6 0.23

T-Bills --- 0.7 ---



19

Measuring Risk-Return Trade-offMeasuring Risk-Return Trade-off

• Mean-Variance Plot

0

2

4

6

8

10

12

14

16

18

20

0 5 10 15 20 25 30 35 40 45

Standard Deviation ()

Exp

ecte

d R

etur

n (E

(r))




BorrowingBorrowing

S

E r rP B

P

where rB is the borrowing rate

y > 1

Usually borrowing rate > lending rate


Borrowing ExampleBorrowing Example

Example: Let the expected return on the risky portfolio, E(rP), be 15%, the return on the risk-free asset, rf, be 7%, the borrowing rate, rB, be 9% and the standard deviation on the risky portfolio, P, be 22%. What is the slope of the CAL for the complete portfolio for points where y > 1?

S=(15%-9%)/22%=6/22

Note: For y 1, the slope is as indicated above if the lending rate is rf.


Investment Opportunity Set with Differential Borrowing and Lending Rates

Investment Opportunity Set with Differential Borrowing and Lending Rates


Passive StrategiesPassive Strategies

•Assumes securities are fairly priced

•Avoids cost of security analysis

•Indexing - value-weighted portfolio

•Assume that the search for mispriced securities (performed by active strategies) keeps prices fair


Capital Market Line (CML)Capital Market Line (CML)

SPECIAL CASE OF CAL (I.e., P=MKT)

The line provided by one-month T-bills and a broad index of common stocks (e.g. S&P500)

Consequence of a passive investment strategy based on stocks and T-bills

25

Which Portfolio to Choose?Which Portfolio to Choose?

E(r)

E(Rm) = 12%

rf = 3%

20%0

M

F

S=0.45P1?

P3?

P2?



26

Key Determinant of Asset Allocation:Attitude towards Risk

Key Determinant of Asset Allocation:Attitude towards Risk

• Risk Preference – Risk averse

» Require compensation for taking risk

– Risk neutral

» No requirement of risk premium

– Risk loving

» Pay to take risk

• Utility Values: A is risk aversion parameter

25.0)( ArEU



27

Utility FunctionUtility Function

U = E ( r ) – 1/2 A 2

Where

U = utility

E ( r ) = expected return on the asset or portfolio

A = coefficient of risk aversion

= variance of returns



28

Utility Scores of Alternative Portfolios for Investors with Varying Risk Aversion

Utility Scores of Alternative Portfolios for Investors with Varying Risk Aversion



29

The Trade-off Between Risk and Returns of a Potential Investment Portfolio

The Trade-off Between Risk and Returns of a Potential Investment Portfolio



30

Utility Values of Possible Portfolios for an Investor with Risk Aversion, A = 4

Utility Values of Possible Portfolios for an Investor with Risk Aversion, A = 4



31

Figure 6.2 The Indifference Curve Figure 6.2 The Indifference Curve



32

Utility Indifference CurvesUtility Indifference CurvesUtility Indifference Curves

0

2

4

6

8

10

12

14

16

18

20

0 5 10 15 20 25 30 35 40 45

Standard Deviation ()

Exp

ecte

d R

etur

n (E

(r))

A=5

A=2

U1

U2

Increasing utility



33

Risk Aversion and Asset AllocationRisk Aversion and Asset Allocation

• Greater levels of risk aversion lead to larger proportions of the risk free rate.

• Lower levels of risk aversion lead to larger proportions of the portfolio of risky assets.

• Willingness to accept high levels of risk for high levels of returns would result in leveraged combinations.



34

Investor’s Willingness to Pay for Catastrophe Insurance Investor’s Willingness to Pay for Catastrophe Insurance



35

Spread Between 3-Month CD and T-bill Rates Spread Between 3-Month CD and T-bill Rates



36

Find the Optimal AllocationFind the Optimal Allocation

• Solve the maximization problem:

• Two approaches:

1. Try different w1

2. Use calculus:

• Solution:

211

2

)(5.0)(

5.0)(Max

mfmf wArrwr

ArEU

01

21 mfm Awrr

dwdU

21m

fm

A

rRw



37

Asset Allocation Rules:Asset Allocation Rules:

• When rm-rf increases, w1 increases

• When A increases, w1 decreases

• When m increases, w1 decreases

• W1 is constant when all three are fixed



38

Illustration of SolutionIllustration of Solution

E(r)

E(Rm) = 12%

rf = 3%

20%0

M

F

S=0.45P1!

P3

P2

Utility indifference curves (A=4)

If A=4 then w1=0.56

11.2%

8%



39

Which Portfolio to Choose?Which Portfolio to Choose?

• For Jack, risk aversion A = 2, the optimal choice is 112.5% (of total capital) in the market, financed by selling short 12.5% (of total capital) in T-bills

• For Jill, risk aversion A = 5, the optimal choice is 45% in the market and 55% in T-bills.

• The weight in the market:

21m

fm

A

rRw



40

Utility ComparisonUtility Comparison

-20

-15

-10

-5

0

5

10

0% 50% 100% 150% 200% 250%

Weight in Market

Uti

lity

Va

lue

Jack

Jill

45% 113%



41

Utility Levels for Positions in Risky Assets for an Investor with Risk Aversion A = 4

Utility Levels for Positions in Risky Assets for an Investor with Risk Aversion A = 4



42

Utility as a Function of Allocation to the Risky Asset, yUtility as a Function of Allocation to the Risky Asset, y



43

Finding the Optimal Complete Portfolio Using Indifference Curves Finding the Optimal Complete Portfolio Using Indifference Curves



44

Expected Returns on Four Indifference Curves and the CALExpected Returns on Four Indifference Curves and the CAL



45

Average Annual Return on Stocks and 1-Month T-bills;

S. Dev. and Reward to Variability of Stocks Over Time Average Annual Return on Stocks and 1-Month T-bills;

S. Dev. and Reward to Variability of Stocks Over Time




SummarySummary

•Capital allocation line (CAL) All combinations of the risky and risk-free asset Slope is the reward-to-variability ratio

•Capital market line (CML) Passive strategy Market index portfolio as the risky asset

•Risk aversion determines position on the capital allocation line

•Next: Market Efficiency

1 Fin 2802, Spring 10 - Tang Chapter 6: Asset Allocation Fina2802: Investments and Portfolio...

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