I.  Portfolio Theory

• investor chooses a group of assets—what affects this decision?
• assume that investors are risk averse
• must be compensated for taking on risk, i.e. there is a risk /return tradeoff
• assets with high expected return also carry a large amount of risk
• assets with low risk also have a low expected return
• an efficient or optimal portfolio either
• maximizes expected return for a given level of risk OR
• minimizes risk for a given level of expected return
• in this chapter we want to measure risk and return, and study asset pricing models that attempt to quantify the risk/return tradeoff
• Measuring Return and Expected Return
• Return, R = (change in value of asset + income)/initial value
• R is ex post, meaning it is based on past data, and is known
• R can be measured over any time interval, although returns are typically annualized
• example 1: Tbill, 1 month holding period, buy for $9488, sell for $9528
• 1 month R = (9528-9488)/9488 = .0042 = .42%
• annualized R: (1.0042)^12 = 1.052 or 5.2% annually
• example 2: 100 shares IBM stock, 9 month holding period, buy for $62, sell for $101.50, .80 dividends
• 9 month R = (101.50 - 62 + .8)/62 = .65 = 65%
• annualized R: (1.65)^(12/9) = 1.95 or 95% annually

• Expected Return, E(R) = SUM (R_i Prob(R_i))
• measuring likely future return, based on the probability distribution, and is a random variable
• example 1:

• E(R) = (.2)10% + (.4)5% + (.4)(-5%) = 2%
• 2% is the center of the distribution, in terms of likelihood
• example 2:

• E(R) = .3(1%) + .4(2%) + .3(3%) = 2%
• same expected value BUT not same return structure
• returns in example one are more variable, deviation from the expected value is more likely
• Risk
• measures likely fluctuation in the value of an asset or portfolio
• how much can actual R vary from E(R)?
• how likely is actual R to vary from E(R)?
• measured by variance (s² ) or standard deviation (s )
• SUM (R_i - E(R))Prob(R_i)
• example 1 (see above)
• s² = (10%-2%)²(.2) + (5%-2%)²(.4) + (-5%-2%)²(.4) =.0036 or s = 6%
• example 2 (see above)
• s² = (1%-2%)²(.3) + (2%-2%)²(.4) + (3%-2%)²(.3) = .00006 or s = .77%
• example one has greater risk, so investors would prefer two, given the E(R)'s are equal
• s is one measure of risk, works best with symmetric distributions

• achieve an optimal portfolio through diversification
• holding a group of assets allows investors to lower risk, often without lowering E(R)
• why does diversification work?
• individual assets do not have same pattern of returns
• combining assets, bad returns cancelled out by good returns of other assets, decreasing overall variation in the return of the portfolio
• two types of risk
• unsystematic risk
• specific to a firm or industry
• can be eliminated through diversification
• examples
• USAir and prospective strike
• Microsoft and antitrust settlement
• systematic risk
• variation in return due to factors affecting all financial assets
• oil prices, interest rates, business cycles, inflation
• present in all portfolios--cannot be eliminated by diversifying
• example:
• about 40-50% of portfolio risk can be eliminated by moving from 1 stock to 20 stocks from NYSE
• Measuring relative risk
• if unsystematic risk can be reduced by diversifying, then s , a measure of total risk is not useful
• more interested in systematic risk of an asset
• measure systematic risk by measuring variation in returns of an asset or portfolio relative to a very well-diversified portfolio
• b = (% variation in asset return/% variation in return of market portfolio)
• market portfolio is portfolio of all financial assets, not really measurable
• use a proxy, like portfolio of S&P 500 or NYSE stocks
• If b = 1, portfolio is very well-diversified
• If b > 1, asset/portfolio is riskier than market (more sensitive to systematic risk factors)
• If b < 1, asset/portfolio is less risky than market
• If b = 0, asset/portfolio is risk free
• measuring b
• estimated by regression:
• problems with estimating b
• length of return interval (weekly)
• choice of market portfolio
• # of observations
• time period

• CAPM, Capital Asset Pricing Model
• 1964, Sharpe, Lintner
• assume:
• investors hold risky and a risk-free asset in their portfolio
• no transactions costs, taxes
• same expectations and time horizon
• risk averse investors
• implications:
• expected return on an asset or portfolio is a function of its b , the risk free return, and the market return
• E(R) = R_f + b [E(R_m) - R_f]
OR
E(R) - R_f = b [E(R_m) - R_f]
• where E(R) - R_f is the risk premium of a portfolio
• where E(R_m) - R_f is the market risk premium
• note that if b > 1, the risk premium of a portfolio is greater than the market risk premium, and thus its expected return will be greater than the expected market return
• more risk, higher expected return
• if b < 1, the risk premium of a portfolio is less than the market risk premium, and thus its expected return will be less than the expected market return
• less risk, lower expected return
• the CAPM tells us the exact size of the risk/return tradeoff; i.e. the price of risk
• Testing the CAPM
• tests of the CAPM have overpredicted returns (return predicted by CAPM greater than actual returns)
• portfolios with higher b have higher realized returns
• tests of the CAPM have found that other factors (other than the market risk premium) are important in determining returns
• January effect, firm size, day of the week, weather,
• problems with testing the CAPM--Roll Critique (1977)
• CAPM is not testable because
• true market portfolio is not observable (use R for S&P 500 or NYSE portfolio)
• true risk-free rate is not observable (use 3 mo. Tbill yield)
• do not observe E(R), only data on past returns
• APT, Arbitrage Pricing Theory
• 1976, Ross
• assume:
• several factors affect E(R) of an asset/portfolio
• does not specify how many factors, or what are the factors
• implication:
• expected return is a function of several factors, each with their own b , measuring the sensitivity of an asset or portfolio to that particular factor
• E(R) - R_f = b₁F₁ + b₂F₂ + . . . + b_NF_N
• note that CAPM is a special case of the APT, with one factor, the market risk premium, and the APT is more general, allowing for multiple unspecified factors
• Testing the APT
• have to test how many factors, and what are the factors
• 1980 Chen, Roll, and Ross
• changes industrial production, inflation, yield curve slope, and yield speads are all important economic factors