Quantitative Methods Flashcards
interest rate (r)
r = real risk-free interest rate + inflation premium + default risk premium + liquidity premium + maturity premium
a rate of return that reflects the relationship between diffrently dated cash flows; can be thought of as 1. required rate of return; 2. discount rate; 3. opportunity cost
nominal risk-free interest rate
rnominal risk-free = real risk-free interest rate + inflation premium
1 + r**nominal risk-free = (1 + rreal risk-free) * (1 + rinflation premium)
often represented by governmental short-term debt interest rate (e.g. 90-day US Treasury bill)
real risk-free interest rate
single-period interest rate for a completely risk-free security if no inflation were expected
reflects time preferences of individuals for current versus future real consumption
inflation premium
compensates investors for expected inflation
reflects average inflation rate expected over the maturity of the debt
default risk premium
compensates investors for possibility that the borrower will fail to make a promised payment at the contracted time and in the contracted amount
liquidity premium
compensates investors for the risk of loss relative to an investment’s fair value if the investment needs to be converted to cash quickly
maturity premium
compensates investors for increased sensitivity of the market value of debt to a change in market interest rates as maturity is extended, in general (ceteris paribus)
simple interest
interest rate times the principal
future value (FV)
FV<em>N</em> = PV(1 + r)N
NB: r and N must be defined in the same time units
FV<em>N</em> = PV(1 + (rs /m))<em>m</em>N
future value factor = (1 + r)N
stated annual interest rate = rs
number of compounding periods per year = m
future value (FV) with continuous compounding
FV<em>N</em> = PVer(s)N
where e = 2.7182818
effective annual rate (EAR)
EAR = (1 + periodic interest rate)<em>m</em> - 1
EAR = er(s) - 1
where m is the number of compounding periods per year
annuity
finite set of level sequential cash flows
ordinary annuity
annuity with first cash flow that occurs one period from now (indexed at t=1)
annuity due
annuity that has first cash flow that occurs immediately (indexed at t=0)
perpetuity
a perpetual annuity, or a set of level never-ending sequential cash flows, with the first cash flow occuring one period from now (indexed at t=1)
examples: dividends from stocks, some government bonds and preferred stocks
general annuity formula
future value factor
(1 + r)N
present value factor
1 / (1 + r)<em>N</em>
present value formula
PV = FVN / (1 + r)N
PV = FVN / (1 + (rs/m)<em>mN</em>
- m* = number of compounding periods per year
- rs =* quoted annual interest rate
- N* = number of years
present value of an ordinary annuity
present value of a perpetuity
PV = A/r
only for perpetuities with level payments
growth rate formula
g = (FV<em>N </em>/ PV)1/<em>N</em> - 1
rule of 72
72 divided by the stated interest rate is the approximate number of years it would take to double an investment at the stated interest rate
converse: it takes 12 years to double an investment at 6% interest rate (6 x 12 = 72)
cash flow additivity principle
dollar amounts indexed at the same point in time can be added
present and future value equivalence
a lump sum can be seen as equivalent to an annuity, and an annuity can be seen as equivalent to its future value
present values, future values, and a series of cash flows can all be considered equivalent if they are indexed at the same point in time
capital budgeting
allocation of funds to relatively long-range projects or investments
capital structure
choice of long-term financing for the investments a company wants to make
working capital management
management of a company’s short-term assets (such as inventory) and short-term liabilities (such as money owed to suppliers)
net present value (NPV)
present value of cash inflows minus present value of cash outflows
considers only incremental cash flows; not sunk costs
account for tax effects by using after-tax cash flows
weighted average cost of capital (WACC)
weighted average of the after-tax required rates of return on the company’s common stock, preferred stock, and long-term debt
weighted by fraction of each source of financing in the company’s target capital structure
NPV rule
if an investment’s NPV is positive, undertake it
if an investment’s NPV is negative, do not undertake it
among mutually exclusive projects, choose the project with the highest positive NPV
if undertaking a NPV = 0 project, the company becomes larger, but shareholders’ wealth does not increase
internal rate of return (IRR)
the discount rate that makes the net present value equal to zero
the rate that equates the present value of the investment’s costs to the present value of its benefits
for bonds, the “yield to maturity”
IRR rule
accept projects or investments for which the IRR is greater than the opportunity cost of capital
hurdle rate
rate that a project’s IRR must exceed for the project to be accepted
problems with the IRR rule
The IRR rule and NPV rule have different results if
- the size or scale of the projects differs (in terms of investment needed to undertake the project)
- the timing of the projects’ cash flows differs
performance measurement
calculating returns of investments in a logical and consistent manner
measured using the money-weighted rate of return measure or the time-weighted rate of return measure
performance appraisal
the evaluation of risk-adjusted performance
the evaluation of investment skill
performance evaluation
the measurement and assessment of the outcomes of investment management decisions
holding period return (HPR)
the return that an investor earns over a specified holding period
money-weighted rate of return in investment management applications
equals the internal rate of return
(because it accounts for the timing and amount of all cash flows into and out of the portfolio)
also known as the dollar-weighted return
(NB: problem in using this to evaluate investment managers is that clients determine when and how much money is given to the investment manager, which affects the money-weighted rate of return and is outside of themanager’s control)
time-weighted rate of return
measures the compound rate of growth of $1 initially invested in the portfolio over a stated measurement period
(preferred performance measure)
money market
market for short-term debt instruments (one-year maturity or less)
pure discount instruments
instruments that pay interest as the difference between the amount borrowed and the amount paid back
e.g. the US Treasury bill (T-bill)
face value of a pure discount instrument
the amount the issuer (e.g. US government) promises to pay back to an investor
discount
the reduction from the face amount that gives the price for the pure discount instrument (e.g. T-bill)
this discount becomes the interest that accumulates
types of money market instruments
pure discount instruments
commercial paper (discount instrument)
bankers’ acceptances (discount instrument)
negotiable certificates of deposit (interest-bearing instruments)
bank discount basis
quoting convention that annualizes, based on a 360-day year, the discount as a percentage of face value (T-bills quoted this way)
- r*BD = (D/F) * (360/t)
- r*BD = annualized yield on a bank discount basis = bank discount yield = discount yield
D = dollar discount = difference between face value of the bill, F, and purchase price, P0
t = number of days remaining to maturity
why bank discount yield is not a meaningful measure of investors’ return (3 reasons)
- bank discount yield is based on the face value of the bond, not on its purchase price
- bank discount yield is annualized based on a 360-day year, not a 365-day year
- bank discount yield annualizes with simple interest, which ignores the opportunity to earn compound interest
holding period yield (HPY)
return that an investor will earn by holding the instrument to maturity in fixed income markets
also known as holding period return, total return, and horizon return
for an instrument that makes one cash payment during its life:
HPY = (P1 - P0 + D1) / P0
- P*0 = initial purchase price of the instrument
- P*1 = price received for the instrument at its maturity
- D*1 = cash distribution paid by the instrument at its maturity (i.e. interest)
accrued interest
coupon interest that the seller earns from the last coupon date but does not receive as a coupon, because the next coupon date occurs after the date of sale
NB: when calculating holding period yield for an interest-bearing instrument (e.g. coupon-bearing bonds), purchase and sale prices must include any accrued interest added to the trade price
full price of an interest-bearing instrument
includes accrued interest in the price
without accrued interest, trade prices are quoted as “clean”
effective annual yield
EAY = (1 + HPY)365/<em>t</em> - 1
NB: the bank discount yield is less than the effective annual yield
the bank discount yield is (greater/less) than the effective annual yield
the bank discount yield is less than the effective annual yield
money market yield
CD equivalent yield
makes the quoted yield on a T-bill comparable to yield quotations on interest-bearing money-market instruments that pay interest on a 360-day basis
- rMM = 360rBD / (360 - (t)(r*BD))
- rMM = HPY * (360/t*)
the money market yield is (larger/smaller) than the bank discount yield
the money market yield is larger than the bank discount yield
yield to maturity for a bond
IRR for a bond
bond equivalent yield
calculation of yield that is annualized using the ratio of 365 to the number of days to maturity
allows for the restatement and comparison of securities with different compounding periods
e.g. semi-annual yield to maturity (YTM) = 4%
bond equivalent yield = 4% * 2 = 8%
statistics
a quantity computed from or used to describe a sample of data
a data or a method
descriptive statistics
study of how data can be summarized effectively to describe the important aspects of large data sets
statistical inference
making forecasts, estimates, or judgments about a larger group from the smaller group actually observed
population
all members of a specified group
parameter
descriptive measure of a population characteristic
sample
a subset of a population
sample statistic
a quantity computed from or used to describe a sample
measurement scales
nominal scales: weakest level; categorize data but do not rank them
ordinal scales: sort data into categories that are ordered by some characteristic
interval scales: rank data and assure that diffrences between scale values are equal (can be added and subtracted meaningfully)
ratio scales: strongest level; interval scales with a true zero point as the origin; can meaningfully compute ratios
frequency distribution
a tabular display of data summarized into a relatively small number of intervals
a list of intervals together with the corresponding measures of frequency
interval
a set of values within which an observation falls
also called classes, ranges, or bins
absolute frequency
the actual number of observations in a given interval
relative frequency
the absolute frequency of each interval divided by the total number of observations
cumulative relative frequency
adds up the relative frequencies as one moves from the first to the last interval
equal to the fraction of observations that are less than the upper limit of each interval
cumulative (absolute) frequency
adds up the absolute frequencies as one moves from the first to the last interval
equal to the total number of observations that are less than the uper limit of each interval
histogram
a bar chart of data that have been grouped into a frequency distribution
frequency polygon
a histogram in line graph form
measure of central tendency
specifies where the data are centered
measures of location
illustrate the location or distribution of data, including meausures of central tendency
arithmetic mean
sum of the observations divided by the number of observations; like the center of gravity of a set of data
population mean: a parameter
sample mean: a statistic
advantage: uses all the information about hte size and magnitude of observations
disadvantage: sensitive to extreme values
good for making investment statements in a forward-looking context
cross-sectional data
observations at a specific point in time
time-series
observations over a period of time
trimmed mean
excludes a stated small percentage of the lowest and highest values, and then computes an arithmetic mean of the remaining values
Winsorized mean
assigns a stated percent of the lowest values equal to one specified low value, and a stated percent of the highest values equal to one specified high value, then computes a mean from the restated data
median
the value of the middle item of a set of items sorted in ascending/descending order
advantage: not affected by extreme values
disadvantage: only focuses on the relative position of ranked observations
mode
the most frequently occuring value in a distribution
can have no mode, or can be unimodal, bimodal, trimodal, etc.
only measure of central tendency that can be used with nominal data
modal interval
the interval with the highest frequency
weighted mean
an average in which each observation is weighted by an index of its relative importance
expected value
the probability-weighted average of the possible outcomes of a random variable
(a weighted average of forward-looking data)
geometric mean
used to average rates of change over time, or to compute the growth rate of a variable
G = (X1X2X3…Xn)1/<em>n</em>
ln G = 1/n * ln(X1X2X3…Xn)
G = eln<em>G</em>
good for making investment statements about past performance
always smaller than or equal to the arithmetic mean
approximately equal to arithmetic return minus half the variance of return
geometric mean return formula
geometric mean calcuated as 1+R for each time period, and then subtracting 1 to get the return rate
the geometric mean is always less than or equal to the arithmetic mean
shows the mutlti-period return of an investment, whereas the arithmetic mean return shows the averager single-period performance
harmonic mean
sum the reciprocals of all the observations, divide it by the number of observations, and take the reciprocal of the average
special type of weighted mean in which an observation’s weight is inversely proportional to its magnitude
the harmonic mean is always less than or equal to the geometric mean, which is less than or equal to the arithmetic mean
quantile/fractile
general term for a value at or below which a stated fraction of the data lies
quartiles - fourths
quintiles - fifths
deciles - tenths
percentiles - hundredths
determining the position of a percentile
Ly = (n + 1) * (y/100)
where Ly is the location of the yth percentile in an array with n entries
linear interpolation
estimating an unknown value on the basis of two known values that surround it, using a straight-line estimate
dispersion
variability around the central tendency
absolute dispersion
amount of variability present without comparison to any reference point or benchmark
range
difference between the maximum and minimum values in a data set
Range = Maximum Value - Minimum Value
mean absolute deviation (MAD)
average of the absolute value of the distances from the mean
variance
average of the squared deviations around the mean
for sample variance (rather than population variance), divide by n-1 instead of n
standard deviation
positive square root of the variance
for sample standard deviation (rather than population standard deviation), divide by n-1 instead of n
standard deviation is always greater than or equal to mean absolute deviation because standard deviation gives greater weight to larger deviations
semivariance
average squared deviation below the mean
(still divide by the total sample size minus 1: n-1)
semideviation
(semistandard deviation)
positive square root of semivariance
target semivariance
average squared deviation below a stated target
(still divided by n-1)
target semideviation
positive square root of the target semivariance
Chebyshev’s Inequality
for any distribution with finite variance, the proportion of the observations within k standard deviations of the arithmetic mean is at least 1 - 1/k2 for all k > 1
(75% within 2 standard deviations; 89% within 3 standard deviations; 94% within 4 standard deviations)
relative dispersion
amount of dispersion relative to a reference value or benchmark
coefficient of variation (CV)
ratio of the standard deviation of a set of observations to their mean value
CV = s/X–
Sharpe Ratio
Sharpe Ratio = (mean return to the portfolio - mean return to a risk-free asset) / standard deviation of the return on the portfolio
the higher the Sharpe Ratio, the better
should only be considered for positive Sharpe ratios
most appropriate for approximately symmetric return distributions; not for strategies with option elements that have asymmetric returns
mean excess return on a portfolio
difference between the mean return to the portfolio and the mean return to a risk-free asset
normal distribution
the mean and median are equal
it is completely described by two parameters: mean and variance
68% within one standard deviation; 95% within two standard deviations; 99% within three standard deviations
skewed/skewness
a quantitative measure of lack of symmetry
the average cubed deviation from the mean standardized by dividing by the standard deviation cubed
sample skewness
SK = [n / ((n-1)(n-2))] * sum of cubed deviations divided by the standard deviation cubed
kurtosis
statistical measure of whether a distribution is more or less peaked than a normal distribution
leptokurtic = more peaked than normal (fatter tails, excess kurtosis greater than 0)
platykurtic = less peaked than normal (thinner tails, excess kurtosis less than 0)
mesokurtic = identical to normal = 3
sample excess kurtosis
KE = [n(n + 1) / ((n-1)(n-2)(n-3))] * sum of deviations from the mean raised to the fourth power / standard deviation raised to the fourth power
– [3(n - 1)2 / (n - 2)(n - 3)]
semilogarithmic
a scale constructed so that equal intervals on the vertical scale represent equal rates of change, and equal intervals on the horizontal scale represent equal amounts of change
