Portfolio theory

State preference theory

 * To an investor a security is a set of possible payoffs, each one associated with a mutually exclusive state of nature.
 * A pure or primitive security (also Arrow-Debreu security) is defined as a security that pays $1 at the end of the period if a given state of nature occurs and nothing if any other state occurs. The expected end-of-period payoff on a pure security s is the price multiplied by the probability of state s occuring
 * Pure security prices are determined by (1) the end-of-period dollar payoff discounted to the present at the riskless rate multiplied by (2) the probability of payoff and (3) a risk adjustment factor: $$ \frac{$1\pi_s}{1+E(R_s)}$$
 * Thus, security prices are determined by the time value of money, the probability beliefs about state-contingent payoffs and individual preferences toward risk
 * If investors are all risk neutral, E(R) on all securities would be equal to the riskless interest rate
 * A security represents a position with regard to each possible future state of nature
 * In state preference theory, uncertainty about securities' future values is represented by a set of possible state-contingent payoffs.
 * Linear combinations of this set of payoffs represent an individuals's opportunity set of state-contingent portfolio payoffs - however, this opportunity set is contingent on whether capital markets are complete (i.e. when # of unique linearly independent securities = total # of alternative future states of nature; vice versa, the market is incompete if not every possible security payoff can be constructed from a portfolio of the existing securities)
 * Time preference for consumption: preference to consume today rather than tomorrow (as measured by interest rates). An individual's marginal rate of time preference for consumption = her marginal rate of substitution of current consumption and certain end-of-period consumption. In a perfect market, the marginal rates of time preference for all individuals are = market interest rate
 * State probability (πs): individual's belief of the relative likelihood of different states occuring
 * Homogenous expectations:If individuals agree on the relative likelihoods of states



Mean variance portfolio theory

 * The return on a portfolio of assets is simply a weighted average of the return on the individual assets
 * The covariance is a measure of how returns on assets move together. Note that the covariance is the expected value of the product of two deviations. The correlation coefficient is simply the covariance standardized (by dividing the covariance between two assets by the product of the standard deviation of each asset)


 * $$\operatorname{cov}(X,Y) = \operatorname{E}[(X-\mu_X)(Y-\mu_Y)],$$


 * The formula for ρ can also be written as$$ \rho_{X,Y}=\frac{\operatorname{E}[(X-\mu_X)(Y-\mu_Y)]}{\sigma_X\sigma_Y} $$


 * The expected return on a portfolio of two assets is given by:$$ \overline{R_P} = X_A \overline{R_A}+X_B \overline{R_B}$$


 * Note that the expected return on the portfolio is a simple weighted average of the expected returns on the individual securities and that the weights add to 1

Variance/STD of a portfolio

 * Variance of a portfolio with two assets: $$\sigma^2_P = X^2_A \sigma^2_A + X^2_B \sigma^2_B + 2X_A X_B \sigma_{AB}$$


 * and hence the STD/risk of a portfolio: $$\sigma_P = (X^2_A \sigma^2_A + X^2_B \sigma^2_B + 2X_A X_B \sigma_{AB})^\frac{1}{2}$$


 * Since the proportions of assets in the total portfolio always add up to one, we can express the proportion of asset X2 simply as 1-X1.


 * $$\sigma_P = (X^2_A \sigma^2_A + (1-X_A)^2 \sigma^2_B + 2X_A(1-X_A) \sigma_{AB})^\frac{1}{2}$$


 * Now recall that the $$ \rho_{AB} = \frac{\sigma_{AB}}{\sigma_A \sigma_B} $$ and hence which re-arranged is the covariance: $$ \sigma_{AB} = \rho_{AB} \sigma_A \sigma_B$$


 * Substituting this formula of the covariance in the equation for the STD of the portfolio we get: $$\sigma_P = (X^2_A \sigma^2_A + (1-X_A)^2 \sigma^2_B + 2X_A(1-X_A) \rho_{AB} \sigma_A \sigma_B)^\frac{1}{2}$$


 * Now assume that the correlation coeffiient = 1:  $$\sigma_P = (X^2_A \sigma^2_A + (1-X_A)^2 \sigma^2_B + 2X_A(1-X_A) 1* \sigma_A \sigma_B)^\frac{1}{2}$$


 * Notice that the term in the brackets is quadratic of the form $$X^2 + 2XY + Y^2$$ and can therefore be re-written as follows: $$(X_A \sigma_A + (1-X_A) \sigma_B)^2$$ and to get the STD we simply take the squre root: $$\sigma_P = X_A \sigma_A + (1-X_A) \sigma_B $$


 * All the while the expected return is: $$ \overline{R_P} = X_A \overline{R_A}+(1-X_A) \overline{R_B}$$


 * Thus with the correlation coefficient equal to 1, both risk and return of the portfolio are simply linear combinations of the risk and return of each security.


 * Variance of a portfolio with any number of assets: General_Variance_of_a_portfolio.jpg where the first part of the equation is (sum of) the variances of the assets multiplied by the (squared) proportion of the portfolio invested in this asset while the second part is simply the respective covariances


 * A general rule: when the return patterns of two assets are independent so that the correlation coefficient and covariance are zero, a portfolio can be found that has a lower variance than either of the assets by themselves.
 * The minimum variance is obtained for very large portfolios and is equal to the average covariance between all stocks in the population
 * Value weighting means that the weight of the portfolio each stock represents is the market value of that stock (price times number of shares) divided by the aggregate market value of all shares in the index. Thus the largest stocks are weighted more heavily
 * Efficient set/efficient frontier: that subset of portfolios that will be preferred by all investors who exhibit risk avoidance and who prefer more return to less
 * Thus the risk of a portfolio of assets is always smaller when the correlation coefficient is -1 than when it is +1.
 * For any value for $$X_C$$ between 0 and 1, the lower the correlation, the lower the standard deviation of the portfolio. The standard deviation reaches its lowest value for $$\rho = -1$$ and its highest value for $$\rho = 1$$. Therefore these two curves should represent the limits within which all portfolios of these two securities must lie for intermediate values of the correlation coefficient.

Finding minimum risk

 * $$ \sigma_P^2 = (X_C^2 \sigma_C^2+ (1-X_C)^2 \sigma_S^2 + 2X_C(1-X_C) \sigma_C \sigma_S \rho_{CS})^\frac{1}{2} $$


 * To find the value of $$X_C$$ that minimizes this equation (i.e. the expression for the fraction of the portfolio to be held in $$X_C$$ to minimize risk), we take the derivative of it with respect to $$X_C$$, set the derivative equal to zero, and solve for $$X_C$$. The derivative is


 * $$ \frac{\partial \sigma_P}{\partial X_C} = (\frac{1}{2}) \frac{2X_C \sigma_C^2 - 2\sigma_S^2 + 2X_C \sigma_S^2 + 2 \sigma_C \sigma_S \rho_{CS} - 4X_C \sigma_C \sigma_S \rho_{CS}} {({X_C^2\sigma_C^2 + (1-X_C)^2\sigma_S^2 + 2X_C (1-X_C) \sigma_C \sigma_S \rho_{CS}})^{0.5}} $$


 * Now setting this equal to 0 and solving for $$X_C$$ in order to find the (global) minimum variance portfolio (i.e. how much to invest in $$X_C$$)  → $$X_C = \frac{\sigma_S^2-\sigma_C \sigma_S \rho_{CS}}{\sigma_C^2 + \sigma_S^2 - 2\sigma_C \sigma_S \rho_{CS}} $$


 * For all assets, there is some value of $$\rho$$ such that the risk of the portfolio can no longer be made less than the risk of the least risky asset in the portfolio
 * Numerous studies indicate that a large fraction of the potential beneWts of diversiWcation are obtained by holding a relatively small number of securities. The marginal benefits of diversification decline rapidly as the number of securities increases.
 * Portfolio possibilities curve: the curve along which all possible combinations of assets must lie in expected return standard deviation space. The portfolio possibility curve that lies above the minimum variance portfolio is concave, whereas that which lies below the minimum variance portfolio is convex.


 * The efficient frontier
 * The global minimum variance portfolio is that portfolio that has the lowest risk of any feasible portfolio
 * The efficient set consists of the envelope curve of all portfolios that lie between the global minimum variance portfolio and the maximum return portfolio. This set of portfolios is called the efficient frontier
 * Combinations of two portfolios must be concave
 * Recall that an efficient set is determined by minimizing the risk for any level of expected return. If we specify the return at some level and minimize risk, we have one point on the efficient frontier. Thus, to get one point on the efficient frontier, we minimize risk subject to the return being some level plus the restriction (in the case of no short selling) that the sum of the proportions invested in each security is 1 and that all securities have positive or zero investment.

Calculating the efficient frontier

 * Short-selling: borrow a security from the broker (against some collateral) for, say, €100 - given that you expect its price to go down - wait until the price went down an buy it back again for only €95 - you made €5.
 * Short-selling allows you to leverage because you get the proceeds from a security sale (the security which you don't own). With short sales, one can sell securities with low expected returns and use the proceeds to buy securities with high expected returns. When an asset is sold short, the covariance term with a long asset is negative, thus reducing risk. It is therefore desirable for a short-sold asset to be highly correlated with an asset held long
 * Borrowing can be considered as selling a riskless security short
 * Call the certain rate of return on the riskless asset $$R_F$$. Because the return is certain, the standard deviation of the return on the riskless asset must be zero
 * Perhaps the most frequent constraints are those that place an upper limit on the fraction of the portfolio that can be invested in any stock. Upper limits on the amount that can be invested in any one stock are often part of the charter of mutual funds. Also, upper limits (and occasionally lower limits) are often placed on the fraction of a portfolio that can be invested in any industry

Short sales allowed & no riskless borrowing

 * BC represents the efficient frontier, while ABC represents the set of minimum-variance portfolios:



Short sales allowed & riskless borrowing/lending
A ($$\overline{R} = 14%$$ and $$\sigma = 6%$$)
 * When riskless lending and borrowing are introduced, the portfolio of risky assets that any investor would hold could be identified without regard to the investor’s risk preferences. This portfolio lies at the tangency point between the original efficient frontier of risky assets and a ray passing through the riskless return (on the vertical axis).
 * The slope of the line connecting a riskless asset and a risky portfolio is the expected return on the portfolio minus the risk-free rate divided by the standard deviation of the return on the portfolio. Thus the efficient set is determined by finding that portfolio with the greatest ratio of excess return (expected return minus risk-free rate) to standard deviation that satisfies the constraint that the sum of the proportions invested in the assets equals 1: $$ \theta = \frac{\overline{R}_P-R_F}{\sigma_P}$$
 * Assume you have three securites with a riskless borrowing rate of 5%:

B ($$\overline{R} = 8%$$ and $$\sigma = 3%$$)

C ($$\overline{R} = 20%$$ and $$\sigma = 15%$$)

Where $$\rho_{AB}=0.5 $$

$$ \rho_{AC}=0.2$$

$$ \rho_{BC}=0.4$$


 * Solving this system of three equations:

$$ \overline{R_1}-R_F = Z_1\sigma_1^2 + Z_2\sigma_{12} + Z_3\sigma_{13}$$

$$ \overline{R_2}-R_F = Z_1\sigma_{12} + Z_2\sigma_2^2 + Z_3\sigma_{23}$$

$$ \overline{R_3}-R_F = Z_1\sigma_{13} + Z_2\sigma_{23} + Z_3\sigma_3^2$$

$$14-5 = Z_1*6^2+Z_20.5*6*3+Z_3*0.2*6*15$$
 * Plug in the values:

$$8-5 = Z_1*0.5*6*3 + Z_2*3^2 + Z_3*0.4*3*15 $$

$$20-5 = Z_1*0.2*6*15 + Z_2*0.4*3*15 + Z_3*15^2$$

$$ 1=4Z_1+Z_2+2Z_3 $$
 * Now simplify to get

$$ 1= 3Z_1 + 3Z_2 + 6Z_3 $$

$$ 5= 6Z_1 + 6Z_2 + 75Z_3 $$

Thus: $$Z_1=\frac{14}{63}, Z_2=\frac{1}{63}, Z_3=\frac{3}{63}$$, now simply scale the Zs so that they add up to 1

$$X_1 = \frac{14}{18}, X_2= \frac{1}{18}, X_3=\frac{3}{18}$$

$$\overline{R_P}=\frac{14}{18}*14+\frac{1}{18}*8+\frac{3}{18}*20 = 14\frac{2}{3}%$$


 * And the variance of the return on the portfolio (see formula above) = $$33\frac{5}{6}$$


 * The efficient set is a straight line with an intercept at the risk-free rate of 5% and a slope equal to the ratio of excess return to standard deviation. The slope is: $$\frac{14\frac{2}{3}-5}{(33\frac{5}{6})^2} = 1.66$$
 * The efficient straight line (see below) is also called capital market line. All investors will end up with portfolios somewhere along the capital market line, and all efficient portfolios would lie along the capital market line. However, not all securities or portfolios lie along the capital market line. In fact, from the derivation of the efficient frontier, we know that all portfolios of risky and riskless assets, except those that are efficient, lie below the capital market line. The slop of the market line is given by: $$ \frac{\overline{R}_M-R_F}{\sigma_M}$$



Where the bullet-shaped curve is called frontier, the upper part of it the efficient frontier and the tangent line the capital market line (i.e. what all efficient capital markets should represent in terms of risk-return-tradeoff)

Short sales not allowed & riskless lending/borrowing

 * This problem is analogous to the case of riskless lending and borrowing with short sales allowed. One portfolio is optimal. Once again, it is the one that maximizes the slope of the line connecting the riskless asset and a risky portfolio.

Input estimation uncertainty

 * Using historical data on risk, return and correlation as a starting point in obtaining inputs for calculations of a efficient frontier
 * Assume an investor is mainly concerned about next month's return, then: $$\sigma_{Pred}^2 = \sigma^2 + \frac{\sigma^2}{T}$$

where $$ \sigma_{Pred}^2 $$ = the predicted variance series, $$\sigma^2$$ = the variance of monthly return and T = # of time periods


 * $$ \sigma_{Pred}^2 $$: the inherent risk in the return


 * $$\sigma^2 + \frac{\sigma^2}{T}$$: the uncertainty that comes from lack of knowledge about the true mean return (notice that predicted variance is always greater than historical variance because of uncertainty as to the future mean)


 * In a Bayesian analysis, the sum of the two terms on the right-hand side of this equation is referred to as the variance of the predictive distribution of returns

Estimating expected returns

 * The valuation of a company’s stock, as well as the valuation of the stock market as a whole, depends on the aggregate of all participants’ expectations
 * The price of securities depends on the average beliefs of investors (where each dollar invested gets one vote)
 * For example, bond prices depend on expectations about future interest rates, and consensus beliefs about future interest rates are impounded in today’s bond prices.
 * Aggregate asset allocation deals with how much to invest in broad categories of securities
 * In the case of many investors (endowmnet managers, pension funds), outside portfolio managers are used for securities selection and the plan manager’s task is to allocate among these portfolio managers
 * Expected returns for asset categories are usually estimated in a three-step procedure by determining:
 * The normal return for the asset category: e.g. by simply using the long-term historical performance of the asset class
 * How much you expect returns in the next period to deviate from normal
 * The expected deviation of the particular manager hired to manage an asset category from the average for that category
 * Market timing/dynamic asset allocation: how much money could be made if one bought stocks or bonds before these asset categories had large positive returns and sold them before periods when returns were negative.
 * The expected 1-year return on a 1-year Treasury bill is its yield to maturity — a quantity that can be observed in the market
 * Bayesian models of expected return: the distribution of return next period (the ‘predictive distribution’) includes uncertainty not only about the possible deviation of returns from expected values but also about these expected values themselves. This additional risk is referred to as estimation risk.
 * In its most basic form, Bayesian estimation begins with a prior about the value to be estimated, in this case, the mean return of an asset class. This prior is updated by empirical data, and the posterior value, used in the mean variance analysis, is a mixture between the prior and the mean of the empirical data.

Capital asset pricing model

 * The CAPM shows that the equilibrium rate of return on all risky assets are a function of their covariance with the market portfolio
 * Assumptions made by the CAPM: risk-averse investors who maximize E(u) of their wealth, are price takers and have homogenous expectations about asset returns, there exists an unlimited supply of a risk-free asset, quantities of assets are fixed, they're marketable & divisible, asset markets are frictionless, information costless and simultaneously available, perfect capital markets, no taxes etc. → hence the market portfolio is efficient
 * In equilibrium the market portfolio will consist of all marketable assets held in proportion to their value weights


 * Explanation graph: M is the point on the efficient frontier where the line from RF is a tangent to the frontier; every investor will maximise his utility by choosing portfolio M and then moving up or down RFMM1 until he reached the point where RFMM1 touched his indifference curve, M1 in the case of one investor, M2 in the case of another.

The security market line

 * Is the line connecting a riskless asset and a risky portfolio
 * Thus the expected return on an efficient portfolio is: (Expected return of an efficient portfolio) = (Price of time) + (Price of risk) x (Amount of risk). Put differently, the required rate of return on any asset is equal to the risk-free rate of return plus a risk premium (which is the price of risk multiplied by the quantity of risk). The price of risk is simply the slope of the security market line
 * Describes the expected return for all assets and portfolios of assets in the economy. The expected return on any asset, or portfolio, whether it is efficient or not, can be determined from this relationship. Notice that E(RM) and E(RF) are not functions of the assets we examine. Thus the relationship between the expected return on any two assets can be related simply to their difference in beta. The higher beta is for any security, the higher must be its equilibrium return.
 * The term $$ \beta_i (\overline{R}_M - R_F)$$ is the risk premium and $$\overline{R}_M - R_F$$ is the market risk premium (i.e. how much more is the market giving you by taking on the extra risk)


 * The SML is given by: $$\mathrm E(R_i) = R_f + \beta_{i}[E(R_M) - R_f]\,$$ ...or... $$\overline{R}_i = R_F + (\frac{\overline{R}_M - R_F}{\sigma_M}) \frac{\sigma_{iM}}{\sigma_M} = R_F + \beta_i (\frac{\overline{R}_M - R_F}{\sigma_M})$$


 * … where we calculate &beta; by dividing the covariance between the security's returns & the benchmark's returns by the variance of the benchmark's returns over a specified period: $$ \beta_i = \frac{\sigma_{iM}}{\sigma^2_M}$$ which simply gives you the impact of a security on the risk of the market portfolio
 * &beta; = the sensitivity of the asset's return to the market variability; a measure of the volatility, or systematic risk, of a security or a portfolio in comparison to the market as a whole; represents the tendency of a security's returns to respond to swings in the market
 * The &beta; on a portfolio is simply the sum of the product of the proportion invested in each stock times the beta on each stock
 * For very well-diversified portfolios, beta was the correct measure of a security’s risk. For very well-diversified portfolios, nonsystematic risk tends to go to zero, and the only relevant risk is systematic risk measured by beta.
 * The ultimate undiversifiable risk is the market risk
 * Assets with no systematic risk have &beta; = 0 and the market portfolio has &beta; = 1; &beta; of less (more) than 1 means that the security is theoretically less (more) volatile than the market
 * Companies in stable industries such as food manufacturers, whose profits and hence returns vary less than the average, will have shares with low &beta;s. Securities with high &beta;s (greater than 1) will be those of companies in cyclical industries, such as construction, with returns fluctuating more than the market average

The capital market line

 * Assumes market equilibrium, i.e. D=S
 * Expresses the expected return of a portfolio which lies on RFMM1 in terms of its risk &sigma;
 * Moving up on the CML means gearing up, moving down means lending


 * Is given by (P stands for your efficient portfolio): $$\overline{R}_P = R_F + \frac{\sigma_P}{\sigma_M}(\overline{R}_M - R_F)$$ …


 * …where the term $$\frac{\sigma_P}{\sigma_M}(\overline{R}_M - R_F)$$ can be thought of as the market price of risk for all efficient portfolios, i.e. the extra return that can be gained by increasing the level of risk (&sigma;) on an efficient portfolio by one unit
 * And where rf is simply the price of time or the return that is required for delaying potential consumption, one period given perfect certainty about the future cash flow.

Difference SML vs. CML
will be the covariance of the security with the market and not its total risk measured by variance or standard deviation
 * CML: x-axis in the case of CML is &sigma;; CML looks at efficient portfolios
 * SML: relates the expected return of any security to its risk; x-axis is &beta;; instead of expressing expected return in terms of its standard deviation (CML), we see that in SML the return of a security is a function of its covariance with the market. If we consider a risk-free security, this covariance will be zero and so E(Ri) = RF. If the security is risky, a premium will be required over and above the risk-free rate, which will increase with the covariance of the security with the market
 * This has serious implications for the investor when considering adding a security to his portfolio. The only risk for which he will be rewarded

CAPM equation

 * $$E(R_i) = R_f + \beta_{i}(E(R_m) - R_f)\,$$


 * Every risk-averse investor will hold a combination between the risk-free and the risky assets along the capital market line: two-fund separation theorem. That is, given the assumptions of homogeneous expectations and unlimited riskless lending and borrowing, all investors will hold the market portfolio to differing degrees
 * Take as a benchmark the zero-risk portfolio z which has a &beta; of 0
 * Risk-free + the risk premium (which is composed of the price (EM - EF and the quantity of risk(&beta;))

Index portfolios

 * Single index
 * To simplify analysis, the single-index model assumes that there is only 1 macroeconomic factor that causes the systematic risk affecting all stock returns and this factor can be represented by the rate of return on a market index, such as the S&P 500. According to this model, the return of any stock can be decomposed into the expected value of the unique return/expected excess return of the individual stock due to firm-specific factors (&alpha;) which is the return that exceeds the risk-free rate, the return due to macroeconomic events that affect the market, and the unexpected microeconomic events that affect only the firm.
 * In the single-index model (and assuming a fair game), the return for stock i is ri = &alpha;i + &beta;irm + &epsilon;i
 * where &beta;irm = the stock's return due to the movement of the market modified by the stock's beta (&beta;i), while &epsilon;i represents the unsystematic risk of the security due to firm-specific factors.


 * Multi-index models
 * Attempt to capture some of the nonmarket influences that cause securities to move together. Simply add these additional influences (along with the stock's sensitivity to these influences) to the general return equation.
 * Assume that the return on any stock is a function of the return on the market, changes in the level of interest rates, and a set of industry indexes that are uncorrelated. If Ri is the return on stock i, then the return on stock i can be related to the influences that affect its return in the following way:
 * $$ R_i = a_i^* + b_{i1}^*I_1^* + b_{i2}^*I_2^* + ... + b_{iL}^*I_L^* + c_i $$


 * Where
 * ai = expected return E(R) of i/expected value of the unique return
 * Ij = actual level of index j
 * bij = measure of the responsiveness of the return on stock i to changes in the index j (i.e. analogous to &beta; in the case of the single-index model)
 * ci = random component of the unique return (with mean=0 and variance = &sigma;2ci)
 * Multi-index models lie in an intermediate position between the full historical correlation matrix itself and the single-index model in ability to reproduce the historical correlation matrix
 * Principal components analysis: extracts from past values of the variance–covariance matrix a set of indexes that best explain (reproduce) the historical matrix itself
 * Adding additional indexes derived from the past correlation matrix to the single–index model led to a decrease in performance. Although adding more indexes led to a better explanation of the historical correlation matrix, it led both to a poorer prediction of the future correlation matrix and to the selection of portfolios that (i.e. random noise rather than useful information)
 * Overall mean model: The most aggregate type of averaging that can be done is to use the average of all pairwise correlation coefficients over some past period as a forecast of each pairwise correlation coefficient for the future. This is equivalent to the assumption that the past correlation matrix contains information about what the average correlation will be in the future but no information about individual differences from this average.

Additional material

 * |Portfolio Theory III & The CAPM and APT I