2015F Investments 04 Portfolio Theory 2

InvestmentsWeek 4: Portfolio Theory

2

Fall 2015Professor Albert Wang

16 October 2015 Agenda

►Admin HW2 Due Today HW3 to-be-posted soon, due in 3 weeks

►Lecture 2 asset examples – graphs not in notes! (some in

text) Numerical Examples Portfolio Theory Examples Portfolio Theory Summary CAPM Introduction/Basics Sharpe Ratio

Current Events

►Index Levels S&P500 2003.69 TAIEX 8567.92

►News EM and Commodities continue recovery Conflicting Fed governor statements, “hawkish”

vs. “dovish”. Market expectations are dovish. China Imports news, media and finance analysts

say this is negative for GDP (i.e. it will be lower). But…

►Y = C + I + G + X – M ►M lower than expected means Y will be?►Why might import costs be lower? China imports

commodities, and commodity prices have recently done what?

Intro Statistics Clarification

►Covariance Matrix vs. Correlation Matrix Covariance Matrix has covariances of

pairs on grid Correlation Matrix has correlation of pairs

on grid Remember, correlation and covariance

differe only by “standardizing” by standard deviations

yx

yxyx

,

,

Intro Statistics Clarification

►Daily return calculations►Standard deviation of daily returns►Annualized standard deviation of daily

returns Stdev(annual) =

Stdev(periodic)*sqrt(periods_per_year)

►Example Stdev(annual) = Stdev(daily)*sqrt(252)

►sqrt(252) = 15.87… useful figure to remember Var(annual) = Var(daily)*252

2 Asset Examples

►Investment Analysis and portfolio theory are a practical, real-world application of statistics

►We’ve gone through initial statistics review, including basic example computations

►2 Asset Example Graphs will give a visual depiction of the calculations, and help to build intuition

2 Asset Examples

►Investments, A and B Identical variances: variance vs. weight

►If correlation = -1 nonlinear curve Min variance = 0 what is the return here?

►If correlation = 0 nonlinear curve Min variance = half of before

►If correlation = 1 Linear… all at same variance as before Min variance = var(a) = var(b)

2 Asset Examples

►Investments, A and B Different variances: stdev vs. weight

►If correlation = -1 Piecewise linear curve Min stdev = 0 what is the return here?

►If correlation = 0 Nonlinear curve

►If correlation = 1 Linear between 2 points 0<w<1 Challenge question: beyond 2 points?

2 Asset Examples

►Investments, A and B Different E(r): return vs. weight

►Linear between 2 points►Recall statistics review, portfolio E(r) is

weighted sum of components

2 Asset Examples

►Investments, A and B Different E(r), different variances: stdev

vs. return►Weights are linear in E(r)►Weight mapping below return axis

Switch x and y axis to get standard “textbook” view of return vs stdev

►Add a risk-free asset (where?)►Find the “OCRA”

2 Asset + Riskfree Example

►One riskfree, one risky Variance vs. weight

►Var(kx) = k2Var(x) square function scaled by var(x)

Stdev vs. weight►Stdev(kx) = Sqrt[k2Var(x)] = kStdev(x)

linear function from origin to Stdev(x) E(r) vs. weight

►Linear in weight

2 Risky Assets

►Minimum Variance Portfolio (MVP) No riskfree asset

►Optimal Combination of Risky Assets (OCRA) With riskfree asset

22 1,2

1 2 12 21 2 1,2

, 12

w w w

21 2 2 1,2

1 2 21 2 2 1 1 2 1,2

2 1

[ ( ) ] [ ( ) ]

[ ( ) ] [ ( ) ] [ ( ) ( ) 2 ]

1

f f

f f f

E r r E r rw

E r r E r r E r E r r

w w

Example: 2-Asset Calculations

► Suppose the annualized volatility of Ebay and Google daily returns are: σEbay = 60% and σGoog = 80%. Correlation of returns for Ebay and Google, σEbay,Goog = .8.

1. What is the annualized volatility of returns for portfolios: 25% EBAY, 75% GOOG 50% EBAY, 50% GOOG 75% EBAY, 25% GOOG

2. What is the MVP using Ebay and Google assuming no riskfree security?

3. What is the OCRA using Ebay and Google assuming a riskfree rate of 4% per year and expected returns of 30% for Ebay and 50% for Google?

Example continued: Portfolio Variance

► Variance of a portfolio:

► ρebay,goog = .8 Correlation(a,b) = Covariance(a,b)/[stdev(a)*stdev(b)] .8 = σebay,goog/(.6*.8) σebay,goog = .384

► Variance of portfolio of 25% EBAY, 75% GOOG (.25)2(.6)2 + (.75)2(.8)2 + 2(.25)(.75)(.384) = 0.5265 Volatility = sqrt(0.5265) = 0.726

► Volatility of portfolio of 50% EBAY, 50% GOOG 0.665

► Volatility of portfolio of 75% EBAY, 25% GOOG 0.622

...222...

...

3,2323,1312,121

23

23

22

22

21

21

2

wwwwww

wwwp

Example continued: MVP and OCRA

2. Minimum Variance Portfolio (MVP) Vol(Ebay) = 60%, Vol(Goog) = 80%, Cov(Ebay,

Goog) = .384 solved in 1. w(ebay) = (.82-.384) / (.62+.82-2*0.384) = 1.1034

= 110.34% w(goog) = 1 – 1.1034 = -10.34%

3. Optimal Combination of Risky Assets (OCRA) E(r,ebay) = 30%, E(r,goog) = 50%, r(f) = 4% w(ebay) = [(.3-.04)*.82 – (.5-.04)*.384] / …… [(.3-.04)*.82 + (.5-.04)*.62 – (.3+.5-2*.04)*.384]

= -0.184 = -18.4% w(goog) = 1 – (-.184) = 1.184 = 118.4%

Graph N Risky Assets + Riskfree

►Same borrowing and lending riskfree rate

►Efficient Frontier►Optimal Combination of Risky Assets

(OCRA)►2-Fund Separation Theory: All mean-

variance efficient portfolios can be created by combining the “OCRA fund” with the “riskfree fund”.

Quantifying the Risk/Return Tradeoff

►Sharpe Ratio Ratio of excess return (expected return less the riskfree

rate) to standard deviation of returns Maximizing this in our portfolio analytics setup will solve

for the OCRA portfolio Quantifiable measure of risk/return tradeoff

►If excess return increases, Sh increases►If risk decreases, Sh increases

One of the primary measures to compare investment portfolios

Can be applied at security or portfolio level

i

fi rrEiSh

)(

)(

Diversification

►Graphical representation of Sharpe Ratio is the slope in this graph Higher slope (Sh) is better

σ

Return

r(f)

xPortfolio i

E[r(i)]

σ(i)

Example: Sharpe Ratio

►Ebay has an expected annual return of 30% and annualized daily return variance of 3600%2 (i.e. 3600%%). The riskfree rate is 4% per year. What is Ebay’s Sharpe ratio? Calculate inputs to Sharpe Ratio

►Vol(Ebay) = Sqrt(3600%%) = 60% Then just plug inputs into formula

►Sh(Ebay) = (.3 - .04)/.6 = 0.433

Diversification

►Diversification is the key takeaway from portfolio theory

►Risk/return tradeoff of a portfolio improves with the number of securities as long as securities are not perfectly correlated Returns of the portfolio are linear in component weights Risk is LESS THAN linear in weights (convex “drooping” risk

graphs)Þ Adding weights of new securities will decrease risk faster

than they decrease return (or increase risk slower than they increase return)

Þ The reward/risk tradeoff improves as new securities are added to a portfolio, if done in the correct weights

As long as correlation of the new securities is less than 1 with the existing set, there can be diversification benefits

Example: Diversification 1

►Diversification (2 securities) Perfect Hedge

►Suppose Mark has 100,000 to invest in securities 1 and 2. 1,2=-1. Mark is very risk averse and wants a portfolio with as little risk as possible:

Find the portfolio weights and tell Mark how much in dollar value and how many shares to invest in each stock.

Report the amount of money he would make and the volatility of his returns

Table below is given information:E(Returns) SD Price

1 0.12 0.12 38.06252 0.26 0.18 26.25

Example: Diversification 1Mark wants the MVP. Since the two securities here that have correlation of -1, we knowwe can achieve a perfect hedge using the MVP formula.

x1=[var(2)-cov(1,2)]/[var(1)+var(2)-2cov(1,2) = 0.6x2=1-x1 0.4

This means we will buy $60,000 in security 1 and $40,000 in security 2. Dividing by price gives number of shares. Summarizing:

Weights $ Amt Invested Shares1 0.6 60000 15762 0.4 40000 1524

The expected profit is :E(Profit)=60000*.12+40000*.26= 17600

The Expected Return of the portfolio is:E(Ret)=x1*R1+x2*R2=.6*.12+.4*.26= 17.60%

Note: This is a risk free return!


Suppose we have a correlation of 0.3. Answer the questions from Diversification

1Mark wants the MVP. Since we have securities here that have rho=.3, we cannot achievea riskfree position. Using the MVP formula:

x1=[var(2)-cov(1,2)]/[var(1)+var(2)-2cov(1,2) = =(.18^2-0.3*.12*.18)/(.12^2+.18^2-2*(.3*.12*.18) = 0.76596x2= 1-x1 = 0.23404

Weights $ Amt Invested Shares1 0.76596 76595.74 20122 0.23404 23404.26 892

The expected profit is :E(Profit)=76595.74*.12+23404.26*.26= 15276.60

The Expected Return of the portfolio is:E(Ret)=x1*R1+x2*R2= 15.28%

Note: This is no longer a risk free return! The variance of the portfolio is:Var(Port) =x1^2*Var(1)+x2^2*Var(2)+2*x1x2*cov(1,2)= 0.012546SD(Port) = 11.20%

which is still less than either of the two individually so there is a diversification effect.


Suppose we have correlation of 0.3 and a riskfree security paying 5% annually

Find the OCRA and the Sharpe ratio of the OCRASince we have securities here that have rho=.3, we now do not have a fully diversifiableportfolio. However, with a riskfree asset we can solve for the optimal combination of risky assets.Use the OCRA formula for 2 risky assets and 1 riskfree asset to find the weight in asset 1.

x1= 0.17143x2= 1-x1 = 0.82857

Weights $ Amt Invested Shares1 0.17143 17142.86 4502 0.82857 82857.14 3156

Sharpe Ratio:[ER(p)-r(f)] / sigma(p)ER(p) 0.23600 Sh(1) 0.58333 =(.12-.05)/.12Sigma(p) 0.15655 Sh(2) 1.16667 =(.26-.05)/.18Sh(p) 1.18813 =(.236-.05)/.15655

==>the maximum Sharpe ratio has increased

Portfolio Theory Summary

►Mapping out possible portfolios Any set of two risky securities/portfolios forms a curve of

possible portfolios, visually depicted on a graph with Return on the y-axis and Standard Deviation on the x axis

Any other set of > 2 securities will push the frontier to the left on the graph, i.e. visually depicting diversification benefits

“Western” of “leftmost” curve is the Minimum Variance Frontier

► investors would rather have less risk so rule out all points to the “east”

stdev

min var frontier

E(r)

Portfolio Theory Summary► “Northern” or “upper” half of the Minimum Variance Frontier is

the Efficient Frontier with only risky assets Except for the Minimum Variance Portfolio, there is a pair of

portfolios on the Minimum Variance Frontier with identical variance but different expected return when drawn in standard textbook view, with return on y-axis and volatility on x-axis

Rather have higher expected return, so rule out the lower points Without a riskfree security, all investors would hold a portfolio on

the Efficient Frontier with only risky assets► If more risk averse, require more return for each unit of risk. That is,

Ret/Risk (Slope) is hi. Their tangent point is farther left. Want lower risk.

M ORE Risk Averse

L ESS Risk Averse

s p

E(R)

M IN V AR P ORT

Portfolio Theory Summary If a risk free asset exists, the tangency point

between it and the risky asset efficient frontier is the Optimal Combination of Risky Assets.

►All mean-variance investors will invest in a weighted (denoted w) combination of the riskfree security and the OCRA

Since ANY portfolio of a riskfree and a risky portfolio will have a linear relationship between E(R) and σ, the graph is a straight line.

The Sharpe ratio at any point on the line is the same, and is the maximum possible given the risky securities

OCRA

E(R)

Rf

s

Borrowing: w[r(f)]<0%, i.e. w(OCRA)>100%

Lending: w[r(f)]>0%

Portfolio Theory Summary

►Other keys to remember Portfolio math can be used whether portfolio components

are individual securities or portfolios, as long as you have the required parameters: Expected Returns, Variances, and Covariances

A riskfree asset has zero variance and zero covariance with all other assets

►This is a hypothetical security that helps the analysis, but in real life there are no securities with these features (even Treasuries)

►There can only be one rate of return with zero variance and zero covariance, regardless whether it is a riskfree asset or combination of perfectly uncorrelated assets

Sharpe Ratio = [E(R)-r(f) ]/Stdev►“Reward to Risk Ratio”►Key measure of portfolio efficiency for mean-variance

investors

Assignments

►Read BKM CH8-9, 11►HW3 to-be-posted, due in 3 weeks

2015F Investments 04 Portfolio Theory 2

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