Quantitative Methods, Essays (high school) of Mathematics

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TIME VALUE OF MONEY

Effective annual rate = 1 + (

m

)

Stated annual rate m

Effective Annual Rate (EAR)

Single Cash Flow (Simplified formula)

FVN = PV x (1 + r)N

PV =

FV N

(1 + r)N

Investments p aying interest more than once a y ear

FVN = PV x (

rs 1 + (^) m )

mN

PV =

(

rs m

)

mN

FV N

rs = Stated annual interest rate m = Number of compounding periods per year N = Number of years

Future Value ( FV) of an Investment w ith Continuous Compounding

FVN = PVersN

Ordinary Annuity

FVN = A x [

(1 + r)N^ - 1 r ]

PV = A x[

(1 + r)N

r ]

Annuity Due

FV ADue =FV A Ordinary x (1 + r) = A x [

(1 + r)N^ - 1 r ]

x (1 + r)

PV ADue =FV A Ordinary x (1 + r) = A x [ x^ (1 + r)

(1 + r)N

r ]

r = Interest rate per period PV = Present value of the investment FVN = Future value of the investment N periods from today

N = Number of time periods A = Annuity amount r = Interest rate per period

A = Annuity amount r = The interest rate per period corresponding to the frequency of annuity paments (for example, annual, quarterly, or monthly) N = Number of annuity payments

STATISTICAL CONCEPTS AND MARKET RETURNS

Geometric Mean G = √ x^1 x^2 x^3 ... xn

n

Sample Mean x =

x (^) i

n

Σ

n

i = 1 ... n

x 1 + x 2 + x 3 + ... + xn n

Harmonic Mean

xn =

x (^) i

n

Σ

n

i = 1 ... n

( )

Interval W idth Interval W idth =

Range k

Relative Frequency Formula Relative f r equency =

Interval frequency Observations in data set

Population Mean μ =

x (^) i

N

Σ

N

i = 1 ... n = x^1 + x^2 + x^3 + ... + xN N

Range = Largest observation number

• Smallest Observation or number k = Number of desired intervals

N = Number of observations in the entire population Xi = the i th^ observation

n = Number of observations

Median f or odd numbers Median =^

(n + 1) { 2 }

Median of even numbers

Median =

(n + 2) { 2 }

Median =

n 2

STATISTICAL CONCEPTS AND MARKET RETURNS

Portfolio Rate of Return rp = wara + wbrb + wc rc + ... + wn rn

Weighted Mean xw = wi x (^) i

n

i = 1 ... n

Position of the Observation at a Given Percentile y

Range Range = Maximum value - Minimum value

Ly = (n + 1)

y

{^100 }

w = Weights X = Observations Sum of all weights = 1

w = Weights r = Returns

y = The percentage point at which we are dividing the distribution Ly = The location (L) of the percentile (Py) in the array sorted in ascending order

Mean Absolute Deviation MAD =

|x (^) i - x |

n

n

i = 1 ... n

Population Variance σ^2 =

(x (^) i - μ)^2

N

N

i = 1 ... n

Population Standard Deviation σ =

(x (^) i - μ)^2

N

N

i = 1 ... n

X = The sample mean n = Number of observations in the sample

μ = Population mean N = Size of the population

μ = Population mean N = Size of the population

Sample Variance s 2 =

(x (^) i - x )^2

n - 1

n

i = 1

X = Sample mean n = Number of observations in the sample

PROBABILITY CONCEPTS

Odds FOR E Odds FOR E =

P(E)

1 - P(E)

Conditional Probability P(A|B) = P(B)

P(A B)

U

Additive Law (The Addition Rule) P(A U B) =^ P(A) + P(B) - P(A^ B)^

U

The Multiplication Rule (Joint Probability)

P(A B) = P(A|B) x P(B)

U

The Total Probability Rule

P(A) = P(A|S 1 ) x P(S 1 ) + P(A|S 2 ) x x P(S 2 ) + ... + P(A|Sn ) x P(Sn )

Expected Value E(X) = P(A)XA + P(B)XB + ... + P(n)Xn

Covariance COVxy = (x - x)(y - y) n - 1

Variance of a Random Variable

Correlation ρ =

covxy σxσy

Portfolio Expected Return E(R P) = E(w 1 r 1 + w 2 r 2 + w 3 r 3 + … + wn rn )

Portfolio Variance

Var(R (^) P) =E[(R (^) p - E(Rp)^2 ] = [w 12 σ 12 + w 22 σ 22 +

• w 32 σ 32 + 2w 1 w 2 Cov(R 1 R 2 ) +
• 2w 2 w 3 Cov(R 2 R 3 ) + 2w 1 w 3 Cov(R 1 R 3 )]

Bayes’ Formula P(A|B) =

P(B|A) x P(A) P(B)

The Combination Formula nCr = (

n c)=

n! (n - r)! r!

The Permutation Formula nP r=

n! (n - r)!

σ^2 X = ∑(x - E(x))^2 x P(x) i = 1 ... n

n

E = Odds for event P(E) = Probability of event

where P(B) ≠ 0

S1, S2, … , Sn are mutually exclusive and exhaustive scenarios or events

P(n) = Probability of an variable Xn = Value of the variable x = Value of x X = Mean of x values y = Value of y y = Means of y n = Total number of values σx = Standard Deviation of x σy = Standard Deviation of y COV (^) xy = Covariance of x and y

w = Constant r = Random variable

Rp = Return on Portfolio

n = Total objects r = Selected objects

The sum is taken over all values of x for which p(x) > 0

COMMON PROBABILITY DISTRIBUTIONS

The Binomial Probability Formula

Safety-First Ratio SFRatio = [

E(Rp) - RL σp ]

FV = PV x ei x t

Continuously Compounded Rate of Return

P(x) = px^ x^ (1 - p)n - x

n! (n - x)! x!

Binomial Random Variable

E(X) = np Variance = np(1 - p)

For a Random Normal Variable X

90% confidence interval for X is x - 1.65s; x + 1.65s

Standard Error of the Sample Mean (Known Population Variance)

SE =

√n

Sampling Error of the Mean Sample Mean - Population Mean

Standard Error of the Sample Mean (Unknown Population Variance)

Z-score Z =

x - μ

Confidence Interval for Population Mean with z

Confidence Interval for Population Mean with t

SE =

S

√n

α ;x -Z 2

x σ √n

x + Zα 2

x σ √n

z or t-statistic?

Z known population, standard deviation σ, no matter the sample size t unknown p opulation, standard deviation s, and sample size b elow 3 0 Z unknown p opulation, standard deviation s, and sample size above 3 0

n = Number of trials x = Up moves px^ = Probability of up moves (1 - p)n - x^ = Probability of down moves

95% confidence interval for X is x - 1.96s; x + 1.96s 99% confidence interval for X is x - 2.58s; x + 2.58s

n = Number of trials p = Probability

s = Standard error 1.65 = Reliability factor x = Point estimate R (^) P = Portfolio Return RL = Threshold level σp = Standard Deviation i = Interest rate t = Time ln e = 1 e = Тhe exponential function, equal to 2.

n = Number of samples σ = Standard deviation

s = Standard deviation in unknown population’s sample x = Observed value σ = Standard deviation μ = Population mean Zα/2 = Reliability factor x = Mean of sample σ = Standard deviation n = Number of trials/size of the sample

x - tα ; 2

x s √n

x + tα 2

x s √n

tα/2 = Reliability factor n = Size of the sample s = Standard deviation

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