hhif lecture series: modern portfolio theory and asset pricing

HHIF Lecture Series:Modern Portfolio Theory and Asset Pricing

Alexander Remorov

University of Toronto

February 18, 2011

Alexander Remorov (University of Toronto) HHIF Lecture Series: Modern Portfolio Theory and Asset Pricing 1 / 23

Portfolio Management

Let’s say after performing some analysis you decided to invest inseveral different securities

Your combined investments in the securities form a portfolio

Question: What are the best weights in this portfolio?

Things to consider:I Expected Return of the whole portfolioI Risk of the whole portfolio and how to measure itI What are your targets for portfolio performance?

To form a portfolio that is expected to meet your targets, need to:I Model the expected return and risk of securitiesI Model the risk of the whole portfolioI Use the model to determine optimal weights


Modern Portfolio Theory

Assumptions:I Standard Deviation of Return should be used to measure RiskI Asset Returns are jointly normally distributedI Investors are risk-averse: Between two assets with same return,

the one with lower risk is chosen

Benefits of Diversification - portfolio risk can be reduced byholding uncorrelated assets

Given assets in a portfolio, can plot all possible risk-returncombinations of portfolios combined of those assets

I Min. Variance portfolio - lowest level of risk possibleI Efficient Frontier - highest return for given level of risk

Investor Goal is to maximize Sharpe Ratio: S =E (R − Rf )

σ(R − Rf )

I Sometimes assume risk-free rate (Rf ) is constant


Modern Portfolio Theory - Optimization

We consider two common optimization problems in portfolio theoryI Consider a portfolio P of n assets A1,A2, . . . ,An

I Asset Ai provides return Ri and has weight wi in portfolioI Let w be the vector of weights, w = (w1,w2, . . . ,wn)I Let Σ be the covariance matrix of r = (R1,R2 . . . ,Rn); Σij = σij

Which weights produce the portfolio with minimum variance?

Minimize: σ2(P) = σ2(w1R1 + w2R2 + . . .+ wnRn)=

∑ni=1 w 2

i σ2(Ri ) +

∑i<j 2wiwjσij = w tΣw

subject to: w1 + w2 + . . .+ wn = w t(1 1 . . . 1) = 1

Which weights produce the portfolio with maximum Sharpe Ratio?I We assume for simplicity Rf is constant

Maximize: S(P) = E(R)−Rf

σ(R) = w t r−Rf√w tΣw

subject to: w1 + w2 + . . .+ wn = w t(1 1 . . . 1) = 1


Modern Portfolio Theory - Optimization

These problems can be explicitly solved in terms of r ,Σ

They can also be solved using software (e.g. Excel, Matlab, R, etc.)

We can consider more complicated problems related to portfoliomanagement

I The portfolio is expected to achieve a specific returnI There are constraints on portfolio weightsI Adjusting weights periodicallyI Many others, depending on portfolio objective

What is the big picture here?

Diversification decreases risk (but also usually expected return)

Assumption: investors hold portfolios to maximize Sharpe Ratio

For a given set of assets, the optimal risky portfolio(with maximum Sharpe Ratio) is unique

I Regardless of risk characteristics of the investorI Invest in optimal risky portfolio and risk-free asset depending

on risk aversion to get optimal complete portfolio


Modern Portfolio Theory - GraphSeries/Efficient.jpg


Capital Asset Pricing Model

Assumptions:I MPT: risk aversion, care about mean and variance onlyI Single period investment horizonI Investors have homogenous expectationsI Perfect Market

F Individual investor cannot affect stock priceF No taxes or transaction costs; everyone is equally informedF Lending and borrowing at risk-free rate

Using these assumptions we can price assets (e.g. stocks)

Market Portfolio: portfolio of all assets on the marketI Exists only in theory, impossible to constructI However, can approximate, e.g. S&P 500 (≈ 75% of US equity)

Total Supply = Total DemandI Net supply of risk-free asset is zeroI Therefore: combine investor portfolios to get market portfolio!I Optimal Risky portfolio for every investor is the same!I Hence optimal risky portfolio is the market portfolio


CAPM - Continued

Consider an asset A with expected return Ra, standard deviation σa,correlation with market σam

Adding A to market portfolio should give same risk-adjusted return

Using basic algebraic manipulations, get following result:

E (Ra) = Rf + βa(E (Rm)− Rf ),

where Rm is market return, βa =σam

σ2m

is beta

This can also be written as E (Ra)− Rf = βa(E (Rm)− Rf )I E (Ra)− Rf is risk premiumI β is riskI E (Rm)− Rf is market risk premium, here ”price of risk”

Usually estimate beta by using historical stock and benchmark pricesI Calculate historical returns (daily/monthly/yearly/etc.)I Find σam and σ2

m, then βa = σam

σ2m

I Or run regression: Ra,t − Rf = β0 + βa(Rm,t − Rf ) + εa,t


CAPM - Systematic and Unsystematic Risk

The CAPM can be rewritten to include an error term εa:

Ra = Rf + βa(Rm − Rf ) + εa

I Assume E (εa) and error is normal (from MPT)I The error is specific to asset, results in unique riskI The βa(Rm − Rf ) term results in systematic (market) risk

By definition of unique risk Rm and εa uncorrelated, so:

σ2a = β2

aσ2m + Var(εa)

I Total risk can be decomposed into market and unique risk

Consider n assets A1,A2, . . .An in a portfolio PI Each asset has weight wi , beta βi and unique risk εiI We can determine unsystematic and unique risk of P

βp =σpm

σ2m

= Cov(w1R1+...+wnRn,rm)σ2

m=

∑ni=1 wi

Cov(Ri ,rm)σ2

m=

∑ni=1 wiβi

Portfolio beta is weighted sum of asset betas


CAPM - Example

Consider a portfolio P of three stocks A,B,CI Assume βA = 0.5,wA = 0.1I βB = 1.1,wB = 0.4I βC = 1.8,wC = 0.4I Remaining part - cash, β = 0

Assume Rf = 1%, market risk premium E (Rm)− Rf = 9%

We can then calculate expected return on whole portfolioI βP = waβA + wbβB + wcβC = 1.21I Then expected return of portfolio is 1 + 1.21 · 9 = 11.89%


Systematic and Unique Risk for a Portfolio

Consider a portfolio P of n assets with same notation as before

Systematic risk is not hard to compute:I systematic risk = β2

pσ2m = (w1β1 + . . .+ wnβn)2σ2

mI Holding low beta stocks results in lower riskI Almost always βi > 0 so βp > 0; very hard to eliminate market risk

Let us look at unique risk:I Recall R = w1R1 + w2R2 + . . .+ wnRn

I According to CAPM Ri = Rf + βi (Rm − Rf ) + εiI So R = Rf + βp(Rm − Rf ) + εp and εp = w1ε1 + . . .+ wnεnI We assume unique risks are uncorrelated, so cov(εi , εj) = 0; then:

Var(εp) = w 21 Var(ε1) + w 2

2 Var(ε2) + . . .+ w 2n Var(εn)

Assume wi = 1n ,Var(εi ) = σ2

I Then Var(εp) = 1nσ

2. This decreases as n (number of assets) increasesI For sufficiently large n, Var(εp) (unique risk) becomes very smallI Note we cannot have n→∞, this is real life


Recap of CAPM and Advantages

Optimal risky portfolio is market portfolio, therefore:

Ra = Rf + βa(Rm − Rf ) + εa

Portfolio beta is weighted sum of asset betas

Using regression can estimate systematic and unique riskI Market risk usually cannot be eliminatedI Unique risk can be greatly reduced with many assets

Advantages

A simple way to estimate asset and portfolio return and risk

Helps explain the benefits of diversification and decomposition of risk

Allows for measurement of portfolio performanceI For a specific period, find average return R on portfolioI Calculate β̂, Rf , Rm based on prices during periodI Perform regression: R − Rf = α + β̂(Rm − R)I α called Jensen’s alpha; if positive - beat the market


Drawbacks of CAPM

CAPM on average explains only 80% of portfolio returnsI Some other models do much better (e.g. 90% for Fama-French)

The market portfolio should include everythingI Stocks, bonds, commodities, real estate, human capital, etc.I Evidence that using a more complete portfolio explains more of returns

Very strong assumptions about distribution of returnsI Stock returns are often not normally distributedI Standard Deviation, Covariance are not constant

Fama and French have shown that small cap value stocks exhibitabnormal positive returns

Not clear that investors have ”homogenous” expectationsI Although many argue that outlying expectations ”cancel out”

Various constraints on portfolio selection and tradingI Institutional Level, e.g. investment grade, margin requirementsI Individual Investor level - risk aversion, irrational behavior?


Arbitrage Pricing Theory

Assumptions:I No taxes or transaction costsI Some agents are smart and can form desired portfoliosI Securities have finite expected values and standard deviations

Assumptions allow a smart investor to execute any trading strategy

Arbitrage: ”free lunch”; there exists a strategy which:

1. Starts with portfolio P containing nothing.2. At the end is guaranteed to produce positive money.

(Self-financing condition - cannot invest external funds in portfolio)

Example: two 1-yr zero-coupon bonds with 3% and 4% yields

APT: whenever there is arbitrage, some smart investor(s) willimmediately exploit it ⇒ Therefore there is no arbitrage!

Note: in real life can have arbitrageI Maybe not worth to exploit due to transaction costs, trading rules, etc.


Multifactor Model

Asset returns are driven by k factorsI e.g. inflation, unemployment, manufacturing, market, etc.

Return Ri on Asset i is given by:

Ri = E (Ri ) + βi ,1f̃1 + βi ,2f̃2 + · · ·+ βi ,k f̃k + εk

I Assume for 1 ≤ j ≤ k , Factor j has return Fj

I Define f̃j = Fj − E (Fj) so we see E (f̃j) = 0I βi,j is factor loading/beta - how return Ri responds to change in f̃jI εk is unique risk (not explained by factors); assume E (εk) = 0

Consider a stock of a copper mining company with a factor loading of1.5 on a US manufacturing index (e.g. Purchasing Managers Index)and a factor loading of 0.6 on inflation.If the manufacturing index increases by 4% and inflation increases by5%, we expect the return on the stock to increase by:1.5 · 4 + 0.6 · 5 = 9%.


APT Model

Assuming the factor model and no arbitrage, we must have:

E (Ri ) = λ0 + βi ,1λ1 + βi ,2λ2 + · · ·+ βi ,kλk (1)

I Here λj is risk premium on factor j ; the extra return you expectto get for an extra unit of exposure to factor j

I Note that risk premium can be negativeI λ0 corresponds to the return on an asset with no factor risks, i.e.

risk-free asset

Where does Arbitrage Pricing Come in?I Two portfolios with the same exposure to each factor should

have the same priceI For assets 1, 2, ..., n in real life, there exist unique numbersλ0, λ1, . . . , λk such that equation (1) holds for every asset

I Otherwise there is arbitrage!


APT - Example

Consider stocks of two companies: Ebay and AmazonI Assume each company will buy back all its shares in one yearI Assume you can buy the stock now and cannot sell until the buybackI Neither company pays dividends so capital gains is all you will get

Consider two cases for the online retail market in one yearI Measure ”online retail market index” I ; now I = 100I Massive Expansion: I = 120 (20% return); probability 70%I Minor Expansion: I = 105 (5% return); probability 30%


APT ExampleEstimate prices for each stock in one year:

EBAY AMZN

I=120 $50.0 $300

I=105 $31.0 $160

E(Price) $44.0 $258

Price Today $34.7 $187.8

Return 27.7% 37.4%Estimate factor loadings βEBAY ,I , βAMZN,I

I For EBAY stock: REBAY = E (REBAY ) + βEBAY ,I · f̃I + εEBAY

I Recall f̃I = FI − E (FI ), NOT the factor return!

Major Expansion: 44.1 = E (REBAY ) + βEBAY ,I · 4.5Minor Expansion: −10.6 = E (REBAY ) + βEBAY ,I · (−10.5)

I Solving get:

E (REBAY ) = 27.7% (already know this) and βEBAY ,I = 3.65

I Do the same thing for AMZN, get:

E (RAMZN) = 37.4% (already know this) and βAMZN,I = 4.97


APT Example

Now that we know factor loadings, can find risk premia

E (REBAY ) = λ0 + βEBAY ,I · λI

For EBAY: 27.7 = λ0 + 3.65λI

For AMZN: 37.4 = λ0 + 4.97λI

Solving, get λI = 7.35%, λ0 = 0.87%

λI - risk premium on the ”Internet Retail” factor

λ0 - return on asset not exposed to factorI If there are no other factors, this is risk-free rate


Applying APT

The previous approach can be done for more than 2 factors

Can apply APT to historical data to price assets

Two general approaches:

Principal Component AnalysisI Statistical procedure to extract factors from historical returns;

estimates βijs and λjs togetherI However factors have no real life significance

Come up with factors yourselfI Could use macroeconomic factors (e.g. inflation, GDP,

manufacturing index. etc.)F See Chen, Roll, Ross 1986 paper

I Could use Fama-French factors - based on size and P/B ratioF See Fama-French three-factor model

I Once you have identified factors, get factor returns, and estimate f̃jsI Perform regression of Ri on f̃js to estimate βijsI Perform cross-sectional regression of E (Ri )s on βijs to estimate λjs


Concluding Remarks

Asset allocation is crucial when it comes to investing

Finding optimal portfolio weights is a very hard problemI Depends on portfolio objectives and constraints

Sharpe Ratio is often used to measure performance

CAPM: a quick way to estimate portfolio return and riskI Makes a lot of assumptions that don’t hold true in real lifeI But does illustrate the benefits of diversification

APT: also used to estimate return and risk, but harder to doI Identifying the factors is a major challenge

F How many factors do we really need?

I Need to be careful about which statistical tools and which datais used to estimate betas and risk premia

These are just models!I Should be used as guidelines, not just blindly followedI Good idea to think about why a model outputs certain results


Any Questions?

Upcoming Events:

HHIF Quarterly MeetingI Friday, March 4, 6:00-8:00 p.m., South Dining Room,

Hart House


References

1. prof. Jiang Wang (MIT) - Arbitrage Pricing Theory Lecturehttp://web.mit.edu/15.407/file/Ch12.pdf

2. prof. Giovanna Nicodano (University of Toledo) -Multi-Factor Models and the APT Lecturehttp://web.econ.unito.it/nicodano/handout_bkm1_apt.pdf

3. Investopedia: Financial Conceptshttp://www.investopedia.com/university/concepts/default.asp


http://web.mit.edu/15.407/file/Ch12.pdf

http://web.econ.unito.it/nicodano/handout_bkm1_apt.pdf

http://www.investopedia.com/university/concepts/default.asp

http://www.investopedia.com/university/concepts/default.asp

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