Quant: TVM & Basic Concepts
Cross-Sectional Data
Data are data on some characteristic of individuals, groups, geographical regions, or companies at a single point in time. The 2014 year-end book value per share for all New York Stock Exchange-listed companies is an example of cross-sectional data.
Estimators
Desirable characteristics of an estimator: Unbiasedness Efficiency
Discrete Uniform Distribution
The cumulative distribution function (cdf) for a Discrete Uniform Distribution rises in steps.
Geometric Mean Return
= [((1 + r)(1 + r1)(1 + r2)(1 + r3)... (1 + rn)) ∧ (1/n)] - 1 n --> total number of returns Geometric Mean will always be smaller than Arithmetic Mean AM > GM Should be used to measure Past performance.
Compound Annual Growth Rate
= [(end value / beginning value) ∧ (1/n)] - 1 n → number of periods, NOT number of dollar values
Continuously Compounded Rate of Return
= [ln(HPY + 1)]/t i.e. An investment of $1,000 appreciates to a value of $1,450 in 3.5 years. What is the continuously compounded annual return on this investment? = [ln(.45 + 1)]/3.5 = 10.62%
PV of Perpetuity
= pmt / r Used to find price of stock with steady (constant) dividend: = dividend/ r where r is the required rate of return on equity
Coefficient of Variation (CV)
= σ ÷ µ = s ÷ x⁻ The ratio of the standard deviation of a set of observations to their mean value. The coefficient of variation measures the amount of risk (standard deviation) per unit of mean return. LOW CV is good; HIGH CV is bad.
Net Present Value (NPV)
= ∑Cash Flow (at time = t) ÷ (1 + r)∧t
Future Value (FV) with more than one compounding period per year
= PV(1 + (r÷m)) ∧ (m×n) n = years m= number of compounding periods per year r = annual interest rate
Effective Annual Yield (EAY)
= ((1 + HPY) ∧ 365/t) - 1 = [BDY/(360/t)] × (365/t) = BEY + (BEY/2)² EAY is an ANNUALIZED return based on a 365 day year with compounding! (compounding = "raised to the power of") EAY will always be higher than BEY.
Holding Period Yield (HPY)
= (P₁ - P₀ + div) ÷ P₀ when EAY is known: = ((1 + EAY) ∧ t/365) - 1 HPY is an UNANNUALIZED return.
Sharpe Ratio
= (return (portfolio) - return (required or risk-free-rate)) ÷ σ (portfolio) The Sharpe Ratio measures excess return per unit of risk. Generally a portfolio with a higher Sharpe Ratio offers a better risk-adjusted return. If standard deviation (risk) increases, the Sharpe Ratio (excess return per unit of risk) decreases.
Bank Discount Rate (aka Bank Discount Yield)
= D/F × 360/t D --> discount F --> face t --> time to maturity
Future Value (FV)
= PV (1 + r)ⁿ ⁿ = number of years r = annual interest rate
Bollinger Bands
A moving average trend line with an upper band at moving average PLUS a set number of standard deviations away from average price, and a lower band at moving average MINUS a set number of standard deviations away from average price. Changes in volatility (standard deviation) will widen or tighten the space between the bands.
Stratified Random Sampling
A population is first divided into subpopulations or strata, based upon one or more classification criteria, then a simple random sample is taken from each strata.
Time Series Data
A sequence of returns collected at discrete and equally spaced intervals of time (such as a historical series of monthly stock returns).
Sample
A subset of a population
Quantiles
A value at, or below which a stated proportion of the observations in a data set lie (e.g. quartile (25% in each); decile (10% in each); percentiles (1% in each); etc.) To determine the position of a percentile in a data set with n observations that are sorted in ascending order: ly = [(n + 1)(y)] ÷ 100 Example: the 4th quintile for the following distribution of returns is closest to: 8%, 6%, 12%, 18%, 25%, 8%, 9%, 17%, 14%, 10% 1) 4th quintile = (100/5) × 4 = 80 2) ly = [(10+1)(80)] ÷ 100 = 8.8 3) rearrange data set into ascending order: 6%, 8%, 8%, 9%, 10%, 12%, 14%, 17%, 18%, 25% The 8th item is 17% and we need .8 of the difference between 17% (the 8th position) and 18% (the 9th position). [18% - 17% = 1; and .8 × 1 = .8] so the 4th quintile contains everything that is at or below 17% + .8% = 17.8%
Bond Equivalent Yield (BEY)
BEY = [((1 + EAY)∧.5) -1] × 2 The bond equivalent yield (BEY) is the semiannual discount rate multiplied by 2. It will always be lower than the EAY.
Stochastic Oscillator
Based on the observation that in uptrends, prices tend to close at or near the high end of their recent range and in downtrends, they tend to close near the low end. Oscillates between 0 and 100 and has a default setting of a 14 day period. Consists of a %K Line and a %D Line: %D is like the long term moving average moving in conjunction with the short term moving average. It is the slower moving smoother line called the signal line. %K is the faster moving short term line When the %K moves from below the %D line to above it, this move is considered a bullish short-term trading signal; When %K moves from above the %D line to below it (when the %D line crosses %K line from below), this pattern is considered bearish.
Money Market Yield (MMY or Rmm)
Convert HPY to MMY: MMY = HPY * (360/t) HPY = (P₁-P₀)/P₀ t →days to maturity Convert BDY to MMY: MMY = (360 * BDY)/(360-(t*BDY)) t → days to maturity Money Market Yield is an annualized yield based on a 360 day year without compounding (without compounding = "multiply by 360/t")
Correlation
Correlation is a number between −1 and +1 for two random variables, X and Y: -1 ≤ ρ(X,Y) ≤ +1 Correlation = 0, means uncorrelated variables -- no linear relationship between return on X and return on Y (Rx, Ry) (The correlation between a risk-free asset and a risky asset is zero) Increasingly positive correlation indicates an increasingly strong positive linear relationship (up to 1, which indicates a perfect linear relationship). Increasingly negative correlation indicates an increasingly strong negative (inverse) linear relationship (down to −1, which indicates a perfect inverse linear relationship).
Effective Annual Rate (EAR)
EAR = (1 + r/n)∧n - 1 r → stated annual rate n→ number of compounding periods in a year r/n → periodic rate EAR with continuous compounding: EAR = e∧rs - 1 Calculator key strokes: [2nd] [2] Enter NOM (stated annual rate) [ENTER] [↓] [↓] Enter C/Y (number of compounding periods (n)) [ENTER] [↓] [↓] [CPT] EFF=
Harmonic Mean
Harmonic Mean = n / ∑(ni/xi) Example: An individual invests $1,000 in a particular security on two different dates ($2,000 total). The price of the security is $10 on the first day and $12 on the second. The investor's average purchase price is = 2000/((1000/10) + (1000/12)) = 10.91% Used to calc average cost of shares purchased over time when the same dollar amount is invested each period.
Convert BDY to EAY
If BDY is given: 1) Find D in the BDY formula; BDY = ((D/F) × (360/t)) 2) P₀ = par - D from step 1 (if par is not given use 100) 3) Calculate HPY using par and P₀ from step 2 4) Calculate EAY using HPY; EAY = [(1+HPY)∧(365/t)] -1
Trend Lines: Ascending Triangle
In an ascending triangle pattern, the trendline connecting the high prices is horizontal and the trend line connecting the low prices is upward sloping. This is a bullish signal indicating prices may continue to rise.
Relative Strength and Intermarket Analysis
In intermarket analysis, technicians often use relative strength analysis to look for inflection points in one market as a warning sign to start looking for a change in trend in a related market. Intermarket analysis assumes that all markets are interrelated and influence each other. Relative strength analysis is utilized for different groups of securities to make allocation decisions.
Moving Average Crossovers (Buy / Sell Signals)
Investors often use moving-average crossovers as a buy or sell signal. GOLDEN CROSS: When a short-term moving average crosses from underneath a longer-term average, this movement is considered bullish and is termed a golden cross. DEAD CROSS: Conversely, when a short-term moving average crosses from above a longer-term moving average, this movement is considered bearish and is called a dead cross.
Market Interest Rate: Components
Market Interest Rate = risk free rate + inflation premium + default risk + liquidity risk + maturity risk
Kurtosis
Mesokurtosis -- normal peak (normal distribution); kurtosis = 3 Leptokurtosis -- more peaked and flatter tails; positive excess kurtosis, or kurtosis greater than 3 Platykurtosis -- less peaked; negative excess kurtosis, or kurtosis less than 3 Excess kurtosis = kurtosis - 3
Types of Scales
N.O.I.R. (in order from least to most useful) Nominal Ordinal Interval Ratio
Longitudinal Data
Observations on characteristic(s) of the same observational unit through time. i.e. Observations on a set of financial ratios for a single company over a 10-year period
Panel Data
Observations through time on a single characteristic of multiple observational units. i.e. the annual inflation rate of the Eurozone countries over a five-year period would represent panel data
Monte Carlo Simulation
Provides a distribution of possible solutions to complex functions. Monte Carlo simulations are used to model the probability of different outcomes.
Roy's Safety First (SF) Ratio
SF Ratio = (E(r) - minimum required return) ÷ σ Minimum required return = threshold level or the risk free rate E(r) = expected return σ = standard deviation of returns
Lognormal Distribution
Stock prices (not stock returns) are represented by a lognormal distribution. The lognormal distribution is truncated at zero and SKEWED to the RIGHT (positively skewed).
Normal Distribution
Symmetrical Kurtosis of 3 Excess Kurtosis of zero Not bound by zero
Covariance
The covariance captures how the co-movements of returns on securities affect portfolio variance. Covariance of returns is negative if, when the return on one asset is above its expected value, the return on the other asset tends to be below its expected value (an average inverse relationship between returns). Covariance of returns is 0 if returns on the assets are unrelated. (As is the case for the covariance of a 'risk-free' -- standard deviation of zero -- and a risky asset). Covariance of returns is positive when the returns on both assets tend to be on the same side (above or below) their expected values at the same time (an average positive relationship between returns).
Continuous Uniform Distribution
The cumulative distribution function (cdf) for a Continuous Uniform Distribution is an upward sloping straight line. The probability of any individual outcome, P(X=x), within or outside the parameters, is 0.
Ordinal Scale
sort data into categories that are ranked according to certain characteristics but tell us nothing about the magnitude of the difference between the categories (i.e. 5 stars vs. 3 stars)
FV w/ Continuous Compounding
[2nd] [CE/C] "r" [2nd] [LN] [y∧x] "N" [×] "$PV" [=] r --> stated rate N --> number of years $PV --> present value of cash
Nominal Scale
categorize or count data but do not rank them (i.e. 1 represents value stocks, 2 represents growth stocks, etc.)
Ratio Scale
contains all of the characteristics of interval scales and have a true zero point as the origin. Meaningful ratios can be computed with ratio scales.
Interval Scale
rank observations such that the difference between scale values is equal; values can be added and subtracted more meaningfully (i.e. temperature scale -- zero does not mean the absence of temperature and 4 degrees isn't "twice as hot" as 2 degrees)
Arithmetic Mean
sum of all observations in a data set divided by number of observations arithmetic mean = ∑n ÷ n Should be used to calculate expected values -- Forward looking context.