課程三：風險與報酬 Risk and Return This note is for lecture use only. 本講義僅供上課教學之用。

報酬Returns Holding period return (HPR)

算術平均Arithmetic average Sum of returns in each period divided by number of periods 幾何平均Geometric average The single per-period return that gives the same cumulative performance as actual returns Required for mutual fund literature Dollar-weighted return Treat cash flows like capital budgeting problem and calculate the internal rate of return (IRR)

例題 算術平均: (0.14 -0.1455 + 0.10) / 3 = 0.0315 = 3.15% 幾何平均: (1 + RG) = (1 + 0.14) x (1 - 0.1455) x (1 + 0.10) RG = [(1 + 0.14) x (1 - 0.1455) x (1 + 0.10)]1/3 - 1 RG = 0.0233 = 2.33%

Annual percentage rate - APR Usually what most people imagine when they think of returns. Ignores compounding of interest on interest Simply (rate per period) x (number of periods) Effective annual rate - EAR Corrects APR for interest on interest compounding EAR= (1 + APR/n)n - 1 當 n趨向無窮大 EAR = eAPR - 1

Risk風險 Returns are important, but they can’t be the sole driver of investment decisions Returns are uncertain and we need a way to quantitatively measure the uncertainty An intuitive measure should take into account how likely are the returns Probability distributions of return

重要統計量 Mean: What is the expected value? Median: What is the middle value? Mode: Which value occurs most frequently? Variance: How compact is the distribution? 偏度Skewness: Is the distribution symmetric? 峰度Kurtosis: What do the tails look like? 常態分配只需均數及變異數即夠描述整個分配。

Expected Returns & Risk Variance

資產配置Asset Allocation 存在兩資產，一為風險性，一為無風險性。 Risky asset Risk-free asset E(ra) = 15% 2(ra) = 22% Risk-free asset rf = 7% What portfolios can we hold? We can invest y(%) in the risky asset and (1-y) in the risk-free asset E(rp)= E(ra) y + rf (1-y)  p = y  a (特例)

資產配置線Capital Allocation Line (CAL) Both the risk premium and the standard deviation of the portfolio increase with weight in the risky asset Varying the weights gives us all portfolio combinations, which fall on a single line - the Capital Allocation Line (CAL) The slope of the (CAL) is the Reward to Variability Ratio

Optimal Portfolio Selection What happened to concept of risk aversion? Investors are assumed to be risk averse so that they only accept risky security if it provides compensation via risk premium. How does that impact our CAL approach? How do we pick the risky portfolio? Active versus passive management Capital Market Line (CML)資本市場線

多角化Efficient Diversification We showed that with an optimal risky portfolio, all investment will be on the CAL How do we select the optimal risky portfolio? Why are portfolios of securities better than single securities? What do we mean by diversification? Why diversify?

Covariance and Correlation What is covariance? Is this important? Is there a difference between covariance and correlation?

Portfolios of Securities Investors’ opportunity set is comprised not only of sets of individual securities but also combinations, or portfolios, of securities The return on a portfolio is the weighted average of returns on component securities: The expected return is also a weighted average

Portfolio Risk HOWEVER, the standard deviation of a portfolio is NOT just a weighted average of securities standard deviations. We also need to account for their covariances. Example with 2 securities: X and Y

Portfolio Risk Will portfolio standard deviation be higher or lower than simple weighted average of standard deviations?

Portfolio Risk Variance of portfolio of TWO securities: What happens to risk if two securities are perfectly positively correlated? Perfectly negatively? What about general case? Intuitively, what implications can we infer for efficient portfolio selection strategies? cov(a,b)=stda *stdb * correl stock--ws=.25, wb=.75 rs=12 std=25% std s=20% std b=10% p=.1 Rp=.25(12)+.75(9)=9.75% var=.25^2(.20)^2+.75^2(.1)^2*+2*.25*.75(.2)*(.1) *.1=.008875 or .88% std=9.42% p140 in book is wrong!!! Lower std--higher return=--power of div. you can’t avg std’s!!!!!!!

For N securities, in general, the formula is: The first term is a complex average of securities variances; the second term captures N(N-1) covariance terms. Intuitively, what happens to the portfolio’s variance as N gets large?

Risk Diversification As N gets large, the number of covariances outnumber variances Covariances among US stocks tend to be lower on average than stock’s own variances (Fisher & Lorie, 1966) Problem: Derive risk formula of an equally-weighted portfolio, i.e.

What Affects Risk? Market risk市場風險，系統風險，不可分散的風險 Risk factors common to the whole economy Systematic or non-diversifiable Firm specific risk公司個別風險，非系統風險，可分散的風險 Risk that can be eliminated by diversification Unique risk Nonsystematic or diversifiable 2 Factors Macroec Factors--infl, bus.Cycle, int. rates, fx rate Firm specific-marketing, product, R&D, location, philosophy Gm and Limited both affected by the business cycle If drought and cotton prices increase it will affect the Limited, but not GM who makes cars

效率前緣 Understanding the return and risk attributes of portfolios of individual securities allows us to construct more efficient combinations which strategically attempt to “reduce risk as much as possible for a given level of expected return.” How do we do it? We work in a mean-variance framework Assumes all investors prefer higher returns, all else equal Assumes all investors prefer lower risk, all else equal

Efficient Portfolios A perfectly negative correlations B --low correlations--say zero here C--perfect positive

效率前緣 For N securities, the problem of identifying efficient portfolios is similar, except that we need special skills in linear programming Efficient Frontier

Optimal Risky Portfolio Efficient Frontier Utility 1 Preferred Direction Tangency Portfolio “T” Assumption: No Riskless Asset Available

Capital Allocation Line “M” Efficient Frontier Preferred Direction Optimal Risky Portfolio “M” Capital Allocation Line “A” A Assumption: Riskless Asset Now Available

Utility 2 Capital Allocation Line Utility 1 Efficient Frontier Optimal Risky Portfolio “M” T Optimal Allocation “C” Assumption: Riskless Asset Now Available

涵義 Optimal Portfolio Selection requires 2 steps: Optimal Risky Portfolio Determination Capital Allocation Decision All rational risk-averse investors will passively index holdings to the market portfolio and the risk free asset. “Two fund separation” Principle 1. optimal port deter-- determine efficient frontier. A technical step. Use estimates of E(r) and STD 2. Pick a spot on the CAL based on Risk Aversion

借貸利率不一樣時 Assumption: rf does not equal rb CAL Efficient Frontier Optimal Risky Portfolio “M” rb T rf Assumption: rf does not equal rb

資本資產定價模式 Capital Asset Pricing Model The CAPM is a centerpiece of modern finance that gives predictions about the relationship between risk & expected return Based on original work on portfolio theory of Harry Markowitz by William Sharpe & John Lintner in 1965-66. Begins with simplistic assumptions for hypothetical world of investors and builds into reasonable & comprehensive model

Assumptions Investors are price takers One-period investment horizon (“myopic”) Fixed quantities of assets and all marketable No taxes, transactions costs, regulations, etc Investors are mean-variance optimizers All investors analyze securities in same way with same probabilistic forecasts for each - homogenous expectations 6 assumptions

Investors hold Market Portfolio All investors will identify same optimal risky portfolio, “M” to combine with riskless asset For supply/demand to clear, the holdings of each security will be by relative market value outstanding M =“Market portfolio” M=“Market”

Passive Indexing is Efficient Market portfolio must be on efficient frontier and it is tangent point for the best feasible capital allocation line Rational investors will passively hold an equity index fund & a money market fund In 1991 75% of $275b of institutional funds were “indexed” Capital Market Line M CML is the best obtainable CAL

Equilibrium Expected Returns CAPM is built on insight that appropriate risk premium on an asset is determined by contribution to risk of investor’s overall portfolio. Portfolio risk is what matters “Market price of risk” or is the benchmark tradeoff for risk & return, because all investors holdings are on CML How does any individual security contribute to the risk of a well-diversified portfolio like the market portfolio?

Equilibrium Expected Returns Since we can diversify away firm-specific risk, should it be rewarded? We use beta as a measure of systematic risk. What about the betas of portfolios?

Equilibrium Expected Returns In equilibrium, all assets (and all portfolios) should have the same reward to risk tradeoff. However, this implies that this should hold for the market portfolio as well. We have a simple expression for expected returns on any asset or portfolio. Beta is the only risk that is priced

Security Market Line證券市場線 SML M CML--function of std SML-function of Beta If a security plots off the Security Market Line, its expected return is different from its “fair” return, or it is “mispriced.”

SML Versus CML Security Market Line Examines individual asset risk premiums against risk measure appropriate for individual assets. With individual assets, the only relevant risk is systematic risk, hence we examine beta. Capital Market Line Examines efficient portfolio risk premiums against appropriate risk measure. With well diversified portfolios, the relevant measure of risk is total risk, hence we examine standard deviation.

Application of CAPM Two professional money managers are being evaluated. One averaged 19% last year and the other only 16%. However, the first manager’s beta was 1.5 and the second manager had a beta of 1.0. Which manager performed better? If the market risk premium were 8% and T-bills were yielding 6%, which is better? What if market risk premium is 12 % and T-bills yield 3%?

The Market Model Alphas and betas are measured statistically using historical returns on the security and the market portfolio proxy, e.g. S&P 500 Simple regression model, known as Market Model, is used (in excess returns): Can we test if CAPM is true doing this? Are there testable implications? Theoretically the CAPM is based on the market portfolio and expected returns We must use past returns and to estimate a beta

市場模式的應用

Arbitrage Pricing Theory套利模式 Steve Ross in 1977 An arbitrage opportunity arises when an investor can construct a zero-investment portfolio that will yield sure profits in future A zero-investment portfolio is one in which some securities are long, others short with no commitment of investor’s money Arbitrage-- creation of profits without risk

CAPM vs APT Expected returns are related to multiple sources of risk (APT) vs only market (beta) risk (CAPM) No special role for market portfolio in APT Equilibrium achieved by arbitrage in APT; CAPM requires rational risk-averse, mean-variance optimizing investors.