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135 Cards in this Set

  • Front
  • Back
Valuation Methods
Time Value of Money

Expected Value

Arbitrage

[#1 assumes no Risk, other two include Risk in the valuation process.]
Basic Time Value of Money
Invest P0 at rate K for 1 year:
F1 = P0 * (1 + K), or P0 + P0* K.
Basic TVM Formula
FN = PV * (1 + K)^N

where:
PV is the present value,
K is the interest or growth rate,
N is the number of periods, and
FN is the future value.
[Given any 3, be able to calc the 4th.]
Two Kinds of Annuities
Ordinary annuity

Annuity Due
Ordinary annuity
equal periodic cash flows at the end of each period
Annuity Due
equal periodic cash flows at the beginning of each period
Internal Rate of Return
Find the interest rate, given PV, FV, PMT, and N periods.

For simple problems, use TVM keys to find i.

For unequal cash flows, use IRR key.
Time-Weighted ROR
the best measure of the portfolio manager performance because it is not affected by clients’ actions (adding or subtracting funds)
Money-Weighted ROR
is affected by clients’ actions, but may be a good measure for each client
Continuous Compounded RoR (Formula)
Log (1 + Discrete RoR)
Three Means Required at Lv I
Arithmetic Mean

Geometric Mean

Harmonic Mean
Geometric Mean
Form value relative: add 1.0 to each annual return expressed as a decimal.

Find the product of the value relatives.

Take the nth root of the product, step 2, then subtract 1.
Measurement Scales
Nominal

Ordinal

Interval

Ratio
Nominal
names but does not order
Ordinal
– orders data but no + or -.
Interval
+ and – allowed but no division since no zero point.
Ratio
includes a true zero, all operations are allowed.
Two Themes of Statistical Measures
central tendency, ex. Mean,
Median, and Mode.

dispersion, ex. for a single variable: Variance,
Skewness, and Kurtosis, and for two variables: Covariance
Harmonic Mean (Formula)
N / ∑1/D

Where
N is number of values and
D is the denominator or divisor
Chebyshev’s Inequality (Formula)
The proportion of data values of x being within k standard deviations of the mean, where k > 1 is:

1 – 1/k2


Note this inequality holds regardless of the shape of the probability distribution.
Coefficient of Variation
CV = s/avg
Risk per unit of average return.
Shortcomings of the Sharpe Ratio
Negative Sharpe Ratio is possible in bear markets. Authors suggest consider some other decision criterion.

Sx may not be the best risk measure. It is OK for symmetric distributions (Normal), but not OK for skewed distributions.
Covariance (Formula)
is a measure of the joint variation of two random variables.

∑ [X – Xmean] * [Y – Ymean] * pi

If equal joint probabilities, then:
for a population, use 1 / N for pi,

but for a sample, use 1 / (N – 1) for pi.
Correlation Analysis
a basic tool in measuring how two variables vary in relation to each other
Correlation Coefficient (Formula)
R(xy) = Covariance(xy) / [Std Dev(x) * Std Dev(y)]

+1: perfect linear relationship
0: no linear relationship
–1: perfect negative linear relationship
Four Moments of a Distribution
Mean or Expected Value

Variance

Skewness

Kurtosis
Kurtosis Platykurtic
less peaked than the Normal Distribution, or flatter. Platy means broad.
Kurtosis Mesokurtic
as peaked as the Normal. Meso means medium
Kurtosis Leptokurtic
more peaked than the Normal. Lepto means slender.
Safety-First Ratio (Formula)
(Exp Rtn – Min Rtn)/Standard Deviation

Similar to the Sharpe Ratio where the Minimum Acceptable Return is the Risk Free Rate.
Mutually Exclusive
no more than one of the events can occur at the same time
Collectively Exhaustive
the group of events represents all possible outcomes.
Empirical Probability
determined by observing the relative frequency when the activity takes place.
A Priori
based on logical analysis.
Subjective
determined based on personal judgment.
Posterior
a revised probability based on some new information about the event.
Dutch Book Theorem
Inconsistent implied probabilities involving two stocks can create a profitable arbitrage opportunity.
Bayes’ Theorem
P(Event | Information) =P(Information | Event ) * P(Event)/P(Information)

where:
P(Event) is the prior probability.

P(Event | Information) is the posterior probability, after the information is available.
Total Probability Rule
P(negative ratio) = P(negative ratio|earnings decline) * P(earnings decline) + P(negative ratio|No earnings decline) * P(No earnings decline)
Monte Carlo simulation
Assume a probability distribution for some critical variables.

Repeat the model of the process many, many times and observe the distribution of results.
Binomial Distribution
Each outcome is in one of two mutually exclusive categories.

Probability of an outcome remains the same from trial to trial, called Bernoulli Trials.

Each trial is independent.

N = number of trials, P = prob of result
occurring
Sampling Methods
Population

Simple Random

Stratified
Parameter Statistic
some characteristic of a population, ex. population mean, (often hard to determine in large populations).
Sample Statistic
a measurable characteristic computed from a sample.
Sampling Error
difference between the parameter and the sample statistic.
Central Limit Theorem
The sampling distribution of sample means is approximately normally distributed regardless of the shape of the population distribution.
Estimator vs. Estimate
Estimator is a formula, ex. sample mean.

The formula produces a resulting number, called the estimate.
Desirable Properties of a Point Estimator
Unbiased

Efficient

Consistent
Confidence Interval
an interval where there is a (1 – α) probability that the population parameter is within the interval.

Usually α is 5%, so the probability of the interval is 95%.
Two Key Properties of a Probability Density Function
The probability density function must be ≥ 0 for all values of the random variable.

The sum or integral over the range of possible random variables must equal 1.0.
Normal Distribution
Two parameters are needed to specify this distribution: mean and standard deviation.

You must memorize the probabilities of:
Mean ± 1 standard deviation.
Mean ± 2 standard deviations.
(Mean ± 3, optional.)
Reliability Factor
Reliability Factor is the number of standard deviations to go out from the mean in each direction to construct a (1 – α) confidence interval.
Standard Normal, or Z value
(X – avg) / std dev of x

Standardized normal distribution has
Mean = 0 and Standard Deviation = 1.
t-Distribution
Similar to a normal distribution, but has:
fatter tails than a normal, and requires three parameters: mean, stnd dev, and Degrees of Freedom.

Degrees of freedom represent the number of independent observations used.
Number of Degrees of Freedom
N – 1 for hypothesis tests involving confidence intervals.
Lognormal Distribution
Used in valuation of options (Lv II) and other pricing models.

If the rate-of-return of a security is modeled well using the Normal Distribution, then the price of the security is often modeled using the Lognormal.
What is a Hypothesis?
a statement about: the value of a population parameter, that is to be tested.
Null Hypothesis
a statement about a parameter of the distribution
Alternative Hypothesis
a statement about the same parameter, but logically opposite of the Null Hypothesis.
Hypothesis Testing Process
The Null Hypothesis is considered true

unless

the sample gives strong evidence that the null should be rejected, and the alternative hypothesis should be accepted.
Accept vs. Reject
Inside the (1 – α) confidence interval, then Accept the Null Hypothesis.

Outside the (1– α) confidence interval, then Reject the Null Hypothesis and accept the alternative hypothesis.
Wrong Decisions
Type I: Reject the null when it is true.

Type II: Accept the null when it is false.
P-Values
P-value is the smallest level of significance at which the null hypothesis is on the borderline of being rejected.

Large p-value → null likely true.

Small p-value → null likely false.
One-Sided t-Test
Follow the multi step process for Two-sided t-Test but use 5% in the tail, rather than the traditional 2.5%.
2 Sided Z-Test
Mean not known, but population standard deviation is known.

Rather than t-test, use a z-test (and z-table).
Real Risk Free Interest Rate
nominal risk-free interest rate - Inflation Premium
Required Rate of Return for an Investment
minimum expected rate of return an investor is willing to receive in order to accept the investment
Opportunity Cost
The highest return that an investor forgoes by choosing whether to invest or not.
Nominal Interest Rate
A started annual rate. Does not get compounded more than once per year.
Effective Annual Interest Rate (Definition)
May have more than 1 interest period and therefore will compound during the year.
Effective Annual Interest Rate (Formula)
(1+ (i/n))^N -1

where:

i:nominal interest rate (annual rate)
n: frequency
Effective Annual Interest Rate (Continuous Formula)
e^(r*t) - 1

where
r: nominal rate
t: years
Perpetual Ordinary Annuity (Formula)
PMT/K

where
PMT: payment
K: Interest Rate
Discounted Present Value of a Perpetual Annuity Due (Formula)
PMT/K * (1+K)

where
PMT: payment
K: Interest Rate
Perpetual Ordinary Annuity-Due to Present Value (Formula)
P * (K/1+K)
Future Value of Future Compound Cash Flows (Formula)
NPV * (1+K)^N
Annuity Due (Numerous Periods Formula)
AD/(1+K)^(N-1)
Holding Period Rate of Return (Formula)
[[P(t)-P(t-1)] + CF(t)] / P(t-1)

where
Money Weighted Rate of Return (Definition)
depends on both the amount invested and the rate of return during various sub-periods.

can be more significantly affected by the actions of clients due to deposits/withdrawals.

can be a good measure for client performance.
Time Weighted Rate of Return (Definition)
depends on the RoR during various sub-periods.

better measure the actions of the Portfolio Manager than the actions of the client.
Time Weighted Rate of Return (Formula)
[(1+RoR(n))^(1/2)] -1
Bank Discount Yield (BDY) Formula
(FV-DV/FV) * 360/DTM

where:
FV: Face Value
DV: Discount Value
DTM: Days to Maturity
Money Market Yield (MMY) Formula
(FV-DV/DV) * 360/DTM

where:
FV: Face Value
DV: Discount Value
DTM: Days to Maturity
Effective Annual Yield (EAY) Formula
(1+HPY)^(365/DTM) -1
Descriptive Statistics
the study of how data can be summarized effectively to describe the important aspects of large data sets
Statistical Inference
involves making forecasts, estimates, or judgments about a Larger set of data form a small set actually observed
Parameter
a characteristic of a Population
Statistic
a characteristic of a Sample
Ratio Scale
measurement scale with includes zero.

Rates of Returns are measure on this scale.
Frequency Distribution
is a display of data values summarized into a moderate to small number of intervals
Geometric Mean (formula)
Add 1 to each value, multiply.

[N(1) +1 * N(2) + 1]^(1/N) -1
Harmonic Mean (formula)
Total Number Sample/[1/N(1) + 1/N(2)]
Determining Percentile Values
N+1/100 * P

where P = the Particular Percentile
Range
Highest Value - Lowest Value
Population Variance (Formula)
(Sum[X(i)-WM]^2)/N

Where:
X(i)=Value,
WM=Weighted Mean,
N=Number of Values in Population
Sample Variance (Formula)
Population Variance * N/(N-1)

or

(Sum[X(i)-WM]^2)/[N] * N/(N-1)

Where:
X(i)=Value,
WM=Weighted Mean,
N=Number of Values in Population
Standard Deviation (Formula)
Sum[(Value - WAM)^2 * Probability]

Where
WAM=Weighted Average Mean
Mean Absolute Deviation (MAD) Formula
Sum[(Value - WAM) * Probablity)

Where:
WAM= Weighted Average Mean
Chebyshev's Inequality
At least 75% of all data values in a distribution fall within 2 standard deviations from the mean
Co Efficient of Variation Formula
SD/RoR

where:
SD=Standard Deviation,
RoR=Rate of Return
Sharpe Ratio (Formula)
(Avg. RoR - RfR)/SD

where:
RfR=Risk Free Rate
SD=Standard Deviation,
RoR=Rate of Return
Harmonic Mean (Formula)
Sum of: N/[1/P(n1) +1/P(n2)]
Relationship between Harmonic Mean and Weighted Average Mean
They are the same value.
Advantage of Mean Absolute Deviation (MAD)
gives relatively less weight to the more extreme data values than Standard Deviation
Co Efficient of Variation Reciprocal Formula
RoR/SD
Sharpe Ratio Definition
the difference between the mean rate of return and the risk-free rate of return divided by the standard deviation.

Measures excess mean reate of return relative to risk (Standard deviation)

If positive, The Higher value, the better

if negative, the higher absolute value is better.
Correlation Co Efficient Formula
CV(xy)/[SD(x) * SD(y)]

Where:
CV(xy)=Coveriance of x and y scenarios,
SD(x)=Standard Deviation of x,
SD(y)= Standard Deviation of y
Interpretation of Correlation Co Efficient
If =1 then perfect linear relationship between x & y

if >0 && <1 then positive linear relationship between x & y

if =0, no indication of linear relationship.

if <0 && >-1 then negative linear relationship between x & y

if ==-1 then perfect linear relationship between x & y
Safety First Ratio Formula
For a Portfolio:

RoR-LR/SD

where:
RoR=Rate of Return,
LR=min. acceptable rate of return,
SD=Standard Deveation

Note: Very similar to the Sharp Ratio if risk free rate=minimum expected rate of return
Odds Against (Formula)
1-P(e)/P(e)

where
P(e) is the probability of the event
Mutually Exclusive
no more than 1 of the events can occur at the same time.
Collectively exhaustive
at least one of the events must occur
A-Priori Probability
based on logical analysis of the relevant activity and the particular event
Empirical Probability
determined by observing the relative frequency of occurrence of that event when the relevant activity takes place
Subjective Probability
determined by personal judgement
Conditional Probability (of B given A) (Formula)
P(A&B)/P(A)

if P(B) !=0
Bayes' Theorem P(B|A) (Formula)
[P(A|B) * P(B)]/P(A)
P(A&B) (Formula)
P(A|B) * P(A)

or

P(B|A) * P(B)
Conditional Probability
P(B|A)
P(A&B)/P(A)
Combination Formula
N!/[(N-R)! * R!]
Permutation Formula
N!/(N-R)!
Normal Distribution Probabilities
1 SD= 68%, 16% on each tail

2 SD=95%, 2.5% on each tail

3 SD=99.7, .03% on each tail
Observation to Z-Value (Formula)
[X-Mean]/SD

where:
X=observation,
SD=Standard Deviation
Normal Distribution Confidence Interval Formula (Individual Values)
Population Mean +/- Z * Pop-SD

where:
Z=Confidence Interval,
Pop-SD=Population Standard Deviation
Normal Distribution Confidence Interval Formula (Sample Values)
Sample Mean +/- Z * (Pop-SD/[N^1/2])

where:
Z=Confidence Interval,
Pop-SD=Population Standard Deviation,
N=Sample Size
T-Distribution
Fatter Tails, and includes degrees of freedom vs Normal Distribution
Degrees of Freedom Formula
N-1

where:
N=Sample Size
t-Distribution Confidence Interval Formula (Sample Values)
Population Mean +/- (N-1) * (SD/[N^1/2]

where:
SD=Sample Standard Deviation
Chi-Squared Distribution
N Degrees of Freedom if independent values are calculated.

N-1 Degrees of Freedom if sample is calculated

if >30 degrees of freedom, Normal Distribution is a good approximation

Starts at 0 and is skewed to the right to positive infinity.
Type 2 Error
when a null hypothesis is incorrectly accepted when it is actually false
Type 1 Error
when the null hypothesis is incorrectly rejected when it is actually true
Power Hypothesis Test (Formula)
1-Probability of Type 2 Error
If P-Value >= Selected Significance Level
Accept Null Hypothesis
If P-Value < Selected Significance Level
Reject Null Hypothesis
P-Values in Hypothesis Testing
the smallest level of significance at which the null hypothesis can be rejected