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37 Cards in this Set

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Correlation formula

p1,2 = Cov1,2


-------------


σ1 σ2

Variance of a two asset portfolio

with covariance:


σ2 = w1^21^2 + w2^22^2 + 2*w1*w2*covariance(1,2)



with correlation:


σ2 = w1^21^2 + w2^2*σ2^2 + 2*w1*w2*p1,2σ1σ2

Expected return on a 2 asset portfolio

E(Rp) = w1E(R1) + w2E(R2)



where E(Rp) is the expected return on portfolio P


w = weighting of that asset


E(R) expected return on that asset



Capital allocation line (CAL)

describes the combinations of expected return and standard deviation of return available by combining an optimal portfolio of risky assets with the risk-free asset; the graph of this starts at the intersection of the RFR return and is tangent to the efficient frontier of risky assets – the line itself represents an optimal portfolio of risky assets

An individual investor’s risk tolerance will determine where on the CAL they will prefer to be

CAL is the maximum slope from the minimum variance portfolio to 100% risky portfolio

If we can invest in a RF asset, CAL represents the best risk-return trade-off achievable

The CAL has a y-intercept = to the RFR

The CAL is tangent to the efficient frontier of risky assets



Y = a +bX


E(Rc) =


[E(Rt) - Rfr]


RfR + --------------------- x σc


σt



Capital market line (CML)

when investors share identical expectations about mean returns, variance of returns, and correlations of risky assets; when the CAL is the same for all investors

CML equation =


E(RA) =


[E(RM) - Rfr] x σA


RfR + ---------------


σM



The slope of this line equation = the market price of risk, because it indicates the market risk premium for each unit of market risk

Relationship between CAL and CML

CML is when the CAL is the same for all investors

Equally weighted portfolio risk

CML and CAPM

CML represents the efficient frontier when the assumptions of the CAPM hold

CAPM = E(Ri) = RFR + Beta * (E(R of Market) – RFR)

Security market line (SML)

is the graph of the CAPM model, or the CAPM equation

is a linear function of beta



CAPM = E(Ri) = RFR + Beta * (E(R of Market) – RFR)

Beta definition as it relates to the market

Beta is a measure of the asset’s sensitivity to movements in the market

Beta =


Covi,m


----------


σm^2




σi


pi,m x ---------


σm




pi,mσiσm


-------------


σm^2

Market risk premium

E(Rm) - RFR

Sharpe Ratio

the ratio of mean return in excess of the RFR to the standard deviation of return

Sharpe Ratio =


(E(Ri) – RFR)


--------------------


sd of Asset i

Adding assets to the portfolio and the Sharpe Ratio

Adding a new asset to your portfolio is optimal if the following condition holds:

1) E(Rnew) – RFR / sd of new > (E(Rport) - RFR / sd of port) * Corr (Rnew, Rport)

2) As long as sharpe of new asset is greater than sharpe of portfolio, (i.e. Corr = 1), should add

Market Model

describes a regression relationship between the returns on an asset and the returns on the market portfolio

Ri = alpha’i’ + beta’i’ * Rm + error’i’

Where alpha is the average return on asset’i’ unrelated to the market return

Market Model assumptions

The expected value of the error term is 0

The market return is uncorrelated with the error term, Cov(Rm, error) = 0

The error terms are uncorrelated among different assets

Adjusted beta

if historical beta is not deemed to be the best predictor, can use adjusted beta

Adjusted beta uses instead a first order autoregression: Bt+1 = alpha initial + alpha 1 * Bt + error

To simply, given mean reverting tendencies, alpha initial = 1/3 and alpha 1 = 2/3

Adjusted Beta = (1/3) + (2/3)*(Beta t)

Historical beta

assumes that beta for each stock is a random walk from one period to the next, and the error term mean is “0” – so Beta t+1 = Beta t + error (or 0)

Multi-factor model

multi-factor models could also address: interest rate movements, inflation, or industry-specific returns

Ri = ai + bi1 * F1 +…+ bik * Fk + error

Where ai is the expected return on Asset i, bi’s are the sensitivities of each factor, and F’s are the surprise in each factor

Factor sensitivity = a measure of the response of return to each unit of increase in a factor, holding all other factors constant

Error is the part that is unexplained by the expected return and factor surprises, is therefore defined as: asset-specific risk

Active return

return on portfolio – return on the benchmark (comparable to the portfolio)

split into two components: active factor sensitivities (sector weightings) and asset selection (stocks per sector)

Active risk

the standard deviation of active returns

Active factor risk

the contribution to active risk squared resulting from the portfolio’s different-than-benchmark exposures relative to factors specified in the risk model

Active selection risk or Active specific risk

the contribution to active risk square resulting from the portfolio’s active weights on individual assets as those weights interact with assets’ residual risk = sum of weight differences and variances of the asset’s returns unexplained by factors

Tracking error

synonym with active risk, but the term “error” is confusing as it is meant to represent “difference” here

Tracking risk

also a synonym of active risk =
sd * (Rportfolio – Rbenchmark)

Make sure the same time periods are used for each return

Can vary from 1% with a passive investment to 6-9% for very active investment management

Separation theorem

everyone holds the same portfolio of risky assets and individual investor’s determine the weight of that portfolio with their domestic RFR “separately”

Real exchange rate movements

are defined as movements in the exchange rates that are not explained by the inflation differential between the two countries

% chg real exchange rate = % chg nominal exchange rate + foreign inflation – domestic inflation

For extended CAPM to hold, there can be no real exchange rate movement; x = 0; appreciation or depreciation must be fully offset by the inflation differential between the two countries

Foreign currency risk premium working in concert with interest rate parity

E(R) – RFR, or the expected movement in the exchange rate less the interest rate differential (domestic RFR – foreign RFR), and after factoring in appreciation/depreciation for the period

Linear approx says that (F – S) / S is approximately equal to RFRdc – RFRfc (interest rate differential) – the best predictor of exchange rates is the interest rate differential

The expected return may be less with currency hedging, as it bears less risk –> difference between expected return no hedging – hedging = foreign currency risk premium

Information ratio

a tool for evaluating mean active returns per unit of active risk

IR = (mean Rportfolio – mean Rbenchmark) / standard deviation * (Rportfolio – Rbenchmark)

Or IR = annualized residual return 𝛂


--------------------------------------- = ------


annualized residual risk w



IR = IC x √BR

Information Coefficient

measures managers forecasting accuracy


if a manager makes N bets on the direction of the market and Nc are correct, the IC is the covariance between forecast and actual direction of the market



IC = 2x (Ncorrect/Ntotal guesses) - 1





when we add another source of information that is correlated, the skill (IC) of the manager does not increase proportionately. ICcom represents the new info.


ICcom = ICorig x √(2/1+r)



where r = correlation


ex-post information ratio

related to the t-stat one obtains for alpha in the regression of portfolio excess returns against benchmark excess returns:


tα t statistic of alpha


----------------------------------------


√n number of years of data

Value added

Objective of active management is to maximize value added



VA = α - (λ x ω^2)



λ = risk aversion


ω = residual risk

Highest achievable value added *

function of optimal level of residual risk and the portfolio managers IR



VA* = ω* x IR


---------- or


2



VA* = IR^2


---------- or





VA* = IC^2 x√BR


-------------


Breadth

# of forecasts made in a year



IR = IC x √BR

Optimal level of residual risk



ω*

ω* = IR IC x √BR


----- = ------------


2λ 2λ




λ = risk aversion

Systematic Risk

reflects factors that have general effect on the securities market as a whole and cannot be diversified away



for example macroeconomic risk




represented by Beta

Unsystematic risk

can be reduced through diversification

The single factor market model covariance calculation

One of the predictions of the single-factor market model is that Cov(Ri,Rj) = bibjsM2. In other words, the covariance between two assets is related to the betas of the two assets and the variance of the market portfolio.