70391 - Finance

Portfolio StatisticsBasics of expected returns, volatility, correlation and diversification

70391 – Finance – Fall 2016Tepper School of BusinessCarnegie Mellon Universityc©2016 Chris Telmer. Some content from slides by Bryan Routledge. Used with permission.

10.24.2016 10:39

Questions

:: Where does the 8.4% come from?

Portfolio Stats 2

Questions

:: You are an investment advisor. Your client is interested in twostocks:

:: Which one is best?

:: Perhaps she should hold some of both?

:: She asks for a particular amount of return?

:: She expands her horizons to include more stocks. What now?

Portfolio Stats 3

Plan

:: Data: what are the facts?:: Bonds, interest rates

:: Stocks, “Equity Premium”

:: Characterizing portfolios

:: Characterizing portfolio risk and return:: Benefits of diversification

:: Characterizing the distribution of returns with a statisticalmodel

:: Next 2 weeks::: Optimal portfolio choice

:: CAPM: Equilibrium asset pricing (where does 8.4% come from)

Portfolio Stats 4

Empirical Facts About Risk and Return(What is the “Equity Premium”)

Portfolio Stats 5

Question

One-year interest rates

:: Now: 0.25%

:: Historical average: 3 or 4%.

How much should you expect to get on stocks?

Portfolio Stats 6

Spreadsheet

See spreadsheet, eq prem data.xlsx

:: Also, AEO data from problem set

Portfolio Stats 7

U.S. Government Bonds

:: U.S. Government bonds are called Treasury Securities

:: Treasury Bills= mature 1, 3, 6, and 12 months

No “coupon” payment (i.e., a “zero coupon”)

:: Treasury Notes= Mature 2 to 10 years

Semi-annual coupons

:: Treasury Bonds= Mature > 10 years

Semi-annual coupons

:: Others

Portfolio Stats 8

Bond Prices[Source: Yahoo http://finance.yahoo.com/bonds/composite_bond_rates]

:: Typically quoted in terms of YTM:

Portfolio Stats 9

Bond Yields (Prices)[Source: St.Louis Fed ]

05

1015

20yi

eld

01jan1960 01jan1970 01jan1980 01jan1990 01jan2000 01jan2010

1−Year 5−Year 10−Year 20−Year

GetTreasuryAndPlot.do

Source: FRED 1962 to 2012−11−19 Constant Maturity US Treasury

Treasury Yields

Portfolio Stats 10

Bond Yields (Prices)[Source: St.Louis Fed ]

02

46

8yi

eld

01jan2000 01jan2002 01jan2004 01jan2006 01jan2008 01jan2010 01jan2012

3−Month 1−Year 5−Year 10−Year 20−Year 30−Year

GetTreasuryAndPlot.do

Source: FRED 2000 to 2012−11−19 Constant Maturity US Treasury

Treasury Yields

Portfolio Stats 11

Interest Rates, Nominal and Real

The Fisher Equation:

Nominal Interest Rate = Real Interest Rate + E(Inflation

)

:: Which interest rate should we use for FCF valuation?

Portfolio Stats 12

Stocks - The Equity Risk Premium

:: Excess Returns

= The return on an asset (e.g., a stock) above the risk-free rate

= r̃i,t − rf ,t

:: Expected Excess Returns

= E [r̃i,t − rf ,t ]

= Compensation for the risk or a “risk premium”

:: Equity Risk Premium

= Expected excess return on a portfolio of all stocks

= All stocks in US; weighted by value; similar to S&P 500

= E [r̃m,t − rf ,t ]

Portfolio Stats 13

Stocks - The Equity Risk Premium

:: Excess Returns

= The return on an asset (e.g., a stock) above the risk-free rate

= r̃i,t − rf ,t

:: Expected Excess Returns

= E [r̃i,t − rf ,t ]

= Compensation for the risk or a “risk premium”

:: Equity Risk Premium

= Expected excess return on a portfolio of all stocks

= All stocks in US; weighted by value; similar to S&P 500

= E [r̃m,t − rf ,t ]

Portfolio Stats 13

Stocks - The Equity Risk Premium

:: Excess Returns

= The return on an asset (e.g., a stock) above the risk-free rate

= r̃i,t − rf ,t

:: Expected Excess Returns

= E [r̃i,t − rf ,t ]

= Compensation for the risk or a “risk premium”

:: Equity Risk Premium

= Expected excess return on a portfolio of all stocks

= All stocks in US; weighted by value; similar to S&P 500

= E [r̃m,t − rf ,t ]

Portfolio Stats 13

Stocks - The Equity Risk Premium[Data: CRSP via Ken French ]

$2816.35

$20.25$12.92

−2

02

46

8

1920 1940 1960 1980 2000 2020T

Market NYSE, AMEX, and NASDAQ [R]Risk Free Rate One−month t−bill [FF] continuously returnConsumer Price Index for All Urban Consumers: All Items

Data: CRSP Indicies :: Monthly

Note: log scale (base e)

Portfolio Value as of month end 2012.8Invest $1 at 1.1.1926

Portfolio Stats 14

Stocks - The Equity Risk Premium

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Stocks - The Equity Risk Premium[Data: CRSP via Ken French ]

−.2

−.1

0.1

.2

1920 1940 1960 1980 2000 2020T

Market all NYSE, AMEX, and NASDAQ [FF]Risk Free Rate One−month t−bill [FF] continuously return

Data: CRSP Daily 1920 to 2012/9

Daily Returns Equity Index

Portfolio Stats 16

Stocks - The Equity Risk Premium[Data: CRSP via Ken French ]

−.1

−.0

50

.05

.1

2000 2005 2010 2015T

Market all NYSE, AMEX, and NASDAQ [FF]Risk Free Rate One−month t−bill [FF] continuously return

Data: CRSP Daily 2000 to 2012/9

Daily Returns Equity Index

Portfolio Stats 17

Stocks - The Equity Risk Premium[Data: CRSP via Ken French ]

$1.32

−.6

−.4

−.2

0.2

.4

2000 2005 2010 2015T

Market NYSE, AMEX, and NASDAQ [R]

Risk Free Rate One−month t−bill [FF] continuously return

Consumer Price Index for All Urban Consumers: All Items

Data: CRSP Indicies :: Monthly

Note: log scale (base e)

Portfolio Value as of month end 2012.8Invest $1 at 1.1.2000

Portfolio Stats 18

Summary

Estimating the discount rate (OCC):

:: Get risk-free “interest rate” from UST securities:

:: rf = 0.02 (10-yr UST spot rate)

:: Estimate equity risk premium:

:: E(r̃ − rf

)≈ 0.06

:: Estimate firm/project β (“beta”) and use:

:: E(r̃i)

= rf + β(E (r̃) − rf

)

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Portfolio Construction

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Portfolio Definitions(today is Year 0. Year 1 notation omitted. Dividends set to zero, for now, for notational simplicity)

:: pi0 = today’s price-per-share, stock i (“ex-dividend”)

:: pi = future price-per-share (random, also “ex-dividend”)

:: ni = number of shares held, stock i

:: V0 =∑N

i=1 pi0ni , total funds invested

:: V =∑N

i=1 pini realized portfolio payoff (random)

:: γi =(pi0ni

)/V0, portfolio share, stock i

:: 1 + ri = pi/pi0 rate-of-return, stock i (random )

:: 1 + rp = V /V0 rate-of-return, portfolio (random )

:: 1 + rf = risk-free return (interest rate)

Portfolio Stats 21

Portfolio Definitions(today is Year 0. Year 1 notation omitted. Dividends set to zero, for now, for notational simplicity)

:: pi0 = today’s price-per-share, stock i (“ex-dividend”)

:: pi = future price-per-share (random, also “ex-dividend”)

:: ni = number of shares held, stock i

:: V0 =∑N

i=1 pi0ni , total funds invested

:: V =∑N

i=1 pini realized portfolio payoff (random)

:: γi =(pi0ni

)/V0, portfolio share, stock i

:: 1 + ri = pi/pi0 rate-of-return, stock i (random )

:: 1 + rp = V /V0 rate-of-return, portfolio (random )

:: 1 + rf = risk-free return (interest rate)

Portfolio Stats 21

Portfolio Definitions(today is Year 0. Year 1 notation omitted. Dividends set to zero, for now, for notational simplicity)

:: pi0 = today’s price-per-share, stock i (“ex-dividend”)

:: pi = future price-per-share (random, also “ex-dividend”)

:: ni = number of shares held, stock i

:: V0 =∑N

i=1 pi0ni , total funds invested

:: V =∑N

i=1 pini realized portfolio payoff (random)

:: γi =(pi0ni

)/V0, portfolio share, stock i

:: 1 + ri = pi/pi0 rate-of-return, stock i (random )

:: 1 + rp = V /V0 rate-of-return, portfolio (random )

:: 1 + rf = risk-free return (interest rate)

Portfolio Stats 21

Portfolio Definitions(today is Year 0. Year 1 notation omitted. Dividends set to zero, for now, for notational simplicity)

:: pi0 = today’s price-per-share, stock i (“ex-dividend”)

:: pi = future price-per-share (random, also “ex-dividend”)

:: ni = number of shares held, stock i

:: V0 =∑N

i=1 pi0ni , total funds invested

:: V =∑N

i=1 pini realized portfolio payoff (random)

:: γi =(pi0ni

)/V0, portfolio share, stock i

:: 1 + ri = pi/pi0 rate-of-return, stock i (random )

:: 1 + rp = V /V0 rate-of-return, portfolio (random )

:: 1 + rf = risk-free return (interest rate)

Portfolio Stats 21

Portfolio Definitions(today is Year 0. Year 1 notation omitted. Dividends set to zero, for now, for notational simplicity)

:: pi0 = today’s price-per-share, stock i (“ex-dividend”)

:: pi = future price-per-share (random, also “ex-dividend”)

:: ni = number of shares held, stock i

:: V0 =∑N

i=1 pi0ni , total funds invested

:: V =∑N

i=1 pini realized portfolio payoff (random)

:: γi =(pi0ni

)/V0, portfolio share, stock i

:: 1 + ri = pi/pi0 rate-of-return, stock i (random )

:: 1 + rp = V /V0 rate-of-return, portfolio (random )

:: 1 + rf = risk-free return (interest rate)

Portfolio Stats 21

Portfolio Definitions(today is Year 0. Year 1 notation omitted. Dividends set to zero, for now, for notational simplicity)

:: pi0 = today’s price-per-share, stock i (“ex-dividend”)

:: pi = future price-per-share (random, also “ex-dividend”)

:: ni = number of shares held, stock i

:: V0 =∑N

i=1 pi0ni , total funds invested

:: V =∑N

i=1 pini realized portfolio payoff (random)

:: γi =(pi0ni

)/V0, portfolio share, stock i

:: 1 + ri = pi/pi0 rate-of-return, stock i (random )

:: 1 + rp = V /V0 rate-of-return, portfolio (random )

:: 1 + rf = risk-free return (interest rate)

Portfolio Stats 21

Portfolio Definitions(today is Year 0. Year 1 notation omitted. Dividends set to zero, for now, for notational simplicity)

:: pi0 = today’s price-per-share, stock i (“ex-dividend”)

:: pi = future price-per-share (random, also “ex-dividend”)

:: ni = number of shares held, stock i

:: V0 =∑N

i=1 pi0ni , total funds invested

:: V =∑N

i=1 pini realized portfolio payoff (random)

:: γi =(pi0ni

)/V0, portfolio share, stock i

:: 1 + ri = pi/pi0 rate-of-return, stock i (random )

:: 1 + rp = V /V0 rate-of-return, portfolio (random )

:: 1 + rf = risk-free return (interest rate)

Portfolio Stats 21

Portfolio Definitions(today is Year 0. Year 1 notation omitted. Dividends set to zero, for now, for notational simplicity)

:: pi0 = today’s price-per-share, stock i (“ex-dividend”)

:: pi = future price-per-share (random, also “ex-dividend”)

:: ni = number of shares held, stock i

:: V0 =∑N

i=1 pi0ni , total funds invested

:: V =∑N

i=1 pini realized portfolio payoff (random)

:: γi =(pi0ni

)/V0, portfolio share, stock i

:: 1 + ri = pi/pi0 rate-of-return, stock i (random )

:: 1 + rp = V /V0 rate-of-return, portfolio (random )

:: 1 + rf = risk-free return (interest rate)

Portfolio Stats 21

Portfolio Definitions(today is Year 0. Year 1 notation omitted. Dividends set to zero, for now, for notational simplicity)

:: pi0 = today’s price-per-share, stock i (“ex-dividend”)

:: pi = future price-per-share (random, also “ex-dividend”)

:: ni = number of shares held, stock i

:: V0 =∑N

i=1 pi0ni , total funds invested

:: V =∑N

i=1 pini realized portfolio payoff (random)

:: γi =(pi0ni

)/V0, portfolio share, stock i

:: 1 + ri = pi/pi0 rate-of-return, stock i (random )

:: 1 + rp = V /V0 rate-of-return, portfolio (random )

:: 1 + rf = risk-free return (interest rate)

Portfolio Stats 21

Portfolio Return: Two-Stock ExampleDividends = 0 for notational simplicty ... incorporated soon

Today’s portfolio value:

V0 = p1,0n1 + p2,0n2

Next period’s value

V = p1n1 + p2n2

V

V0=

p1,0n1

V0

p1

p1,0+

p2,0n2

V0

p2

p2,0

1 + rp = γ1

(1 + r1

)+ γ2

(1 + r2

)

rp = γ1 r1 +(1 − γ1

)r2

:: “Return on portfolio is value-weighted average of return onportfolio components”

Portfolio Stats 22

Portfolio Return: Two-Stock ExampleDividends = 0 for notational simplicty ... incorporated soon

Today’s portfolio value:

V0 = p1,0n1 + p2,0n2

Next period’s value

V = p1n1 + p2n2

V

V0=

p1,0n1

V0

p1

p1,0+

p2,0n2

V0

p2

p2,0

1 + rp = γ1

(1 + r1

)+ γ2

(1 + r2

)

rp = γ1 r1 +(1 − γ1

)r2

:: “Return on portfolio is value-weighted average of return onportfolio components”

Portfolio Stats 22

Portfolio Return: Two-Stock ExampleDividends = 0 for notational simplicty ... incorporated soon

Today’s portfolio value:

V0 = p1,0n1 + p2,0n2

Next period’s value

V = p1n1 + p2n2

V

V0=

p1,0n1

V0

p1

p1,0+

p2,0n2

V0

p2

p2,0

1 + rp = γ1

(1 + r1

)+ γ2

(1 + r2

)

rp = γ1 r1 +(1 − γ1

)r2

:: “Return on portfolio is value-weighted average of return onportfolio components”

Portfolio Stats 22

Portfolio Return: Two-Stock ExampleDividends = 0 for notational simplicty ... incorporated soon

Today’s portfolio value:

V0 = p1,0n1 + p2,0n2

Next period’s value

V = p1n1 + p2n2

V

V0=

p1,0n1

V0

p1

p1,0+

p2,0n2

V0

p2

p2,0

1 + rp = γ1

(1 + r1

)+ γ2

(1 + r2

)

rp = γ1 r1 +(1 − γ1

)r2

:: “Return on portfolio is value-weighted average of return onportfolio components”

Portfolio Stats 22

Many Stocks

1 + rp = γ1

(1 + r1

)+ γ2

(1 + r2

)+ . . .+ γN

(1 + rN

)

rp = γ1 r1 + γ2r2 + . . .+ γN rN

=N∑

i=1

γi ri

Note:

:: Portfolio weights sum to one:∑N

i=1 γi = 1

:: If γi < 0 we call this “short selling” stock i

:: If γi is restricted to be positive, we call this a “no short salesrestriction.”

Portfolio Stats 23

Many Stocks

1 + rp = γ1

(1 + r1

)+ γ2

(1 + r2

)+ . . .+ γN

(1 + rN

)

rp = γ1 r1 + γ2r2 + . . .+ γN rN

=N∑

i=1

γi ri

Note:

:: Portfolio weights sum to one:∑N

i=1 γi = 1

:: If γi < 0 we call this “short selling” stock i

:: If γi is restricted to be positive, we call this a “no short salesrestriction.”

Portfolio Stats 23

Many Stocks

1 + rp = γ1

(1 + r1

)+ γ2

(1 + r2

)+ . . .+ γN

(1 + rN

)

rp = γ1 r1 + γ2r2 + . . .+ γN rN

=N∑

i=1

γi ri

Note:

:: Portfolio weights sum to one:∑N

i=1 γi = 1

:: If γi < 0 we call this “short selling” stock i

:: If γi is restricted to be positive, we call this a “no short salesrestriction.”

Portfolio Stats 23

Dividends

For stocks 1 and 2, dividends are d1 and d2. Nothing changes,except returns incorporate dividend income. Note that dividendsare (typically) random. Today’s value:

V0 = p1,0n1 + p2,0n2

Next period’s value

V = (p1 + d1)n1 + (p2 + d2)n2

V

V0=

p1,0n1

V0

(p1 + d1

p1,0

)+

p2,0n2

V0

(p2 + d2

p2,0

)

1 + rp = γ1

(1 + r1

)+ γ2

(1 + r2

)

rp = γ1 r1 +(1 − γ1

)r2

:: Same as before: “Return on portfolio is value-weightedaverage of return on portfolio components”

Portfolio Stats 24

Dividends

For stocks 1 and 2, dividends are d1 and d2. Nothing changes,except returns incorporate dividend income. Note that dividendsare (typically) random. Today’s value:

V0 = p1,0n1 + p2,0n2

Next period’s value

V = (p1 + d1)n1 + (p2 + d2)n2

V

V0=

p1,0n1

V0

(p1 + d1

p1,0

)+

p2,0n2

V0

(p2 + d2

p2,0

)

1 + rp = γ1

(1 + r1

)+ γ2

(1 + r2

)

rp = γ1 r1 +(1 − γ1

)r2

:: Same as before: “Return on portfolio is value-weightedaverage of return on portfolio components”

Portfolio Stats 24

Dividends

For stocks 1 and 2, dividends are d1 and d2. Nothing changes,except returns incorporate dividend income. Note that dividendsare (typically) random. Today’s value:

V0 = p1,0n1 + p2,0n2

Next period’s value

V = (p1 + d1)n1 + (p2 + d2)n2

V

V0=

p1,0n1

V0

(p1 + d1

p1,0

)+

p2,0n2

V0

(p2 + d2

p2,0

)

1 + rp = γ1

(1 + r1

)+ γ2

(1 + r2

)

rp = γ1 r1 +(1 − γ1

)r2

:: Same as before: “Return on portfolio is value-weightedaverage of return on portfolio components”

Portfolio Stats 24

Dividends

For stocks 1 and 2, dividends are d1 and d2. Nothing changes,except returns incorporate dividend income. Note that dividendsare (typically) random. Today’s value:

V0 = p1,0n1 + p2,0n2

Next period’s value

V = (p1 + d1)n1 + (p2 + d2)n2

V

V0=

p1,0n1

V0

(p1 + d1

p1,0

)+

p2,0n2

V0

(p2 + d2

p2,0

)

1 + rp = γ1

(1 + r1

)+ γ2

(1 + r2

)

rp = γ1 r1 +(1 − γ1

)r2

:: Same as before: “Return on portfolio is value-weightedaverage of return on portfolio components”

Portfolio Stats 24

Numerical Examples

:: You have 120 shares of Stock 1 and 50 shares of Stock 2. Today’s prices

are p1,0 = 25 and p2,0 = 40. Returns are 15% for Stock 1 and 2% for

Stock 2. Dividends are zero.:: What are the portfolio shares. What is the portfolio return. What are the

realized (next period) stock prices. What is today’s wealth and tomorrow’swealth? (Answers: 60%, 40%, 9.8%, $28.75, $40.80, $5,000, $5,490)

:: Dividends are now 0.5 and 1.6.

:: Same questions. (Answers: 60%, 40%, 9.8%, $28.25, $39.2 $5,000,

$5,490). Also, dividend yields are d1/p1,0 = 0.02 and d2/p2,0 = 0.04 (this

is a common convention for expressing dividends ... “dividend per dollar

of share price”)

:: Same data as previous question, but future prices are 32 and 35 and you

must solve for returns.:: (Answers: 60%, 40%, 14.6%, 30%, −8.50%, $5,000, $5,730, for portfolio

shares, portfolio return, stock returns, initial and terminal wealth,respectively)

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Statistical Reminder

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Notation and Statistical Reminder

:: x and y are random variables.

:: Notation (µ, σ, σxy , ϕxy )

:: Expected value: µx = E(x), µy = E

(y)

:: Variance: σ2x = Var

(x)

= E(x − µx

)2= E

(x2)− µ2

x

:: Standard Deviation: σx =√σ2x

:: Covariance: σxy = E(x − µx

)(y − µy

)= E

(x y)− E

(x)E(y)

:: Correlation: ϕxy = σxy/(σxσy

)

Portfolio Stats 27

Reminder

:: Linear combinations of random variables: if x and y are random, and a,b, c are constants, and z = a + bx + cy , then

E(z)

= a + bE(x)

+ cE(y)

= a + bµx + cµy

Var(z)

= Var(a + bx + cy

)= b2Var

(x)

+ c2Var(y)

+ 2bcCov(x , y

)= b2σ2

x + c2σ2y + 2bcσxy

= b2σ2x + c2σ2

y + 2bcσxσyϕxy

:: “Linear combinations of normals are normal.” If x and y are normalrandom variables,

x ∼ N(µx , σ

2x

)y ∼ N

(µy , σ

2y

)then, for z = a + bx + cy ,

z ∼ N(µz , σ

2z

)∼ N

(a + bµx + cµy , b

2σ2x + c2σ2

y + 2bcσxy

)Portfolio Stats 28

Reminders

:: Important special cases of z = a + bx + cy :

:: b = c = 0: Var(z)

= Var(a)

= 0

:: c = 0: Var(z)

= Var(a + bx

)= b2σ2

x

:: We typically work with data on correlations not covariances.But, we sometimes need to use the covariance to docalculations. Just remember:

Correlation : ϕxy =σxy(σxσy

)

:: What is the standard deviation?:: Sometimes called “volatility”:: Given an intuitive answer, in terms of the normal distribution. i.e.,

if x ∼ N(µx , σ

2x

), and µx = 10 and σx = 2, what does σx = 2

mean? (answer: the realized value of x is very unlikely to be outside of the interval (6, 14).)

Portfolio Stats 29

Discrete Random VariablesThere are S states of nature, each indexed with s. The probability of state sbeing realized is ps . x and y are discrete-valued random variables than take onvalues xs and ps , respectively. We refer to the means, standard deviations andthe correlation of these random variables as (some of their) “moments.”Formulae for the moments are below.

This is review from your stats class. In order to check that you remember whatyou need to remember, here is a test. There are two states of nature, State 1and State 2. Probs are 0.4 and 0.6, respectively. x and y take on values (18,9)and (25,10) in States 1 and 2, respectively. What are the means and thestandard deviations of x and y and what is their correlation?Here is spreadsheet and a “coding-up” of the formulae that appear below.

I Mean: E(x) =∑

s psxsI Standard Deviation:√

Var(x) =√

E(x2) − E(x)2 =√∑

s psx2s − (

∑s psxs)

2

I Covariance:Cov(x , y) = E(xy) − E(x)E(y) =

∑s psxsys −

∑s psxs

∑s psys

I Correlation: Corr(x , y) = Cov(x , y)/(σxσy )

I Notation: µx ≡ E(x), σx ≡ Stdev(x) =√

Var(x), σ2x ≡ Var(x),

σxy ≡ Cov(x , y), ϕxy ≡ Corr(x , y) = σxy/(σxσy ).

Portfolio Stats 30

Statistical Properties of Portfolio Returns

Portfolio Stats 31

Portfolio Behavior

:: We now know what portfolio returns are.

:: What next? Consider an investor. What does she care about?

:: Higher returns are better than lower ones?

:: But returns are random, uncertain, stochastic ...

:: We’ll focus on “moments” of the return distribution:

:: Mean (“expected return”)

:: Variance, standard deviation (“volatility”)

:: Why? Graph: tradeoff

Portfolio Stats 32

Graph: Expected Return vs VolatilitySame as “Risk versus Return.” µ− −σ space. Consider portfolios, first dominant then not

Portfolio Stats 33

Data

r̃k : Random rate of return stock rk

E[r̃k] = r̄k =X

i

Prob(rk = i) ⇥ i

�2k =

X

i

Prob(rk = i) ⇥ (i � r̄k)2

�k,l =X

i,j

Prob(rk = i, rl = j) ⇥ (i � r̄k)(j � r̄l)

Formulas

Historical Means and Standard Deviations: 1972-2010

Fixed Income

Intermediate Term Bond

Long Term Bond

High Yield BondsInternational Govt Bonds

Commodities

Large Cap Equity

Mid Cap Equity

Small Cap Equity

International Equity

Emerging Market Equity

REITs

05

1015

Aver

age

0 5 10 15 20 25Std_Dev

Inflation Adjusted Returns: Arithmetic Returns

Source: Burton Hollifield

Portfolio Stats 34

Optimal Portfolio Choice

:: Choose portfolio with maximal expected return, conditional ongiven amount of risk (volatility).

:: Sometimes called:

:: Mean-Variance Analysis

:: Markowitz Diversification

:: The property of investors desiring mean-variance-efficientportfolios

:: Maximize the Sharpe Ratio

Portfolio Stats 35

Portfolio Moments

rp = γ1 r1 +(1 − γ1

)r2

:: Expected return:

E rp = γ1E r1 +(1 − γ1

)E r2

µp = γ1µ1 +(1 − γ1

)µ2

:: Variance and volatility (using γ2 = (1 − γ1) to avoid clutter)

Var(rp)

= γ21Var

(r1

)+ γ2

2Var(r2

)+ 2γ1γ2Cov

(r1 , r2

)

σ2p = γ2

1σ21 + γ2

2σ22 + 2γ1γ2σ12

= γ21σ

21 + γ2

2σ22 + 2γ1γ2σ1σ2ϕ12

σp =(γ2

1σ21 + γ2

2σ22 + 2γ1γ2σ1σ2ϕ12

)1/2

Portfolio Stats 36

Portfolio Moments

rp = γ1 r1 +(1 − γ1

)r2

:: Expected return:

E rp = γ1E r1 +(1 − γ1

)E r2

µp = γ1µ1 +(1 − γ1

)µ2

:: Variance and volatility (using γ2 = (1 − γ1) to avoid clutter)

Var(rp)

= γ21Var

(r1

)+ γ2

2Var(r2

)+ 2γ1γ2Cov

(r1 , r2

)

σ2p = γ2

1σ21 + γ2

2σ22 + 2γ1γ2σ12

= γ21σ

21 + γ2

2σ22 + 2γ1γ2σ1σ2ϕ12

σp =(γ2

1σ21 + γ2

2σ22 + 2γ1γ2σ1σ2ϕ12

)1/2

Portfolio Stats 36

Portfolio Moments

rp = γ1 r1 +(1 − γ1

)r2

:: Expected return:

E rp = γ1E r1 +(1 − γ1

)E r2

µp = γ1µ1 +(1 − γ1

)µ2

:: Variance and volatility (using γ2 = (1 − γ1) to avoid clutter)

Var(rp)

= γ21Var

(r1

)+ γ2

2Var(r2

)+ 2γ1γ2Cov

(r1 , r2

)

σ2p = γ2

1σ21 + γ2

2σ22 + 2γ1γ2σ12

= γ21σ

21 + γ2

2σ22 + 2γ1γ2σ1σ2ϕ12

σp =(γ2

1σ21 + γ2

2σ22 + 2γ1γ2σ1σ2ϕ12

)1/2

Portfolio Stats 36

Portfolio Moments

rp = γ1 r1 +(1 − γ1

)r2

:: Expected return:

E rp = γ1E r1 +(1 − γ1

)E r2

µp = γ1µ1 +(1 − γ1

)µ2

:: Variance and volatility (using γ2 = (1 − γ1) to avoid clutter)

Var(rp)

= γ21Var

(r1

)+ γ2

2Var(r2

)+ 2γ1γ2Cov

(r1 , r2

)

σ2p = γ2

1σ21 + γ2

2σ22 + 2γ1γ2σ12

= γ21σ

21 + γ2

2σ22 + 2γ1γ2σ1σ2ϕ12

σp =(γ2

1σ21 + γ2

2σ22 + 2γ1γ2σ1σ2ϕ12

)1/2

Portfolio Stats 36

Portfolio Moments

rp = γ1 r1 +(1 − γ1

)r2

:: Expected return:

E rp = γ1E r1 +(1 − γ1

)E r2

µp = γ1µ1 +(1 − γ1

)µ2

:: Variance and volatility (using γ2 = (1 − γ1) to avoid clutter)

Var(rp)

= γ21Var

(r1

)+ γ2

2Var(r2

)+ 2γ1γ2Cov

(r1 , r2

)

σ2p = γ2

1σ21 + γ2

2σ22 + 2γ1γ2σ12

= γ21σ

21 + γ2

2σ22 + 2γ1γ2σ1σ2ϕ12

σp =(γ2

1σ21 + γ2

2σ22 + 2γ1γ2σ1σ2ϕ12

)1/2

Portfolio Stats 36

Portfolio Moments

rp = γ1 r1 +(1 − γ1

)r2

:: Expected return:

E rp = γ1E r1 +(1 − γ1

)E r2

µp = γ1µ1 +(1 − γ1

)µ2

:: Variance and volatility (using γ2 = (1 − γ1) to avoid clutter)

Var(rp)

= γ21Var

(r1

)+ γ2

2Var(r2

)+ 2γ1γ2Cov

(r1 , r2

)

σ2p = γ2

1σ21 + γ2

2σ22 + 2γ1γ2σ12

= γ21σ

21 + γ2

2σ22 + 2γ1γ2σ1σ2ϕ12

σp =(γ2

1σ21 + γ2

2σ22 + 2γ1γ2σ1σ2ϕ12

)1/2

Portfolio Stats 36

Graphs: Mean and Variance vs. γ1Mean is linear, variance (stdev) are quadratic ... unless ϕ12 = 1. Diversification reduces risk

Portfolio Stats 37

Graph: Distribution of Portfolio ReturnsHow does diversification reduce portfolio risk? Suppose two stocks have same marginal distribution.

Portfolio Stats 38

Dow Hits Record HeightsMarch 2013, Dow hits new record. Note that many Dow stocks went down!

3/6/13 The Dow’s Movers - Graphic - NYTimes.com

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Published: March 5, 2013

The Dow’s MoversChange in each company’s stock price since the Dow’s previous high on Oct. 9, 2007. Related Article »

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Portfolio Stats 39

Graph of Variance vs. Number of SecuritiesDiversification reduces risk ... but typically only up to a point

Portfolio Stats 40

DataStandard deviation of an equally-weighted portfolio against number of stocks, σ = 0.40, ϕ = 0.28

We can increase the diversification benefit by adding

more risky investments

Volatility of an equally weighted portfolio vs. the number of stocks

std=40%, correlation=0.28

Source: Burton Hollifield

Portfolio Stats 41

Systematic vs Idiosyncratic RiskWhy are there limits to risk-reducing diversification?

Portfolio Stats 42

Statistical Model of Prices/Returns(Moments versus Models)

Portfolio Stats 43

Why Have a Model?

:: Moments:

:: Statistical moments are things like mean, variance, covariance,skewness, kurtosis, etc.

:: They are properties of statistical distributions.

:: Up to now, we’ve just studied moments of returns

:: We can use data to estimate them

:: For some questions it is necessary to develop a full model ofthe statistical distribution.

:: Example: Value-at-Risk. “What is the most I might lose onmy portfolio, with 95% probability?”

Portfolio Stats 44

Continuous DistributionsNormal, Lognormal, Fat-Tailed, Skewed, etc.

Portfolio Stats 45

Discrete Distributions

There are three states of nature and three assets. The asset’s prices and thedistribution of the asset’s payoffs are as follows

State of Nature (s) 1 2 3 PriceProbability (ps) 0.3 0.4 0.3

Bond 100 100 100 97.00Asset 1 (c1s) 70 102 120 91.27Asset 2 (c2s) 88 96 110 93.91

:: Moments: mean, standard deviation, covariance (i = 1, 2)

E(ci ) =∑s

pscis

Stdev(ci ) =(E(c2

i ) − E(ci )2)1/2

=(∑

s

psc2is − E(ci )

2)1/2

Cov(c1 c2) = E(c1c2) − E(c1)E(c2) =∑s

psc1sc2s − E(c1)E(c2)

Portfolio Stats 46

Moments of Discrete Distributions

E(c) Stdev(c) Cov(c1 c2) Corr(c1 c2)

Asset 1 97.80 19.67 159.96 0.94Asset 2 97.80 8.65

Portfolio Stats 47

Returns

Distribution:

State of Nature (s) 1 2 3Probability (ps) 0.3 0.4 0.3

Bond 0.031 0.031 0.031Asset 1 (r1s) −0.233 0.118 0.315Asset 2 (r2s) −0.063 0.022 0.171

Moments:1

E(r) E(r) − rf Stdev(r) Cov(r1 r2) Corr(r1 r2)

Asset 1 0.072 0.041 0.215 0.019 0.941Asset 2 0.041 0.010 0.092

1rf = 0.031 is the risk-free interest rate, so that E(r) − rf is the equitypremium: the excess expected return on equity (excess means “in excess of therisk-free rate)

Portfolio Stats 48

Summary

Don’t confuse moments with models

:: Moments:Expected Return Standard Deviation

Riskless Bond 0.0500 0.00Stock 1 0.10 0.18Stock 2 0.16 0.22Correlation −0.50

:: What is the expected return and volatility of a portfolio?

:: Models (here are two examples):

r1 ∼ N(µ1 , σ

21

)r2 ∼ N

(µ2 , σ

22

)State of Nature 1 2Probability 0.5 0.5Asset 1 110 110Asset 2 98 118

:: What is Prob(rp < −0.10

)?

Portfolio Stats 49

Summary: Risk and Return, Diversification

Portfolio Stats 50

Summary

:: Historical equity premium is about 6%.

:: Compute OCC by adding risk-adjusted equity premium to today’s

riskless rate

:: Construct portfolio returns as value-weighted averages

:: Portfolio statistics (moments)

:: Mean and variance of linear combinations ... correlations show up.

:: Diversification reduces portfolio risk .... but only up to a point:

systematic risk

:: Modeling the distribution, versus estimating the moments.

:: Discrete distributions.:: Moments are probability-weighted averages of realizations.

Portfolio Stats 51

Importance of “Principle of Diversification”

Phillips from FT, Aug 30 2016

Portfolio Stats 52

Expected Returns

Don’t confuse expected returns with realized returns

:: Example: Sony’s Blu-Ray wins technological contest againstToshiba’s HD-DVD.

Portfolio Stats 53

Spaces

It is important to understand the various “spaces” that the variousgraphs appear in.

:: Time-series space (return against time)

:: Frequency space (return frequency distribution)

:: Mean-variance (or mean-standard deviation) space

:: Captures the risk-return tradeoff

:: See spreadsheets

Portfolio Stats 54