Quantitative Methods- Definitions and Formulas - CFA L1

Equilibrium Interest Rate

Required rate of return for a particular investment, in the sense that the market rate of return is the return that investors and savers require to get them to willingly lend their funds.

Real risk-free rate

theoretical rate on a single-period loan that contains no expectation of inflation and zero probability of default.

Time preference

the degree to which current consumption is preferred to equal future consumption. Represented by the real-risk free rate

Nominal risk free rate

Nominal risk-free rate = real risk free rate + expected inflation rate + default risk premium + liquidity risk premium + maturity premium

Default risk

This is the risk that a borrower will not make the promised payments in a timely manner.

Liquidity Risk

This is the risk of receiving less than fair value for an investment if it must be sold quickly for cash.

Maturity risk

the prices of longer-term bonds are more volatile than those of shorter-term bonds. Longer-maturity bonds have more maturity risk than shorter-term bonds and require a maturity risk premium.

Holding period return (HPR)

the percentage increase in the value of an investment over a given period

Arithmetic mean return

the simple average of a series of periodic returns

Geometric mean return

a compound rate. When periodic rates of return vary from period to period, the geometric mean return will have a value less than the arithmetic mean return

Harmonic mean

used for certain computations, such as the average cost of shares purchased over time

Money weighted return

applies the concept of the internal rate of return (IRR) to investment portfolios. An IRR is the interest rate at which a series of cash inflows and outflows sum to zero when discounted to their present value. That is, they have a net present value (NPV) of zero. defined as the IRR on a portfolio, taking into account all cash inflows and outflows. The beginning value of the account is an inflow, as are all deposits into the account. All withdrawals from the account are outflows, as is the ending value.

Time weighted rate of return

measures compound growth and is the rate at which $1 compounds over a specified performance horizon. Time-weighting is the process of averaging a set of values over time. The annual time-weighted return for an investment may be computed by performing the following steps:

Step 1:	Value the portfolio immediately preceding significant additions or withdrawals. Form subperiods over the evaluation period that correspond to the dates of deposits and withdrawals.
Step 2:	Compute the holding period return (HPR) of the portfolio for each subperiod.
Step 3:	Compute the product of (1 + HPR) for each subperiod to obtain a total return for the entire measurement period [i.e., (1 + HPR)1 × (1 + HPR)2 . . . (1 + HPR)n] – 1. If the total investment period is greater than one year, you must take the geometric mean of the measurement period return to find the annual time-weighted rate of return.

Annualized Return (formula)

Present value of a future cash flow (formula)

because an increase in the frequency of compounding increases the effective interest rate, it also increases the future value of a given cash flow and decreasesthe present value of a given cash flow

Continuously Compounded Return (formula)

Price relative

end-of-period value divided by the beginning-of-period value

Gross return

total return on a security portfolio before deducting fees for the management and administration of the investment account

Net return

return after these fees have been deducted

Pre-tax nominal return

return before paying taxes

After-tax nominal return

return after the tax liability is deducted

Real return

nominal return adjusted for inflation

Leveraged return

a return to an investor that is a multiple of the return on the underlying asset

Pure Discount Instrument

the investor pays less than the face value to buy the instrument and receives the face value at maturity. For example a zero coupon bond

Yield to Maturity

the discount rate applied to the face value

Fixed coupon bond

the investor receives a cash interest payment each period in addition to the face value at maturity

Coupon rate

percentage of the face value and determines the amount of the interest payments

Perpetual bonds (perpetuities)

bonds that have no maturity date

PV of a perpetuity (formula)

Annuity instrument

An instrument that pays a level amount each period, including its maturity period. The difference between an amortizing bond and a fixed-coupon bond is that for an amortizing bond, each payment includes some portion of the principal

Annuity payment (formula)

Preferred stock (definition and formula)

pays a fixed dividend that is stated as a percentage of its par value (similar to the face value of a bond)

Common stock

a residual claim to a company's assets after it satisfies all other claims

Constant growth DDM / Gordon growth model (definition and formula)

Assume a constant future dividend and a constant growth rate of dividends

Multistage DDM

Assume changing dividend growth rates

Cash flow additivity principle

the fact that the PV of any stream of cash flows equals the sum of the PVs of the cash flows

No-arbitrage principle (law of one price)

if two sets of future cash flows are identical under all conditions, they will have the same price today (or if they don't, investors will quickly buy the lower-priced one and sell the higher-priced one, which will drive their prices together)

Forward interest rates

the interest rate for a loan to be made at some future date. 1y1y is the rate for a 1-year loan to be made one year from now; 2y1y is the rate for a 1-year loan to be made two years from now; 3y2y is the 2-year forward rate three years from now; and so on

Spot interest rate

interest rate for a loan to be made today

Exchange rate

the price of one country's currency in terms of another country's currency

Price currency

Numerator in exchange rate

Base currency

Denominator in interest rate

Option

the right, but not the obligation, to buy or sell an asset on a future date for a specified price

Binomial model

Approach to valuing options. based on the idea that, over the next period, some value will change to one of two possible values. To construct a one-period binomial model for pricing an option, we need the following:

• A value for the underlying asset at the beginning of the period

• An exercise price for the option; the exercise price can be different from the value of the underlying, and we assume the option expires one period from now

• Returns that will result from an up-move and a down-move in the value of the underlying over one period

• The risk-free rate over the period

Arithmetic Mean

sum of the observation values divided by the number of observations

Median

midpoint of a dataset

Mode

value that occurs most frequently in a dataset

Unimodal

When a distribution has one value that appears most frequently.

Trimmed Mean

excludes a stated percentage of the most extreme observations. A 1% trimmed mean, for example, would discard the lowest 0.5% and the highest 0.5% of the observations.

Winsorized Mean

Instead of discarding the highest and lowest observations, we substitute a value for them. To calculate a 90% winsorized mean, for example, we would determine the 5th and 95th percentile of the observations, substitute the 5th percentile for any values lower than that, substitute the 95th percentile for any values higher than that, and then calculate the mean of the revised dataset.

Quantile

general term for a value at or below which a stated proportion of the data in a distribution lies (i.e quartile, quintile, decile, percentile).

Interquartile Range

difference between the third quartile and the first quartile.

Box and Whisker Plot

In a box and whisker plot, the box represents the central portion of the data, such as the interquartile range. The vertical line represents the entire range

Dispersion

the variability around the central tendency.

Range

distance between the largest and the smallest value in the dataset

Mean Absolute Deviation (MAD)

the average of the absolute values of the deviations of individual observations from the arithmetic mean

Sample Variance (S²)

the measure of dispersion that applies when we are evaluating a sample of n observations from a population

Standard Deviation (Formula/Answer)

Relative Dispersion

the amount of variability in a distribution around a reference point or benchmark. Relative dispersion is commonly measured with the coefficient of variation (CV)

Coefficient of Variation (CV) (Formula)

Measure of relative dispersion

Target Downside Deviation / Target Semideviation (Formula)

A measure of only the downside risk

Skewness

the extent to which a distribution is not symmetrical

Positively Skewed Distribution

characterized by outliers greater than the mean (in the upper region, or right tail). A positively skewed distribution is said to be skewed right because of its relatively long upper (right) tail

Negatively Skewed Distribution

has a disproportionately large amount of outliers less than the mean that fall within its lower (left) tail. A negatively skewed distribution is said to be skewed left because of its long lower tail.

Sample Skewness

the sum of the cubed deviations from the mean divided by the cubed standard deviation and by the number of observations

Kurtosis

a measure of the degree to which a distribution is more or less peaked than a normal distribution. The computed kurtosis for all normal distributions is three.

Leptokurtic

a distribution that is more peaked than a normal distribution

Platykurtic

a distribution that is less peaked, or flatter than a normal one

Mesokurtic

has the same kurtosis as a normal distribution

Excess Kurtosis

A distribution is said to exhibit excess kurtosis if it has either more or less kurtosis than the normal distribution. The computed kurtosis for all normal distributions is three. Statisticians, however, sometimes report excess kurtosis, which is defined as kurtosis minus three. Thus, a normal distribution has excess kurtosis equal to zero, a leptokurtic distribution has excess kurtosis greater than zero, and platykurtic distributions will have excess kurtosis less than zero

Sample Kurtosis (Formula)

Covariance

measure of how two variables move together

Correlation

A standardized measure of the linear relationship between two variables

Spurious Correlation

correlation that is either the result of chance or present due to changes in both variables over time that is caused by their association with a third variable

Expected Value

the weighted average of the possible outcomes for the variable

Conditional Expected Values

expected values contingent on the outcome of some other event

Bayes Formula

Used to update a given set of prior probabilities for a given event in response to the arrival of new information

Expected Return of a Portfolio

when composed of n assets with weights, w_i, and expected returns, R_i, can be determined using the following formula

Portfolio Variance (formula)

Shortfall Risk

the probability that a portfolio value or return will fall below a particular target value or return over a given period

Roy’s Safety First Criterion

the optimal portfolio minimizes the probability that the return of the portfolio falls below some minimum acceptable level. This minimum acceptable level is called the threshold level

Safety First Ration (formula)

Lognormal Distribution

generated by the function e^x, where x is normally distributed. Because the natural logarithm, ln, of e^x is x, the logarithms of lognormally distributed random variables are normally distributed

Monte Carlo Simulation

technique based on the repeated generation of one or more risk factors that affect security values to generate a distribution of security values

Resampling

another method for generating data inputs to use in a simulation. To conduct resampling, we start with the observed sample and repeatedly draw subsamples from it, each with the same number of observations

Bootstrap Resampling

draw repeated samples of size n from the full dataset, replacing the sampled observations each time so that they might be redrawn in another sample

Probability Sampling

selecting a sample when we know the probability of each sample member in the overall population

Random Sampling

each item is assumed to have the same probability of being selected

Nonprobability Sampling

based on either low cost and easy access to some data items, or on using the judgment of the researcher in selecting specific data items

Systematic Sampling

selecting every nth member from a population

Stratified Random Sampling

uses a classification system to separate the population into smaller groups based on one or more distinguishing characteristics. From each subgroup, or stratum, a random sample is taken and the results are pooled. The size of the samples from each stratum is based on the size of the stratum relative to the population.

Cluster Sampling

assuming that each subset (cluster) is representative of the overall population with respect to the item we are sampling. For example, we may have data on personal incomes for a state's residents by county. The data for each county is a cluster

One-Stage Cluster Sampling

a random sample of clusters is selected, and all the data in those clusters comprise the sample

Two-Stage Cluster Sampling

random samples from each of the selected clusters comprise the sample

Convenience Sampling

selecting sample data based on ease of access, using data that are readily available

Judgmental Sampling

samples for which each observation is selected from a larger dataset by the researcher, based on one's experience and judgment

Central Limit Theorem

for simple random samples of size n from a population with a mean µ and a finite variance σ², the sampling distribution of the sample mean −x approaches a normal probability distribution with mean µ and a variance equal to σ²/n as the sample size becomes large

Standard Error of the Sample Mean

the standard deviation of the distribution of the sample means

Jackknife Resampling

calculates multiple sample means, each with one of the observations removed from the sample

Hypothesis

a statement about the value of a population parameter developed for the purpose of testing a theory or belief