CFA Module 9

Holding Period Return formula

R = (P_t - P_(t - 1) + D_t) /(P_(t - 1)), where P_t is price at time t, P_(t - 1) is price at time t - 1, and D_t is dividend earned at time t

Holding Period Return formula over time

HPR = (1 + R_1)(1 + R_2)...(1+R_n) - 1

Arithmetic Mean formula

mean(R_i) = 1/T * summation i = 1 to T of R_i

Geometric Mean formula

R_(G_i) = [pi i = 1 to T (1 + R_(t_i))]^(1/T) - 1

Define Money Weighted Rate of Return

the IRR that equates the present value of all cash flows to the ending value of an investment

Money Weighted Return formula

summation t = 0 to T (CF_t) / (1 + IRR)^t = 0

What does money weighted return tell about an investor?

accurately reflects what a specific investor earned because it accounts for the size and timing of all cash flows

Why can't money weighted return be compared to that of another portfolio?

because size and timing of cash flows are different between portfolios

define Time Weighted Return

measures the compounded rate of growth of $1 over the measurement period

Time Weighted Return formula

(1 + TWRR)^n = (1 + HPR_1)(1 + HPR_2)...(1 + HPR_n)

Annualized Return formula?

r_ann = (1 + r_days)^(365/days)

Define Portfolio Return

weighted avg. of the individual returns

Portfolio Return formula?

R_p = summation i = 1 to N (w_i * R_i), where summation i = 1 to N (w_i) = 1

Define Gross Returns

Total Return - trading fees (basis for comparing manager performance)

Define Net Returns

Gross Returns - mgmt. fees (what the investor returns)

Real Return formula

(1 + r) = (1 + r_f) * (1 + pi) + (1 + RP), where 1 + r is nominal, 1 + r_f is real risk-free, 1 + pi is inflation premium and 1 + RP is risk premium

define Variance

a measure of the dispersion of returns

Variance of portfolio formula

Var(R_p) = summation i = 1 to n (w_i)^2 Var(R_i) + summation i = 1 to n summation j = 1 to n w_i w_j * Cov(R_i, R_j)

Correlation formula

Cov(X, Y) / (σ_x * σ_y)

what are the bounds of correlation?

-1 <= ρ_(x,y) <= 1

what does high correlation between two stocks mean?

their returns tend to move together either in the same direction (high positive correlation) or in the opposite direction (high negative correlation)

Expected Return formula?

1 + E(R) = (1 + r_f) (1 + E(pi)) (1 + E(RP))

Define Risk Tolerance

level of risk willingly accepted to achieve investment goals

define Utility Theory

investors derive satisfaction (utility) from particular choices (relative to others)

Utility Theory: Risk Aversion formula?

U = E(R) - 1/2 * Aσ^2, where A is weight of risk aversion for individual, σ^2 is variance/risk

Describe A > 0, A = 0 and A < 0

A > 0 is risk averse, A = 0 is risk neutral, A < 0 is risk seeking

Describe plotting utility over E(R) and σ

utility increase in non-linear shape meaning need an increase rate of return for each additional unit of risk

keeping utility constant, describe the curve with differing As over E(R) and σ

all curves non-linear with high risk aversion having the steepest slope (fast increasing rate of E(R) for each additional unit of σ), risk-seeking has slopes downward non-linearly (each additional unit of σ results in a faster decreasing rate of E(R))

define Capital Allocation Line

graphical representation of risk-return combinations available to an investor by mixing a risk-free asset and a risky asset

what is the expectation and variance of a risk-free asset and risky asset

E(R_p) = w_1 r_f + (1 - w_1)E(R_i), σ^2_p = (w_1)^2(σ_1)^2 + (1 - w_1)^2(σ_2)^2 + 2w_1 (1 - w_1ρ_(1, 2)σ_1σ_2

define Indifference Curve

graph that shows different combinations of risk and return between which an investor is equally satisfied or indifferent

describe any point on an indifference curve below the CAL

undesirable, move up to get higher return for same risk

describe point of intersection on indifference curve tangent to CAL

optimal portfolio to investor

describe point strictly above point of intersection on indifference curve tangent to CAL

unattainable

describe other points of intersection to CAL on other indifference curves not tangent

these points on same curve that has other points that are suboptimal => these same points are also suboptimal

lowly correlated portfolio says what about risk

lower σ => lower risk

Describe curve when correlation between 0 and 1 when plotted over E(R_p) and σ_p

at first, there is actually higher return for less risk and then return increases for each additional unit of risk (curve resembles top 60% of horizontal parabola)

less correlated implies what about diversification?

less correlated, the greater the benefit of diversification

define Sharpe Ratio

measures the excess return earned on an investment per unit of total risk

Sharpe Ratio formula?

E(R_i) - R_f / (σ_i)

Risk-Adjusted Return formula

(E(R_i) - R_f) / (σ_i) > ((E(R_p) - R_f) / σ_p) * ρ_(i, p)

Define Minimum Variance Frontier

set of all portfolios that offer the lowest possible risk (variance or standard deviation) for a given level of expected return, based on all possible combinations of risky assets, looks like a sideways parabola when graphed

describe points below the global minimum of Min. Variance Frontier

high risk., low return => inefficient

Define Markowitz Efficient Frontier

any point on the Minimum Variance Frontier above the global minimum

Define The Two Fun Separation Theorem

all investors regardless of taste, risk, preferences, wealth, will hold a combination of 2 portfolios, a risk-free asset and a risky portfolio

Expected return on portfolio formula in terms of risk-free asset and risky portfolio

E(R_p) = r_f + (E(R_rp) - r_f) / (σ_rp) * σ_p, where r_f is the y-intercept, E(R_rp - r_f) / σ_rp is the slope

describe points on CALs that lie left of the Markowitz Efficient Frontier

known as Lending Portfolio, because holding some combination of risk-free asset and risky portfolio (holding risk-free asset means lending money to govt.)

describe points on CALs that lie right of the Markowitz Efficient Frontier

known as Borrowing Portfolio, because beyond 100% weighting on risky portfolio so need to borrow money

formula for slope of CAL(i)

(E(R_i) - r_f ) / σ_i

define Homogeneity of Expectations

Assuming markets are informationally efficient, all investors have the same economic expectations => only 1 optimal risky portfolio

what if expectations are not homogenous?

multiple optimal risky portfolios

what if markets are not informationally efficient?

active investing may deliver excess return

define Capital Market Line

A CAL where the risky portfolio is the market portfolio

what would the slope of a CML represent?

Market Price of Risk

define Non-systematic Risk

unique to individual investments, can be diversified away

define Systematic Risk

Market-wide, cannot be diversified away (ex. interest rate changes, inflation, recession)

Describe Non-Systematic and Systematic Risk as the # of securities increases

Non-Systematic decreases at a decreasing rate and Systematic Risk is unchanged

Total Variance formula in terms of risk

Total Variance = Nonsys. Var. + Sys. Var.

can excess return be attained by diversifying away nonsys. risk?

excess return cannot be obtained because very investor will do it, thereby driving up the price of individual asserts, decreasing potential return

Describe sys. and nonsys. risk of T-bill

risk-free asset => Total Var. = 0 => 0 sys. risk and 0 nonsys. risk

Describe sys. and nonsys. risk of S&P 500

sys. risk = market risk and nonsys. risk = 0 because it has been diversified away

Consider two assets A (15% sys, 15% nonsys), B (17% sys, 0% nonsys). Which has higher E(R)?

Asset B has higher E(R) because only get paid for sys. risk and sys. risk of B = 17% > sys. risk of A = 15%

define Capital Market Theory

The market will expect a higher return on the investment that has a higher level of systematic risk, regardless of total risk (nonsystematic risk is not rewarded by an efficient market)

Multi-Factor Model formula

E(R_i) - r_f = summation j = 1 to k β_ij * E(F_j), where E(R_i) - r_f is excess return, β_ij is factor weights, F_j is factor j

Single Factor Model formula

E(R_i) - r_f = β_i * [E(R_m) - r_f]

derive β_i as a weight

β_i = σ_i / σ_m = total security risk / total market risk = security sys. risk / sys. risk = β * σ_m / σ_m = β

Market Model formula

R_i = α + β R_m + ε_i, where α = r_f (1 - β)

define Security Market Line

shows expected return of an asset as a function of its systematic risk, measured by β

β_p formula

β_p = w_1β_1 + w_2 β_2 + ... + w_n * β_n

what does slope of SML represent?

market price of risk

β_i formula?

β_i = ρ_im * σ_i / σ_m

Define Beta

a measure of how sensitive an asset's return is to the market as a whole, captures an asset's systematic risk

what is β_m?

β_m = ρ_mm * σ_m / σ_m = ρ_mm = 1

β_m being 1 says what about the avg. β of the stock in the market

avg. beta of the stocks is also 1

define Capital Asset Pricing Model (CAPM)

financial model that describes the relationship between the expected return of an investment and its systematic risk, measured by beta

assumptions of CAPM?

investors are utility maximizing, risk-averse, rational, markets are frictionless, no transaction costs, no taxes, all investors have the same single-period investment horizon, investors have homogenous expectations, all investments are infinitely divisible, investors are price takers

CAPM formula?

E(R_i) = r_f + β_i * [E(R_m) - r_f]

Portfolio Performance Evaluation: Sharpe Ratio

(R_p - r_f) / σ_p, where σ_p is total risk

Portfolio Performance Evaluation: Treynor Ratio

(R_p - r_f) / β, where β is sys. risk

Portfolio Performance Evaluation: M^2

(R_p - r_f) * σ_m / σ_p - (R_m - r_f), where R_p - r_f is excess return on portfolio and R_m - r_f is excess return on market, and σ_m / σ_p is a measure of total risk

Portfolio Performance Evaluation: Jensen's Alpha

α_p = R_p - [r_f + β_p (R_m - r_f)], where R_p is actual portfolio return and [r_f + β_p (R_m - r_f)] is what the return should have been, and β_p is a measure of sys. risk

What ratios should be used for portfolios with high nonsys. risk?

Sharpe and M^2 because they use total risk in their formulas

What ratios can be used for highly diversified portfolios?

highly diversified => low nonsys. risk, so Treynor and Jensen's Alpha should be used as they use sys, risk in their formulas

define Security Characteristic Line

regression line that shows the relationship between the return of a security and the return of the market, same formula on the Market Model: R_i - r_f = α_i + β_i(R_m - r_f)

What does the Security Characteristic Line say about securities with α > 0 and α < 0

Select/overweight securities with α > 0 and deselect/underweight securities with α < 0

if two portfolios have the same expected return, which one should you choose?

choose the one with lower variance (risk)

define Fix Drift

prices will drift from the asset allocation mix

define Dynamic Rebalancing

get back to original mix

define Tactical Rebalancing

intentional deviations from the mix

Mutual Funds: define Open-Ended Funds

accepts new funds and issue new units at Net Asset Value (NAV), must have ready liquidity (cannot be 100% invested)

Mutual Funds: define Closed-End Funds

fixed number of units/shares, which are exchange-traded

Mutual Funds: define Load Funds

annual fee + buy/sell fees

Mutual Funds: define No-Load Funds

annual fee based on NAV

define Strategic Asset Allocation (SAA)

% allocated to each asset class in order to achieve the client's objectives

What are the two overarching types of cognitive errors?

Belief Preservation Biases and Processing Errors

define Belief Preservation Biases

tendency to cling to prior beliefs by committing statistical, information-processing, or memory errors

define Processing Errors

information being processed and used illogically/irrationally

Belief Preservation Biases: define Conservation Bias

maintain prior views or forecasts by inadequately incorporating new, conflicting information

Belief Preservation Biases: define Confirmation Bias

people tend to look for and notice what confirms their beliefs and ignore or undervalue what contradicts their beliefs

Belief Preservation Biases: define Representative Bias

tendency to classify new information based on past experiences and classifications