Ch 6.Risk, Return and the CAPM

Goals: To understand return and risk To understand portfolio To understand diversifiable risks and market (systematic) risks To understand CAPM

1.Investment returns Dollar return = amount received –amount invested Problems: Scale effect Times spent or holding period Rate of return = (amount received – amount invested)/amount invested

2. Stand alone risk Risk: the chance that some unfavorable event will occur. It is measured by variance or standard deviation – distance from the average. (Stand-alone risk: the risk an investor would face if he or she only held one asset) Ex) Historical return and risk

Historical return and risk

No investment will be undertaken unless the expected rate of return is high enough to compensate the investor for the perceived risk To deal with unknown futures, a probability will be applied.

1) Probability distributions Probability is the chance that the event will occur. Probability distribution: A listing of all possible outcomes or events with a probability assigned to each outcome. Discrete & Continuous probability distribution 68.26% of probability that actual return will be within a mean + or – 1 *standard deviation % of probability that actual return will be within a mean + or – 2 *standard deviation 99.74% of probability that actual return will be within a mean + or – 3 *standard deviation

Ex) State Probability (P) S B Strong Normal Weak ) Expected Rates of Returns and Risks: Stand-alone (1)Expected return

E(k s ) = 0.3* * *0.25=0.15 E(k B ) = (2) Variance: risk measurement The tighter the probability distribution, the more likely it is that the actual return will be close to the expected value. Thus the tighter the probability distribution, the lower the risk assigned. It is measured by variance

How to calculate? Standard Deviation

(3) The coefficient of variation: standardized measure of the risk per unit of return. It provides a more meaningful basis for comparison CV= CVs= /0.15 =0.707

(4) Risk Aversion and Required Returns Risk Aversion: Investors dislike risk and require the higher rate return as an inducement to buy riskier securities. It is commonly assumed in finance. Therefore, if other things held constant, the higher a security’s risk, the lower its price and the higher its required (expected) returns.

3) Portfolios: Return and Risks Portfolios: combination of assets or securities to minimize total risks and to improve returns (1) Portfolio Returns: Simply the weighted average of the expected returns on individual assets in the portfolio

Equation

Ex) using the previous example with an assumption that weights for S and B are 40% and 60%. Expected returns = 0.4(0.15)+0.6(0.175)=0.165

Realized Rate of Return (k bar): The return actually earned during some past period. It differs from the expected returns (2) Risks Covariance: measurement of the degree which two assets covary Correlation coefficient: Standardization of covariance

Formulas

Portfolio variance (Risks)

In our example, Ws=0.4 W B =0.6 r SB = σ s = σ B = Portfolio risk is

Lessons from Portfolio variance calculation (1) If the numbers of securities composing a portfolio increase, the number of covariance will be greater than number of variances. Thus, the variance of well diversified portfolio reflects the covariance. (2) If the correlation is less than 1, the covariance will reduce, leading to lower portfolio risks. Diversification effects (3) If the correlation is 1, the covariance will increase to the maximum value

(4) If the correlation is -1, then the variance will reduce to the minimum value. Diversification effects (5) In reality, securities’ correlations are between 0 and 1. 4) Diversifiable Risk and Market Risk Systematic risk Unsystematic risk Total Risk =Market risk+Firm specific risk Undiversifiable riskDiversifiable risk Nondiversifiable risk

Diversifiable risk: That part of a security’s risk associated with random events, diversified away Market risk: The part of a security’s risk that can not be eliminated by diversification If the diversifiable risks can be diversified away, investors are mainly concerned about the systematic (un-diversifiable) risks. They may consider the systematic risk in individual stocks or contribution to risks of well diversified portfolio to be risks compensated.

The systematic risk is a market risk inherent in market 4) The concept of beta How to measure the systematic risk? It can be measured by the degree to which a given stock tends to move up or down with the market (stock market). It is called “beta”

Three meanings of beta to changes in the return of the market portfolio. [the comovement of an asset i's return (price) to the market portfolio's return (price)] beta is a measure of the undiversifiable (market, systematic) risk of the asset i.

Graphically, it is a slop of a line in the scatter plot composed of market returns and individual stock returns Beta of a portfolio 5) CAPM (Capital Asset Pricing Model) Based on the beta estimate, we are able to come up with CAPM that will estimate required (expected) rate of return in individual stock.

CAPM: = risk free rate + market risk premium*beta k RF : risk free rate k M : market return Ex) Currently, T-bill and S&P 500 are offering 3% and 5%. F503 stocks’ beta is 0.7. Required rate of return = ( )

SML (security market line): Equilibrium relationship between the expected returns and the systematic risk of assets (individual stocks or portfolios). Graphical description of CAPM. Underpriced: when an asset lies above the SML. Then demand will increase and price will increase, leading the current return to a required return on SML (CAPM).

.. Repo. Alta T-bills. Am. Foam r M = 15 r RF = r i (%) Risk, b i Market

Overpriced: when an asset lies below the SML. Then demand will decrease and price will decrease, leading the current return to a required return on SML (CAPM). Therefore, in equilibrium, all assets lies on the SML. 6) Impacts of Inflation. It will not change market risk premium but increase risk free rate, shifting SML upward

Thus, it will increase a required rate of return from CAPM 7) Impact of risk aversion. The slope of SML reflects the extent to which investors are averse to risk. The steeper the slope of the line, the greater the average investor’s risk aversion. The increasing risk aversion will increase market premium but not change risk free rates

SML rotates counterclockwise. Equilibrium required rate of return according to the CAPM increases for assets with positive . Equilibrium required rate of return according to the CAPM decreases for assets with negative . 8) Impact of changing betas. Without changing risk free rate and slop of SML, it will increase a required rate of return