Portfolio Theory Finance - Pedro Barroso1

Motivation Mean-variance portfolio analysis – Developed by Harry Markowitz in the early 1960’s (1990 Nobel Prize in Economics) – Foundation of modern finance Used by all mutual funds, pension plans, wealthy individuals, banks, insurance companies,... There is an industry of advisors (e.g. Wilshire Associates) and software makers (e.g. BARRA, Quantal) that implement what we will learn in the next few classesWilshire AssociatesBARRA Quantal Finance - Pedro Barroso2

Modern Investment Advice The optimal portfolio of risky assets should contain a large number of assets – it should be well diversified – and is the same for all investors Investors should control the risk of their portfolio not by re-allocating among risky assets, but through the split between risky assets and the risk-free asset Finance - Pedro Barroso3

Individual Securities Characteristics of individual securities that are of interest are the: – Expected Return – Variance and Standard Deviation – Covariance and Correlation (to another security or index) Finance - Pedro Barroso4

Expected (mean) return where r s :return if state s occurs p s : probability of state s happening Finance - Pedro Barroso5

Example Consider the following two risky asset world. There is a 1/3 chance of each state of the economy, and the only assets are stock A and stock B Finance - Pedro Barroso6

Expected Return Finance - Pedro Barroso7

Variance and Standard Deviation Shortcut Standard Deviation is the square root of variance Finance - Pedro Barroso8

Variance Finance - Pedro Barroso9

Standard Deviation Finance - Pedro Barroso10

Covariance Measure of movement in tandem Correlation: Finance - Pedro Barroso11

Covariance and Correlation Finance - Pedro Barroso12

Correlation Correlation measures relationship between the return on a stock and the return on another: –Perfect positive correlation: 1 –No correlation: 0 –Perfect negative correlation: -1 Finance - Pedro Barroso13

Correlation Finance - Pedro Barroso14

Estimating Means and Covariances In real life we do not know probability of each state of the world and the return that corresponds to it We need to use historical data to estimate average returns, variance and covariance of returns Finance - Pedro Barroso15

Estimating Means and Covariances We can use the functions average(), var(), stdev(), covar(), correl() in Excel We are implicitly assuming that the returns came from the same probability distribution in each year of the sample The estimated mean and variance are themselves random variables since there is estimation error that depends on the particular sample of data used (sampling error) – We can calculate the standard error of our estimates and figure out a confidence interval for them – This contrasts with the true (but unknown) mean and variance which are fixed numbers, not random variables Finance - Pedro Barroso16

Annualizing Mean and Covariances Annual return is approximately equal to the sum of the 12 monthly returns; assuming monthly returns are independently distributed (a consequence of market efficiency) and have same variance If mean, standard deviation or covariance are estimated from historic monthly returns, estimates will be per month To annualize: – mean, variance, covariance: multiply by 12 – standard deviation: multiply by sqrt(12) Finance - Pedro Barroso

Portfolios Weights: fraction of wealth invested in different assets – add up to 1.0 – denoted by w Example – $100 MSFT, $200 in GE Total investment: $100+$200=$300 – Portfolio weights MSFT: $100/$300 = 1/3 GE: $200/$300 = 2/3 Can we have negative portfolio weights? Finance - Pedro Barroso18

Portfolios You can have negative weights if you short sell a stock – Borrow stock from broker – Sell stock and get proceeds (stock price) – Buy stock back later to give it back – Profit/loss = sell price - buy price Example – $500 MSFT (buy), $200 in GE (short sell) Total investment: $500-$200=$300 – Portfolio weights MSFT: $500/$300 = 5/3 GE: -$200/$300 = -2/3 Finance - Pedro Barroso19

Portfolio Expected Return Portfolio return – Average of returns on individual securities weighted by their portfolio weights Then expected return on the portfolio Remember from stats that E(aX+bY)=aE(X)+bE(Y) Finance - Pedro Barroso20

Portfolios (60% Stock A, 40% Stock B) Expected rate of return on the portfolio is a weighted average of the expected returns on stocks in portfolio: Finance - Pedro Barroso21

Portfolio Variance Variance of a portfolio is Remember Var(aX+bY)=a 2 Var(X)+b 2 Var(Y)+2abCov(X,Y) Finance - Pedro Barroso22

Portfolios Variance of the rate of return on the two stock portfolio : Finance - Pedro Barroso23

Portfolios Observe the decrease in risk that diversification offers Portfolio with 60% in stock A and 40% in stock B has less risk than either stock in isolation Finance - Pedro Barroso24

Efficient Frontier - Two Stocks We can consider other portfolio weights besides 60% in stock A and 40% in stock B … 100% A 100% B

Efficient Frontier - Two Stocks Note that some portfolios are “better” than others; they have higher returns for the same level of risk or less Portfolios in the frontier above the MVP are efficient 100% A 100% B MVP

Minimum Variance Portfolio (MVP) Portfolio with lowest possible variance Finance - Pedro Barroso27

Portfolios with Different Correlations 100% B E(return)  100% A  = 0.2  = 1.0  = -1.0 Relationship depends on correlation coefficient – If  = +1.0 no diversification effect – If  < 1.0 some diversification effect – If  = –1.0 diversification can eliminate all the risk Finance - Pedro Barroso28

Portfolios with Different Correlations Standard deviation of a portfolio is – If  = +1.0 – If  < 1 – If  = -1.0 Finance - Pedro Barroso29

Portfolio with Many Stocks For portfolio with N stocks, we need: – N expected returns (one for each asset) – N variances (one for each asset) – N(N-1)/2 covariances (for each pair of assets) Finance - Pedro Barroso30

Opportunity Set for Many Stocks Consider a world with many risky assets; we can still identify the opportunity set of risk-return combinations of various portfolios (it is an area rather than a line) E(return) PP Individual Assets Finance - Pedro Barroso31

Efficient Frontier for Many Stocks Section of opportunity set above minimum variance portfolio is efficient frontier (north-west edge): -offers minimum risk for a given expected return -offers maximum expected return for a given risk E(return) PP Minimum Variance Portfolio Efficient frontier Individual Assets Finance - Pedro Barroso32

Diversification and Portfolio Risk Diversification can substantially reduce the variability of returns without an equivalent reduction in expected returns This reduction in risk arises because worse than expected returns from one asset are offset by better than expected returns from another However, there is a minimum level of risk that cannot be diversified away, and that is the systematic portion Finance - Pedro Barroso33

Portfolio Risk and Number of Stocks Systematic Risk Market Risk, N on-diversifiable risk Idiosyncratic Risk Diversifiable Risk, Nonsystematic Risk Firm Specific Risk Number of stocks  In a large portfolio the variance terms are effectively diversified away, but the covariance terms are not Portfolio risk

Limits to Diversification Consider equal-weighted portfolio w i =1/N portfolio variance depends on average stock variance and average correlation among stocks When N grows: Average correlation is 0.20, average standard deviation is 50%, so minimum portfolio volatility about 22% Finance - Pedro Barroso35

Systematic Risk Risk factors that affect a large number of assets Also known as non-diversifiable risk or market risk Includes such things as changes in GDP, inflation, interest rates, etc. Finance - Pedro Barroso36

Idiosyncratic Risk Risk factors that affect a limited number of assets Includes such things as labor strikes, part shortages, earnings announcements Risk that can be eliminated by combining assets into a portfolio (need about 60 stocks) If we hold only one asset, or assets in the same industry, then we are exposing ourselves to risk that we could diversify away Finance - Pedro Barroso37

Total Risk Total risk = systematic risk + idiosyncratic risk Standard deviation of returns is a measure of total risk For well-diversified portfolios, idiosyncratic risk is very small Consequently, the total risk for a diversified portfolio is essentially equivalent to the systematic risk Finance - Pedro Barroso38

Riskless Borrowing and Lending Allow for lending and borrowing at risk-free rate (T-bill) – zero risk and zero covariance with stock returns Capital allocation line (CAL): feasible combinations of stock(s) and riskless asset CAL Slope is Sharpe ratio : excess return per unit of risk E(return) PP Efficient Frontier rfrf Finance - Pedro Barroso39

Efficient Frontier with Riskless Asset Efficient frontier is the capital allocation line (CAL) with the steepest slope Investors allocate their wealth between riskless asset and tangency portfolio rfrf E(return) Tangency portfolio T PP Finance - Pedro Barroso40

Efficient Frontier with Riskless Asset If E(r P )< E(r T ): riskless lending, w T < 100% If E(r P ) > E(r T ): riskless borrowing, w T > 100% rfrf E(return) Tangency portfolio T PP Finance - Pedro Barroso41

Tangency Portfolio – Two Stocks Portfolio weights of tangency portfolio (max Sharpe) Finance - Pedro Barroso42

Tangency Portfolio – Two Stocks Using our previous example of two stocks (A and B) and a riskless rate of 8%: – stock A weight 0.69, stock B weight 0.31 – expected return 10.17%, std.dev. 9.54%, Sharpe Investor wants to form portfolio with an expected return of 9%: – Combine tangency portfolio with riskless asset – E(r P ) = 9% = 10.17% x w T + 8% x (1 – w T ) w T = 46% (w A = 0.46 x 0.69 = 32%, w B =0.46 x 0.31 = 14%) w f = 54% –  P = 0.46 x 9.54% = 4.4% Finance - Pedro Barroso43

Optimal Portfolio Investor with quadratic utility function – Only cares about mean and variance of returns – How do indifference curves plot? – Where  is coefficient of risk aversion: e.g.  = 4 – How do we determine  ? With questionnaires How much would you pay to avoid a chance of doubling or losing x dollars? Finance - Pedro Barroso

Optimal Portfolio To find optimal portfolio choice Finance - Pedro Barroso45

Optimal Portfolio: Example Optimal combination of tangency and risk-free asset for an investor with risk aversion γ = 4 Optimal weight on the tangency portfolio: 0.6 So the weight on the risk-free asset is 0.4 To find the weights on stocks, multiply the weight on T by the weights that stocks have in T – Weight on stock A: 0.6 × 0.69 = 0.41 – Weight on stock B: 0.6 × 0.31 = 0.19 Expected return 9.30%, standard deviation 5.69% What if the risk aversion coefficient was 2? Finance - Pedro Barroso46

Tangency portfolio – Many Stocks Portfolio weights of tangency portfolio must be solving the optimization problem (max Sharpe): We can obtain weights using Solver in Excel Finance - Pedro Barroso47