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267 Cards in this Set
- Front
- Back
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Shortfall risk
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value of port goes below the target value over a given period
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Roys Safety Ratio
what does it say in words |
optimal risk minimizes the shortfall risk
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cov of a portfolio
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ps x (ra- ERa) x (rb - ERb)
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Monte Carlo simulation
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repeated genereation of risk to calc the value of sec
Computer are used |
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Discrete random variable also give eg.
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possible outcomes can be counted no. of days it rained last month
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Multivariate distribution
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specifs the prob ass with group of rand variable
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Multivariate distribution can be both
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continuos and discrete , for cont it uses prob dist and for discrete it use the joint prob tables
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Multivarite dist give an eg. of contiuous dist
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if ind asset in a port are norm dist than the port is also norm dist
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Correlation and multivarite
give eg. give formula for no. of covariances |
4 asset port with 4 means 4 variances and [n*(n-1)]/2 no. of covariances
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Discrete uniform random variable
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px= is same for all
fx =npx range b/w 2 and 8 kpx fx is also a cummulative dist function |
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Binomial Dist (Variance formula)
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n p (1-P)
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Exp value of a portfolio
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Wa ERa + Wb ERb+........
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Probabilty dist function
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All possible outcomes of random variable like dice has 1/6+1/6.... and for continuos dist....
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Monte carlo simulation process
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specifcy parameter
Genereate random risk Calc the value of option Calc mean option value |
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Correlation and Multivariate Dist what is req.
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when building a portflio of assets assets with low corr are required req so the less variance comes out
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EAR for continuous comp
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e to the power stated rate minus 1
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Confidence interval
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range of values around a expected outcome
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Confidence interval of most interest are
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90
95 99% |
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Confidence interval Formula for like 90% interval
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90% interval
= average return - 1.65 (s) |
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1 std dev =
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68% prob
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2 std dev=
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95% prob
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Standard normal Dist properties
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mean of 0
std dev =1 |
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Standardization
process |
convert observed value to z value
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Standardization
formula |
observation - pop mean
____________________ Std Dev |
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what does z = -2 means in standardization process
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means the eps of 2 is 2 std dev below the mean.
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Roys safety first fomula
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ERp-RL
--------------- std dev of port |
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Roys saftey first similar to
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it is similr to sharpe ratio
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Roys Saftey first steps
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calc the SFR Ratio
choose the port with larger SFR no. |
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Roys Saftey first is used for and use
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choosing amount the 2 diff port with diff ret and devations
and uses z tables see eg on pg 259 |
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Additive property is
used in |
used in continuos compounding
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Additive property is defined
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mulitple periods the
[e^2(0.1)]-1 |
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by looking at ztable what is happening
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it converts the std dev values to probablity values so that we understand
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Binomial Dist p(x)=
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n!
-------- p^x (1-p)^(n-x) (n-x)!x! |
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Binomial Dist
give eg. |
5 trials
3 black balls prob of black ball is 0.6 |
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Compounding
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calculating fv of the cashflows
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Discounting
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calculating pv of the cashflows
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liquidty risk
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risk of receiving less amount of money if sold for cash quickely
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req rate on a sec has
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nom rate+mat risk pre+liq risk premium+defalut risk prem
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Effective annual rate means
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return actually earing after compounding adjustments are made
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Effective annual rate formula
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[(1+stated rate/n)^n]-1
(1+0.1/12)^12 - 1 |
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EAR increases
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when the compounding freq increases
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simple random sampling
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each pop person has same likely hood of being selected
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systematic sampling
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another way of random sampling
seclecting ever nth member of population |
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sampling error define
give eg. |
diff b/w sample parameter and corresponding pop parameter
eg. sampling error mean is sample mean- the population mean |
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Sampling distribution
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important to recognize that sample statistic of rv can have a probabitity distribution
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sampling dist of mean
eg. |
sampling dist of mean
100 bonds out of 1000 select mean of sample , repeat the process many time to get many means the dist of these mean is sampling dist of mean |
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stratified random sampling
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uses a classification system
to sep population into sep groups, samples are extracted from the groups used in bond indexing |
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stratified random sampling
give eg |
1000 bonds
first classify like maturity/rates then select random samples from cells then individual sample combined to make population |
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stratified random sampling
guarntees |
bonds are selected from each cat of pop, otherwise in random one can select none from one cat and 10 from another category instead of 5
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Times series Data
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taken over a period of time spaced interval, eg. monthly ret of microsoft from 1999 to 2004
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cross sectional data
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single point in time like EPS of all nasdaq companies ON DEC 31 2005
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longitudness data
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over time and multiple characteristics like gdp, inflation, unemployment of a country over 10 years
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panel data
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over time but same characteristic like debt/equity ratio of 20 companies for last 4 quarters
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Central Limit Theorm define
useful why |
states if the sample size is large ie 30 or more than the dist of the sample mean approaches a normal probabiliy
useful b/c norm dist is easy to apply hypothesis testing to and construction of confidence intervals |
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Central limit theorm ki
properties batao |
if sample 30 or more the dist of sample mean distribution approaches normal
mean of pop and mean of sample same variance of dist mean is variance/n, ie the pop variance divided by sample size |
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Point estimate
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single value of sample to estimate the pop parameters
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formula to calc point estimate is
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_
x = Σ x ....... ___ ........ n |
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confidence interval
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range of values in which the pop parementer is expected to lie
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Standard error of sample mean what is it
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measure of variability or in other words it is the std dev of the distribution of sample means
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standard error formula
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x=σ of pop /√n sample size |
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what is practically unknown in the following
_ x=σ of pop /√n |
std dev of pop so instead std error is estimated dividing std dev of sample mean with √n
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what happens to the standard error as the sample size increases
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as sample size go up from 30 to 300 as in eg. value of std error decreases b/c sample means on av gets closer to true mean or in other words distribution of sample mean around pop means gets smaller
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Desirable properties of an estimator
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Unbiased b/c the expted value of the estimator is equal to the parameter you are trying to estimate like value of sample mean = the pop mean
Efficient unbiased if also efficient if the "variance" samp dist of the estimator is very small than all the other unbiased estimators Consistent accuracy increase as the sample size goes up |
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For a consistent estimator if the sample size goes to infinity
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the standard error goes to zero
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confidence interval estimates
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THE RANGE of values within which the actual parameter lies
1- alpha |
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1-alpha
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is the level of significance
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and prob of 1-alpha
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is the degree of confidence
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eg of confidence interval
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like the pop mean will range b/w b/w 15 and 25 with a 95% degree of confidence
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Confidence interval formula =
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point estimate+- (relibility factor x standard error)
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Continuous compounding
define |
when discrete compounding go small small it becomes continuous
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Continuous compounding
formula |
(e^0.1) -1
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discrete compounding
formula |
[(1+0.10/12)^12] - 1
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More frequent discrete compounding meas rates
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go up
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Joint probability of indepenet events
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p(a)xp(b)
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Univariate Distributions
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is distbtn of single variable
not very common in practice |
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Historical simulation
uses limitations |
no estimation required
less frequent may not be reflected past is not indicator of future cannot address what if |
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Normal distribution
properties plays central role in |
described by its mean and variance
skewness 0 kurtosis 3 outcomes above and below means get less and less centrl role in port mgt theory |
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Price relatives
formula |
s1/s0 s1 is end price this is equal to 1+ HPR
as 1.01 in the binomial tree |
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Price relatives are used to get
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The end price of a security
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0 in price relative means
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-100% HPR return and the price of the asset gets to 0
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Lognormal distribution
properties |
is skewed to right
min value restricted to 0 on the left ln of e^x =x |
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Lognormal distribution is used to get what
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used to model the price relatives
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Lognormal dist is generated
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by the function of e^x
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Continuous uniform distribution
properties |
upper lower limits a&b
all x1,x2 lie within boundries prob of x outside a&b is zero range within boundries formula (8-4)/(12-2) |
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Monte Carlo simulation
uses limiatations |
VAR calcs
simulate pL value complex secur value non normal distributions fairly complex statistic and not analytical assumptions lots of them |
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Historical Simulation
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based on the actual changes in risk rather than model the dist
is actually randomly seleting one of the past risk and calc the value of sec |
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Expected return of ind asset in a portfolio
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ERA=P (a1|b1) Ra1+ P (a2|b2) Ra2+....
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Variance of a portfolio
cov terms |
wa^2 * variance of a+wb^2 * variance of b+2 wa wb cov (a,b)
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Variance of portfolio
corr terms |
wa^2 * variance of a+wb^2 * variance of b+2 wa wb std a std b corrleation (a,b)
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Tracking Error or Tracking risk
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Diff b/w port retrn and benchmark return like port r is 4 and bench market is 7 so T error is -3%
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Continuous Random variable
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no of poss outcomes infinite upper lower bounds exists amount of rain fall in the last month
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Probability function
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px decribes that prob of a rv is eq to a specific value px=x/10 when x can be (1,2,3,4)
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P (A and B) for
independent events |
PAxPB
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Joint prob of 2 or more indepent events
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is like 1/6*1/6*1/6=.00463
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for independent evens Pa|b or Pb|a is
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Pa and Pb respectively
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Total Probablity or an unconditional probability is the
(recall diagram) pg 205 |
sum of all joint probablitys
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delete this ard
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discrete and continuous
for discrete we use joint prob tables and continus we use the normal distributions |
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Joint prob of two dependent events
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p(ab)=p (a|b) p(b)
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prob to odds like 12.5% prob
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0.125/1-0.125 or 1/8 / 7/8= 1/7 or 1 to 7
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Priori prob
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formal reasoning
inspection process eg. dice coin roll |
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prob of at 1 of the 2 events happen
dependent events |
p(a or b)= pa+pb-p(ab)
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empirical probability
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based on past data like stk mkt went up 2 of the last 3 days to tomm the prob of stock mkt rising is 2/3
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odds to prob like 1 to 6
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1 to 6 = 1/1+6 or 1/7 or 14.29%
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Subjective prob
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Based on judgement like a feeling that tomm is going to rain
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Binomial Dist define
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is the prob of success in the no. trials, its a discrete dist, 2 poss outcomes, if only trial then bernolli RV
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Binomial Dist formula
card 5 |
n!/(n-x)!x! * (p)^x* (1-p)^(n-x)
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Binomial dist eg
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5 trials
3 black out what is the prob when prob of blacks out is 0.6 |
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Binomial dist expected value
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Ex= n*p where n=no. of trials and p=prob of sucess
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Binomial Tree
Recall Diagram card 4 |
two possible outcomes
with prob of (p) 0.6 and (1-p) 0.4 uus price is equal to 1.01x 1.01 x 50 and prob is equal to 0.6*.06 |
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Permutation nPr 8P3
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n!/(n-r)! 8p3=336
order matters no. of ways of choosing r from n |
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Combination nCr or 8C3
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n!/(n-r)!r! 8C3=56
order not matter ways of choosing r from n |
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Labelling or Factorial
eg. |
8 stocks have to label long s
short s and sell s what the no. of possible ways=8!/4!3!1!=280 |
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Cummulative dist function
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cummulative value upto and including that specified outcome f(3)= 0.1+0.2+0.3
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Cov to Corr formula
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Cov a,b=σ a * σ b P (a,b)
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Cov of a portfolio
eg. on page 210 |
Σ PSx(Ra-ERa)x(Rb-ERb) it means for all the scnarios poor , normal , good
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Unique combi of 4 asset portfolio when calculating the variance of the portfolio
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n(n-1)/2
for eg 4*3/2=6 |
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Possiblity in normal dist vs lognormal dist
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that the asset retrn can go even go down below -100% means asset price is below 0 so modelling price relatives using log norm dist avoids this problem
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Probability density function
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upper lower bounds
2 possible outcome used on the continuous dists denoted by fx integration and calculas is used |
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fx is what
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probabitly density function
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Total Probability
Formula |
PA=P(A|B1)*P(B1)+P(A|B2)*P(B2)+P(A|B3)*P(B3) where b1,b2 are the mutually ex events and exhaustive events
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Exhaustive events
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that includes all possible outcomes
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Random variable
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uncertain number
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outcome
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observed value of the random variable
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Event
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is a singe or set of outcomes
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Students T distribution
properties 2 |
bell shapped prob dist that is symmetrical about its mean
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Student T Distribution
appropriate when |
sample size is less than 30 and pop with unknown variance, or large ie more than 30 when pop var is unknow
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Student T Distribution
properties recall card 3 |
symmetrical
defined by degree of freemdom fatter tails as n increases the t dist move closer to normal dist |
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Student T Distribution
degree of freedom n-1 b/c |
given mean the unique observations are only n-1
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Student T Distribution
more observation on the |
tails ie more outliers
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Student T Distribution becomes normal dist as the
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as the degree of freedom increases
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confidence interval should be wider when degree of freedm in t dist is/ and should be narrower when dof is
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less
more |
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Confidence interval of the pop mean formula
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recall card 1
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90 % interval value
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1.645
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95 % interval value
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1.96
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99 % interval value
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2.575
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Confidence interval formula when the pop mean is unknow
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recall card 2
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step of calculating the confidence of interval
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first the degree of freedom
find the app level of alpha or sigfcnce ie one tailed alpha or 2 ie alpha /2 look up t table to get the relbility factor |
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confidence interval for pop mean when pop variance is not known and sample size is large
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use t statistic
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pop mean unknown dist normal
which statistic is used |
t statistic
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pop mean unknown dist non- normal
which statistic is used |
t statistic
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pop mean known dist normal
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z statistic
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pop mean know non normal dist
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z statistic
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issues regarding sample selection
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size large
data mining sample selection bias survivorship bias look ahead bias time period bias |
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why large sample which seem good can have limitations
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consideration of cost
different population (observations fm) |
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Data mining
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using same database
to searching for patterns until one is discovered |
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how to avoid data mining
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test a rule a data set diff from the one which is used to develop the rule
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sample selection bias
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data is systematically excluded may be b/c of lack of av
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surviourship bias
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using only surviving mutual funds not including funds close at that time
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look ahead bias
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data not av on the test date ie price to book ratio fr eg.
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time period bias
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short phenomina
long econmic relations may have changed |
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Hypothesis testing
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statistical assesment of a statment
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Null hypothesis is denoted by
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Ho
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Probability distribution function
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lays out all the possible outcomes of a random variable like 1/6---> 1/6 Σ to 1
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Expected value of a portfolio
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Check card no. 7
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Correlation is nothing but the
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standardized form of the covaranice see card 8
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bayes theorm tree
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see card 9
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Bayes theorm forumla easy one
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see card 10 thanks
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labelling formula applies to
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3 or more sub groups
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permutation and combination applies to only
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2 sub groups
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Inferential statistics
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procedures to make forecasts
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Descriptive statistics
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summarize important characteristics of large data
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Ratio Scale
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true zero point, most defined, measurement of money is a good example
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interval scale
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provide relative ranking like temperatue is a prime eg.
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ordinal scales
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every observation is assigned a category like 1000 small cap growth stocks 1 to best performing 10 to the worst performing stocks
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Nominal scales
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no particular order, 1 to bond fund, 2 to corporate bond, 3 to equity etc
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NOIR means black in french is also what
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types of measurement scales
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sample statistic is used to
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measure a characteristic of a sample
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Process for frequency distribution
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1.define the intervals
2.tally the observations 3.count the observations |
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Relative frequency how is calc
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divide absolue freq of an interval with total no. of observations
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cummulative absolute/relative frequency how is that calc
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summing the absolute or relative freq starting at lower interval and progressing thru see pg 164 book 1 see card 20
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Histogram
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graphical representation of relative frequency, allows to quickely see where most obs are concentrated
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Frequency polygon
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where the mid point of each interval is plotted
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Arithmatic mean formula
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sum of obs/no of observations
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Weighted mean formula
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recognized that diff obs may have diff weights
wa ra+wb rb+wc rc |
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Median is
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the mid point when arranged in ascending or descending order
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median is impt b/c
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arithmatic mean can be affected by extremely large values
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geomatric mean
used for formula |
used in calc return OVER period
see card 11 pg 170 of book 1 |
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Harmonic mean
used for formula |
is used to calc avg cost shares
see eg on pg 171 book 1 also see card 21 |
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order the means
geomatric,harmonic and arithmaic which is larger |
aritmatic then geomatric and then harmonic
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Mean Absolute deviation MAD
define formala |
is the average of "absolute" deviations from the arithmatic mean formula card 12
see example 2 of pg 173 book 1 |
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Population Variance formula
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see card 13
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Sample variance formula
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see card 14
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Chebyshev's inequality
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for any set of obs the % of obs lie within k std dev is at least 1-1/k^2 for all k>1
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what is chebysheve ineq for +- 4 std dev
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see card 15
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Relative dispersion
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amount of variability in a distribution relative to a reference point.
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Coefficient of variation
what is measured by it |
relative dispersion is measured by cV, just to remember cv is the measure of variation (risk) per unit of return
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CV formula for an asset x
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see card 16
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CV important in investment b/c
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it is used to measure the risk per unit (variability) of expected return( mean)
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Sharpe ratio or reward to variablity ratio
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measure the excess return ie excess return over the risk free rate/unit of risk
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sharpe ratio formula
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see card 17
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if researcher wants to test return on option is differnet from zero
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than 2 tailed test
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if the res want to test the retrun on option is greater than or less than zero than
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one tailed test is used
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Hypothesis testing process two in my own language also see card 22
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state the hypothesis ie ho and ha
select one or two tail based on = or >< get the level of sig make a decicison rule on the basis of level of sig ie if test statis is > or< etc get the standard error compare the decision rule and result of test statistic accept or reject ho |
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how is test statistic calculated
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by comparing the point estimate with the hyopthesized value of the parameter
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test statistic is acually difference
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diff b/w sample statistic and hypothsized value scaled by std dev of sample statistic see card 18
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what is this type 1 and type 2 error
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we make wrong inferences about the population computed by the sample drawn from that population
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type one error
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rejection of null hypothesis when it is true
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type 2 error
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failure to rejected null hypothesis when it is actually false
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significance level
relation with the error |
is the possibility of commiting type 1 error
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significance level is req hypo testing b/c
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in order to identify critical values to evaluate the test statistic
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Define decision rule delete
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is rejecting for failing to reject the Ho
based on the distribution of test statistic |
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Define power of a test
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rejecting null hypo when it is false or 1- type II error
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As oppose to sig level (prob of rejected the Ho when it is true) POWER OF A TEST is
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correctly rejecting the null hypothesis when it is false or 1-prob of type 2 error (fail to reject the null hypo when it is false)
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Define relation b/w confidence interval and hypothesis test
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CRITICAL VALUE
confidence interval also use critical value as well as does the hypothesis tests see formula on card 19 |
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statistical signifcance doest mean
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economic significance
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statistical sig is not eq to economic significance b/c
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a. cost
b. tax. c. risk (short sale is risk, variation from year to year |
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p-value define
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probability of obtaining a test statistic that would lead to rejecting null hypo when it is true
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when to use t test
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when variance of pop unknown and sample is less than 30 or when variance of pop unknown and sample is more than 30
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test statistic based on t statistic formula see card 23
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see card 23
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t statistic enjoys
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world wide application b/c pop mean is normally not known
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z statistic formula
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see card 24
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when pop size is large but variance is unknown what can be done
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can use z or t statistic but t statistic prefered b/c more conservative
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in a statistcal world when is a situation nothing can be done
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when sample size is small and it is not normally distributed then we have no reliable test
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10% level of sig give the two tail and one tail z values
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1.65 and 1.28
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5% level of sig give the two tail and one tail z values
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1.96 and 1.65
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1% level of sig give the two tail and one tail z values
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2.58 and 2.33
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when is sample size is tooo large what happens to the t and z critical values
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the become almost identical
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when we want to compare if there is any diff b/w the two means of two diff INDEPENDENT pop
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there are two tests
1. when pop var (unknown) is equal 2. when pop var (unknown) is unequal |
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when pop var (uknown) is EQUAL what should be done in test when we want to see if there is any diff b/w the two means
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pooled variances ie both added sp are used in the denominator of t-statistic
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when the pop var unknown is not equal what should be done if we want to see if there is any difference b/w the two means
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then the denominator is based on the in sample variances for each sample ie s1 and s2
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comparison b/w two means equal or not is only useful
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for two populations
independent normally distributed |
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when we want to compare if there is any diff b/w the two means of two DEPENDENT pop samples
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we construct a paired comparison test
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eg. how a paired comparison test is used
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for eg. to check if the return of two steel firms have been equal for the 5 year period Note we cant use diff of means formula
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The paired comparison test is used to check if the av differnce of means b/w the two DEPENDENT companies
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is significantly diff from zero
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Test statistic for paired comparison test
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see card 25
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when diff b/w means is required and the samples are depenedent we use
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paired comparison test the av diff in the paired obs is divided by the standard error
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when diff b/w two means is required and the samples are independent we use
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the difference of means test, there are two of them also
a. when variances unknown are equal b. when variances unknown are unequal |
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chi square properties
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Asymmetrical
approaches to normal dist when deg of freedom increases since bounded below the chi square values cannot be negative |
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chi square test statistic formula
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see card 26
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when we use chi distribution
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when we test a hypothesis about the population variance for normally distributed population
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when do we use f distribution
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when we are compare if the two populations have the same varainces ///// comparing two variances based on different independent samples from normally distributed populations
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when we are trying to find if two populations have the same variance what do we use
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F distrtibution this is becasue diff b/w two chi squared random variable does not follow a chi sq distribution
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f distribution test statistic formula
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see card 27
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Parametric test
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like t test , f test , chi square test etc these make assumptions abou the distribution of population from which sample is drawn like z relies on mean and std dev to define normal dist
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Non Parametric tests
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1.assumptions of parametric tests cannot be supported like ranked observations
2. Data are ranks like ordinal scale measurements 3.eg. non parametric test is runs test provide series of change ie +_+_+_+ are random |
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Spearman rank correlation test is eg. of
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eg. of non parametric test, when pop is not normal eg. 20 mutual funds high spearmen corr eg .85 means high rank in one year is associated with the high rank in the second year
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what does a researcher wants to reject
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a null hypothesis
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what does a researcher wants to prove
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alternate hypothesis
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test statistic forumula
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see card 28
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define p value of 7% in terms of level of significance b/w 5% and 10%
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p value of 7% means that hypotheses with 10% level of signifcance can be rejected but cannot be rejected at 5% sig level
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Inter market analysis
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refer to analysis of interrelationships amoung the various asset CLASSES like currencies and stocks, bonds etc
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Relative strengh ratio
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is applied to the various assets classes to see which asset class is outperforming other
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Relative strength analysis
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refers to analysis which asset among these classes is OUTPERFORMING OTHERS
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Elliott Wave theory
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market prices can be described by interconnected set of cycles.
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up trend Elliott wave consists of
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5 upward moves and three downward moves
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down trend elliott wave consist of
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5 downward move and three up ward move recall diagram on pg 351
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Size of elliott waves are thought to correspond to
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fibnoacci ratios
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Fibnoacci nos
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st from 0 and 1 and then previous 2 numbers are added to get the next no.
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Ratio of consective fibonacci nos converge to which two nos
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0.618 and 1.618 used to project prices
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Cycle theory
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STUDY of process that occur in cycles like 4 year presidential cycle,decennial patters or 10 year
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54 year cycle is called
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Kondratieff wave
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Price bases indicators list them
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Moving average lines
Bollinger bands |
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Oscillators define
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also a tool to identify overbought or oversold markets
based on prices but scaled so that ehy oscillate around 0 and 100, high value means mkt overbought |
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Give eg. oscillators
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Rate of change oscillator
Relative strength index MACD Stocastic oscillator |
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Non price based indicators
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describe investor sentiment based on sentiment and capital flows rather than on price and volume
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Eg of Non price based indicatiors
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SENTIMENT
Put call ratio volatility index Margin debt short interest ratio FLOW OF FUNDS Arms index or TRIN Margin debt Mutual fund cash position New equity issuance |
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Bollinger bands
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based on std dev of closing prices
analyst draw bands above and below n period moving average mostly 2 std dev |
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Common chart patterns
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REVERSAL PATTERNS
Head and shoulder double top triple top CONTINUATION PATTERNS Triangles Rectangles |
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What are flags and pennants
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refer to rectangles and triangles that appear on the short term charts
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Up trend line
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is connecting the increasing lows with a straight line
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down trend line what does it connect
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it connects the decreasing highs with a straight line
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Define support and resistance
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price levels or ranges at which buying selling pressure is expected to limit price movement
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Change in polarity principle
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breached resistance levels become support and breached support levels become new resistance levels
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What accompanies the price charts for technical analysis
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volume charts
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Point and figure chart
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gives Changes in the direction of price trends
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Technical analysis is based on the which assumptions
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1 prices are determined by investor supply/demand
2 rational and irrational investor both drive demand and supply 3 prices (mkt) show actual shift in supply/demand 4 tend to repeat overtime Price levels |
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Relative strength analysis ratio
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ratio asses close price/benchmark prices
ITS high value show asset outperfoms mkt |
