fi8000 valuation of financial assets

Fi8000Fi8000Valuation ofValuation of

Financial AssetsFinancial Assets

Spring Semester 2010Spring Semester 2010

Dr. Isabel TkatchDr. Isabel TkatchAssistant Professor of FinanceAssistant Professor of Finance

TodayToday

☺Portfolio TheoryPortfolio Theory

☺ The Mean-Variance CriterionThe Mean-Variance Criterion

☺ Capital AllocationCapital Allocation

☺ The Mathematics of Portfolio TheoryThe Mathematics of Portfolio Theory

Nation’s Financial Industry Nation’s Financial Industry Gripped by FearGripped by Fear

NY Time, September 15, 2008NY Time, September 15, 2008

By BEN WHITE and JENNY ANDERSONBy BEN WHITE and JENNY ANDERSON

‘‘Fear and greed are the stuff Fear and greed are the stuff that Wall Street is made ofthat Wall Street is made of.’.’

The Mean-Variance CriterionThe Mean-Variance Criterion(M-V or (M-V or μμ--σσ criterion) criterion)

STD(R) – “fear”

E(R) -“greed” ☺

☺

Capital Allocation - DataCapital Allocation - Data

There are three (risky) assets and one risk-free There are three (risky) assets and one risk-free asset in the market. The risk-free rate is asset in the market. The risk-free rate is rf = 1%,rf = 1%, and the distribution of returns of risky assets is and the distribution of returns of risky assets is normal with the following parametersnormal with the following parameters

AssetAsset AA BB CC

Expected Return Expected Return 5.6%5.6% 4.2%4.2% 1.7%1.7%

Standard Deviation Standard Deviation of the Returnof the Return 2.5%2.5% 5.0%5.0% 2.1%2.1%

Capital Allocation:Capital Allocation: n mutually exclusive assets n mutually exclusive assets

State all the possible investments.State all the possible investments.

Assuming you can use the Mean-Variance Assuming you can use the Mean-Variance (M-V) rule, which investments are M-V (M-V) rule, which investments are M-V efficient (i.e. which assets can not be thrown efficient (i.e. which assets can not be thrown out of the set of desirable investments by a out of the set of desirable investments by a risk-averse investor who uses the M-V rule)?risk-averse investor who uses the M-V rule)?

Present your results on the Present your results on the μμ--σσ (mean – (mean – standard-deviation) plane.standard-deviation) plane.

The Mean-Variance CriterionThe Mean-Variance Criterion(M-V or (M-V or μμ--σσ criterion) criterion)

Let A and B be two (risky) assets. All risk-Let A and B be two (risky) assets. All risk-averse investors prefer asset A to B ifaverse investors prefer asset A to B if

{ { μμA A ≥ ≥ μμBB and and σσAA < < σσB B }}

or ifor if

{ { μμA A > > μμBB and and σσAA ≤ ≤ σσBB } }

Note that these rules apply only when we assume that the Note that these rules apply only when we assume that the distribution of returns is normal.distribution of returns is normal.

The Expected Return andThe Expected Return andthe STD of Return (the STD of Return (μμ--σσ plane) plane)

0.0%

2.0%

4.0%

6.0%

8.0%

0.0% 2.0% 4.0% 6.0% 8.0%

STD(R)

E(R)

rfC

A

B

Capital Allocation:Capital Allocation: n mutually exclusive assets n mutually exclusive assets

The The investment opportunity setinvestment opportunity set::

{rf, A, B, C}{rf, A, B, C}

The The Mean-Variance (M-V or Mean-Variance (M-V or μμ--σσ ) efficient ) efficient investment setinvestment set::

{rf, A, C}{rf, A, C}

Note that investment B is not in the efficient set since investment Note that investment B is not in the efficient set since investment A dominates it (one dominant investment is enough).A dominates it (one dominant investment is enough).

Capital Allocation:Capital Allocation:One Risky Asset (A) and One Risk-free AssetOne Risky Asset (A) and One Risk-free Asset

State all the possible investments – how State all the possible investments – how many possible investments are there?many possible investments are there?

Assuming you can use the Mean-Variance Assuming you can use the Mean-Variance (M-V) rule, which investments are M-V (M-V) rule, which investments are M-V efficient?efficient?


The Expected Return and STD of Return The Expected Return and STD of Return of the Portfolio of the Portfolio

αα = the proportion invested in the risky asset A = the proportion invested in the risky asset Ap = the portfolio with p = the portfolio with αα invested in the risky asset Ainvested in the risky asset A and (1- and (1- αα) ) invested in the risk-free asset invested in the risk-free asset rfrf

RRpp = the return of portfolio p = the return of portfolio p

μμpp = the expected return of portfolio p= the expected return of portfolio p

σσpp = the standard deviation of return of portfolio p= the standard deviation of return of portfolio p

RRpp = = αα··RRAA + (1- + (1-αα)·rf)·rf

μμpp = E[ = E[ αα··RRAA + (1- + (1-αα)·rf ] = )·rf ] = αα··μμAA + (1- + (1-αα)·rf)·rf

σσ22pp = V[ = V[ αα··RRAA + (1- + (1-αα)·rf ] = ()·rf ] = (αα··σσAA))2 2 Or Or σσpp = = αα··σσAA

Capital Allocation:Capital Allocation:One Risky Asset and One Risk-free AssetOne Risky Asset and One Risk-free Asset

The investment opportunity set:The investment opportunity set:

{ all portfolios with proportion { all portfolios with proportion αα invested in A invested in A and (1-and (1-αα) invested in the risk-free asset rf }) invested in the risk-free asset rf }

The Mean-Variance (M-V or The Mean-Variance (M-V or μμ--σσ ) ) efficient investment set:efficient investment set:

{ all the portfolios in the opportunity set }{ all the portfolios in the opportunity set }

The Capital Allocation LineThe Capital Allocation Line

( )( ) ( )

( )

or

Ap p

A

Ap p

A

E R rfE R rf STD R

STD R

rfrf


rfC

A

B

0.0%

2.0%

4.0%

6.0%

8.0%

0.0% 1.0% 2.0% 3.0% 4.0%

STD(R)

E(R)

rf

A

The Capital Allocation Line (CAL):The Capital Allocation Line (CAL):Four Basic Investment StrategiesFour Basic Investment Strategies

rfC

A

B

0.0%

2.0%

4.0%

6.0%

8.0%

0.0% 1.0% 2.0% 3.0% 4.0%

STD(R)

E(R)

A

rf

P1

P2

Portfolios on the CALPortfolios on the CAL

PortfolioPortfolio αα E(RE(Rpp) = ) = μμpp Std(RStd(Rpp) = ) = σσpp

rfrf 00 1.00%1.00% 0.00%0.00%

PP11 0.250.25 2.15%2.15% 0.625%0.625%

AA 11 5.60%5.60% 2.50%2.50%

PP22 1.51.5 7.90%7.90% 3.75%3.75%

Capital Allocation: Capital Allocation: n Mutually n Mutually Exclusive Risky Asset and One Risk-free AssetExclusive Risky Asset and One Risk-free Asset





0.0%

2.0%

4.0%

6.0%

8.0%

0.0% 2.0% 4.0% 6.0% 8.0%

STD(R)

E(R)

rf C

A

B

Capital Allocation:Capital Allocation:One Risky Asset and One Risk-free AssetOne Risky Asset and One Risk-free Asset


{all the portfolios with proportion {all the portfolios with proportion αα invested in the invested in the risky asset j and (1-risky asset j and (1-αα) invested in the risk-free asset, ) invested in the risk-free asset,

(j = A or B or C)}(j = A or B or C)}

The Mean-Variance (M-V or The Mean-Variance (M-V or μμ--σσ ) efficient ) efficient investment set:investment set:

{all the portfolios with proportion {all the portfolios with proportion αα invested in the invested in the risky asset A and (1-risky asset A and (1-αα) invested in the risk-free asset ) invested in the risk-free asset

– (why A?)}– (why A?)}

Capital Allocation:Capital Allocation:Two Risky AssetsTwo Risky Assets




The Expected Return and STD of The Expected Return and STD of Return of the PortfolioReturn of the Portfolio

wwAA = the proportion invested in the risky asset A = the proportion invested in the risky asset A

wwBB = (1- = (1-wwAA) = the proportion invested in the risky asset B) = the proportion invested in the risky asset B

pp = the portfolio with = the portfolio with wwAA invested in the risky asset A andinvested in the risky asset A and

(1-(1-wwAA) ) invested in the risky asset Binvested in the risky asset B

RRpp = the return of portfolio p = the return of portfolio p

μμpp = the expected return of portfolio p= the expected return of portfolio p

σσpp = the standard deviation of the return of portfolio p= the standard deviation of the return of portfolio p

RRpp = w = wAA·R·RAA + (1-w + (1-wAA)·R)·RBB

μμpp = E[ w = E[ wAA·R·RAA + (1-w + (1-wAA)·R)·RB B ]]

σσ22pp= V[ w= V[ wAA·R·RAA + (1-w + (1-wAA)·R)·RB B ]]

Two Risky Assets:Two Risky Assets:

The Investment OpportunityThe Investment Opportunity SetSet

STD(Rp)

E(Rp)

B

A

Two Risky Assets:Two Risky Assets:The M-V Efficient Set (Frontier)The M-V Efficient Set (Frontier)

STD(Rp)

E(Rp)

B

A

Two Mutually Exclusive Risky Assets:Two Mutually Exclusive Risky Assets:

The M-V Efficient Set The M-V Efficient Set

STD(R)

E(R)

B

A


STD(R)

E(R)

B

A


STD(R)

E(R)

B

AP

Capital Allocation:Capital Allocation: Two Risky AssetsTwo Risky Assets


{all the portfolios on the frontier: with {all the portfolios on the frontier: with proportion proportion wwAA invested in the risky asset A and invested in the risky asset A and

(1-(1-wwAA)) invested in the risky asset B}invested in the risky asset B}

The Mean-Variance (M-V or The Mean-Variance (M-V or μμ--σσ ) ) efficient investment set:efficient investment set:

{all the portfolios on the efficient frontier}{all the portfolios on the efficient frontier}


STD(R)

E(R)

B

AP1

P2

P3

Pmin

Portfolios on the Efficient FrontierPortfolios on the Efficient Frontier

wwAA = the proportion invested in the risky asset A = the proportion invested in the risky asset AwwBB = (1- = (1-wwAA) = the proportion invested in the risky asset B) = the proportion invested in the risky asset B

What is the value of What is the value of wwA A for each one of the for each one of the portfolios indicated on the graph? - Assume that portfolios indicated on the graph? - Assume that μμAA=10%; =10%; μμBB=5%; =5%; σσAA=12%; =12%; σ σ BB=6%; =6%; ρρABAB=(-0.5).=(-0.5).

What is the investment strategy that each portfolio What is the investment strategy that each portfolio represents?represents?

How can you find the minimum variance portfolio? How can you find the minimum variance portfolio? What is the expected return and the std of return of What is the expected return and the std of return of that portfolio?that portfolio?

Portfolios on the FrontierPortfolios on the Frontier

PortfolioPortfolio wwAA E(RE(Rpp) = ) = μμpp Std(RStd(Rpp) = ) = σσpp

PP11 1.31.3 11.50%11.50% 16.57%16.57%

AA 11 10.00%10.00% 12.00%12.00%

PP22 0.350.35 6.75%6.75% 4.06%4.06%

PPminmin ?? ?? ??

BB 00 5.00%5.00% 6.00%6.00%

PP33 -0.5-0.5 2.50%2.50% 13.08%13.08%

The Minimum Variance PortfolioThe Minimum Variance Portfolio

2 2 2 2 2

The variance of a portfolio on the frontier

(2 risky assets, A and B) is

( ) 2

If you differentiate this expression with respect to

and set the derivative equal to zero

p p A A B B A B A B AB

A

V R w w w w

w

2

2 2

,

you will get the minimum variance portfolio:

and 12

B A B ABA B A

A B A B AB

w w w

The Minimum Variance PortfolioThe Minimum Variance Portfolio

2

2 2

2

2 2

min min

The minimum variance portfolio in our case is:

2

(6%) 12% 6% ( 0.5) 0.2857

(12%) (6%) 2 12% 6% ( 0.5)

Therefore,

6.43% and 3.93%

B A B ABA

A B A B AB

w

Practice ProblemsPractice Problems

BKM 7th Ed. Ch. 6:BKM 7th Ed. Ch. 6:

15-18, 20-21, 25, 32, 34-35;15-18, 20-21, 25, 32, 34-35;

BKM 8th Ed. Ch. 6:BKM 8th Ed. Ch. 6:

15-18, 26-27, 21, CFA: 6, 8-9;15-18, 26-27, 21, CFA: 6, 8-9;

Mathematics of Portfolio Theory:Mathematics of Portfolio Theory:

Read and practice parts Read and practice parts 6-106-10..

fi8000 valuation of financial assets

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