CHAPTER 5 Risk and Rates of Return Return Return Stand-alone risk Stand-alone risk Portfolio risk:Portfolio Theory Portfolio risk:Portfolio Theory Risk & return: CAPM/SML Risk & return: CAPM/SML Arbitrage Pricing Theory(APT) Arbitrage Pricing Theory(APT)

第一節 報酬率

第二節 風險 1. 企業面臨之風險 圖 8-2 企業面臨之風險

Measuring Risk and Return? Risk refers to the chance that some unfavorable event will occur. An asset’s risk can be analyzed in two ways:  On a stand-alone basis, where the asset is considered in isolation; and  On a portfolio basis,where the asset is held as one of a number of asset in a portfolio.

What is investment risk? Investment risk pertains to the probability of earning less than the expected return. The greater the chance of low or negative returns, the riskier the investment.

Probability Distribution  An asset’s probability is defined as the chance that the asset will occur.  If all possible events,or outcomes,are listed,and if a probability is assigned to each event,the listing is called a probability distribution. (P.D.F.:prob. Density function)

Probability distribution Expected Rate of Return Rate of return (%) Firm X Firm Y

Annual Total Returns, AverageStandard ReturnDeviationDistribution Large-company stocks 12.7% 20.3% Small-company stocks Long-term corporate bonds Long-term government Intermediate-term government U.S. Treasury bills Inflation % 0% 90%

Investment Alternatives (Given in the problem) EconomyProb.T-BillHTCollUSRMP Recession %-22.0% 28.0% 10.0%-13.0% Below avg Average Above avg Boom

Why is the T-bill return independent of the economy? Will return the promised 8% regardless of the economy.

Do T-bills promise a completely risk-free return? No, T-bills are still exposed to the risk of inflation. However, not much unexpected inflation is likely to occur over a relatively short period.

Calculate the expected rate of return on each alternative: k = expected rate of return. k HT = (-22%)0.1 + (-2%) (20%) (35%) (50%)0.1 = 17.4%. ^ ^

k HT17.4% Market15.0 USR13.8 T-bill8.0 Coll.1.7 HT appears to be the best, but is it really? ^

What ’ s the standard deviation of returns for each alternative?

Prob. Rate of Return (%) T-bill USR HT

Standard deviation (s i ) measures total, or stand-alone, risk. Standard deviation (s i ) measures total, or stand-alone, risk. The larger the s i, the lower the probability that actual returns will be close to the expected return. The larger the s i, the lower the probability that actual returns will be close to the expected return.

Expected Returns vs. Risk Security Expected return Risk,  HT 17.4% 20.0% Market USR 13.8* 18.8* T-bills Coll. 1.7* 13.4* *Seems misplaced.

Coefficient of Variation (CV) Standardized measure of dispersion about the expected value: Shows risk per unit of return. CV = =. Std dev  ^ k Mean

0 A B  A =  B, but A is riskier because larger probability of losses. = CV A > CV B.  ^ k

Portfolio Risk and Return Assume a two-stock portfolio with $50,000 in HT and $50,000 in Collections. Calculate k p and  p. ^

Portfolio Return, k p k p is a weighted average: k p = 0.5(17.4%) + 0.5(1.7%) = 9.6%. k p is between k HT and k COLL. ^ ^ ^ ^ ^^ ^^ k p =   w i k i  n i = 1

Portfolio Risk, σ p σ 2 p =E((k p -k p ^)) ^^ k p =   w i k i  n i = 1

 證券組合 AB 圖 8-3(A) A 與 B 證券之報酬率 圖 8-3(B) AB 證券投資組合之報酬率

 證券組合 AC 圖 8-4(A) A 與 C 證券之報酬率 圖 8-4(B) AC 證券投資組合之報酬率

3. 證券的相關性 表 8.3 A 、 B 證券組合之報酬率 圖 8-5 A 、 B 證券組合之報酬率與風險

表 8.4 A 、 C 證券組合之報酬率 圖  A 、 C 證券組合之報酬率與風險

圖 8-7 兩種證券之相關係數與風險

Portfolio Risk, σ p σ 2 p =E((k p -k p ^)) So two key concepts in the portfolio analysis are: 1.covariance:is a measure which combines the variance of a stock ’ s returns with the tendency of those returns to more up or down at the same time other stocks move up or down.

Portfolio Risk, σ p 2.correlation coefficient (r) : a measure the degree of relationship measure the degree of relationship between two variables (is generally between two variables (is generally used to measure the degree of used to measure the degree of co-movement between two variables), co-movement between two variables), which can vary from +1.0 to – 1 which can vary from +1.0 to – 1 r ab = cov(a,b)/ σ a σ b r ab = cov(a,b)/ σ a σ b

Portfolio Risk, σ p  Diversification What would happen if we included more than two stocks in the portfolio? What would happen if we included more than two stocks in the portfolio? As a rule,the riskiness of a portfolio will decline as the number of stocks in the portfolio increases. As a rule,the riskiness of a portfolio will decline as the number of stocks in the portfolio increases.

Portfolio Risk, σ p So the total risk can be divided into two parts:  Diversifiable risk :sometimes called company- specific or unsystematic risk) specific or unsystematic risk)  Nondiversifiable risk :sometimes called systematic or market risk) systematic or market risk) so that, total risk=diversifiable + nondiversifiable risk

Portfolio Risk, σ p The only risk a well-diversified portfolio has is the nondiversifiable or systematic portion. Therefore,the contribution of any one asset to the riskiness of a portfolio is its nondiversifiable or systematic risk. relevant risk=nondiversifiable risk relevant risk=nondiversifiable risk

General statements about risk Most stocks are positively correlated. r k,m » Most stocks are positively correlated. r k,m »   »  35% for an average stock. Combining stocks generally lowers risk. Combining stocks generally lowers risk.

Returns Distribution for Two Perfectly Negatively Correlated Stocks (r = -1.0) and for Portfolio WM Stock WStock MPortfolio WM

Returns Distributions for Two Perfectly Positively Correlated Stocks (r = +1.0) and for Portfolio MM’ Stock M Stock M ’ Portfolio MM ’

Large 0 15 Prob. 2 1 Even with large N,  p »  20%

# Stocks in Portfolio ,000+ Company Specific Risk Market Risk 20 0 Stand-Alone Risk,  p  p (%) 35

證券數目及風險 圖  投資組合之證券數目與風險

As more stocks are added, each new stock has a smaller risk-reducing impact. As more stocks are added, each new stock has a smaller risk-reducing impact.   p falls very slowly after about 40 stocks are included. The lower limit for  p is about 20% =  M.

Stand-alone risk=Market risk + Firm-specific risk Market risk is that part of a security ’ s stand-alone risk that cannot be eliminated by diversification. Firm-specific risk is that part of a security ’ s stand-alone risk which can be eliminated by proper diversification.

If you chose to hold a one-stock portfolio and thus are exposed to more risk than diversified investors, would you be compensated for all the risk you bear?

NO! NO! Stand-alone risk as measured by a stock ’ s  or CV is not important to a well-diversified investor. Stand-alone risk as measured by a stock ’ s  or CV is not important to a well-diversified investor. Rational, risk averse investors are concerned with  p, which is based on market risk. Rational, risk averse investors are concerned with  p, which is based on market risk.

There can only be one price, hence market return, for a given security. Therefore, no compensation can be earned for the additional risk of a one- stock portfolio. There can only be one price, hence market return, for a given security. Therefore, no compensation can be earned for the additional risk of a one- stock portfolio.

The Capital Asset Pricing Model In the port.,how should the riskiness of an individual stock be measured?  One answer is provided by the CAPM, an important tool used to analyze the relationship between risk and return.  The primary conclusion of the CAPM is this: the relevant riskiness of an individual stock is its contribution to the riskiness of a well-diversified port. the relevant riskiness of an individual stock is its contribution to the riskiness of a well-diversified port.  And the relevant risk can be measured by the degree to which a given stock tends to move up or down with the market.

The concept of beta Beta coefficient, β,is the tendency of a stock to move up or down with the market.  Beta measures a stock ’ s market risk. It shows a stock ’ s volatility relative to the market. β = σ im /σ 2 m  Beta shows how risky a stock is if the stock is held in a well-diversified portfolio.

The relationship between risk and rates of return. Security Market Line(SML) Security Market Line(SML) K i =K rf + β i (K m -K rf ) K i =K rf + β i (K m -K rf ) required return =risk-free + market risk 。 stocki ’ s required return =risk-free + market risk 。 stocki ’ s on stock i rate premium beta on stock i rate premium beta

證券市場線： E(kj) = kf + (E(km) - kf) ． bj

How are betas calculated? Run a regression of past returns on Stock i versus returns on the market. Returns = D/P + g. Run a regression of past returns on Stock i versus returns on the market. Returns = D/P + g. The slope of the regression line is defined as the beta coefficient. The slope of the regression line is defined as the beta coefficient.

Yeark M k i 115% 18% kiki _ kMkM _ Illustration of beta calculation: Regression line: k i = k M ^^

Calculator. Enter data points, and calculator does least squares regression: k i = a + bk M = k M. r = corr. coefficient = Calculator. Enter data points, and calculator does least squares regression: k i = a + bk M = k M. r = corr. coefficient = In the real world, we would use weekly or monthly returns, with at least a year of data, and would always use a computer or calculator. In the real world, we would use weekly or monthly returns, with at least a year of data, and would always use a computer or calculator. Find beta

If beta = 1.0, average stock. If beta = 1.0, average stock. If beta > 1.0, stock riskier than average. If beta > 1.0, stock riskier than average. If beta < 1.0, stock less risky than average. If beta < 1.0, stock less risky than average. Most stocks have betas in the range of 0.5 to 1.5. Most stocks have betas in the range of 0.5 to 1.5.

List of Beta Coefficients Stock Beta America Online 1.80 Mirage Resorts 1.40 General Electric 1.20 Coca-Cola 1.15 IBM 1.10 Microsoft Corp Procter & Gamble 1.05 Heinz 0.90 Energen Corp Empire District Electric 0.60

Use the SML to calculate the required returns. Assume k RF = 8%. Assume k RF = 8%. Note that k M = k M is 15%. (Equil.) Note that k M = k M is 15%. (Equil.) RP M = k M - k RF = 15% - 8% = 7%. RP M = k M - k RF = 15% - 8% = 7%. SML: k i = k RF + (k M - k RF )b i. ^

Required Rates of Return k HT = 8.0% + (15.0% - 8.0%)(1.29) = 8.0% + (7%)(1.29) = 8.0% + 9.0%= 17.0%. k M = 8.0% + (7%)(1.00)= 15.0%. k USR = 8.0% + (7%)(0.68)= 12.8%. k T-bill = 8.0% + (7%)(0.00)= 8.0%. k Coll = 8.0% + (7%)(-0.86)= 2.0%.

Expected vs. Required Returns ^ ^ ^ ^ k k HT 17.4% 17.0% Undervalued: k > k Market Fairly valued USR Undervalued: k > k T-bills Fairly valued Coll Overvalued: k < k

.. Coll.. HT T-bills. USR SML k M = 15 k RF = SML: k i = 8% + (15% - 8%) b i. k i (%) Risk, b i

Calculate beta for a portfolio with 50% HT and 50% Collections b p = Weighted average = 0.5(b HT ) + 0.5(b Coll ) = 0.5(1.29) + 0.5(-0.86) = 0.22.

The required return on the HT/Coll. portfolio is: k p = Weighted average k = 0.5(17%) + 0.5(2%) = 9.5%. Or use SML: k p = k RF + (k M - k RF ) b p = 8.0% + (15.0% - 8.0%)(0.22) = 8.0% + 7%(0.22) = 9.5%.