1

Financial Engineering and Portfolio Optimization

Dr. A. Ravi RavindranProfessor of Industrial Engineering

Pennsylvania State University

March-April 2015

Agenda• Portfolio selection problem

• Diversification to reduce risk– Examples

• Markowitz’s Bi-criteria QP model– Example

– Efficient portfolios

• Sharpe’s Bi-criteria LP model

• Asset Allocation Principles

2

Modern Portfolio Theory

• Developed by Harry Markowitz in the 50’s.

• Further refined by William Sharpe in the 60’s.

• Both shared the Nobel Prize in Economics in 1990 for this work.

4

Investment Basics

• Liquidity – How accessible is your money?

• Risk – What is the safety involved?

• Return – How much profit will you be able to expect from your investment?

3

5

Investment Strategies

• Trade-Off between Risk and Return– Cash: the least risky with the lowest returns– Bond (Income): moderately risky with

moderate returns– Stocks (Equities): the most risky but

offering the greatest payoff

• Broader diversification (Asset allocation) reduces risk and increases return

6

Investing in Stocks

• Stocks: Ownership shares in a corporation

• Ownership: If a company issues 1M shares, and you buy 10,000 shares, you own a 1% of the company.

• Valuation: (1) cash dividend and (2) share appreciation at the time of sale

4

7

Investing in Bond

• Bonds: Loans that investors make to corporations and governments.

• Face (par) value: Principal amount

• Coupon rate: yearly interest payment

• Maturity: the length of the loan

Portfolio Selection Problem:

• N Possible Securities (stocks, bonds, treasury notes, banks, mutual funds, etc.)

• C capital available for investment

• Problem: To determine an optimal investment policy

• Decision Variables: xJ - Investment in security J, where J=1,...,N

• x1+ x2 +...+ xN <= C,

• xJ >= 0 for all j

5

Return on Investment

• Historical data for T years available

• pJ ( t ) = Price of security J at the end of year t.

• dJ ( t ) = Dividends/Interests paid in year t.

• rJ ( t ) = Total return per dollar invested in security J in year t.

r tp t p t d t

p tJJ J J

J

( )( ) ( ) ( )

( )

11

Note: rJ (t) can be positive,negative or zero. Let μJ = Average annual return per dollar invested in security j.

trT

T

tJJ

1

1

6

MODEL 1(Simple Linear Programming Model)

21

1=

1=

:sConstraintOther

0

Subject to

= MAX

bxb

x

Cx

xz

Jjj

J

N

JJ

J

N

JJ

Drawbacks of the LP Model

• Investment risk is ignored

• No diversification ("All eggs in one basket")

• Mean values mask the variability in returns

• Illustration of Risk

• Historical returns of securities

7

13

What is Risk?

Scenario 1:

• Option1

– Pay fixed sum of Rs. 100

• Option 2

– Toss a coin; if Head you get Rs. 1000; if Tail, you get nothing

• Which option would you prefer?

14

What is Risk (Continued)

Scenario 2:

• Option1

– Pay fixed sum of Rs. 100

• Option 2

– Toss a coin; if Head you get Rs. 2000; if Tail, you have to pay me Rs. 1000.

• Which option would you prefer now?

8

20 Year Returns from Various Investment Securities (1993-2012)

Average Annual Return

Best Year Worst year

U.S. Stocks 8.2% 37.5%

(1995)

-22%

(2002)

International Stocks

6.1% 38.6%

(2003)

-21.4%

(2001)

90-day US Treasury bills

3.2% 6.3%

(2000)

0.1%

(2012)

U.S. Bonds 6.3% 18.5%

(1995)

-1.0%

(1999)

Markowitz’s Mean Variance Model

• Diversify the portfolio to reduce risk– Variance of security returns– Correlations between returns

• Investment Risk in Security j – Variance of return from its average value

2

1

2 1

T

tJJJJ tr

T

9

Risk due to Correlation of Returns between Securities

• Securities in similar industries such as auto, utilities etc. would rise and fall together

• General impact of economy

• Interest rate changes

σij2 = Covariance of return between securities

i and j

T

tjJiIij trtr

T 1

2 1

Matrix Q = [qij] = [σ2ij]: an NxN variance-

covariance matrix of returns

Excel Functions

• Variance:– VAR(A1:A10)

– Here, T=10 years and A1 to A10 contain annual returns over 10 years for security A

• Covariance– COVAR(A1:A10,B1:B10)

– Gives the covariance of the returns between securities A and B

10

Correlations With US Stocks(1998-2008)

• Foreign Stocks: 0.87

• Emerging Markets: 0.79

• Commodities: 0.19

• Government Bonds: -0.16

(Long Term)

Impact of Diversification

• Example 1– Mix of bonds and stocks

• Example 2– Mix of high risk stock categories

• Markowitz models for portfolio risk

11

Impact of Diversification: Example 1

PORTFOLIOS(Stock/Bond)

Year Stock Bond 1(0/100) 2(25/75) 3(50/50) 4(75/25) 5(100/0) 6(5/95) 7(10/90)

1 30.00% 10.00% 10.00% 15.00% 20.00% 25.00% 30.00% 11.00% 12.00%

2 30.00% 0.00% 0.00% 7.50% 15.00% 22.50% 30.00% 1.50% 3.00%

3 -10.00% 10.00% 10.00% 5.00% 0.00% -5.00% -10.00% 9.00% 8.00%

4 -10.00% 0.00% 0.00% -2.50% -5.00% -7.50% -10.00% -0.50% -1.00%

Mean 10.00% 5.00% 5.00% 6.25% 7.50% 8.75% 10.00% 5.25% 5.50%

STD 23.09% 5.77% 5.77% 7.22% 11.90% 17.38% 23.09% 5.61% 5.69%

Impact of Diversification: Example 1 (contd..)

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

0.00% 5.00% 10.00% 15.00% 20.00% 25.00%

Ret

urn

Standard Deviation

Efficient Frontier

Series1

12

Impact of Diversification: Example 2

• U.S. stock performance (1979 – 2008)

Source: T.Rowe Price

Average Annual Return

Standard Deviation

Large Cap stocks(S&P 500)

11% 17%

Mid/Small cap stocks (Russell 2500)

12% 20.7%

International stocks (MSCI – EAFE)

9.4% 19%

Diversified Portfolio60% Large cap, 20% Mid/small cap, 20% International

11.1% 16.5%

Markowitz’s Mean Variance Models for Portfolio Selection

• Single objective Quadratic Programming Model– Minimize Risk for a certain minimum return

• Bi-criteria Optimization Model– Minimize Risk

– Maximize Return

13

Quadratic Programming Model (Model 2)

Minimize variance of portfolio

Ret Subject to1

1 1

N

JJJ

N

I

N

JJIIJ

T

x

xxqQxx

+ Other Constraints

• Ret = Minimum portfolio return required

Bi-Criteria Model: (Model 3)Combines the LP model which maximizes return and the QP model which minimizes risk.

Minimize Risk (Portfolio Variance) = xTQx Maximize average Annual Return

xx TN

JJJ

1

• In general, there will be no portfolio which simultaneously maximizes return and minimizes risk.

• Need for a "trade-off" analysis.

14

RISK - RETURN CURVE

RISK

RETURN

EfficientPortfolios

FeasiblePortfolios

Maximize return

Minimize risk

EFFICIENT PORTFOLIO:

• An efficient portfolio or investment plan is such that there exists no other plan which has

– A higher return with no greater risk

or

– The same return with a lesser risk

• PROBLEM: Determine all the efficient portfolios from which to choose from.

15

Markowitz’s Bi-criteria Model: Example 3

An investment company can invest in threestocks. From past data, the means andstandard deviations of annual returns havebeen estimated as shown in Table 1. Thecorrelations between the annual returns onthe stocks are listed in Table 2.Table 1 Means & Std. deviations Table 2 Correlations

Means STDEV Correlation

Stock 1 0.14 0.20 Stocks 1 & 2 0.6

Stock 2 0.11 0.15 Stocks 1 & 3 0.4

Stock 3 0.10 0.08 Stocks 2 & 3 0.7

Markowitz’s Bicriteria Model: Example 3 (contd..)

The company has $100,000 to invest with the following requirements:

(i) No more than 50% should be invested on any stock.(ii) Invest all $100,000 among the three stocks.(iii) Invest at least $10,000 each in stocks 1 and 3.(iv) Achieve maximum portfolio return.(v) Achieve minimum portfolio variance.

a) Formulate the above problem as a bicriteria problem.b) Reformulate the problem if the objective were to find a minimum variance portfolio that yields an average portfolio return of at least 12%.

16

Markowitz’s Bicriteria Model: Example 3 - Solution

a) Let xi = $ invested in stock i, i=1, 2, 3. The bicriteria math programming model is the following,

Max z1 = 0.14x1 + 0.11x2 + 0.10x3 (Maximize portfolio return)

Min z2 = xtQx = 0.04x12 + 0.0225x2

2 + 0.0064x32 + 0.036x1x2 +

0.0128x1x3 + 0.0168x2x3 (Minimize portfolio variance)

Subject to:(No more than 50% invested in any stock)(Invest all $100,000)

(Invest at least $10,000 each in stocks 1 and 3)(Non-negativity)

  3,2,1000,50 iforxi

000,100321 xxx

  000,101 x 000,103 x

3,2,10 iforxi

Markowitz’s Bicriteria Model: Example 3 – Solution (contd..)

Note: The variance-covariance matrix is given by Q=

b) The formulation remains the same, except that the objective function, Max z1, is now converted to a constraint as follows:

Note: By varying the minimum return (RHS) and solving each QP problem, we can generate the entire efficient frontier

 

0064.00084.00064.0

0084.00225.0018.0

0064.0018.004.0

  000,1210.011.014.0 321 xxx

17

Example 3 –Markowitz’s QP Model SolutionsLower

limit on return

Return (%)

Risk(%)

12000 12.00% 12.17%11750 11.75% 11.54%11500 11.50% 11.05%11250 11.25% 10.70%11000 11.00% 10.53%10870 10.87% 10.500%

Harry Markowitz (1950) developed the Quadratic Programming Model.

• Drawback: Lot of data collection due to QP Model. Need (NxN) matrix of N(N+1)/2 elements.

• Extension by William Sharpe (1963).• Concept of Market Risk for each security.• Showed price fluctuations of securities are

correlated with stock market performance. Hence, covariance with respect to a broad market index is sufficient to capture the investment risk.

• Used Standard & Poor 500 Stock Index.

18

Market Risk of a Security

Let βj = Beta risk of security j representing the market risk.

• Beta for S&P 500 = 1.0

• For security J, βj = 1.3 means it is 30% more volatile than the market.

• Requires only N elements to compute as opposed to N(N+1)/2 for Markowitz's model.

Market Indices

• Measures trends in performance of stocks and bonds– Dow Jones Industrial Average

– Standard & Poor 500 Index

– Russell 2000 Index

– NASDAQ

– Morgan Stanley EAFE Index

– Barclay’s Bond Index

19

2012 Market Returns• DJIA 7.3%

• S&P 500 15.3%

• Russell 2000 16.4%

• EAFE Index 17.9%

• NASDAQ 15.9%

• Barclay’s Bond Index 4.2%

• Cash (90-day T-Bills) 0.1%

20

Model 4: (Sharpe’s Bi-criteria LP Model)Achieve portfolio diversification by mixing low beta with high beta securities

xT N

1=JJJx=Risk Portfolio

Now the portfolio selection or asset allocation problem becomes

Minimize Portfolio’s Beta Risk Maximize Portfolio Return

+Other investment constraints

Both Objectives are linear and we have a Bi-criteria Linear Program!

Sharpe Ratio = Return per unit of risk

J

JJ

TS

T= Average return from a "safe" investment (e.g. Treasury bills with zero risk) μJ - T = Excess Return over no risk investment Model 5:

Max z S xJ JJ

N

1

+ other investment restrictions

This is a single objective LP model!

21

Security Returns

• Average Return (Statistical)

• Annualized Return (Compounded)

• Illustrative Example– Google stock

• Uses

Google Stock• Went public on Aug. 2004

• Annual returns– 2005: 115%; 2006: 11%; 2007: 50%

– 2008: -56%; 2009: 102%; 2010: -4%;

– 2011: 9%; 2012: 10%; 2013: 58%

• For 2005-2013 (9 years):– Average annual return: 33%

– Compounded annualized return: 22%

22

Investment Strategies

Creating a Diversified Portfolio

Advice on Selecting Stocks for Investment

“I try to buy stock in businesses that are so wonderful that an idiot can run them. Because, sooner or later, one will”

-------Warren Buffet

23

Asset Allocation• Equities, Bonds and Cash• Equities

– US Stocks• Large, Medium and Small Cap• Growth and Value

– International Stocks• Developed Countries• Emerging Markets

– Real Estate

• Bonds– Long, Intermediate, Short Term– Government, Corporate and High Yield

Asset Allocation/Portfolio Selection

• Allocation of investment funds in Equities, Bonds and Cash.

• Two conflicting objectives– Maximize Return

– Minimize Risk

• More than 90% of portfolio’s performance is tied to asset allocation strategies

24

Asset Allocation: Illustrations

• Historical Performance of Asset Classes – Annual Returns (1993-2012)

• Impact of Diversification– 20-Year Return (1993-2012)

25

Impact of Diversification(20-Year Returns, 1993-2012)

Average Annual Return

Standard Deviation

Large Cap stocks (S&P 500) 8.2% 15.1%

Mid/Small Cap Stocks (Russell 2000) 8.4% 19.6%

International Stocks (MSCI-EAFE) 6.1% 17.0%

Bonds (Barclay’s Bond Index) 6.3% 3.7%

Cash (T-bills, 90 days) 3.2% 0.6%

Diversified Portfolio45% Large Cap, 10% mid/small cap, 10%International and 35% bonds)

7.9% 9.9%

Investment Advice

• Set up an Emergency fund to cover 6 months of living expenses.

• Save at least 10% of your net pay each month, beginning with the first pay check.

• Pay your credit card balances in FULL at the end of each month.

26

Investment Advice (Contd.)

• Follow the Golden Rule of investment:

“BUY LOW AND SELL HIGH”To achieve this, follow Dollar Cost Averaging strategy: – Invest a fixed amount at regular intervals (monthly

• Use the “Birthday Rule” for asset allocation: “Own your age in bonds”

Investment Advice (Contd.)

• Check your portfolio’s asset allocation twice a year. Make adjustments if necessary.

• Do not pay attention to the daily ups and downs of the stock market. You are in for the long haul. RELAX and enjoy your investment grow!

27

Modern Portfolio TheoryReferences:

• Heching, A.R. and A.J. King, “Financial Engineering”, Chap. 21, Operations Research and Management Science Handbook, A. Ravi Ravindran (Ed.), CRC Press, 2008.

• Reklaitis, Ravindran, and Ragsdell, (2006). Engineering Optimization, Wiley, Second Edition, New York, pp. 494-498.

• Markowitz, H. M., (1952). "Portfolio Selection", J. of Finance, Vol. 12, 77-91.

• Markowitz, H. M., (1956). "The optimization of a Quadratic Function Subject to Linear Constraints", Naval Res. Log. Qtly, Vol.3, 111-133.

Modern Portfolio TheoryReferences: (cont..)

• Markowitz, H. M., (1959). Portfolio Selection, Efficient Diversification of Investments, Wiley, New York,.

• Sharpe, W. F., (1963). "A simplified Model for Portfolio Analysis", Management Science, 9(2), 277-293.