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CFA Level 1 > Quantitative Methods > Flashcards

Quantitative Methods Flashcards

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1
Q

interest rate (r)

A

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

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2
Q

nominal risk-free interest rate

A

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)

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3
Q

real risk-free interest rate

A

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

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4
Q

inflation premium

A

compensates investors for expected inflation

reflects average inflation rate expected over the maturity of the debt

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5
Q

default risk premium

A

compensates investors for possibility that the borrower will fail to make a promised payment at the contracted time and in the contracted amount

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6
Q

liquidity premium

A

compensates investors for the risk of loss relative to an investment’s fair value if the investment needs to be converted to cash quickly

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7
Q

maturity premium

A

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)

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8
Q

simple interest

A

interest rate times the principal

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9
Q

future value (FV)

A

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

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10
Q

future value (FV) with continuous compounding

A

FV<em>N</em> = PVer(s)N

where e = 2.7182818

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11
Q

effective annual rate (EAR)

A

EAR = (1 + periodic interest rate)<em>m</em> - 1

EAR = er(s) - 1

where m is the number of compounding periods per year

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12
Q

annuity

A

finite set of level sequential cash flows

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13
Q

ordinary annuity

A

annuity with first cash flow that occurs one period from now (indexed at t=1)

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14
Q

annuity due

A

annuity that has first cash flow that occurs immediately (indexed at t=0)

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15
Q

perpetuity

A

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

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16
Q

general annuity formula

A
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17
Q

future value factor

A

(1 + r)N

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18
Q

present value factor

A

1 / (1 + r)<em>N</em>

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19
Q

present value formula

A

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
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20
Q

present value of an ordinary annuity

A
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21
Q

present value of a perpetuity

A

PV = A/r

only for perpetuities with level payments

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22
Q

growth rate formula

A

g = (FV<em>N </em>/ PV)1/<em>N</em> - 1

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23
Q

rule of 72

A

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)

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24
Q

cash flow additivity principle

A

dollar amounts indexed at the same point in time can be added

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25
Q

present and future value equivalence

A

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

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26
Q

capital budgeting

A

allocation of funds to relatively long-range projects or investments

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27
Q

capital structure

A

choice of long-term financing for the investments a company wants to make

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28
Q

working capital management

A

management of a company’s short-term assets (such as inventory) and short-term liabilities (such as money owed to suppliers)

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29
Q

net present value (NPV)

A

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

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30
Q

weighted average cost of capital (WACC)

A

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

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31
Q

NPV rule

A

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

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32
Q

internal rate of return (IRR)

A

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”

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33
Q

IRR rule

A

accept projects or investments for which the IRR is greater than the opportunity cost of capital

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34
Q

hurdle rate

A

rate that a project’s IRR must exceed for the project to be accepted

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35
Q

problems with the IRR rule

A

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
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36
Q

performance measurement

A

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

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37
Q

performance appraisal

A

the evaluation of risk-adjusted performance

the evaluation of investment skill

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38
Q

performance evaluation

A

the measurement and assessment of the outcomes of investment management decisions

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39
Q

holding period return (HPR)

A

the return that an investor earns over a specified holding period

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40
Q

money-weighted rate of return in investment management applications

A

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)

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41
Q

time-weighted rate of return

A

measures the compound rate of growth of $1 initially invested in the portfolio over a stated measurement period

(preferred performance measure)

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42
Q

money market

A

market for short-term debt instruments (one-year maturity or less)

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43
Q

pure discount instruments

A

instruments that pay interest as the difference between the amount borrowed and the amount paid back

e.g. the US Treasury bill (T-bill)

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44
Q

face value of a pure discount instrument

A

the amount the issuer (e.g. US government) promises to pay back to an investor

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45
Q

discount

A

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

46
Q

types of money market instruments

A

pure discount instruments

commercial paper (discount instrument)

bankers’ acceptances (discount instrument)

negotiable certificates of deposit (interest-bearing instruments)

47
Q

bank discount basis

A

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

48
Q

why bank discount yield is not a meaningful measure of investors’ return (3 reasons)

A
  1. bank discount yield is based on the face value of the bond, not on its purchase price
  2. bank discount yield is annualized based on a 360-day year, not a 365-day year
  3. bank discount yield annualizes with simple interest, which ignores the opportunity to earn compound interest
49
Q

holding period yield (HPY)

A

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)
50
Q

accrued interest

A

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

51
Q

full price of an interest-bearing instrument

A

includes accrued interest in the price

without accrued interest, trade prices are quoted as “clean”

52
Q

effective annual yield

A

EAY = (1 + HPY)365/<em>t</em> - 1

NB: the bank discount yield is less than the effective annual yield

53
Q

the bank discount yield is (greater/less) than the effective annual yield

A

the bank discount yield is less than the effective annual yield

54
Q

money market yield

CD equivalent yield

A

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*)
55
Q

the money market yield is (larger/smaller) than the bank discount yield

A

the money market yield is larger than the bank discount yield

56
Q

yield to maturity for a bond

A

IRR for a bond

57
Q

bond equivalent yield

A

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%

58
Q

statistics

A

a quantity computed from or used to describe a sample of data

a data or a method

59
Q

descriptive statistics

A

study of how data can be summarized effectively to describe the important aspects of large data sets

60
Q

statistical inference

A

making forecasts, estimates, or judgments about a larger group from the smaller group actually observed

61
Q

population

A

all members of a specified group

62
Q

parameter

A

descriptive measure of a population characteristic

63
Q

sample

A

a subset of a population

64
Q

sample statistic

A

a quantity computed from or used to describe a sample

65
Q

measurement scales

A

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

66
Q

frequency distribution

A

a tabular display of data summarized into a relatively small number of intervals

a list of intervals together with the corresponding measures of frequency

67
Q

interval

A

a set of values within which an observation falls

also called classes, ranges, or bins

68
Q

absolute frequency

A

the actual number of observations in a given interval

69
Q

relative frequency

A

the absolute frequency of each interval divided by the total number of observations

70
Q

cumulative relative frequency

A

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

71
Q

cumulative (absolute) frequency

A

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

72
Q

histogram

A

a bar chart of data that have been grouped into a frequency distribution

73
Q

frequency polygon

A

a histogram in line graph form

74
Q

measure of central tendency

A

specifies where the data are centered

75
Q

measures of location

A

illustrate the location or distribution of data, including meausures of central tendency

76
Q

arithmetic mean

A

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

77
Q

cross-sectional data

A

observations at a specific point in time

78
Q

time-series

A

observations over a period of time

79
Q

trimmed mean

A

excludes a stated small percentage of the lowest and highest values, and then computes an arithmetic mean of the remaining values

80
Q

Winsorized mean

A

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

81
Q

median

A

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

82
Q

mode

A

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

83
Q

modal interval

A

the interval with the highest frequency

84
Q

weighted mean

A

an average in which each observation is weighted by an index of its relative importance

85
Q

expected value

A

the probability-weighted average of the possible outcomes of a random variable

(a weighted average of forward-looking data)

86
Q

geometric mean

A

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

87
Q

geometric mean return formula

A

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

88
Q

harmonic mean

A

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

89
Q

quantile/fractile

A

general term for a value at or below which a stated fraction of the data lies

quartiles - fourths

quintiles - fifths

deciles - tenths

percentiles - hundredths

90
Q

determining the position of a percentile

A

Ly = (n + 1) * (y/100)

where Ly is the location of the yth percentile in an array with n entries

91
Q

linear interpolation

A

estimating an unknown value on the basis of two known values that surround it, using a straight-line estimate

92
Q

dispersion

A

variability around the central tendency

93
Q

absolute dispersion

A

amount of variability present without comparison to any reference point or benchmark

94
Q

range

A

difference between the maximum and minimum values in a data set

Range = Maximum Value - Minimum Value

95
Q

mean absolute deviation (MAD)

A

average of the absolute value of the distances from the mean

96
Q

variance

A

average of the squared deviations around the mean

for sample variance (rather than population variance), divide by n-1 instead of n

97
Q

standard deviation

A

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

98
Q

semivariance

A

average squared deviation below the mean

(still divide by the total sample size minus 1: n-1)

99
Q

semideviation

(semistandard deviation)

A

positive square root of semivariance

100
Q

target semivariance

A

average squared deviation below a stated target

(still divided by n-1)

101
Q

target semideviation

A

positive square root of the target semivariance

102
Q

Chebyshev’s Inequality

A

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)

103
Q

relative dispersion

A

amount of dispersion relative to a reference value or benchmark

104
Q

coefficient of variation (CV)

A

ratio of the standard deviation of a set of observations to their mean value

CV = s/X–

105
Q

Sharpe Ratio

A

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

106
Q

mean excess return on a portfolio

A

difference between the mean return to the portfolio and the mean return to a risk-free asset

107
Q

normal distribution

A

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

108
Q

skewed/skewness

A

a quantitative measure of lack of symmetry

the average cubed deviation from the mean standardized by dividing by the standard deviation cubed

109
Q

sample skewness

A

SK = [n / ((n-1)(n-2))] * sum of cubed deviations divided by the standard deviation cubed

110
Q

kurtosis

A

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

111
Q

sample excess kurtosis

A

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)]

112
Q

semilogarithmic

A

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

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