410 - Exam 2 Flashcards
Covariance
btwn 2 securities is positive = securities tend to move together, when it is negative = 2 securities tend to move in opposite directions
take the historical returns and find ave. to get E[r] for each security
- -find st. dev. by subtraction E[r] from the actual (actual - E[r]
- -prev. find var. for on seuciry by taking ace. sq. forecase error - Excel st. dev. function
COVAR = the average PRODUCT of the 2 forcast errors (multiply the single deviations of security 1 and 2 then find ave. product)
MAIN POINT - covariance indicates the SIGN of the relationship btwn 2 random variables
–positive dependence, neg. dep. or independent if covar = 0
graphing covariance and testing postive/negative dependence
- -each dot on graph represents a JOINT outcome (x coord. isoutcome for A and Y coor. is utcome for Y)
- -x and y are CENTERED ON THE MEANS
- -shows that every dot above x axis, y return is alos positive and dots below, Y is also below = move together so POSITIVE DEPENDENCE
bc products of dev. x and y are positive in Quad II and st. dev. both nef. in Quad III but (neg * neg) = positive —> so estimated covar would be positive
indepndence = if A does well B is equally liekly to do well or poorly
correlation
estimates the STRENGTH of the dependence –> how closely the scatter plot fits around a striaght line
Px,y = Cov(X,y) / (stdev. x* st. dev.y)
bound to be btwn -1 and +1
find by using covar and st. dec. functions in excel
regression
condional expectation - given ___ what do you expect ___?
conditional E[r] is conditional on some set of info - estimate conditional E(stock r) given someoutcome of a portfolio its in
ex. holding an auto portfolio with ford stock inside, E[r] Fprd = 8% and E[rp] = 10% –> if heard E[r] was 18%, what would you expect Ford’s return to be?
- -expected movement of one given expected movfement of another (children ex,)
- -even if 2 indiv. have a lot of energy, group as a whole does not move much
- if expect stock to do really well if p does well, then stock helps portfolio move a lot –> so tilting towards stock will amplify portfolio volatility bc move in same direction
- if expect stock to do poorly if portfolio does well, then stock and portfolio pull against each other - stock slows portfolio down - so if tilt oward stock it will help diversify risk and act as a hedge (bc if portfolio does poorly our stock will do well)
regression 2
process of estimating the parameters necessary to get a conditional expectation
–conditional exp. also helps us understand FORECAST ERROR (st.dev.) of securities - forecast error realized return minus forecast E[r]
regression parameters help understand forcast error ecamples:
- price related to news that affects portfolip
- unrelated news about same portfolio
- ex. Eccon mobile jumps upu so forecast error = 8% one day - why? bc good news about industry or good news boaut Exxon mobile?
- -with regression we understand that 6% of forecast error related to oil industry news and 2% to exxon movile news only
- -helps us inderstand what KIND of risk is diversified away in our portfolio (systematic/firm specific)
regression also helps us create statistical models:
conditional E[R] = (E(ra | rp = x) – expected return on stock A given portfolio return is x
in linear function
E(ra | rp =x) = a + bx
called a linear regression model
ex. linear regression model
a = .01, b = 1.2, rp = 25%, A return =
.01 + 1.2*.25 = .31
a +bx gives the return on stock A given rp = x
a = regression intercept
b = regression slope
rp = explanatory var. + depentand var.
ex. Autoliv and portfolio - want to know if tiling toward autolivve will impact portfolio volatility
- -graph showed positive dependence - so tilting towards A will inc. portfolio volatility bc move togehter
run regression to fit scatter plot with line of best fit - line gives an estimate of conditional e[r] for autoliv for any possible return on portfolio
solve and get y = .002 + 1.87x
- -bc b is positive, know that tiliting toward autoliv will amplify portfolio
- -the HIGHER and morepositive the regression SLOPE the more volatility will inc. if we tilt portfolio toward autoliv (BECAUSE B IS BETA!!!!)
not always accurate = it is the expected!!
residual
diff. btwn realized return for a given month and the expected return on Autoliv (stock A) given the return on portfolio that month
zt = yt - (a + bx)
yt = a + bxt + zt
- -shows variation in “yt” is driven by variation in xt AND zt
- -var (yt) = var (bxt) + var(zt)
estimating regression parameters
- -understand residuals helps us estimate parameters a and b
- -if line does not fit data well, get very large residuals
- -so fit line choose a and b to MINIMIZE SQUARED RESIDUALS
using STAT RULE #3 can find regression parameters - says for any regressiong with dependent variable Y and independent (explanatory) variable X, the slope (b) and intercept (a) of regression line are:
b = COV(x,y) / Var(x)
a= average Y - b(average X)
MEAN Y - bmean x
Y= dependent variable (stock) X = independent (portfolio)
OR TO CALCULATE B AND A JUST DO THE SLOPE AND INTERCEPT FUNCTIONS IN EXCEL
slope of regression line
slope where Y is return on security and X is return portfolio is called BETA
–diff. than mkt. beta - this is beta for a security in a specific portfolio (diff. if use other portfolio)
By,x - x = explantory and Y = dependent
BetaA,P = beta tells us how we expect dep. variable to change given a change in explanatory variaible
–if B = 2, then if portfolio jumps, the stock jumps twice as much
two kinds of variation in Yt
- variation in x can explain variation in y
- -the fraction of variation in yt that can be explained by variation in xt is called R^2 (fraction of risk that is systematic)
R^2 = b^2*var(x) / var(y)
- -this # depends on REGRESSION SLOPE
- -if b = 0 then none of the variation in y is explained by x, but if FAR from 0 then lots is explained
- but other factors that explain y variation aren’t captures in x changes are explained by RESIDUAL
- -Residual explains firm specific risk
portion of risk that is firm specific = var(Residual) / var(y)
2 rules of the regression is well fit
- the ave. value of the residual is 0!! (bc should have even positive and negative)
- the covariance btwn residual z and explanatory variable x is 0
- -means that z and x are independent - have no relation and means var. in z that has impact on y has no relation with x changes (impact y differently)
the residual represents other factors that influenced Y that were independent of the explanatory variable
–if covar btwn x and Y is not 0 then line does not fit well (means we calculated parameters wrong and there is either positive or negative dependence of z and x that shouldnt exist)
estimating regression parameters 2
residual helps us estimate a and b - if line does NOT fit data well then we will have large residuals
–to find best fitting line - MINIMIZE squared residuals
S = sum(y - a - bx)^2
ex. in AppB-Reg2
y = % wins, x = points per game scored by top scorer
- -calc. residuals –> represents the compononet of % wins (y variable) that had nothing to do with points per game scored by top scorer
- -residual represents other factors INDEPENDENT of x (points per game score by top scorer)
if find wrong line of best fit - can have dependence
- -negative dependence shows that when rp is low, residuals are high and when rp = high, residuals are low
- so covar and corr. are neg. when should be = 0
positive dependene - covar btwn x and z is positive and should be 0 - low alues of rp and negative residuals and high va for rp and positive residuals
if y and x are dependent and explanatory vairables then fraction of variation in y that can be explained by var. in factor UNRELATED to x are:
VAR(z) / VAR(Y)- variation due to unsystematic risk
last formulas check to decide which portion is systematic and which portion of variance is firm-specific
firm specific = VAR(z) / (VAR y)
systematic = 1 - above^^^
OR b^2*VAR(X) / VAR (y)
look at APPb -Reg 4 in examples for practice!!
using excel to run regression
AppB-reg6
date –> data analysis –> regression
–input Y range then input X range
-“output range” and select where you want it placed
gives you box that is in my notes - the 2 rows are intercept and x var. 1
- -coeff. column gives intercept first then slope(beta)
- -at end gives 95% confidence interval boundaries
- -tells you you are 95% confident where beta will fall
regressions with multiple explanatory variables - MULTIPLE BETAS
y = a + b1x1 + b2x2 + z
- -y is dependent
- -x1 and x2 are explanatory
- -a is regression interepy
- -b determines influence of 1st exp. var. and b2 detrmines influence of 2nd exp. var.
- -a + 5b1 + 2b2 –> gives expected value of y givex x1 = 5 and x2 = 2
- -z is residual and explains variation in Y unlreatled to x1 and x2
z = y - (a + b1x1 + b2x2)
firm specific = VAR(Z) / VAR(Y)
systematic = 1 - VAR(Z) / VAR(y) = R^2
COVARIANCE(Z,X1) AND covar(Z, X2) both must = 0!!!
EX. AppB - Reg7
- -find a and b1, b2 by running a regression OR by doing INDEX(LINEST,Y range, Xrange){3,2,1}) - command shift enter
- -gives in order the intercept, B1, B2
