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


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1 CAS Exam 8 Notes  Parts A&B Portfolio Theory and Equilibrium in Capital Markets Fixed Income Securities
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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 risklinked securities Additional Notes 113
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5 A Portfolio Theory and Equilibrium in Capital Markets
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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 longterm bonds 2. Deciding how much to invest in that risky portfolio vs. safe assets such as shortterm Tbills Fundamental part of asset allocation problem: Characterize risky portfolio riskreturn tradeoff Then, the fundamental decision is capital allocation between the riskfree 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 tradeoffs 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 > riskfree 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 riskaverse investors for the risks they bear A risky investment with a risk premium of zero, aka fair game, amounts to a gamble A riskaverse 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 riskaverse 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 riskreturn 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 riskaverse investors (larger values of A) penalize risky investments more severely Investors select the investment portfolio providing the highest utility level Riskfree 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 riskfree 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 > riskfree alternative Riskneutral investors (A = 0) judge risky prospects solely by their expected rates of return The level of risk is irrelevant to the riskneutral 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 tradeoff 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 riskaverse 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 Meanvariance (MV) criterion: Portfolio A dominates B if: ¾ E(r A ) E(r B ) and σ A σ B In the Eσ 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 meanstandard 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 riskaversion 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 stockindex portfolio 4
9 Capital allocation across risky and riskfree 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/riskfree securities When we shift wealth from the risky portfolio to the riskfree 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 riskfree 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 riskfree asset The riskfree asset There are no true riskfree assets Only the government can issue defaultfree bonds Even the defaultfree guarantee by itself is not sufficient to make the bonds riskfree in real terms The only riskfree asset in real terms would be a perfectly priceindexed bond Moreover, a defaultfree 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 riskfree asset Their short term nature makes their values insensitive to interest rate fluctuations An investor can lock in a shortterm 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 riskfree 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) Tbills, (ii) Bank certificates of deposit (CDs), and (iii) Commercial paper (CP), differing slightly in their default risk Portfolios of one risky asset and a riskfree 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 riskfree 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 riskfree 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 = riskfree 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 riskfree 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/riskfree asset in the Eσ 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 returnstandard 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 riskreturn 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 rewardtovolatility 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, nongovernment investors cannot borrow at the riskfree rate Then in the borrowing range, the rewardtovolatility 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 tradeoff 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 riskfree 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 riskaverse investors have steeper indifference curves than less riskaverse 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 expectedreturn riskfree 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 welldiversified 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 lmonth Tbills 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. Freerider 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 welldiversified 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 riskfree shortterm Tbills (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 topheavy: True, but S&P 500 not so narrow focused (77.2% of US stockmarket 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
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13 BKM  Ch. 7: Optimal risky portfolios Introduction The investment decision can be viewed as a topdown process: 1. Capital allocation between the risky portfolio and riskfree assets 2. Asset allocation across broad classes (US stocks, international stocks, longterm 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 riskreturn tradeoff 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 riskreturn tradeoff 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 firmspecific, 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/firmspecific/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 twoasset 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 lowrisk asset to the highrisk 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 minimumvariance 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 tradeoff 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 riskfree 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 rewardtovolatility ratio Thus, the tangency portfolio (P in Fig. 3) is the optimal risky portfolio to mix with Tbills 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 MV Portfolio P CAL(E) CAL(P) E Efficient Frontier Opportunity Set of Risky Assets MinimumVariance 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 riskfree 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 rewardtovolatility 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 riskfree asset: (a) Calculate the fraction y of the complete portfolio allocated to portfolio P (the risky portfolio) and to T bills (the riskfree 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 Tbills 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 rewardtovolatility 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 riskfree asset 1. Identify the riskreturn 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 riskfree asset with optimal risky portfolio Security selection In the riskreturn 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) offdiagonal 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 MV 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 MV 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 rewardtovolatility 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 rewardtovolatility 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 riskaverse clients invest more in riskfree asset and less in optimal risky portfolio than less riskaverse 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 Tbills 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 firmspecific, portfolio variance can be driven to zero Hence when security returns are uncorrelated, the power of diversification is unlimited However, usually, economywide 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 assetclass portfolio independently At the same time, top management continually updates the asset allocation of the organization, adjusting the investment budget allotted to each assetclass 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 10,000policy 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 onepolicy 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 riskreturn tradeoff 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
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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 firmspecific 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 singlefactor 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 systemwide versus firmspecific 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 firmspecific 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 firmspecific 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 singlefactor 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 singleindex 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 Singleindex model singlefactor model because market index = proxy for common factor The regression equation of the singleindex 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 zeromean, firmspecific surprise in the security return in time t, aka the residual The expected returnbeta relationship Taking expected values, we obtain the expected returnbeta relationship of the singleindex 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 nonmarket premium Risk and covariance in the singleindex model Both variances/covariances are determined by security betas/properties of market index: Total risk = Systematic risk + Firmspecific 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 singleindex 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 singleindex 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 firmspecific 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 realworld 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 singleindex model therefore can be significantly inferior to that of the fullcovariance (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 crosssecurity 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 firmspecific 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 firmspecific e i Because e i s are independent and have zero expected value, as more stocks are added to the portfolio, firmspecific components cancel out, resulting in eversmaller 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 singleindex 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 Rsquare 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 Rsquare (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 tstatistic 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 firmspecific 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 tdistribution From a table of the tdistribution, we can find the probability that the true α is actually zero This is called the level of significance or the probability or pvalue. Conventional cutoff for statistical significance is a probability of less than 5%, which requires tstatistic 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 5year alpha values do not persist over time Virtually no correlation between estimates from one sample period to the next 20
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