Quantitative

Weight on the exam = 12%

Number of formulas = 27

Effective annual rate

Effective annual rate is an annual rate of interest when compounding occurs more often than once a year.

EAR - effective annual rate

i - annual interest rate

n - number of compounding periods

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Future value

Future value measures nominal value of money at a specified time in the future

FV - future value

PV - present value

r - periodic rate

k - number of compounding periods

n - number of years

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Present value

Present value is a current value of some future cash flow

PV - present value

FV - future value

r - periodic rate

k - number of compounding periods

n - number of years

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Annuity due
An annuity-due is an annuity when payments are made at the beginning of each period (not in the end as for usual annuity)

Present value of an annuity due

Future value of an annuity due

PVdue - present value of an annuity due

PVord - present value of an ordinary annuity

FVdue - future value of an annuity due

FVord - future value of an ordinary annuity

r - periodic rate

k - number of compounding periods

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Perpetuity

Perpetuity is an annuity in which the periodic payments continue indefinitely

PVperp - present value of a perpetuity

P - periodic payment

i - periodic interest rate

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Net present value

Net present value is a sum of present values of individual cash flows.

NPV - net present value

CF_{0} - initial investment in the project

CF_{1} - cash flow in period 1

CF_{2} - cash flow in period 2

CF_{n} - cash flow in period n

r - discount rate

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Holding period yield

Holding period yield (or holding period return) is a total return on an assets over the period during which it was held

P_{1} - current price

P_{0} - purchase price

D_{1} - dividend paid during a holding period

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Bank discount yield

Bank discount yield is the annualized percentage discount from face value.

BDY - bank discount yield

$discount - dollar discount from face value

face value - face value of the bond

days - days until maturity

360 - days in a year

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Effective annual yield
Effective annual yield converts holding period yield (HPY) to a compound annual yield based on a 365-day year to make the yield comparable with other investments.

EAY - effective annual yield

HPY - holding period yield

t - days to maturity

365 - days in a year

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Money market yield

Money market yield is an annualized holding period yield without compounding based on a 360-days year

MMY - money market yield

HPY - holding period yield

t - days to maturity

360 - days in a year

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Geometric mean return

The geometric mean return formula is used to calculate the average rate per period on an investment that is compounded over multiple periods.

Rg - geometric mean return

R1 - return in period 1

R2 - return in period 2

Rn - return in period n

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Weighted mean

Weighted mean is a mean in which different parameters have different weights. The sum of weights is equal to 1.

Xw - weighted mean value

X_{1} - value of parameter 1

X_{2} - value of parameter 2

X_{n} - value of parameter n

w_{1} - weight of parameter 1

w_{2} - weight of parameter 2

w_{n} - weight of parameter n

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Variance
Variance is the average of the squared deviations from the mean. Variance is used to measure how far a set of numbers are spread out from each other.

Population variance

σ² - population variance

X_{i} - observation value

μ - population mean

N - number of observations

Sample variance

s² - sample mean

X_{i} - observation value

X - sample mean

n - number of observations

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Coefficient of variation

Coefficient of variation measures variability in relation to the mean and is used to compare the relative dispersion in one type of data with the relative dispersion in another type of data.

CV - coefficient of variation

σ - standard deviation of returns

μ - mean return

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Sharpe ratio

Sharpe ratio measures excess return per unit of risk

SR - sharpe ratio

R_{p} - return of the portfolio

R_{rf} - risk-free rate

σ_{p} - portfolio risk

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Multiplication rule

The probability of A and B occurs together (joint probability) is the probability that A occurs, given that probability B occurs, multiplied by probability B.

P(AB) - joint probability of two events

P(A|B) - the probability that A occurs, given that probability B occurs

P(B) - the probability of B occurs

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Addition rule

The probability that either A or B occurs equal to the sum of probability that A occurs and probability that B occurs minus the probability that A and B occurs at the same time.

P(A or B) - the probability that either A or B occurs

P(A) - the probability that A occurs

P(B) - the probability that B occurs

P(AB) - joint probability of two events (A and B occurs at the same time)

If A and B are mutually exclusive, then P(AB)=0 and P(A or B) = P(A) + P(B).

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Total probability rule

The total probability rule explains an unconditional probability of an event, in terms of that event's conditional probabilities in a series of mutually exclusive, exhaustive scenarios.

P(R) - the probability that R occurs

P(R|I) - the probability that R occurs, given that probability I occurs

P(I) - the probability that I occurs

P(R|I^{C} ) - the probability that R occurs, given that probability I^{C} occurs

P(I^{C} ) - the probability that I^{C} occurs

I and I^{C} are mutually exclusive and exhaustive set of events

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Correlation coefficient

Correlation coefficient measures the strength and the direction of a linear relationship between two variables. The correlation coefficient can range from -1 to +1

COR(R_{i} ,R_{j} ) - correlation coefficient between R_{i} and R_{j}

Cov(R_{i} ,R_{j} ) - covariation coefficient between R_{i} and R_{j}

σ(R_{i} ) - standard deviation of R_{i}

σ(R_{j} ) - standard deviation of R_{j}

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Portfolio expected return

Expected return of portfolio of two assets is the weighted average of the expected returns on the assets which comprise the portfolio.

E(R_{p} ) - expected portfolio return

w_{A} - weight of asset A in the portfolio

R_{A} - return of asset A

w_{B} - weight of asset B in the portfolio

R_{B} - return of asset B

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Portfolio variance

Portfolio variance is a measure of the risk (volatility) of a portfolio. Variance of a portfolio of two assets is a combination of the return variance and co-variance of each security and its proportion in that portfolio.

Var_{p} - portfolio variance

w_{A} - weight of asset A in the portfolio

σ_{A} - standard deviation of asset A

σ_{A} ^{2} - dispersion of asset A

w_{B} - weight of asset B in the portfolio

σ_{B} - standard deviation of asset B

σ_{B} ^{2} - dispersion of asset B

p_{A,B} - correlation coefficient between assets A and B

Cov_{A,B} - covariation coefficient between assets A and B

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Binomial distribution

Binomial distribution is a statistical distribution giving the probability of obtaining a specified number of successes in a specified number of independent trials of an experiment with a constant probability of success in each. The probability of success is p while the probability of failure equals 1-p. The formula above calculates the probability of x successes in n trials.

p(x) - the probability of x successes in n trials

x - number of successes

n - number of trials

p - probability of success in each trial

1-p - probability of failure in each trial

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Z-value

Z-value indicates how many standard deviations an observation is above or below the mean.

x - observation

μ - population mean

σ - standard deviation

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Standard error

Standard error of the sample mean is the standard deviation of the distribution of the sample means. If the standard deviation of the population (σ) is known the standard error of the sample mean is calculated with the formula above.

SE - standard error of the sample mean

σ - standard deviation of the population

n - size of the sample

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Confidence interval

Confidence interval is a particular kind of interval estimate of a population parameter and is used to indicate the reliability of an estimate.Confidence interval with a known variance (formula above) is very important for hypothesis testing.

CI - confidence interval

x - mean value

z_{α/2} - reliability factor, a standard normal random variable for which the probability in the right-hand tale of the distribution is &alpha/2.

σ - standard deviation of the population

n - size of the sample

σ/n - standard error

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Safety-first ratio

Roy's safety-first ratio states that the optimal portfolio minimizes the probability that the return of the portfolio falls below some minimum acceptable level (threshold level).

SFR - safety-first ratio

R_{p} - portfolio return.

R_{L} - threshold level return.

σ_{p<} - portfolio standard deviation

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Continuously compounded returns
Descretely compounded returns are returns with a given compounding period such as quaterly (4) or monthly (12). As we increase the compounding period the effective annual return also increases. The limit of this increase is continuously compounded return.

CCR -continuously compounded return

HPR - holding period return

S_{1} - ending value

S_{0<} - beginning value

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Sampling error of the mean
Sampling error is a difference between a sample statistic (mean, variance, standard deviation of the sample) and its corresponding population parameter.

SEM - sampling error of the mean

x - sample mean

μ - population mean

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