Study Session 2

Front 1

Define Interest Rate with an equation of premiums.

Back 1

r = Real risk-free rate + Inflation premium + Default risk premium + Liquidity premium + Maturity premium

Front 2

Future Value equation

Back 2

FV = PV(1+r)^n

Front 3

Future Value from Stated Annual Interest Rate

Back 3

SAR = Quoted Interest Rate; with more than one compounding period per year, FV = PV(1+r/m)^mn
where m = periodicity

Front 4

Future Value from Continuous Compounding

Back 4

FVn = PVe^(r*N)

Front 5

finding Effective Annual Rate (EAR) from Periodic interest rate

Back 5

EAR = (1 + Periodic interest rate)^m - 1

Front 6

finding EAR from continuous compounding

Back 6

EAR = e^r - 1

Front 7

define: Annuity, Ordinary Annuity, Annuity Due, and Perpetuity

Back 7

Annuity - a finite set of level sequential CFs
Ordinary Annuity - first CF that occurs one period from now (t=1)
Annuity Due - first CF occurs immediate (t=0)
Perpetuity - set of level never-ending sequential CFs, with first CF occuring at t=1

Front 8

Future Value of an Annuity

Back 8

FV = A*[((1+r)^n - 1)/r]

Front 9

How does PV relate to discount rate and number of periods?

Back 9

1. For a given discount rate, the farther in the future the amount to be received, the smaller that amount's present value
2. Holding time constant, the larger the discount rate, the smaller the present value of a future amount

Front 10

Present Value of a Perpetuity

Back 10

PV = A/r as long as interest rates are positive

Front 11

solve for interest Rates and Growth Rates

Back 11

g = (FV/PV)^(1/n) - 1

Front 12

Cash Flow Additivity

Back 12

dollar amounts indexed at the same point in time are additive. use when dealing with uneven cash flows

Front 13

NPV Rule and equations

Back 13

if the investment's NPV is positive, an investor should undertake it, if NPV is negative, investor should not take it.
NPV = sum(CFt/(1+r)^t)

Front 14

IRR rule

Back 14

Accept projects for which the IRR is greater than the opportunity cost of capital or the hurdle rate. IFF is only correct if CFs are reinvested at IRR.

Front 15

Problems with IRR rule

Back 15

IRR and NPV rules project differently when:
1. size or scale of the projects differ
2. the timing of the projects' cash flows differ
when IRR and NPV rules differ, follow NPV b/c NPV represents the expected addition to shareholder wealth from an investment.

Front 16

Holding Period Return (Holding Period Yield)

Back 16

HPR = (P1 - P0 + D1)/P0

Front 17

Money-Weighted vs. Time-Weighted Returns

Back 17

MW accounts for the timing and amount of all dollar flows into and out of the portfolio. Generally, clients determine when money is given to the manager and how much money is given - which may influence the manager's money-weighted ROR.
TW measures the compound rate of growth of $1 initially invested in the portfolio over a stated measurement period. Not sensitive to additions and withdrawals of funds; returns are averaged over time. To compute TW; price portfolio, compute HPR for each subperiod, obtain annual ROR, and take geometric mean of annual ROR for average.

Front 18

define Money Market, Pure Discount Instrument, Discount, Bank Discount Basis

Back 18

MM = market for short-term debt instruments.
Pure Discount - pay interest as the difference b/w the amount that gives the price for the T-bill.
Discount - reduction from the face amount that gives the price for the T-bill.
Bank Discount Basis - used to quote T-bills, based on 360-day year, a percentage of face value.

Front 19

equation for Yield on a Bank Discount Basis and why is it not a meaningful measure

Back 19

r = D/F * 360/t
1. yield based on face value of bond, not purchase price
2. yield annualized on 360-day year
3. bank discount yield annualized with simple interest ignores opportunity to earn interest on interest

Front 20

Effective Annual Yield (EAY)

Back 20

EAY = (1 + NPY)^(365/t) - 1

does account for interest-on-interest.

Generally more than the bank discount yield.

Front 21

Money Market Yield

Back 21

r = 360(bank discount rate)/[360 - (t * bank discount rate)]

Front 22

define Descriptive Statistics and Statistical Inference

Back 22

Descriptive - study of how data can be summarized effectively to describe the important aspects of large data sets.
Statistical - making forecasts, estimates, or judgements about a larger group from the smaller group actually observed.

Front 23

define Population, Parameter, Sample, Variance, Sample Statistic.

Back 23

Population - all members of a specified group
Parameter - any descriptive measure of a population.
Sample - a subset of a population
Variance - range of investment returns.
Sample Statistic - a quantity computed from or used to describe a sample.

Front 24

define Measurement Scales: Nominal, Ordinal, Interval, and Ratio Scales.

Back 24

Nominal Scale - the weakest level of measurement, categorize data but do not rank them.
Ordinal - stronger level of measurement, categories that are ordered with respect to some characteristics
Interval - ranks and assures that the differences b/w scale values are equal.
Ratio - strongest level, interval scale plus a true zero point as the origin.

Front 25

Define Frequency Distribution and explain how to construct.

What is Absolute Frequency, Relative Frequency, and Cumulative Relative Frequency?

Back 25

Frequency Distribution - tabular display of data summarized into relatively small number of intervals.

determine range and number of intervals (k), Divide range/k for interval end points, count # points falling into each range, construct table to show observations.

Absolute Frequency is the actual number of observations in a given interval.
Relative Frequency is the absolute frequency of each interval divided by the total number of observations.
Cumulative Relative Frequency adds the relative frequency as we move from the first to last interval.

Front 26

define Histogram and Frequency polygon

Back 26

Histogram - bar chart of data that have been grouped into frequency distribution.
Frequency Polygon - plot midpoint of each interval on x-axis and absolute frequency for that interval on y-axis; connect points with straight line.

Front 27

define arithmetic mean, population mean, and sample mean, median, mode.

What is important about arithmetic means, drawbacks?

Back 27

Arithmetic - sum of observations divided by total number of observations
Population - arithmetic mean value of a population
Sample - (average) arithmetic mean value of a sample.
Median - value of the middle item of a set
Mode - most frequently occurring value in a distribution.

Deviations from arithmetic means indicate risk. A drawback is its sensitivity to extreme values.
Can also compute a weighted mean.

Front 28

define Geometric Mean and what would it be used for? Formula for Geometric Mean Return?

Back 28

the most frequently used to average rates of change over time or to compute the growth rate of a variable.

Compute by multiplying all rates, take the nth root of the product.

Rg = [product(1+Rt)^t]^(1/t)-1

Front 29

define Harmonic Mean and what is it used for?

Back 29

the value obtained by summing the reciprocals of the observations - terms of the form 1/X - then averaging that sum by dividing it by the number of observations, n, then taking the reciprocal of the average.

X = n/sum(1/X)

Special type of weighted mean in which the weight is inversely proportional to its magnitude.

Most appropriate when averaging ratios when the ratios are repeatedly applied to a fixed quantity to yield a variable number of units (EX: cost averaging).

Front 30

define Quartiles, Quintiles, Deciles, Percentiles.

How can they be used in investment practice?

Back 30

QUARTILES divide the distribution into quarters, QUINTILES into fifths, DECILES into tenths, PERCENTILES into hundredths.

Used in portfolio performance evaluation, strategy development & research. characterize an investment in terms of quartiles in which they fall relative to the performance of peers.

Front 31

define Dispersion, Mean Absolute Deviation, Absolute Dispersion, Range, Variance, Standard Deviation, Population Variance, and Population Standard Deviation.

Back 31

Dispersion - variability around central tendency
Mean Absolute Deviation - (MAD) variance & SD
MAD = sum(abs(x-xi))/n
Absolute Dispersion - the amount of variability present without comparison to any reference point or benchmark.
Range - the difference b/w the max and min values in a data set.
Variance - average of the squared deviation around the mean.
Standard Deviation - positive square root of the variance.
Population Variance - σ^2 = sum(Xi-u)^2/n where u=population mean

Population SD = σ = sqrt(σ^2)

Front 32

define Semivariance, Semideviation, Chebyshev's Inequality

Back 32

Semivariance - defined as the average squared deviation below the mean.
Semideviation - the positive square root of semivariance.
Semivariance = (Xi-X)^2/(n-1)
Chebyshev's Inequality - the proportion of the observations within k standard deviations of the arithmetic mean is at least 1 - 1/k^2 for all k>1.

Front 33

define Relative Dispersion, Coefficient of Variation (and formula), Sharpe Ratio (and formula, and limitation)

Back 33

Relative Dispersion - the amount of dispersion relative to a reference value or benchmark.
Coefficient of Variation - ratio of the SD of a set of observations to their mean value; CV = SD/X
Sharpe Ratio - S = (Rp-Rf)/SDp
Rf risk free return
Rp portfolio return
SDp SD of portfolio
measures excess return over risk taken, but Sharpe is limited to only considering one aspect of risk, the SD of return.

Front 34

Symmetry and Skewness in Return Distributions. Describe characteristics of Normal Distribution.

Define Skewness.

Back 34

Normal Distribution: 1. its mean and median are equal.
2. Completely described by its mean and variance
3. roughly 68% of observations b/t +/- 1 SD, 95% b/t +/- 2 SDs, 99% b/t +/- 3 SDs.

Skewness - a non-symmetrical distribution. Computed as the average cubed deviation from the mean standardized by dividing by the SD cubed to make the measure free of scale.
Sk = [n/(n-1)(n-2)]*[sum(Xi-X)^3/SD^3]

Front 35

define Kurtosis, Leptokurtic, Platykurtic, Mesokurtic.

What do they describe. Write equation to compute excess kurtosis.

Back 35

Kurtosis - statistical measure that telss when a distribution is more or less peaked than a normal distribution.
Leptokurtic - more peaked than normal
Platykurtic - less peaked than normal
Mesokurtic - distribution identical to the normal distribution.

Describe the depth and width of a normal distribution curve.

Sample Excess Kurtosis
Ke={n(n+1)/(n-1)(n-2)(n-3)*(Xi-X)^4/SD^4}-[3(n-1)^2/(n-1)(n-2)]

Front 36

Contrast Variance with Semivariance and Target Semivariance

Back 36

Variance - average of the squared deviation around the mean.
Semivariance - average squared deviation below the mean
Target Variance - average squared deviation below a stated target.

Variance is often used to measure an asset's risk, but since investors are only concerned with downside risk, use semivariance to determine the downside risk. Target semivariance can be used to determine the risk below a stated target.

Front 37

define Probability

Back 37

the 2 defining properties are as follows:
1. the probability of any event E is a number b/t 0 and 1
2. the sum of the probabilities of any set of mutually exclusive and exhaustive events equals 1.

mutually exclusive means that only one event can occur at a time, exhaustive means that the events cover all possible outcomes.

Front 38

Name and Define the 3 types of Probability.

Back 38

Empirical - probability of an event as a relative frequency of occurrence based on historical data.
Subjective - probability drawing on personal or subjective judgment - of great importance to investors.
A Priori - baed on logical analysis rather than on observation or personal judgment.

A Priori and Empirical probabilities are usually referred to as Objective Probabilities.

Front 39

Probabilities stated as Odds of event E happening and not happening.

Back 39

1. Odds of E = P(E)/[1-P(E)]
2. Odds against E = [1-P(E)]/P(E) the reciprocal of odds for E.

Front 40

define Unconditional, Conditional, and Joint probabilities. Show examples and equations of each.

Back 40

Joint - answers "What is the probability of both A and B happening?"
Unconditional - answers straight-forward questions such as "What is the probability of this event A?"
Conditional - answers the question "What is the probability of A, given B has occurred?" This is equal to the joint probability of A and B divided by the probability of B.
P(AǀB)=P(AB)/P(B),P(B)≠0

Front 41

define Multiplication Rule of Probability, and Addition Rule.

Back 41

Multiplication - joint probability of A and B: P(AB) = P(AǀB)P(B)

Addition Rule - probability that A or B occurs, or both occur, is equal to the probability that A occurs, plus the prob that B occurs, plus the prob that B occurs minus prob that A and B occur.
P(AorB) = P(A)+P(B)-P(AB)

Front 42

Define Independent Events, and Multiplication Rule for Independent Events.

Back 42

Independent Event - two events A and B are independent if and only if P(AǀB)=P(A) or P(BǀA)=P(B)

Multiplication Rule - when 2 events are independent, the joint prob of A and B equals the product of individual prob of A and B.
P(AB)=P(A)P(B)

Front 43

define Total Probability Rule

define Expected Value, Variance, and Standard Deviation

Back 43

Total Probability Rule:
1. P(A)=P(AS)+P(ASc)
=P(AǀS)P(S)+P(AǀSc)P(Sc)
2. P(A)=P(AS1)+P(AS2)+...+P(ASn)
P(A)=P(AǀS1)P(S1)+P(AǀS2)P(S2)+...+P(AǀSn)P(Sn)
where Sn are mutually exclusive and exhaustive scenarios.

Expected Value of a random variable is the prob-weighted average of the possible outcomes of the random variable. Denoted as E(X)
E(X)=P(X1)X1+P(X2)X2+...+P(Xn)Xn

Variance of random variable is the expected value of squared deviations from the random variable's expected value.
σ^2(X)=E[(X-E(X))^2]

Standard Deviation is the positive square root of variance.

Front 44

define Conditional Expected Values and Total Probability Rule for Expected Value.

Back 44

Conditional E(X)=expected value of random variable X given an event or scenario S is denoted E(XǀS).
E(XǀS)=P(X1ǀS)X1+P(X2ǀS)X2+...+P(XnǀS)Xn

Total Probability Rule for E(X): principle for stating unconditional expected values in terms of conditional expected values:
1. E(X)=E(XǀS)P(S)+E(XǀSc)P(Sc)
2. E(X)=E(XǀS1)P(S1)+...+E(XǀSn)P(Sn)

Front 45

Example: use Total Probability Rule for Expected Value to determine expected EPS when rates can decline or stay stable.

Back 45

E(EPS) = E(EPSǀdeclining rates)* P(declining rates) + E(EPSǀstable rates)* P(stable rates)

Front 46

Properties of Portfolio Expected Value

Back 46

1. E(w1R1)=w1*E(R1)
2. E(w1R1+w2R2+...+wnRn) = w1*E(R1)+w2*E(R2)+...+wn*E(Rn)

to calculate the Expected Return of a Portfolio:
E(Rp) = sum(wn*E(Rn)

Front 47

define Covariance. What does a positive, negative, and 0 covariance mean.

define Correlation and its properties.

Back 47

Cov(Ri,Rj)=E[(Ri-E(Ri))(Rj-E(Rj))]

if Covariance is negative, return on one asset is above its expected value, return on the other is below expected value.
Covariance is 0 if returns on assets are unrelated.
Covariance is pos if returns on both assets tend to be on the same side (above or below) their expected values at the same time.

Correlation b/t 2 random variables Ri and Rj defined as p(Ri,Rj)=Cov(Ri,Rj)/σ(Ri)σ(Rj).

Properties of Correlation:
1. Corr is a number b/t -1 and 1 for 2 variables X and Y.
2. Corr of 0 indicates an absence of any linear relationship b/t the variables. Pos Corr indicates an increasingly strong pos. linear relationship, and neg corr indicates negative linear relationship.

Front 48

When are two random variables independent? What is the expected values of two random uncorrelated variables?

Back 48

X and Y are independent if and only if P(X,Y)=P(X)P(Y).

EV of the product of uncorrelated random variables is the product of their expected values.
E(XY) = E(X)E(Y)

Front 49

Bayes' Formula

Back 49

given a set of prior probabilities for an event of interest, if you receive new information, the rule for updating your probability of the event is:
Updated probability of event given new info = (Prob of new info given event/Unconditional prob of new info)*Prior prob of event. OR

P(EventǀInfo)=[P(infoǀEvent)/P(Info)]*P(Event

Front 50

define multiplication rule of counting.

Multinomial Formula
Combination Formula
Permutation Formula

Back 50

Multiplication: if one task can be done in n1 ways, and a second task, given the first, can be done in n2 ways...for k tasks, then the number of ways k tasks can be done is (n1)(n2)...(nk)

Multinomial: n!/(n1!n2!...nk!)
Combination: C = (n!/(n-r)!r!)
Permutation: P = n!/(n-r)!

Multinomial: number of ways that n objects can be labeled with k different labels, with n1 of the first type...

Combination: number of ways to choose r objects from a total of n objects, when order in which th r objects does not matter.

Permulation - number of ways to choose r objects from a total of n objects, when the order in which the r objects are listed does matter.