CAS Exam 8 Notes - Parts A&B Portfolio Theory and Equilibrium in Capital Markets Fixed Income Securities

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "CAS Exam 8 Notes - Parts A&B Portfolio Theory and Equilibrium in Capital Markets Fixed Income Securities"

Transcription

1 CAS Exam 8 Notes - Parts A&B Portfolio Theory and Equilibrium in Capital Markets Fixed Income Securities

2

3 Part I Table of Contents A Portfolio Theory and Equilibrium in Capital Markets 1 BKM - Ch. 6: Risk aversion and capital allocation to risky assets BKM - Ch. 7: Optimal risky portfolios BKM - Ch. 8: Index models BKM - Ch. 9: The Capital Asset Pricing Model (CAPM) BKM - Ch. 10: Arbitrage pricing theory and multifactor models of risk and return BKM - Ch. 11: The efficient market hypothesis BKM - Ch. 12: Behavioral finance and technical analysis BKM - Ch. 13: Empirical evidence on security returns B Fixed Income Securities 69 BKM - Ch. 14: Bond prices and yields Hull - Ch. 4: Interest rates BKM - Ch. 15: The term structure of interest rates Hull - Ch. 6.1: Day count and quotation conventions Hull - Ch Part 1: Credit risk Altman: Measuring corporate bond mortality and performance Cummins: CAT Bonds and other risk-linked securities Additional Notes 113

4

5 A Portfolio Theory and Equilibrium in Capital Markets

6

7 BKM - Ch. 6: Risk aversion and capital allocation to risky assets Introduction The process of constructing an investor portfolio can be viewed as a sequence of two steps: 1. Selecting the composition of one s portfolio of risky assets such as stocks and long-term bonds 2. Deciding how much to invest in that risky portfolio vs. safe assets such as short-term T-bills Fundamental part of asset allocation problem: Characterize risky portfolio risk-return trade-off Then, the fundamental decision is capital allocation between the risk-free and the risky portfolio Two themes in portfolio theory: 1. Investors will avoid risk unless they can anticipate a reward for engaging in risky investments 2. Utility model allows to quantify investors trade-offs between portfolio risk/expected return Risk and risk aversion Risk, speculation, and gambling Speculation: The assumption of considerable investment risk to obtain commensurate gain Considerable risk : The risk is sufficient to affect the decision Commensurate gain : Positive risk premium, i.e., expected profit > risk-free alternative Gamble: To bet or wager on an uncertain outcome The central difference with speculation is the lack of commensurate gain To turn a gamble into a speculative prospect requires an adequate risk premium to compensate risk-averse investors for the risks they bear A risky investment with a risk premium of zero, aka fair game, amounts to a gamble A risk-averse investor will reject it In some cases a gamble may appear to the participants as speculation: Economists call this case of differing beliefs heterogeneous expectations Risk aversion and utility values Risky assets command a risk premium in the marketplace Most investors are risk averse Investors who are risk averse reject investment portfolios that are fair games or worse A risk-averse investor penalizes the expected rate of return of a risky portfolio by a certain % (or penalizes the expected profit by a dollar amount) to account for the risk involved The greater the risk, the larger the penalty We will assume that each investor can assign a welfare, or utility, score to competing investment portfolios based on the expected return and risk of those portfolios Higher utility values are assigned to portfolios with more attractive risk-return profiles Portfolios have higher utility scores for higher expected returns/lower scores for higher volatility E.g.,utility score for portfolio with expected return E(r) and variance of returns σ 2 : utility score: U = E(r) 1 2 Aσ2 (1) Where U is the utility value and A is an index of the investor s risk aversion How variance of risky portfolios lowers utility depends on A, the investor s degree of risk aversion More risk-averse investors (larger values of A) penalize risky investments more severely Investors select the investment portfolio providing the highest utility level Risk-free portfolios utility score = their (known) rate of return (no penalty for risk) We can interpret the utility score of risky portfolios as a certainty equivalent rate of return Certainty equivalent rate: The rate that risk-free investments would need to offer to provide the same utility score as the risky portfolio Natural way to compare the utility values of competing portfolios A portfolio is desirable only if its certainty equivalent return > risk-free alternative Risk-neutral investors (A = 0) judge risky prospects solely by their expected rates of return The level of risk is irrelevant to the risk-neutral investor: There is no penalty for risk For this investor, a portfolio s certainty equivalent rate is simply its expected rate of return 3

8 E (r p ) E(r ) Northwest (preferred direction) I P Indifference curve Q II III IV ¾ p Figure 1: The trade-off between risk and return of a potential investment portfolio P A risk lover (A < 0) is willing to engage in fair games and gambles: This investor adjusts the expected return upward to take into account the fun of confronting the prospect s risk Portfolio P (expected return E(r p ), standard deviation σ p ) is preferred by risk-averse investors to any portfolio in quadrant IV because it has an expected return any portfolio in that quadrant and a standard deviation any portfolio in that quadrant Conversely, any portfolio in quadrant I is preferable to portfolio P Mean-variance (M-V) criterion: Portfolio A dominates B if: ¾ E(r A ) E(r B ) and σ A σ B In the E-σ plane in Fig. 1, the preferred direction is northwest, because we simultaneously increase the expected return/decrease the variance of the rate of return Indifference curve: Equally preferred portfolios will lie in the mean-standard deviation plane on a curve called the indifference curve that connects all portfolio points with the same utility value Estimating risk aversion One way is to observe individuals decisions when confronted with risk Consider an investor with risk aversion A whose entire wealth is in a piece of real estate Suppose that in any given year there is a probability p of a disaster that will wipe out the investor s entire wealth. Such an event would amount to a rate of return of 100% With probability 1 p, real estate remains intact, and rate of return is zero The expected rate of return of this prospect is: E(r) = p ( 1) + (1 p) 0 = p The variance of the rate of return equals the expectation of the squared deviation: σ 2 (r) = p (p 1) 2 + (1 p) p 2 = p(1 p) Utility score: U = E(r) 1 2 Aσ2 (r) = p 1 2Ap(1 p) (2) We can relate the risk-aversion parameter to the amount that an individual would be willing to pay for insurance against the potential loss. Suppose an insurance company offers to cover any loss over the year for a fee of ν dollars per dollar of insured property Such a policy amounts to a sure negative rate of return of ν, with a utility score: U = ν Maximum value of ν the investor is willing to pay? Equate the utility score of the uninsured property to that of the insured property, and solve for ν ν = p[ A(1 p)] (3) Square brackets in Eq. 3 = multiple of expected loss p the investor is willing to pay Economists estimate that investors exhibit degrees of risk aversion in the range of 2 to 4 More support for the hypothesis that A is somewhere in the range of 2 to 4 can be obtained from estimates of the expected rate of return and risk on a broad stock-index portfolio 4

9 Capital allocation across risky and risk-free portfolios The most straightforward way to control the risk of the portfolio is through the fraction of the portfolio invested in Treasury bills and other safe money market securities versus risky assets This capital allocation decision is an example of an asset allocation choice - a choice among broad investment classes, rather than among the specific securities within each asset class Most investment professionals: Asset allocation = most important part of portfolio construction Take composition of risky portfolio as given and focus on allocation between it/risk-free securities When we shift wealth from the risky portfolio to the risk-free asset, we do not change the relative proportions of the various risky assets within the risky portfolio Rather, we reduce the relative weight of the risky portfolio as a whole in favor of risk-free assets As long as we do not alter the weights of each security within the risky portfolio, the probability distribution of the rate of return on the risky portfolio remains unchanged by the asset reallocation What will change is the probability distribution of the rate of return on the complete portfolio that consists of the risky asset and the risk-free asset The risk-free asset There are no true risk-free assets Only the government can issue default-free bonds Even the default-free guarantee by itself is not sufficient to make the bonds risk-free in real terms The only risk-free asset in real terms would be a perfectly price-indexed bond Moreover, a default-free perfectly indexed bond offers a guaranteed real rate to an investor only if the maturity of the bond is identical to the investor s desired holding period Even indexed bonds are subject to interest rate risk: Real interest rates change unpredictably Nevertheless, it is common practice to view Treasury bills as the risk-free asset Their short term nature makes their values insensitive to interest rate fluctuations An investor can lock in a short-term nominal return by buying a bill and holding it to maturity Inflation uncertainty over a few weeks/months uncertainty of stock market returns In practice, most investors use a broader range of money market instruments as a risk-free asset All the money market instruments are virtually free of interest rate risk because of their short maturities and are fairly safe in terms of default or credit risk Most money market funds hold three types of securities: (i) T-bills, (ii) Bank certificates of deposit (CDs), and (iii) Commercial paper (CP), differing slightly in their default risk Portfolios of one risky asset and a risk-free asset The concern is with the proportion of the investment budget y to be allocated to the risky portfolio P. The remaining proportion 1 y is to be invested in the risk-free asset F Denote the risky rate of return of P by r p, its expected rate of return by E(r p ) and its standard deviation by σ p. The rate of return on the risk-free asset is denoted as r f The risk premium on the risky asset is: E(r p ) r f The rate of return on the complete portfolio C is r c = yr p + (1 y)r f E(r c ) = r f + y[e(r p ) r f ] Base rate of return = risk-free rate. The portfolio is also expected to earn a risk premium that depends on risk premium of risky portfolio E(r p ) r f and investor s position y in P When combining risky and risk-free assets, standard deviation σ c of complete portfolio = standard deviation σ p of risky asset multiplied by weight y of risky asset: σ c = yσ p (4) Investment opportunity set with risky/risk-free asset in the E-σ plane Equation for the straight line between F and P : E(r c ) = r f + y[e(r p ) r f ] = r f + σ c σ p [E(r p ) r f ] (5) 5

10 E(r p ) E(r) Capital Allocation Line (CAL) P S(y > 1) B rf S(y < 1) E(r p ){r f rf F Slope S = E(r p ) { r f ¾ p Capital Allocation Line (CAL) B with borrowing rate r f ¾ p Figure 2: The investment opportunity set in the expected return-standard deviation plane Investment opportunity set The set of feasible expected return and standard deviation pairs of portfolios resulting from different values of y The Capital Allocation Line (CAL) and the Sharpe ratio The CAL depicts all the risk-return combinations available to investors The slope S of the CAL equals the increase in the expected return of the complete portfolio per unit of additional standard deviation, i.e. incremental return per incremental risk The slope is called the reward-to-volatility ratio or the Sharpe ratio S = E(r p) r f σ p If investors can borrow at r f, they can construct portfolios to the right of P on the CAL However, non-government investors cannot borrow at the risk-free rate Then in the borrowing range, the reward-to-volatility ratio (i.e. the slope of the CAL) will be lower The CAL will therefore be kinked at point P Risk tolerance and asset allocation Investor confronting the CAL must choose one optimal portfolio C from set of feasible choices This choice entails a trade-off between risk and return Differences in risk aversion Different investors choose different positions in risky asset Investors attempt to maximize utility by choosing the best allocation to the risky asset y As allocation to risky asset increases (y ), expected return increases, but so does volatility Solving the utility maximization problem: max U = E(r c ) 1 y 2 Aσ2 c = r f + y[e(r p ) r f ] 1 2 Ay2 σp 2 Setting the derivative of this expression to zero and solving for y yields the optimal position: ¾ (6) y = E(r p) r f Aσ 2 p (7) The optimal position in the risky asset is inversely proportional to the level of risk aversion and the level of risk (variance) and directly proportional to the risk premium offered by the risky asset Indifference curve analysis First calculate the utility value of a risk-free portfolio yielding r f Then, find the expected return the investor would require to maintain the same level of utility when holding a risky portfolio for a given σ This yields all combinations of expected return/volatility with a given constant utility level Any investor prefers a portfolio on higher indifference curve (higher certainty equivalent) Portfolios on higher indifference curves offer a higher return for any given level of risk Higher indifference curves correspond to higher levels of utility 6

11 E(r) Higher U Complete portfolio maximizing U E(r p ) U = r f P CAL E(r c ) C r f ¾ c Figure 3: Indifference curve analysis More risk-averse investors have steeper indifference curves than less risk-averse investors Steeper curves: Investors require greater increase in expected return for increase in risk Investors thus attempt to find the complete portfolio on the highest possible indifference curve Superimpose indifference curves on the investment opportunity set represented by the CAL as in Fig. 3, and identify highest possible indifference curve that still touches CAL That indifference curve is tangent to the CAL, and the tangency point corresponds to the standard deviation and expected return of the optimal complete portfolio The choice for y the fraction of overall investment funds to place in the risky portfolio versus the safer but lower expected-return risk-free asset, is in large part a matter of risk aversion Passive strategies: The capital market line Passive strategy Describes a portfolio decision that avoids any direct or indirect security analysis Natural candidate for passively held risky asset is a well-diversified portfolio of common stocks Because a passive strategy requires that we devote no resources to acquiring information on any individual stock or group of stocks, we must follow a neutral diversification strategy Select diversified stock portfolio that mirrors the value of the US corporate sector This results in a portfolio in which, e.g., the proportion invested in Microsoft stock will be the ratio of Microsoft s total market value to the market value of all listed stocks The Capital Market Line (CML) Defined as the CAL provided by l-month T-bills and a broad index of common stocks A passive strategy generates an investment opportunity set that is represented by the CML How reasonable is it for an investor to pursue a passive strategy? 1. The alternative active strategy is not free, Whether you choose to invest the time and cost to acquire the information needed to generate an optimal active portfolio of risky assets, or whether you delegate the task to a professional who will charge a fee 2. Free-rider benefit: Another reason to pursue a passive strategy If there are many active investors who quickly bid up prices of undervalued assets and force down prices of overvalued assets, then at any time most assets will be fairly priced Therefore, a well-diversified portfolio of common stock will be a reasonably fair buy, and the passive strategy may not be inferior to that of the average active investor To summarize, a passive strategy involves investment in two passive portfolios: (i) Virtually risk-free short-term T-bills (or a money market fund) and (ii) A fund of common stocks that mimics a broad market index. The capital allocation line representing such a strategy is called the capital market line Criticisms of index funds don t hold up They are undiversified: The same complaint could be leveled at actively managed funds They are top-heavy: True, but S&P 500 not so narrow focused (77.2% of US stock-market value) They are chasing performance: This is what all investors do You can do better: As a group, investors in can t outperform market (collectively, they = market) ¾ p ¾ 7

12 8

13 BKM - Ch. 7: Optimal risky portfolios Introduction The investment decision can be viewed as a top-down process: 1. Capital allocation between the risky portfolio and risk-free assets 2. Asset allocation across broad classes (US stocks, international stocks, long-term bonds) 3. Security selection of individual assets within each asset class The optimal capital allocation is determined by risk aversion as well as expectations for the risk-return trade-off of the optimal risky portfolio In principle, asset allocation and security selection are technically identical: Both aim at identifying that optimal risky portfolio, namely, the combination of risky assets that provides the best risk-return trade-off In practice, however, asset allocation and security selection are typically separated into two steps: 1. The broad outlines of the portfolio are established first (asset allocation) 2. Details concerning specific securities are filled in later (security selection) Diversification and portfolio risk When all risk is firm-specific, diversification can reduce risk to arbitrarily low levels With all risk sources independent, exposure to any specific source of risk reduced to negligible level The insurance principle: Reduction of risk to very low levels for independent risk sources Risk eliminated by diversification: Unique/firm-specific/nonsystematic/diversifiable risk When common sources of risk affect all firms, even extensive diversification cannot eliminate risk The risk that remains even after extensive diversification is called market risk, risk that is attributable to marketwide risk sources (aka systematic risk, or nondiversifiable risk) Portfolio of two risky assets Efficient diversification: Constructing risky portfolios to provide the lowest possible risk for any given level of expected return Consider two mutual funds: A bond portfolio D, and a stock fund E w D is invested in the bond fund, and the remainder 1 w D = w E is invested in the stock fund Denoting r D, r E the rate of return on debt/equity funds, the rate of return r p on portfolio is: r p = w D r D + w E r E (1) The expected return on the portfolio is a weighted average of expected returns on the component securities with portfolio proportions as weights: E(r p ) = w D E(r D ) + w E E(r E ) (2) The variance of the two-asset portfolio is: σ 2 p = w 2 Dσ 2 D + w 2 Eσ 2 E + 2w D w E Cov(r D, r E ) (3) Variance is reduced if the covariance term is negative Even if covariance term 0, the portfolio standard deviation is less than weighted average of individual security σ s, unless the two securities are perfectly positively correlated The covariance can be computed from the correlation coefficient ρ DE : Cov(r D, r E ) = ρ DE σ D σ E σ 2 p = w 2 Dσ 2 D + w 2 Eσ 2 E + 2w D w E σ D σ E ρ DE (4) A hedge asset has negative correlation with the other assets in the portfolio Such assets will be particularly effective in reducing total risk Expected return is unaffected by correlation between returns Always prefer to add to portfolio assets with low or negative correlation with existing position 9

14 How low can portfolio standard deviation be? With perfect negative correlation, ρ = 1: σ 2 p = (w D σ D w E σ E ) 2 (5) When ρ = 1, a perfectly hedged position can be obtained by choosing the portfolio proportions to solve w D σ D w E σ E = 0. Then: w D = σ E σ D + σ E and w E = σ D σ D + σ E = 1 w D (6) What happens when w D > 1 and w E < 0? Strategy: Sell the equity fund short and invest proceeds of short sale in debt fund This will decrease the expected return of the portfolio The reverse happens when w D < 0 and w E > 1 Strategy: Sell bond fund short and use proceeds to buy more of equity fund Portfolio Standard Deviation E ½= {1 ½= 0 ½= :3 ½= 1 10 D Weight in Stock Fund Figure 1: Portfolio standard deviation as a function of investment proportions As the portfolio weight in the equity fund increases from zero to 1, portfolio standard deviation first falls with the initial diversification from bonds into stocks, but then rises again as the portfolio becomes heavily concentrated in stocks, and again is undiversified Pattern generally holds as long as the correlation coefficient between funds is not too high If ρ < σ D /σ E, volatility initially falls when we start with all bonds and move into stocks For a pair of assets with a large positive correlation of returns, the portfolio standard deviation will increase monotonically from the low-risk asset to the high-risk asset. Even in this case, however, there is a positive (if small) value from diversification What is the minimum level to which portfolio standard deviation can be held? The portfolio weights that solve this minimization problem turn out to be: w min (D) = σ 2 E Cov(r D, r E ) σ 2 D + σ2 E 2Cov(r D, r E ) The minimum-variance portfolio has a standard deviation smaller than that of either of the individual component assets Effect of diversification Portfolio opportunity set Pair of investment weights (w D, w E ) Resulting pair of expected return/standard deviation These constitute the portfolio opportunity set that can be constructed from the two available assets Fig. 2 shows the portfolio opportunity set for other values of the correlation coefficient ρ The solid black line connecting the two funds shows that there is no benefit from diversification when the correlation between the two is perfectly positive (ρ = 1) The dashed colored line demonstrates the greater benefit from diversification when the correlation coefficient is lower than.30 10

15 Expected Return (%) ½= {1 E 10 9 ½= 0 ½= :3 ½= 1 8 D Standard Deviation (%) Figure 2: Portfolio expected return as a function of standard deviation For ρ = 1, the portfolio opportunity set is linear, but now it offers a perfect hedging opportunity and the maximum advantage from diversification The lower the correlation, the greater the potential benefit from diversification Suppose now an investor wishes to select the optimal portfolio from the opportunity set The best portfolio will depend on risk aversion Portfolios to the northeast in Fig. 2 provide higher rates of return but impose greater risk The best trade-off among these choices is a matter or personal preference Investors with greater risk aversion prefer southwest portfolios (lower expected return, lower risk) Given a level of risk aversion, determine the portfolio that provides highest level of utility Using U = E(r p ) 1 2 Aσ2 p and the portfolio mean/variance determined by the portfolio weights in the two funds w E and w D, the optimal investment proportions in the two funds are: w D = E(r D) E(r E ) + A(σ 2 E σ Dσ E ρ DE ) A(σ 2 D + σ2 E 2σ Dσ E ρ DE ) and w E = 1 w D Asset allocation with stocks, bonds, and bills The optimal risky portfolio with two risky assets and a risk-free asset Graphical solution Ratchet the CAL upward until it reaches point of tangency with investment opportunity set This must yield the CAL with the highest feasible reward-to-volatility ratio Thus, the tangency portfolio (P in Fig. 3) is the optimal risky portfolio to mix with T-bills We can read the expected return and standard deviation of portfolio P from the graph Expected Return (%) Optimal Complete Portfolio = 5% r f Indifference Curve C G Global M-V Portfolio P CAL(E) CAL(P) E Efficient Frontier Opportunity Set of Risky Assets Minimum-Variance Frontier Standard Deviation (%) Figure 3: The opportunity set of the debt and equity funds with the CAL Portfolio construction with only two risky assets and a risk-free asset The objective is to find the weights w D and w E that result in the highest slope of the CAL (i.e., the weights that result in the risky portfolio with the highest reward-to-volatility ratio) 11

16 Thus our objective function is the slope (equivalently, the Sharpe ratio) S p : S p = E(r p) r f σ p For portfolio with two risky assets, expected return and standard deviation of portfolio P are: E(r p ) = w D E(r D ) + w E E(r E ) σ p = [w 2 Dσ 2 D + w 2 Eσ 2 E + 2w D w E Cov(r D, r E )] 1/2 Therefore, we solve an optimization problem formally written as (subject to w i = 1): max S p = E(r p) r f w i σ p In the case of two risky assets, the solution for the weights of the optimal risky portfolio P, using excess rates of return R rather than total returns r, is: w D = E(R D )σ 2 E E(R E)Cov(R D, R E ) E(R D )σ 2 E + E(R E)σ 2 D [E(R D) + E(R E )]Cov(R D, R E ) and w E = 1 w D (7) The steps to arrive at the complete portfolio are: 1. Specify the return characteristics of all securities (expected returns, variances, covariances) 2. Establish the risky portfolio: (a) Calculate the optimal risky portfolio P [Eq. (7)] (b) Calculate properties of P using weights determined in step (a) and Eqs. (2) and (3) 3. Allocate funds between the risky portfolio and the risk-free asset: (a) Calculate the fraction y of the complete portfolio allocated to portfolio P (the risky portfolio) and to T -bills (the risk-free asset) y = E(r p) r f Aσ 2 p (8) (b) Calculate the share of the complete portfolio invested in each asset and in T-bills Our two risky assets, the bond and stock mutual funds, are already diversified portfolios. The diversification within each of these portfolios must be credited for a good deal of the risk reduction compared to undiversified single securities Optimizing the asset allocation between bonds and stocks contributed incrementally to the improvement in the reward-to-volatility ratio of the complete portfolio The CAL with stocks, bonds, and bills shows that the standard deviation of the complete portfolio can be further reduced while maintaining the same expected return as the stock portfolio The Markowitz portfolio selection model Generalizing the portfolio construction problem to the case of many risky securities and a risk-free asset 1. Identify the risk-return combinations available from the set of risky assets 2. Identify optimal portfolio of risky assets by finding portfolio weights resulting in steepest CAL 3. Choose appropriate complete portfolio by mixing risk-free asset with optimal risky portfolio Security selection In the risk-return analysis, the portfolio manager needs as inputs a set of estimates for the expected returns of each security and a set of estimates for the covariance matrix Hence, we have n estimates of E(r i ) and the n n estimates of the covariance matrix in which the n diagonal elements are estimates of the variances σi 2 and the n 2 n = n(n 1) off-diagonal elements are the estimates of the covariances between each pair of asset returns 12

17 The expected return/variance of any risky portfolio with weights in each security w i is: n E(r p ) = w i E(r i ) (9) σ 2 p = i=1 n i=1 j=1 n w i w j Cov(r i, r j ) (10) Markowitz model is precisely step one of portfolio management: The identification of the efficient set of portfolios, or the efficient frontier of risky assets M-V frontier: Graph of the lowest possible variance for a given portfolio expected return All individual assets lie to the right inside the frontier, at least when short sales are allowed. When short sales are prohibited, single securities may lie on the frontier The part of the frontier above the global M-V portfolio is the efficient frontier The principal idea behind the frontier set of risky portfolios is that, for any risk level, we are interested only in that portfolio with the highest expected return The frontier is the set of portfolios that minimizes variance for any target expected return Some clients may be subject to additional constraints. E.g., prohibited from taking short positions For these clients the portfolio manager will add to the optimization program constraints that rule out negative (short) positions in the search for efficient portfolios In this special case, single assets may be, in and of themselves, efficient risky portfolios E.g., asset with highest expected return is a frontier portfolio because, without short sales, the only way to obtain that rate of return is to hold the asset as one s entire risky portfolio Some may want to ensure minimal level of expected dividend yield from optimal portfolio In this case the input list will be expanded to include a set of expected dividend yields d 1,, d n and the optimization program will include an additional constraint that ensures that the expected dividend yield of the portfolio will equal or exceed the desired level d Any constraint carries a price tag in the sense that an efficient frontier constructed subject to extra constraints will offer a reward-to-volatility ratio inferior to that of a less constrained one Another type of constraint is aimed at ruling out investments in industries or countries considered ethically or politically undesirable. This is referred to as socially responsible investing Capital allocation and the separation property We ratchet up the CAL by selecting different portfolios until we reach portfolio P which is the tangency point of a line from F to the efficient frontier Portfolio P maximizes the reward-to-volatility ratio and is the optimal risky portfolio The most striking conclusion is that a portfolio manager will offer the same risky portfolio P to all clients regardless of their degree of risk aversion The degree of risk aversion comes into play only in the selection of desired point along CAL More risk-averse clients invest more in risk-free asset and less in optimal risky portfolio than less risk-averse clients. However, both use portfolio P as their optimal risky investment vehicle Separation property: Portfolio choice problem may be separated into 2 independent tasks: 1. Determination of the optimal risky portfolio (purely technical) 2. Allocation of the complete portfolio to T-bills vs. the risky portfolio (personal preference) In practice, different managers estimate different input lists, thus deriving different efficient frontiers, and offer different optimal portfolios Source of disparity lies in security analysis This analysis suggests that a limited number of portfolios may be sufficient to serve the demands of a wide range of investors Theoretical basis of the mutual fund industry The power of diversification Consider the naive diversification strategy in which an equally weighted portfolio is constructed w i = 1/n for each security. In this case Eq. (10) becomes: σ 2 p = 1 n n i=1 1 n σ2 i + n n j=1,j i i=1 1 n 2 Cov(r i, r j ) (11) 13

18 Define the average variance and average covariance of the securities as: σ 2 = 1 n n σi 2 and Cov = i=1 1 n(n 1) n j=1,j i i=1 n Cov(r i, r j ) (12) Then, the portfolio variance is: σp 2 = 1 n σ2 + n 1 n Cov Effect of diversification When the average covariance among security returns is zero, as it is when all risk is firm-specific, portfolio variance can be driven to zero Hence when security returns are uncorrelated, the power of diversification is unlimited However, usually, economy-wide risk factors impart positive correlation among stocks The irreducible risk of a diversified portfolio depends on covariance of the returns of component securities, which is a function of the importance of systematic economic factors Suppose that all securities have a common σ and all security pairs have a common ρ The covariance between all pairs of securities is ρσ 2 and Eq. (13) becomes: σ 2 p = 1 n σ2 + n 1 n ρσ2 (14) When ρ = 0, we again obtain the insurance principle: σ p 0 as n For ρ > 0, however, portfolio variance remains positive For ρ = 1, portfolio variance equals σ 2 regardless of n Diversification is of no benefit In the case of perfect correlation, all risk is systematic More generally, as n becomes greater, Eq. (14) shows that systematic risk becomes ρσ 2 For diversified portfolios, the contribution to portfolio risk of a particular security depends on the covariance of that security s return with other securities, and not on the security s variance Asset allocation and security selection Why distinguish between asset allocation and security selection? 3 reasons: 1. As a result of greater need and ability to save (for college, recreation, longer life, health care needs, etc.), the demand for sophisticated investment management has increased enormously 2. The widening spectrum of financial markets and financial instruments has put sophisticated investment beyond the capacity of many amateur investors 3. There are strong economies of scale in investment analysis The end result is that the size of a competitive investment company has grown with the industry, and efficiency in organization has become an important issue The practice is therefore to optimize the security selection of each asset-class portfolio independently At the same time, top management continually updates the asset allocation of the organization, adjusting the investment budget allotted to each asset-class portfolio Risk pooling, risk sharing, and risk in the long run Risk pooling and the insurance principle The insurance principle Suppose an insurer sells 10,000 uncorrelated policies, each with a standard deviation σ Because the covariance between any two insurance policies is zero and σ is the same for each policy, the standard deviation of the rate of return on the 10,000-policy portfolio is: σ 2 p = 1 n σ2 σ p = σ n (15) σ p could be further decreased by selling even more policies This is the insurance principle (13) 14

19 It seems that as the film sells more policies, its risk continues to fall. Flaw in this argument: Probability of loss = inadequate measure of risk: Does not account for the magnitude of loss If 10,000 policies are sold, maximum possible loss is 10,000 bigger and comparison with a one-policy portfolio cannot be made based on means/standard deviations of rates of return Similar flaw as the argument that investing in stocks for the long run reduces risk Increasing the size of the bundle of policies does not make for diversification! Diversifying a portfolio means dividing a fixed investment budget across more assets When we combine n uncorrelated insurance policies, each with an expected profit $π, both expected total profit and standard deviation (SD) grow in direct proportion to n: E(nπ) = ne(π) V ar(nπ) = n 2 V ar(π) = n 2 σ 2 SD(nπ) = nσ The ratio of mean to standard deviation does not change when n increases The risk-return trade-off therefore does not improve with the assumption of additional policies Risk sharing If risk pooling (sale of additional independent policies) does not explain insurance, what does? The answer is risk sharing, the distribution of a fixed amount of risk among many investors An underwriter will contact other underwriters who each will take a piece of the action: Each will choose to insure a fraction of the project risk Each underwriter has a fixed amount of equity capital. Underwriters engage in risk sharing. They limit their exposure to any single source of risk by sharing that risk with other underwriters Underwriter diversifies its risk by allocating its budget across many projects that are not perfectly correlated One underwriter will decline to u/w too large a fraction of any single project This is the proper use of risk pooling: Pooling many sources of risk in a portfolio of given size The bottom line is that portfolio risk management is about the allocation of a fixed investment budget to assets that are not perfectly correlated In this environment, rate of return statistics (expected returns, variances, and covariances) are sufficient to optimize the investment portfolio Choices among alternative investments of a different magnitude require that we abandon rates of return in favor of dollar profits This applies as well to investments for the long run 15

20 16

21 BKM - Ch. 8: Index models Introduction The Markowitz procedure suffers from two drawbacks: 1. The model requires a huge number of estimates to fill the covariance matrix 2. The model does not provide any guideline to the forecasting of the security risk premiums Index models simplify estimation of covariance matrix and enhance analysis of risk premiums By allowing to explicitly decompose risk into systematic and firm-specific components, these models also shed considerable light on both the power and limits of diversification Further, they allow to measure these components of risk for particular securities and portfolios Despite simplification, index models remain true to concepts of efficient frontier/portfolio optimization Empirically, index models are as valid as the assumption of normality of rates of return on securities A single-factor security market The input list of the Markowitz model The success of a portfolio selection rule depends on the quality of the input list The Markowitz model necessitates n expected returns, n variances, and n(n 1)/2 covariances Markowitz model: Errors in estimation of correlation coefficients can lead to nonsensical results This can happen because some sets of correlation coefficients are mutually inconsistent Introducing a model that simplifies the way we describe the sources of security risk allows us to use a smaller, consistent set of estimates of risk parameters and risk premiums The simplification emerges because positive covariances among security returns arise from common economic forces that affect the fortunes of most firms Examples of common economic factors: Business cycles, interest rates, cost of natural resources The unexpected changes in these variables cause, simultaneously, unexpected changes in the rates of return on the entire stock market By decomposing uncertainty into these system-wide versus firm-specific sources, we vastly simplify the problem of estimating covariance and correlation Normality of returns and systematic risk Decompose rate of return on security i into its expected plus unanticipated components: r i = E(r i ) + e i (1) Where the unexpected return e i has a mean of zero and a standard deviation of σ i When security returns can be well approximated by normal distributions that are correlated across securities, we say that they are joint normally distributed At any time, security returns are driven by one or more common variables Suppose the common factor m that drives innovations in security returns is some macroeconomic variable that affects all firms Decompose sources of uncertainty into uncertainty about economy as a whole (captured by m) and uncertainty about firm in particular (captured by e i ) r i = E(r i ) + m + e i (2) The macroeconomic factor m measures unanticipated macro surprises m has a mean of zero (over time, surprises average out) with standard deviation of σ m m and e i are uncorrelated, because e i is firm-specific Independent of shocks to the common factor that affect the entire economy The variance of r i thus arises from two uncorrelated sources, systematic and firm specific σ 2 i = σ 2 m + σ 2 (e i ) (3) The common factor m generates correlation across securities Because all securities will respond to the same macroeconomic news 17

22 Since m is uncorrelated with any firm-specific surprises, the covariance between any i and j is: Cov(r i, r j ) = Cov(m + e i, m + e j ) = σ 2 m (4) We recognize that some securities are more sensitive than others to macroeconomic shocks: Capture this refinement by assigning each firm a sensitivity coefficient to macro conditions This leads to the single-factor model: r i = E(r i ) + β i m + e i The systematic risk of security i is β 2 i σ2 m, and its total risk is: σ 2 i = β 2 i σ 2 m + σ 2 (e i ) (6) The covariance between any pair of securities also is determined by their betas: Cov(r i, r j ) = Cov(β i m + e i, β j m + e j ) = β i β j σ 2 m (7) Normality of security returns alone guarantees that portfolio returns are also normal and that there is a linear relationship between security returns and the common factor Seek a variable that can proxy for common factor. To be useful, variable must be observable, so we can estimate its volatility/sensitivity of individual securities returns to variation in its value The single-index model A reasonable approach is to assert that the rate of return on a broad index of securities such as the S&P 500 is a valid proxy for the common macroeconomic factor Single-index model single-factor model because market index = proxy for common factor The regression equation of the single-index model Denote the market index by M, with excess return of R M = r M r f and standard deviation of σ M We regress the excess return of a security R i = r i r f on the excess return of the index R M We collect a historical sample of paired observations R i (t) and R M (t) where t denotes the date of each pair of observations. The regression equation is: R i (t) = α i + β i R M (t) + e i (t) (8) The intercept α i is the security s expected excess return when the market excess return is zero The slope coefficient β i is the security s sensitivity to the index e i is the zero-mean, firm-specific surprise in the security return in time t, aka the residual The expected return-beta relationship Taking expected values, we obtain the expected return-beta relationship of the single-index model: E(R i ) = α i + β i E(R M ) (9) Part of a security s risk premium is due to the risk premium of the index The market risk premium is multiplied by the relative sensitivity β of the individual security This is the systematic risk premium because it derives from the risk premium that characterizes the entire market, which proxies for the condition of the full economy or economic system α is the non-market premium Risk and covariance in the single-index model Both variances/covariances are determined by security betas/properties of market index: Total risk = Systematic risk + Firm-specific risk σ 2 i = β 2 i σ 2 M + σ 2 (e i ) (10) Covariance = Product of betas Market index risk Cov(r i, r j ) = β i β j σm 2 Correlation = Product of correlations with the market index Corr(r i, r j ) = β iβ j σ 2 M σ i σ j = Corr(r i, r M ) Corr(r j, r M ) (12) (5) (11) 18

23 The set of parameter estimates needed for the single-index model consists of only α, β, and σ(e) for the individual securities, plus the risk premium and variance of the market index The set of estimates needed for the single-index model 1. α i : Stock s expected return if the market is neutral, i.e. r M r f = 0 2. β i (r M r f ): The component of return due to movements in the overall market 3. e i : The unexpected component of return due to unexpected firm specific events 4. βi 2σ2 M : The variance attributable to the uncertainty of the common macroeconomic factor 5. σ 2 (e i ): The variance attributable to firm-specific uncertainty Advantages of the model The index model is a very useful abstraction because for large universes of securities, the number of estimates is only a small fraction of what otherwise would be needed The index model abstraction is crucial for specialization of effort in security analysis: If a covariance term had to be calculated for each security pair, then no specialization by industry Disadvantages Cost of the model: Restrictions it places on structure of asset return uncertainty Classification of uncertainty into dichotomy - macro vs. micro risk - oversimplifies sources of real-world uncertainty and misses important sources of dependence in stock returns E.g., dichotomy rules out industry events (affect an industry but not the broad macroeconomy) The optimal portfolio derived from the single-index model therefore can be significantly inferior to that of the full-covariance (Markowitz) model when stocks with correlated residuals have large alpha values and account for a large fraction of the Portfolio If many pairs of the covered stocks exhibit residual correlation, it is possible that a multiindex model, which includes additional factors to capture those extra sources of cross-security correlation, would be better suited for portfolio analysis and construction The index model and diversification Suppose that we choose an equally weighted portfolio of n securities. The excess rate of return on each security is given by: R i = α i + β i R M + e i Similarly, we can write the excess return on the portfolio of stocks as: R p = α p + β p R M + e p As the number of stocks included in this portfolio increases, the part of the portfolio risk attributable to nonmarket factors becomes ever smaller: This part of the risk is diversified away In contrast, market risk remains, regardless of the number of firms combined into the portfolio ( ) n R p = w i R i = 1 n (α i + β i R M + e i ) = 1 n 1 n α i + β i R M + 1 n e i (13) n n n n i=1 i=1 Portfolio s sensitivity to market, nonmarket return and avg. of firm-specific components: β p = 1 n β i α p = 1 n α i e p = 1 n e i (14) n n n i=1 Hence the portfolio s variance is: i=1 i=1 σ 2 p = β 2 pσ 2 M + σ 2 (e p ) (15) The systematic risk component of the portfolio variance, which depends on marketwide movements, is βpσ 2 M 2 and depends on the sensitivity coefficients of the individual securities. This part of the risk will persist regardless of the extent of portfolio diversification The nonsystematic component of the variance is σ 2 (e p ) and is attributable to firm-specific e i Because e i s are independent and have zero expected value, as more stocks are added to the portfolio, firm-specific components cancel out, resulting in ever-smaller nonmarket risk Such risk is thus termed diversifiable: Because the e i s are uncorrelated, n ( ) 1 2 σ 2 (e p ) = σ 2 (e i ) = 1 n n σ2 (e) (16) i=1 i=1 i=1 i=1 19

24 Estimating the single-index model The security characteristic line for stock i Regression: Line with intercept α i, slope β i = security characteristic line (SCL) for stock i: R i (t) = α i + β i R S&P 500 (t) + e i (t) The explanatory power of the SCL for stock i The R-square tells us the percentage of the variation in the stock i series that is explained by the variation in the S&P 500 excess returns The adjusted R 2 (slightly smaller) corrects for upward bias in R 2 that arises because we use fitted values of parameters (slope β and intercept α) rather than true, but unobservable values In general, the adjusted R-square (RA 2 ) is derived from the unadjusted by: RA 2 = 1 (1 R2 ) (n 1) (n k 1) where k is the number of independent variables An additional degree of freedom is lost to the estimate of the intercept Analysis of variance The sum of squares (SS) is the portion of the variance of the dependent variable (stock i s return) that is explained by the independent variable (S&P 500 return); it is equal to βi 2σ2 S&P 500 The MS column for the residual shows the variance of the unexplained portion of stock i s return, i.e. the portion of return that is independent of the market index. The square root of this value is the standard error (SE) of the regression Dividing the total SS of the regression by the number of dof (59 for 60 observations), we obtain the estimate of the variance of the dependent variable (stock i), per month R 2 (ratio of explained/total variance) = [explained (regression) SS]/[total SS] R 2 = β 2 i σ2 S&P 500 β 2 i σ2 S&P σ2 (e i ) = 1 σ 2 (e i ) β 2 i σ2 S&P σ2 (e i ) The estimate of alpha The intercept is the estimate of stock i s alpha for the sample period (per month) The standard error of the estimate is a measure of the imprecision of the estimate. If the standard error is large, the range of likely estimation error is correspondingly large We can relate the standard error of α to the standard error of the residuals as: SE(α i ) = σ(e i ) 1 n + (Avg. S&P 500) 2 V ar(s&p 500) (n 1) The t-statistic is the ratio of the regression parameter to its standard error This statistic equals the number of standard errors by which our estimate exceeds zero, and therefore can be used to assess the likelihood that the true but unobserved value might actually equal zero rather than the estimate derived from the data For α, we are interested in avg. value of stock i s return net of market movements Define the nonmarket component of return as actual return minus the return attributable to market movements. Call this stock i s firm-specific return R fs = R i β i R S&P 500 If R fs were normally distributed with a mean of zero, the ratio of its estimate to its standard error would have a t-distribution From a table of the t-distribution, we can find the probability that the true α is actually zero This is called the level of significance or the probability or p-value. Conventional cut-off for statistical significance is a probability of less than 5%, which requires t-statistic 2.0 Even if α was both economically and statistically significant within the sample, we still would not use that α as a forecast for a future period Overwhelming empirical evidence shows that 5-year alpha values do not persist over time Virtually no correlation between estimates from one sample period to the next 20

1. a. (iv) b. (ii) [6.75/(1.34) = 10.2] c. (i) Writing a call entails unlimited potential losses as the stock price rises.

1. a. (iv) b. (ii) [6.75/(1.34) = 10.2] c. (i) Writing a call entails unlimited potential losses as the stock price rises. 1. Solutions to PS 1: 1. a. (iv) b. (ii) [6.75/(1.34) = 10.2] c. (i) Writing a call entails unlimited potential losses as the stock price rises. 7. The bill has a maturity of one-half year, and an annualized

More information

CHAPTER 6: RISK AVERSION AND CAPITAL ALLOCATION TO RISKY ASSETS

CHAPTER 6: RISK AVERSION AND CAPITAL ALLOCATION TO RISKY ASSETS CHAPTER 6: RISK AVERSION AND CAPITAL ALLOCATION TO RISKY ASSETS PROBLEM SETS 1. (e). (b) A higher borrowing is a consequence of the risk of the borrowers default. In perfect markets with no additional

More information

Review for Exam 2. Instructions: Please read carefully

Review for Exam 2. Instructions: Please read carefully Review for Exam 2 Instructions: Please read carefully The exam will have 25 multiple choice questions and 5 work problems You are not responsible for any topics that are not covered in the lecture note

More information

CHAPTER 7: OPTIMAL RISKY PORTFOLIOS

CHAPTER 7: OPTIMAL RISKY PORTFOLIOS CHAPTER 7: OPTIMAL RIKY PORTFOLIO PROLEM ET 1. (a) and (e).. (a) and (c). After real estate is added to the portfolio, there are four asset classes in the portfolio: stocks, bonds, cash and real estate.

More information

Lesson 5. Risky assets

Lesson 5. Risky assets Lesson 5. Risky assets Prof. Beatriz de Blas May 2006 5. Risky assets 2 Introduction How stock markets serve to allocate risk. Plan of the lesson: 8 >< >: 1. Risk and risk aversion 2. Portfolio risk 3.

More information

Lecture 1: Asset Allocation

Lecture 1: Asset Allocation Lecture 1: Asset Allocation Investments FIN460-Papanikolaou Asset Allocation I 1/ 62 Overview 1. Introduction 2. Investor s Risk Tolerance 3. Allocating Capital Between a Risky and riskless asset 4. Allocating

More information

CHAPTER 6 RISK AND RISK AVERSION

CHAPTER 6 RISK AND RISK AVERSION CHAPTER 6 RISK AND RISK AVERSION RISK AND RISK AVERSION Risk with Simple Prospects Risk, Speculation, and Gambling Risk Aversion and Utility Values Risk with Simple Prospects The presence of risk means

More information

Review for Exam 2. Instructions: Please read carefully

Review for Exam 2. Instructions: Please read carefully Review for Exam Instructions: Please read carefully The exam will have 1 multiple choice questions and 5 work problems. Questions in the multiple choice section will be either concept or calculation questions.

More information

Capital Allocation Between The Risky And The Risk- Free Asset. Chapter 7

Capital Allocation Between The Risky And The Risk- Free Asset. Chapter 7 Capital Allocation Between The Risky And The Risk- Free Asset Chapter 7 Investment Decisions capital allocation decision = choice of proportion to be invested in risk-free versus risky assets asset allocation

More information

Lecture 2: Equilibrium

Lecture 2: Equilibrium Lecture 2: Equilibrium Investments FIN460-Papanikolaou Equilibrium 1/ 33 Overview 1. Introduction 2. Assumptions 3. The Market Portfolio 4. The Capital Market Line 5. The Security Market Line 6. Conclusions

More information

Solution: The optimal position for an investor with a coefficient of risk aversion A = 5 in the risky asset is y*:

Solution: The optimal position for an investor with a coefficient of risk aversion A = 5 in the risky asset is y*: Problem 1. Consider a risky asset. Suppose the expected rate of return on the risky asset is 15%, the standard deviation of the asset return is 22%, and the risk-free rate is 6%. What is your optimal position

More information

CHAPTER 9: THE CAPITAL ASSET PRICING MODEL

CHAPTER 9: THE CAPITAL ASSET PRICING MODEL CHAPTER 9: THE CAPITAL ASSET PRICING MODEL PROBLEM SETS 1. E(r P ) = r f + β P [E(r M ) r f ] 18 = 6 + β P(14 6) β P = 12/8 = 1.5 2. If the security s correlation coefficient with the market portfolio

More information

The CAPM (Capital Asset Pricing Model) NPV Dependent on Discount Rate Schedule

The CAPM (Capital Asset Pricing Model) NPV Dependent on Discount Rate Schedule The CAPM (Capital Asset Pricing Model) Massachusetts Institute of Technology CAPM Slide 1 of NPV Dependent on Discount Rate Schedule Discussed NPV and time value of money Choice of discount rate influences

More information

PART II RISK AVERSION AND CAPITAL ALLOCATION TO RISKY ASSETS CHAPTER SIX

PART II RISK AVERSION AND CAPITAL ALLOCATION TO RISKY ASSETS CHAPTER SIX PART II 6 CHAPTER SIX RISK AVERSION AND CAPITAL ALLOCATION TO RISKY ASSETS THE PROCESS OF constructing an investor portfolio can be viewed as a sequence of two steps: (1) selecting the composition of one

More information

CFA Examination PORTFOLIO MANAGEMENT Page 1 of 6

CFA Examination PORTFOLIO MANAGEMENT Page 1 of 6 PORTFOLIO MANAGEMENT A. INTRODUCTION RETURN AS A RANDOM VARIABLE E(R) = the return around which the probability distribution is centered: the expected value or mean of the probability distribution of possible

More information

Optimal Risky Portfolios Chapter 7 Investments Bodie, Kane and Marcus

Optimal Risky Portfolios Chapter 7 Investments Bodie, Kane and Marcus Optimal Risky ortfolios Section escription 7.0 Introduction 7.1 iversification and ortfolio Risk 7. ortfolios of Two Risky Assets 7.3 Asset Allocation with Stocks, Bonds and Bills 7.4 The Markowitz ortfolio

More information

AFM 472. Midterm Examination. Monday Oct. 24, 2011. A. Huang

AFM 472. Midterm Examination. Monday Oct. 24, 2011. A. Huang AFM 472 Midterm Examination Monday Oct. 24, 2011 A. Huang Name: Answer Key Student Number: Section (circle one): 10:00am 1:00pm 2:30pm Instructions: 1. Answer all questions in the space provided. If space

More information

Models of Risk and Return

Models of Risk and Return Models of Risk and Return Aswath Damodaran Aswath Damodaran 1 First Principles Invest in projects that yield a return greater than the minimum acceptable hurdle rate. The hurdle rate should be higher for

More information

Chapter 7 Portfolio Theory and Other Asset Pricing Models

Chapter 7 Portfolio Theory and Other Asset Pricing Models Chapter 7 Portfolio Theory and Other sset Pricing Models NSWERS TO END-OF-CHPTER QUESTIONS 7-1 a. portfolio is made up of a group of individual assets held in combination. n asset that would be relatively

More information

SAMPLE MID-TERM QUESTIONS

SAMPLE MID-TERM QUESTIONS SAMPLE MID-TERM QUESTIONS William L. Silber HOW TO PREPARE FOR THE MID- TERM: 1. Study in a group 2. Review the concept questions in the Before and After book 3. When you review the questions listed below,

More information

Module 7 Asset pricing models

Module 7 Asset pricing models 1. Overview Module 7 Asset pricing models Prepared by Pamela Peterson Drake, Ph.D., CFA Asset pricing models are different ways of interpreting how investors value investments. Most models are based on

More information

1 Capital Allocation Between a Risky Portfolio and a Risk-Free Asset

1 Capital Allocation Between a Risky Portfolio and a Risk-Free Asset Department of Economics Financial Economics University of California, Berkeley Economics 136 November 9, 2003 Fall 2006 Economics 136: Financial Economics Section Notes for Week 11 1 Capital Allocation

More information

Chapter 11, Risk and Return

Chapter 11, Risk and Return Chapter 11, Risk and Return 1. A portfolio is. A) a group of assets, such as stocks and bonds, held as a collective unit by an investor B) the expected return on a risky asset C) the expected return on

More information

Mean-Variance Portfolio Analysis and the Capital Asset Pricing Model

Mean-Variance Portfolio Analysis and the Capital Asset Pricing Model Mean-Variance Portfolio Analysis and the Capital Asset Pricing Model 1 Introduction In this handout we develop a model that can be used to determine how a risk-averse investor can choose an optimal asset

More information

Lecture 2: Delineating efficient portfolios, the shape of the meanvariance frontier, techniques for calculating the efficient frontier

Lecture 2: Delineating efficient portfolios, the shape of the meanvariance frontier, techniques for calculating the efficient frontier Lecture 2: Delineating efficient portfolios, the shape of the meanvariance frontier, techniques for calculating the efficient frontier Prof. Massimo Guidolin Portfolio Management Spring 2016 Overview The

More information

Chapter 5. Conditional CAPM. 5.1 Conditional CAPM: Theory. 5.1.1 Risk According to the CAPM. The CAPM is not a perfect model of expected returns.

Chapter 5. Conditional CAPM. 5.1 Conditional CAPM: Theory. 5.1.1 Risk According to the CAPM. The CAPM is not a perfect model of expected returns. Chapter 5 Conditional CAPM 5.1 Conditional CAPM: Theory 5.1.1 Risk According to the CAPM The CAPM is not a perfect model of expected returns. In the 40+ years of its history, many systematic deviations

More information

Answers to Concepts in Review

Answers to Concepts in Review Answers to Concepts in Review 1. A portfolio is simply a collection of investments assembled to meet a common investment goal. An efficient portfolio is a portfolio offering the highest expected return

More information

CHAPTER 11: ARBITRAGE PRICING THEORY

CHAPTER 11: ARBITRAGE PRICING THEORY CHAPTER 11: ARBITRAGE PRICING THEORY 1. The revised estimate of the expected rate of return on the stock would be the old estimate plus the sum of the products of the unexpected change in each factor times

More information

Chapter 5 Risk and Return ANSWERS TO SELECTED END-OF-CHAPTER QUESTIONS

Chapter 5 Risk and Return ANSWERS TO SELECTED END-OF-CHAPTER QUESTIONS Chapter 5 Risk and Return ANSWERS TO SELECTED END-OF-CHAPTER QUESTIONS 5-1 a. Stand-alone risk is only a part of total risk and pertains to the risk an investor takes by holding only one asset. Risk is

More information

Lecture Notes 8. Index Models. I. Readings and Suggested Practice Problems. III. Why the Single Index Model is Useful?

Lecture Notes 8. Index Models. I. Readings and Suggested Practice Problems. III. Why the Single Index Model is Useful? Prof. Alex Shapiro Lecture Notes 8 Index Models I. Readings and Suggested Practice Problems II. A Single Index Model III. Why the Single Index Model is Useful? IV. A Detailed Example V. Two Approaches

More information

CHAPTER 10. Capital Markets and the Pricing of Risk. Chapter Synopsis

CHAPTER 10. Capital Markets and the Pricing of Risk. Chapter Synopsis CHAPE 0 Capital Markets and the Pricing of isk Chapter Synopsis 0. A First Look at isk and eturn Historically there has been a large difference in the returns and variability from investing in different

More information

This is the trade-off between the incremental change in the risk premium and the incremental change in risk.

This is the trade-off between the incremental change in the risk premium and the incremental change in risk. I. The Capital Asset Pricing Model A. Assumptions and implications 1. Security markets are perfectly competitive. a) Many small investors b) Investors are price takers. Markets are frictionless a) There

More information

Holding Period Return. Return, Risk, and Risk Aversion. Percentage Return or Dollar Return? An Example. Percentage Return or Dollar Return? 10% or 10?

Holding Period Return. Return, Risk, and Risk Aversion. Percentage Return or Dollar Return? An Example. Percentage Return or Dollar Return? 10% or 10? Return, Risk, and Risk Aversion Holding Period Return Ending Price - Beginning Price + Intermediate Income Return = Beginning Price R P t+ t+ = Pt + Dt P t An Example You bought IBM stock at $40 last month.

More information

CHAPTER 13: EMPIRICAL EVIDENCE ON SECURITY RETURNS

CHAPTER 13: EMPIRICAL EVIDENCE ON SECURITY RETURNS CHAPTER 13: EMPIRICAL EVIDENCE ON SECURITY RETURNS Note: For end-of-chapter-problems in Chapter 13, the focus is on the estimation procedure. To keep the exercise feasible the sample was limited to returns

More information

Practice Set #4 and Solutions.

Practice Set #4 and Solutions. FIN-469 Investments Analysis Professor Michel A. Robe Practice Set #4 and Solutions. What to do with this practice set? To help students prepare for the assignment and the exams, practice sets with solutions

More information

CAPM, Arbitrage, and Linear Factor Models

CAPM, Arbitrage, and Linear Factor Models CAPM, Arbitrage, and Linear Factor Models CAPM, Arbitrage, Linear Factor Models 1/ 41 Introduction We now assume all investors actually choose mean-variance e cient portfolios. By equating these investors

More information

Diversification and the Risk-Return Trade-Off

Diversification and the Risk-Return Trade-Off Diversification and the Risk-Return Trade-Off Dr. Patrick Toche References : Zvi Bodie, Alex Kane, Alan Marcus, Investments, McGraw-Hill/Irwin. The relevant chapters of this textbook will be followed closely

More information

15.401 Finance Theory

15.401 Finance Theory Finance Theory MIT Sloan MBA Program Andrew W. Lo Harris & Harris Group Professor, MIT Sloan School Lecture 13 14 14: : Risk Analytics and Critical Concepts Motivation Measuring Risk and Reward Mean-Variance

More information

The Tangent or Efficient Portfolio

The Tangent or Efficient Portfolio The Tangent or Efficient Portfolio 1 2 Identifying the Tangent Portfolio Sharpe Ratio: Measures the ratio of reward-to-volatility provided by a portfolio Sharpe Ratio Portfolio Excess Return E[ RP ] r

More information

FE670 Algorithmic Trading Strategies. Stevens Institute of Technology

FE670 Algorithmic Trading Strategies. Stevens Institute of Technology FE670 Algorithmic Trading Strategies Lecture 6. Portfolio Optimization: Basic Theory and Practice Steve Yang Stevens Institute of Technology 10/03/2013 Outline 1 Mean-Variance Analysis: Overview 2 Classical

More information

Risk Aversion and Capital Allocation to Risky Assets

Risk Aversion and Capital Allocation to Risky Assets Chapter 6 Risk Aversion and Capital Allocation to Risky Assets The process of constructing an investor portfolio can be viewed as a sequence of two steps: (1) selecting the composition of one s portfolio

More information

Lecture 3: CAPM in practice

Lecture 3: CAPM in practice Lecture 3: CAPM in practice Investments FIN460-Papanikolaou CAPM in practice 1/ 59 Overview 1. The Markowitz model and active portfolio management. 2. A Note on Estimating β 3. Using the single-index model

More information

The CAPM & Multifactor Models

The CAPM & Multifactor Models The CAPM & Multifactor Models Business Finance 722 Investment Management Professor Karl B. Diether The Ohio State University Fisher College of Business Review and Clarification In the last few lectures

More information

15.433 Investments. Assignment 1: Securities, Markets & Capital Market Theory. Each question is worth 0.2 points, the max points is 3 points

15.433 Investments. Assignment 1: Securities, Markets & Capital Market Theory. Each question is worth 0.2 points, the max points is 3 points Assignment 1: Securities, Markets & Capital Market Theory Each question is worth 0.2 points, the max points is 3 points 1. The interest rate charged by banks with excess reserves at a Federal Reserve Bank

More information

Chapter 7 Risk, Return, and the Capital Asset Pricing Model

Chapter 7 Risk, Return, and the Capital Asset Pricing Model Chapter 7 Risk, Return, and the Capital Asset Pricing Model MULTIPLE CHOICE 1. Suppose Sarah can borrow and lend at the risk free-rate of 3%. Which of the following four risky portfolios should she hold

More information

2. Mean-variance portfolio theory

2. Mean-variance portfolio theory 2. Mean-variance portfolio theory (2.1) Markowitz s mean-variance formulation (2.2) Two-fund theorem (2.3) Inclusion of the riskfree asset 1 2.1 Markowitz mean-variance formulation Suppose there are N

More information

Chapter 13 Composition of the Market Portfolio 1. Capital markets in Flatland exhibit trade in four securities, the stocks X, Y and Z,

Chapter 13 Composition of the Market Portfolio 1. Capital markets in Flatland exhibit trade in four securities, the stocks X, Y and Z, Chapter 13 Composition of the arket Portfolio 1. Capital markets in Flatland exhibit trade in four securities, the stocks X, Y and Z, and a riskless government security. Evaluated at current prices in

More information

CHAPTER 10 RISK AND RETURN: THE CAPITAL ASSET PRICING MODEL (CAPM)

CHAPTER 10 RISK AND RETURN: THE CAPITAL ASSET PRICING MODEL (CAPM) CHAPTER 10 RISK AND RETURN: THE CAPITAL ASSET PRICING MODEL (CAPM) Answers to Concepts Review and Critical Thinking Questions 1. Some of the risk in holding any asset is unique to the asset in question.

More information

FORECASTING THROUGH SINGLE INDEX MODEL: A STUDY ON SELECTED INDIAN BANKS

FORECASTING THROUGH SINGLE INDEX MODEL: A STUDY ON SELECTED INDIAN BANKS FORECASTING THROUGH SINGLE INDEX MODEL: A STUDY ON SELECTED INDIAN BANKS ABSTRACT A model of stock returns that decomposes influences on returns into a systematic factor, as measured by the return on the

More information

Risk and Return Models: Equity and Debt. Aswath Damodaran 1

Risk and Return Models: Equity and Debt. Aswath Damodaran 1 Risk and Return Models: Equity and Debt Aswath Damodaran 1 First Principles Invest in projects that yield a return greater than the minimum acceptable hurdle rate. The hurdle rate should be higher for

More information

Portfolio Performance Measures

Portfolio Performance Measures Portfolio Performance Measures Objective: Evaluation of active portfolio management. A performance measure is useful, for example, in ranking the performance of mutual funds. Active portfolio managers

More information

Chapter 6 - Practice Questions

Chapter 6 - Practice Questions Chapter 6 - Practice Questions 1. If a T-bill pays 5 percent, which of the following investments would not be chosen by a risk-averse investor? A) An asset that pays 10 percent with a probability of 0.60

More information

The Capital Asset Pricing Model (CAPM)

The Capital Asset Pricing Model (CAPM) Prof. Alex Shapiro Lecture Notes 9 The Capital Asset Pricing Model (CAPM) I. Readings and Suggested Practice Problems II. III. IV. Introduction: from Assumptions to Implications The Market Portfolio Assumptions

More information

Chapter 7 Risk and Return: Portfolio Theory and Asset Pricing Models ANSWERS TO END-OF-CHAPTER QUESTIONS

Chapter 7 Risk and Return: Portfolio Theory and Asset Pricing Models ANSWERS TO END-OF-CHAPTER QUESTIONS Chapter 7 Risk and Return: Portfolio Theory and Asset Pricing odels ANSWERS TO END-OF-CHAPTER QUESTIONS 7-1 a. A portfolio is made up of a group of individual assets held in combination. An asset that

More information

CHAPTER 6. Topics in Chapter. What are investment returns? Risk, Return, and the Capital Asset Pricing Model

CHAPTER 6. Topics in Chapter. What are investment returns? Risk, Return, and the Capital Asset Pricing Model CHAPTER 6 Risk, Return, and the Capital Asset Pricing Model 1 Topics in Chapter Basic return concepts Basic risk concepts Stand-alone risk Portfolio (market) risk Risk and return: CAPM/SML 2 What are investment

More information

Econ 422 Summer 2006 Final Exam Solutions

Econ 422 Summer 2006 Final Exam Solutions Econ 422 Summer 2006 Final Exam Solutions This is a closed book exam. However, you are allowed one page of notes (double-sided). Answer all questions. For the numerical problems, if you make a computational

More information

1 Portfolio mean and variance

1 Portfolio mean and variance Copyright c 2005 by Karl Sigman Portfolio mean and variance Here we study the performance of a one-period investment X 0 > 0 (dollars) shared among several different assets. Our criterion for measuring

More information

A Review of Cross Sectional Regression for Financial Data You should already know this material from previous study

A Review of Cross Sectional Regression for Financial Data You should already know this material from previous study A Review of Cross Sectional Regression for Financial Data You should already know this material from previous study But I will offer a review, with a focus on issues which arise in finance 1 TYPES OF FINANCIAL

More information

Executive Summary of Finance 430 Professor Vissing-Jørgensen Finance 430-62/63/64, Winter 2011

Executive Summary of Finance 430 Professor Vissing-Jørgensen Finance 430-62/63/64, Winter 2011 Executive Summary of Finance 430 Professor Vissing-Jørgensen Finance 430-62/63/64, Winter 2011 Weekly Topics: 1. Present and Future Values, Annuities and Perpetuities 2. More on NPV 3. Capital Budgeting

More information

2. Capital Asset pricing Model

2. Capital Asset pricing Model 2. Capital Asset pricing Model Dr. Youchang Wu WS 2007 Asset Management Youchang Wu 1 Efficient frontier in the presence of a risk-free asset Asset Management Youchang Wu 2 Capital market line When a risk-free

More information

FIN 432 Investment Analysis and Management Review Notes for Midterm Exam

FIN 432 Investment Analysis and Management Review Notes for Midterm Exam FIN 432 Investment Analysis and Management Review Notes for Midterm Exam Chapter 1 1. Investment vs. investments 2. Real assets vs. financial assets 3. Investment process Investment policy, asset allocation,

More information

is dead in the context of empirical models of assets returns. Rhys Frake Word count: 2997

is dead in the context of empirical models of assets returns. Rhys Frake Word count: 2997 Present a critique of the Capital Asset Pricing Model, and hence discuss the claim that beta is dead in the context of empirical models of assets returns. Rhys Frake 0937708 Word count: 2997 1 P a g e

More information

Chapter 9 Interest Rates

Chapter 9 Interest Rates Chapter 9 Interest Rates Concept Questions 1. Short-term rates have ranged between zero and 14 percent. Long-term rates have fluctuated between about two and 13 percent. Long-term rates, which are less

More information

Chapter 6 The Tradeoff Between Risk and Return

Chapter 6 The Tradeoff Between Risk and Return Chapter 6 The Tradeoff Between Risk and Return MULTIPLE CHOICE 1. Which of the following is an example of systematic risk? a. IBM posts lower than expected earnings. b. Intel announces record earnings.

More information

Instructor s Manual Chapter 12 Page 144

Instructor s Manual Chapter 12 Page 144 Chapter 12 1. Suppose that your 58-year-old father works for the Ruffy Stuffed Toy Company and has contributed regularly to his company-matched savings plan for the past 15 years. Ruffy contributes $0.50

More information

Wel Dlp Portfolio And Risk Management

Wel Dlp Portfolio And Risk Management 1. In case of perfect diversification, the systematic risk is nil. Wel Dlp Portfolio And Risk Management 2. The objectives of investors while putting money in various avenues are:- (a) Safety (b) Capital

More information

1 Capital Asset Pricing Model (CAPM)

1 Capital Asset Pricing Model (CAPM) Copyright c 2005 by Karl Sigman 1 Capital Asset Pricing Model (CAPM) We now assume an idealized framework for an open market place, where all the risky assets refer to (say) all the tradeable stocks available

More information

Chapter 5 Uncertainty and Consumer Behavior

Chapter 5 Uncertainty and Consumer Behavior Chapter 5 Uncertainty and Consumer Behavior Questions for Review 1. What does it mean to say that a person is risk averse? Why are some people likely to be risk averse while others are risk lovers? A risk-averse

More information

The Term Structure of the Risk- Return Tradeoff

The Term Structure of the Risk- Return Tradeoff The Term Structure of the Risk- Return Tradeoff Luis M. Viceira Harvard Business School Netspar Opening Conference Tilburg University Tilburg, March 2005 New Research on Asset Allocation Portfolio choice

More information

TPPE17 Corporate Finance 1(5) SOLUTIONS RE-EXAMS 2014 II + III

TPPE17 Corporate Finance 1(5) SOLUTIONS RE-EXAMS 2014 II + III TPPE17 Corporate Finance 1(5) SOLUTIONS RE-EXAMS 2014 II III Instructions 1. Only one problem should be treated on each sheet of paper and only one side of the sheet should be used. 2. The solutions folder

More information

Economic Feasibility Studies

Economic Feasibility Studies Economic Feasibility Studies ١ Introduction Every long term decision the firm makes is a capital budgeting decision whenever it changes the company s cash flows. The difficulty with making these decisions

More information

The Capital Asset Pricing Model. Capital Budgeting and Corporate Objectives

The Capital Asset Pricing Model. Capital Budgeting and Corporate Objectives The Capital Asset Pricing odel Capital Budgeting and Corporate Objectives Professor Ron Kaniel Simon School of Business University of Rochester 1 Overview Utility and risk aversion» Choosing efficient

More information

Value-Based Management

Value-Based Management Value-Based Management Lecture 5: Calculating the Cost of Capital Prof. Dr. Gunther Friedl Lehrstuhl für Controlling Technische Universität München Email: gunther.friedl@tum.de Overview 1. Value Maximization

More information

CHAPTER 9 THE CAPITAL ASSET PRICING MODEL

CHAPTER 9 THE CAPITAL ASSET PRICING MODEL CHAPTER 9 THE CAPITAL ASSET PRICING MODEL THE CAPITAL ASSET PRICING MODEL The Capital Asset Pricing Model Extensions of the APM The CAPM and Liquidity THE CAPITAL ASSET PRICING MODEL The capital asset

More information

Chapter 5. Risk and Return. Copyright 2009 Pearson Prentice Hall. All rights reserved.

Chapter 5. Risk and Return. Copyright 2009 Pearson Prentice Hall. All rights reserved. Chapter 5 Risk and Return Learning Goals 1. Understand the meaning and fundamentals of risk, return, and risk aversion. 2. Describe procedures for assessing and measuring the risk of a single asset. 3.

More information

Paper 2. Derivatives Investment Consultant Examination. Thailand Securities Institute November 2014

Paper 2. Derivatives Investment Consultant Examination. Thailand Securities Institute November 2014 Derivatives Investment Consultant Examination Paper 2 Thailand Securities Institute November 2014 Copyright 2014, All right reserve Thailand Securities Institute (TSI) The Stock Exchange of Thailand Page

More information

Session X: Lecturer: Dr. Jose Olmo. Module: Economics of Financial Markets. MSc. Financial Economics. Department of Economics, City University, London

Session X: Lecturer: Dr. Jose Olmo. Module: Economics of Financial Markets. MSc. Financial Economics. Department of Economics, City University, London Session X: Options: Hedging, Insurance and Trading Strategies Lecturer: Dr. Jose Olmo Module: Economics of Financial Markets MSc. Financial Economics Department of Economics, City University, London Option

More information

Capital Asset Pricing Model Homework Problems

Capital Asset Pricing Model Homework Problems Capital Asset Pricing Model Homework Problems Portfolio weights and expected return 1. Consider a portfolio of 300 shares of firm A worth $10/share and 50 shares of firm B worth $40/share. You expect a

More information

RISKS IN MUTUAL FUND INVESTMENTS

RISKS IN MUTUAL FUND INVESTMENTS RISKS IN MUTUAL FUND INVESTMENTS Classification of Investors Investors can be classified based on their Risk Tolerance Levels : Low Risk Tolerance Moderate Risk Tolerance High Risk Tolerance Fund Classification

More information

Investment Statistics: Definitions & Formulas

Investment Statistics: Definitions & Formulas Investment Statistics: Definitions & Formulas The following are brief descriptions and formulas for the various statistics and calculations available within the ease Analytics system. Unless stated otherwise,

More information

CHAPTER 5 HISTORY OF INTEREST RATES AND RISK PREMIUMS

CHAPTER 5 HISTORY OF INTEREST RATES AND RISK PREMIUMS CHAPTER 5 HISTORY OF INTEREST RATES AND RISK PREMIUMS HISTORY OF INTEREST RATES AND RISK PREMIUMS Determinants of the Level of Interest Rates Risk and Risk Premium The Historical Record Real Versus Nominal

More information

I.e., the return per dollar from investing in the shares from time 0 to time 1,

I.e., the return per dollar from investing in the shares from time 0 to time 1, XVII. SECURITY PRICING AND SECURITY ANALYSIS IN AN EFFICIENT MARKET Consider the following somewhat simplified description of a typical analyst-investor's actions in making an investment decision. First,

More information

3. You have been given this probability distribution for the holding period return for XYZ stock:

3. You have been given this probability distribution for the holding period return for XYZ stock: Fin 85 Sample Final Solution Name: Date: Part I ultiple Choice 1. Which of the following is true of the Dow Jones Industrial Average? A) It is a value-weighted average of 30 large industrial stocks. )

More information

CML is the tangent line drawn from the risk free point to the feasible region for risky assets. This line shows the relation between r P and

CML is the tangent line drawn from the risk free point to the feasible region for risky assets. This line shows the relation between r P and 5. Capital Asset Pricing Model and Factor Models Capital market line (CML) CML is the tangent line drawn from the risk free point to the feasible region for risky assets. This line shows the relation between

More information

DESCRIPTION OF FINANCIAL INSTRUMENTS AND INVESTMENT RISKS

DESCRIPTION OF FINANCIAL INSTRUMENTS AND INVESTMENT RISKS DESCRIPTION OF FINANCIAL INSTRUMENTS AND INVESTMENT RISKS A. General The services offered by Prochoice Stockbrokers cover a wide range of Financial Instruments. Every type of financial instrument carries

More information

Chapter 8 Risk and Return

Chapter 8 Risk and Return Chapter 8 Risk and Return LEARNING OBJECTIVES (Slides 8-2 & 8-3) 1. Calculate profits and returns on an investment and convert holding period returns to annual returns. 2. Define risk and explain how uncertainty

More information

The number of mutual funds has grown dramatically in recent

The number of mutual funds has grown dramatically in recent Risk-Adjusted Performance of Mutual Funds Katerina Simons Economist, Federal Reserve Bank of Boston. The author is grateful to Richard Kopcke and Peter Fortune for helpful comments and to Jay Seideman

More information

Chapter 5. Risk and Return. Learning Goals. Learning Goals (cont.)

Chapter 5. Risk and Return. Learning Goals. Learning Goals (cont.) Chapter 5 Risk and Return Learning Goals 1. Understand the meaning and fundamentals of risk, return, and risk aversion. 2. Describe procedures for assessing and measuring the risk of a single asset. 3.

More information

Capital Asset Pricing Model. Joel Barber. Department of Finance. Florida International University. Miami, FL 33199

Capital Asset Pricing Model. Joel Barber. Department of Finance. Florida International University. Miami, FL 33199 Capital Asset Pricing Model Joel Barber Department of Finance Florida International University Miami, FL 33199 Capital Asset Pricing Model Mean-variance efficient risky portfolio For each asset j =1, 2,...,

More information

MODERN PORTFOLIO THEORY AND INVESTMENT ANALYSIS

MODERN PORTFOLIO THEORY AND INVESTMENT ANALYSIS MODERN PORTFOLIO THEORY AND INVESTMENT ANALYSIS EIGHTH EDITION INTERNATIONAL STUDENT VERSION EDWIN J. ELTON Leonard N. Stern School of Business New York University MARTIN J. GRUBER Leonard N. Stern School

More information

An introduction to Value-at-Risk Learning Curve September 2003

An introduction to Value-at-Risk Learning Curve September 2003 An introduction to Value-at-Risk Learning Curve September 2003 Value-at-Risk The introduction of Value-at-Risk (VaR) as an accepted methodology for quantifying market risk is part of the evolution of risk

More information

Journal of Exclusive Management Science May 2015 -Vol 4 Issue 5 - ISSN 2277 5684

Journal of Exclusive Management Science May 2015 -Vol 4 Issue 5 - ISSN 2277 5684 Journal of Exclusive Management Science May 2015 Vol 4 Issue 5 ISSN 2277 5684 A Study on the Emprical Testing Of Capital Asset Pricing Model on Selected Energy Sector Companies Listed In NSE Abstract *S.A.

More information

DO NOT COPY PORTFOLIO SELECTION AND THE CAPITAL ASSET PRICING MODEL

DO NOT COPY PORTFOLIO SELECTION AND THE CAPITAL ASSET PRICING MODEL UVA-F-1604 Rev. Mar. 4, 011 PORTFOLIO SELECTION AND THE CAPITAL ASSET PRICING MODEL What portfolio would you recommend to a 8-year-old who has just been promoted to a management position, and what portfolio

More information

Lecture 6: Arbitrage Pricing Theory

Lecture 6: Arbitrage Pricing Theory Lecture 6: Arbitrage Pricing Theory Investments FIN460-Papanikolaou APT 1/ 48 Overview 1. Introduction 2. Multi-Factor Models 3. The Arbitrage Pricing Theory FIN460-Papanikolaou APT 2/ 48 Introduction

More information

Note: There are fewer problems in the actual Final Exam!

Note: There are fewer problems in the actual Final Exam! HEC Paris Practice Final Exam Questions Version with Solutions Financial Markets Fall 2013 Note: There are fewer problems in the actual Final Exam! Problem 1. Are the following statements True, False or

More information

M.I.T. Spring 1999 Sloan School of Management 15.415. First Half Summary

M.I.T. Spring 1999 Sloan School of Management 15.415. First Half Summary M.I.T. Spring 1999 Sloan School of Management 15.415 First Half Summary Present Values Basic Idea: We should discount future cash flows. The appropriate discount rate is the opportunity cost of capital.

More information

ANALYSIS AND MANAGEMENT

ANALYSIS AND MANAGEMENT ANALYSIS AND MANAGEMENT T H 1RD CANADIAN EDITION W. SEAN CLEARY Queen's University CHARLES P. JONES North Carolina State University JOHN WILEY & SONS CANADA, LTD. CONTENTS PART ONE Background CHAPTER 1

More information

Chapter 7. Principle 2: There is a Risk-Return Tradeoff. We will expect to receive higher returns for assuming more risk.

Chapter 7. Principle 2: There is a Risk-Return Tradeoff. We will expect to receive higher returns for assuming more risk. Chapter 7 1. Calculate Realized and Expected Rates of Return and Risk. 2. Explain Efficient Market Hypothesis and Why it is Important to Stock Prices. Principle 2: There is a Risk-Return Tradeoff. We will

More information

Chap 3 CAPM, Arbitrage, and Linear Factor Models

Chap 3 CAPM, Arbitrage, and Linear Factor Models Chap 3 CAPM, Arbitrage, and Linear Factor Models 1 Asset Pricing Model a logical extension of portfolio selection theory is to consider the equilibrium asset pricing consequences of investors individually

More information

The Capital Asset Pricing Model

The Capital Asset Pricing Model Finance 400 A. Penati - G. Pennacchi The Capital Asset Pricing Model Let us revisit the problem of an investor who maximizes expected utility that depends only on the expected return and variance (or standard

More information