2: Quantitative Methods Flashcards
keystrokes for compounding
either adjust the interest rate (i/y) OR adjust the periods per year (P/y). But NOT both.
calculating EAR
determine the “periodic rate” (interest/compounding period). add 1, then multiply by $1, to get growth of the $1, but then “raise” that value by the amount of compounding periods. this will give EAR PLUS 1, so subtract 1.
EAR with continuous compounding
use 365. Interest as a decimal, divide by 365, to get periodic rate. add 1, multiply by $1. “raise” by 365, subtract 1 to get EAR.
annuity (as a series of cash flows)
a finite set of LEVEL SEQUENTIAL cash flows
ordinary annuity (as a series of cash flows)
has a first cash flow that occurs one period from now (indexed at t=1)
annuity due (as a series of cash flows)
has a first cash flow that occurs immediately (indexed as t=0). Change to BGN on calculator.
perpetuity (as a series of cash flows)
a perpetual annuity, or a set of level never-ending sequential cash flows, with the first cash flow occurring one period from now. END on calculator
computing PV or FV of unequal cash flows
Use CF function for uneven cash flows. enter interest rate, then CPT NPV. this is the PV.
To find FV: Do previous steps first, then use PV above to find FV, using same periods and interest rate.
calculating the projected present value of an annuity
solve for PV first. then use PV to find FV for period of time in the future
if an annuity begins today, t=0, then…
choose BGN on calculator
calculating PV of perpetuities
use 500 for N
Rule of 72
divide 72 by the stated interest rate to get the approximate number of years it would take to double an investment at the interest rate.
EAR equation
(1+Periodic Rate)^m - 1
steps for calculating Perpetuity PV (as it relates to stocks)
- PV=A/r, give value at t=1.
2. change P/y, then solve for PV where N=1
2 things to remember on unequal cash flows/annuities…
make note of timeline… and CLEAR WORK before advancing in the math
annuity calculations t=x
adjust the timeline or N if annuity payment is 1 year away, ordinary annuity
college planning with inflation steps
- inflate each year to FV
- discount each year back to ordinary annuity time frame using RoR
- add each year for total cost at ordinary annuity timeframe
- solve for PMT, total in step 3 is FV
nominal risk-free rate
sum of the real risk-free rate and inflation premium
mutually exclusive projects
an investor has two candidates for investment but can only invest in one
money-weighted rate of return
aka IRR, aka dollar-weighted return. it accounts for the timing and amount of all cash flows into and out of the portfolio.
time-weighted rate of return
does not take into account cash inflows/outflows. better measurement for investment management performance.
HPR (holding period return) equation
(ending-beginning)/beginning
time-weighted return calculation considers…
dividend payments as cash flows
bank discount yield (T-bills)
r=(D/F)(360/t); where D=the dollar discount, which is equal to the difference between the face and the purchase price, F=face value, t=actual number of days remaining to maturity
effective annual yield (EAY)
takes the quantity 1 plus the holding period yield and compounds it forward to one year, then subtracts 1 to recover an annualized return that accounts for the effect of interest on interest.
the bank discount yield is always…
less than the effective annual yield.
EAY formula
think EAR…
(1+periodic rate or HPY)^m or 365/t - 1.
just like EAR, except m=365/t
money market yield
aka 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.
in general, the money market yield is equal to…
the annualized holding period yield; assuming a 360-day year. Compared to the bank discount yield, the mm yield is computed on the purchase price.
money market yield formula
(360rBD)/360-(t)(rBD)
HPY formula
(P1-P0+D1)/P0
bond equivalent yield
doubling the semiannual YTM, ignoring compounding
the money market yield is equal to the…
annualized HPY, assuming a 360 day year.
rMM=(HPY)(360/t).
No need to know the rBD.
the EAY is…
365
central tendency
where the returns are centered
dispersion
how far returns are dispersed from their center
skew-ness
whether the distribution of returns is symmetrically shaped or lopsided
kurtosis
whether extreme outcomes are likely
nominal scales
represent the weakest level of measurement: they categorize data but do not rank them
ordinal scales
stronger level of measurement with regard to nominal scales. they sort data into categories that are ordered with respect to some characteristic
interval scales
provide not only ranking, but also assurance that the differences between scale values are equal. as a consequence of the absence of a true zero point, we cannot meaningfully form ratios on interval scales.
ratio scales
represent the strongest level of measurement. they have all the characteristics of interval measurement scales as well as a true zero point as the origin.
measurement scales, listed worst to best
nominal, ordinal, interval, ratio
measurement scales:
credit ratings for bond issues
ordinal scale. a rating places a bond issue in a category, and the categories are ordered with respect to the expected probability of default. but the difference in the expected probability of default between ratings is not equal. letter credit ratings are not measured on an interval scale
measurement scales:
cash dividends per share
measured on a ratio scale. for this variable, 0 represents the complete absence of dividends; it is a true zero point.
measurement scales:
hedge fund classification types
nominal scale. each type groups together hedge funds with similar investment strategies. hedge fund classification schemes do not involve a ranking.
measurement scales:
bond maturity in years
ratio scale. true zero point.
cross-sectional data
examining the characteristics of some units at a specific point in time. the mean of these observations is called a cross-sectional mean
time-series data
if our sample consists of historical returns. the mean of these observations is called a time-series mean
the geometric mean is always…
less than or equal to the arithmetic mean
the only time geometric and arithmetic mean will be equal is when…
there is no variability in the observations- when all the observations in the series are the same
the difference between the arithmetic and geometric means increases with…
the variability in the period-by-period observations
the geometric mean represents…
the growth rate or compound rate of return on an investment.
geometric mean formula and key strokes
value=(1+R1)(1+R2)(1+R3)…
=value [y^x] number of returns Rn’s above [1/x] [=]
then subtract 1
locating the position of a percentile formula
=(n+1)(y/100)
harmonic mean is appropriate…
when averaging ratios (amount per unit) when the ratios are repeatedly applied to a fixed quantity to yield a variable number of units. Ex. cost averaging
mathematical fact concerning harmonic, geometric, and arithmetic means is…
unless all the observations in a data set have the same value, the harmonic mean is less than the geometric mean, which in turn is less than the arithmetic mean
the lower the quantile…
the lower the value in ranking
coefficient of variation formula
sd/mean
sharpe ratio formula
(rP-rF)/sdP; where rP is portfolio return, rF is risk free rate, and sdP is standard deviation of portfolio
positive skewness
has frequent small losses and a few extreme gains
negative skewness
has frequent small gains and a few extreme losses
normal distribution
68 percent within 1 standard deviation, 95 percent within 2 standard deviations, 98 percent within 3 standard deviations
for the continuous positively skewed distribution…
the mode is less than the median, which is less than the mean
for the continuous negatively skewed distribution…
the mean is less than the median, which is less than the mode
leptokurtic
distribution that has fatter tails than the normal distribution. has excess kurtosis greater than 0
platykurtic
distribution that has thinner tails than the normal distribution. has excess kurtosis less than 0.
mesokurtic
distribution that is identical to the normal distribution as it relates to the weight of the tails. has excess kurtosis equal to 0.
if we want to estimate the average return over a one-period horizon, use
arithmetic mean because it is the average of one-period returns
if we want to estimate the average returns over more than one period, use
geometric mean because it captures how the total returns are linked over time
a priori probability
one based on logical analysis rather than on observation or personal judgment
pairs arbitrage trade
a trade in two closely related stocks involving the short sale of one and the purchase of the other
multiplication rule for probability
P(AB)=P(A/B)*P(B)
probability of both happening (joint probability)=conditional probability (probability of a, given b) multiplied by unconditional probability (probability of b)
addition rule for probability
P(A or B)=P(A)+P(B)-P(AB)
multiplication rule for independent events
P(AB)=P(A)*P(B)
total probability rule
explains the unconditional probability of the event in terms of probabilities conditional on the scenarios
total probability rule formula
P(A)=P(A/S)P(S)+P(A/Sc)P(Sc)
covariance matrix formula
variance(A+B)=(weightA)^2(varianceA)+(weightB)^2(varianceB)+2(weightA)(weightB)[covariance(A,B)]
