CFA 4_Portfolio Management

Question 1

Holding period return (HPR)

Answer 1

HPR = (end of period value / beginning of period value) - 1

HPR = (Price at end of period + Dividend over period / Price at beginning of period) - 1

Question 2

Arithmetic mean return

Answer 2

Simple average of series of periodic returns (average of HPRs)

AMR = (R1 + R2 + …) / n

Question 3

Geometric mean return

Answer 3

Compound annual return which adjusts for when periodic returns vary over periods.

GMR = Square root[(1+ R1) * (1 + R2) …] -1

Question 4

Money-weighted rate of return

Answer 4

IRR on a portfolio based on all cash inflows and outflows.

Enter CFs into CF on calculator and solve for IRR.

Question 5

Gross vs net return on portolfio

Answer 5

Gross: return before deducting fees for management and administration of investment account. Gross still accounts for trading commissions and fees necessary to generate returns.

Net: Return after fees have been deducted.

Question 6

Variance (Sample variance)

Answer 6

Avg squared deviation from the mean.

Variance:delta^2 = (Sum of each year’s deviation from mean of population squared) / (Years in series)

Sample variance (nb years - 1):delta^2 = (Sum of each year’s deviation from mean of observations squared) / (Years in series - 1)

Question 7

Covariance

Answer 7

Extent to which two variables move together over time.

Cov = Sum[(Ra - Ra(m))+(Rb - Rb(m))] / n - 1

Cov = (Sum of [each year’s deviation of stock A from mean for Stock A * Sum of each year’s deviation of stock B from mean for Stock B] ) / n - 1

n = number of periods.

Eg:MeanX = (7 + 9 + 10 + 10) / 4 = 9;

MeanY = (5 + 8 + 11 + 8) / 4 = 8

CovX,Y = [(7 − 9)(5 − 8) + (9 − 9)(8 − 8) + (10 − 9)(11 − 8) + (10 − 9)(8 − 8)] / (4 − 1) = 3.0

Question 8

Correlation

Answer 8

Standardized measure of covariance produced by dividing product of SD of two securities. Adding ANY stock that is not PERFECTLY correlated with the portfolio (less than +1) will reduce the risk of the portfolio.

Correlation coefficient = Covariance of A and B / SD[A] *SD[B]

therefore: Covariance = Correlation coefficient * SD[A] * SD[B]

Between -1 and 1. If -1, a zero-variance portfolio can be created. If +1, the SD will be equal to the weighted avg SD’s in portfolio. If 0, there is no relationship and they are uncorrelated. Anything less than +1 reduces portfolio variance.

Question 9

Standard deviation of portfolio

Answer 9

SD Portfolio = Square root [(Wa^2 * SDa^2) + (Wb^2 * SDb^2) + (2 * Wa * Wb * Correlation coefficient * SDa * SDb)]

Variance of portfolio is this, but not squared in total

If corellation coefficient is +1, then SD is weighted average of the SDs of the individual assets.Remember: Covariance = Correlation coefficient * SDa * SDb)

Question 10

Variance of portfolio

Answer 10

Variance = (Wa^2 * SDa^2) + (Wb^2 * SDb^2) + (2 * Wa * Wb * Correlation coefficient * SDa * SDb)*SD of portfolio is the square root of this

Remember: Covariance = Correlation coefficient * SDa * SDb)

Question 11

Minimum variance portfolios

Answer 11

Those portfolios that have the lowest SD of all portfolios with a given expected return

Question 12

Efficient frontier

Answer 12

Those portfolios that have the greatest expected return for each SD. The set of portfolios that gives investors the highest return for a given level of risk or the lowest risk for a given level of return

The optimal Capital Allocation Line is tangent to the efficient frontier.The line that represents possible combinations of a risky asset and the risk-free asset is referred to as a capital allocation line (CAL). Portfolios to the right are inefficient. Portfolios to the left are unachievable. The efficient frontier will shift left if correlation is reduced (therefore reducing risk).

Question 13

Utility function

Answer 13

U = E(r) - 1/2ASD^2E(r) = expected returnA = measure of risk aversionSD^2 = variance

The optimal portfolio for an investor is determined as the point where the investor’s highest utility curve is tangent to the efficient frontier. The optimal portfolio occurs when the investor achieves the diversified portfolio with the highest utility.Investors who are less risk averse will have flat utility curves, meaning they are willing to take on more risk for a slightly higher return. Investors who are more risk averse require a much higher return to accept more risk, producing a steep utility curve.

Question 14

Two-fund separation theorem

Answer 14

All investors’ optimum portfolios will be made up of some combination of an optimal portfolio of risky assets and the risk-free asset.

E(r portfolio) = WaE(Ra) + WbE(Rb)

SDportfolio = Wa * SDa

This is because the SD of the risk-free asset is 0. The optimal portfolio may be different for different investors, and is NOT just the portfolio that gives the investor the maximum return, it also has risk.

Question 15

Capital allocation line (CAL)

Answer 15

The line representing possible combinations of risk-free assets and the “optimal risky asset portfolio” (two-fund separation theorem). ie the set of possible efficient portfolios.The line that represents possible combinations of a risky asset and the risk-free asset is referred to as a capital allocation line (CAL).

Slope of CAL = (E(r) - RFR) / SD

This equation is also the calculation for the Sharpe ratio.Can move further up CAL by borrowing at risk-free rate and investing in risky assets. Combine CAL with indifference curve to decide what portfolio to recommend. Assume all investors face same efficient frontier, therefore optimal CAL is tangent to efficient frontier. The risky portfolio used for this is the market portfolio. This line is now referred to the CML (Capital Market Line). The capital market line (CML) represents possible combinations of the market portfolio (rather than risky asset) with the risk-free asset.

The market portfolio contains all risky assets in existence.

Question 16

Capital Market Line (CML)

Answer 16

The capital market line (CML) represents possible combinations of the market portfolio (all risky assets in existence) with the risk-free asset. A straight line drawn from the risk-free rate of return (on the Y axis) through the market portfolio. It is the optimal CAL for all investors tangent to the efficient frontier. Assume all investors face same efficient frontier, therefore optimal CAL is tangent to efficient frontier. The risky portfolio used for this is the market portfolio. The CML intercepts the y-axis at the RFR.

Portfolios above this line are unachievable.Portfolios below this line are inefficient.By definition, all stocks and portfolios (other than the market portfolio) fall below the CML. (Only the market portfolio is efficient).

E(Rp) = Rf + (E(Rm) - Rf) * (SDportfolio / SDmarket)

Compare with: The line that represents possible combinations of a risky asset and the risk-free asset is referred to as a capital allocation line (CAL).

Question 17

Unsystematic risk

Answer 17

Risk that can be eliminated through diversification. AKA unique, diversifiable, or firm-specific risk. Unsystematic risk is not compensated for in equilibrium because we assume it can be diversified away for free. Equilibrium security returns depend on a portfolio’s systematic risk, NOT its total risk.

Total risk = systematic risk + unsystematic risk

Question 18

Systematic risk

Answer 18

Risk that cannot be diversified away. AKA nondiversifiable, or market risk. Equilibrium security returns depend on a portfolio’s systematic risk, NOT its total risk. Assumes diversification is free.

Total risk = systematic risk + unsystematic risk

A positive beta indicates return of asset follows same direction as market. Lower beta = less market risk. Market beta is 1, RF beta is 0.

Question 19

Single-index model (including market model)

Answer 19

A return model in which the only risk factor is the return on the market (Rm, written as Rm - Rf). Market model: simplified form, based on actual market return.

R = intercept + Beta*Rm + abnormal return on asset

Question 20

Beta

Answer 20

Sensitivity of asset’s return to return on the market index. Standardized measure of covariance of asset’s excess return to market excess return.

B = Covariance of Asset A and market / variance (SD^2) of market return

This is also the slope of the security characteristic line. Or:

B = Correlation coefficient * (SDa / SDmarket)

Correlation coefficient = Covariance of A and B / (SD[A] *SD[B])

Slope of least squares regression line (the security characteristic line) is our estimate of beta. A positive beta indicates return of asset follows same direction as market. Lower beta = less market risk. Market beta is 1, RF beta is 0.

Cov = Sum[(Ra - Ra(m))+(Rb - Rb(m))] / n - 1

Cov = (Sum of [each year’s deviation of stock A from mean for Stock A * Sum of each year’s deviation of stock B from mean for Stock B] ) / n - 1

Question 21

Security market line vs Security characteristic line

Answer 21

SML = graphical representation of CAPM. CAPM shows relationship between beta and expected return. Uses beta on the x-axis, and therefore is a measure of SYSTEMATIC risk. In CAPM, all properly priced portfolios will plot on the SML. If expected return is greater than required (CAPM) return, a stock is above the SML.

Graphically: If you plot a stock’s expected return on the SML and it falls below the line, it indicates that the stock is currently overpriced, causing its expected return to be too low. If the plot is above the line, it indicates that the stock is underpriced. If the plot falls on the SML, it indicates the stock is properly priced. Mathematically: In the context of the SML, a security is underpriced if the required return is less than the holding period (or expected) return, is overpriced if the required return is greater the holding period (or expected) return, and is correctly priced if the required return equals the holding period (or expected) return.

SCL = Least squares regression line of asset’s return on that of market index. Slope is beta.

Question 22

CAPM

Answer 22

E(Ra) = Rf + Ba[E(Rmkt) - Rf]

The relationship between beta and expected return. In equilibrium, the expected return on risky asset E(Ra) is the RFR (Rf) plus a beta-adjusted market risk premium, Ba[E(Rmkt) - Rf].

Question 23

Capital market line (CML) 2

Answer 23

the tangent line drawn from the point of the risk-free asset to the feasible region for risky assets. The CML results from the combination of the market portfolio and the risk-free asset (the point L). All points along the CML have superior risk-return profiles to any portfolio on the efficient frontier, with the exception of the Market Portfolio, the point on the efficient frontier to which the CML is the tangent. From a CML perspective, this portfolio is composed entirely of the risky asset, the market, and has no holding of the risk free asset, i.e., money is neither invested in, nor borrowed from the money market account.

Question 24

Sharpe ratio

Answer 24

Sharpe ratio = (R(portfolio) - Rf) / SDmarket

Based on total risk. Treynor measure is based on market risk.

Similar to slope of CAL. Sharpe ratio is excess returns per unit of total portfolio risk. Tells us whether a portfolio’s returns are due to smart investment decisions or a result of excess risk. Higher ratio indicates better risk-adjusted portfolio performance. Uses total risk. Sharpe ratios of all portfolios along CML are the same. Value is only useful for comparing against other Sharpe ratios.CAL is the line representing possible combinations of risk-free assets and the “optimal risky asset portfolio” (two-fund separation theorem). ie the set of possible efficient portfolios.

Question 25

M-squared

Answer 25

= (R(portfolio) - Rf) * (SDmarket / SDportfolio) - (R(portfolio) - Rf)

Produces same rankings as Sharpe ratio, but in percentage terms.

Question 26

Treynor measure

Answer 26

Risk-adjusted return based on market risk (beta) rather than total risk (like Sharpe ratio). relates excess return over the risk-free rate to the additional risk taken; however, systematic risk is used instead of total risk. The higher the Treynor ratio, the better the performance of the portfolio under analysis.

(R(portfolio) - Rf) / B(portfolio)

Question 27

Jensen’s alpha

Answer 27

A risk-adjusted performance measure that represents the average return on a portfolio over and above that predicted by the capital asset pricing model (CAPM), given the portfolio’s beta and the average market return.

a(portfolio) = R(portfolio) - [Rf + Bp * (Rm - Rf)]