Formulas Quantative Flashcards
The Future Value of a Single Cash Flow
FVN = PV (1+ r) ^ N
The cash flow can be discounted back to a present value by using a discount rate that accounts for the factors mentioned above. Conversely, cash flows in the present can be compounded to arrive at an expected future cash flow.
PV- the present value (or initial principal) FVn- future value at the end of n periods i- the interest rate paid each period n- the number of periods
The Future Value of a Single Cash Flow
The cash flow can be discounted back to a present value by using a discount rate that accounts for the factors mentioned above. Conversely, cash flows in the present can be compounded to arrive at an expected future cash flow.
PV- the present value (or initial principal) FVn- future value at the end of n periods i- the interest rate paid each period n- the number of periods
The Present Value of a Single Cash Flow
PV = FV/ (1+ r) ^ N
PVAnnuity Due = PVOrdinaryAnnuity x (1 + r)
FVAnnuity Due = FVOrdinaryAnnuity x (1 + r)
The Present Value of a Single Cash Flow
We will first look at discounting a single cash flow or amount. The cash flow can be discounted back to a present value by using a discount rate that accounts for the factors mentioned above (present consumption preference, risk, and inflation). Conversely, cash flows in the present can be compounded to arrive at an expected future cash flow.
The present value of a single cash flow can be written as follows:
Present Value of a Perpetuity
PV(perpetuity) = PMT / (I/Y)
Present Value of a Perpetuity
Sometimes annuities last forever – or so we pretend. What is the value of payments that are received indefinitely, like proverbial AT&T dividends? On first reflection, you might think that a perpetual annuity would be infinite. But remember that $1 paid in 10 years is worth only pennies today (assuming normal inflation and the inherent time value of money) and $1 paid in 100 years is not worth picking up from the sidewalk.
PV
the present value (or initial principal)
Pymtn- the payment made at the end of each of an infinite number of periods
i- the discount rate for each period (assumed equal throughout)
The present value of a perpetual stream of future payments eventually reaches a limit. And, it turns out that the formula for an infinite series of equal payments, discounted by a constant discount rate, is simplicity itself:
Continuous Compounding and Future Values
FVn = PVe ^ rs * N
Continuous Compounding and Future Values
The future value (FV) of an annuity with continuous compounding formula is used to calculate the ending balance on a series of periodic payments that are compounded continuously. Understanding the future value of annuity with continuous compounding formula requires the understanding of two specific financial and mathematical concepts, which are future value of an annuity and continuous compounding.
Effective Annual Rates
EAR = (1 + Periodic interest rate) ^ N- 1
Effective Annual Rates
An investment’s annual rate of interest when compounding occurs more often than once a year. i= stated intered rate- n= number of coumpounding periods.
Net Present Value
NPV = CFt / (1 + r) ^ t
where: CFt = the expected net cash flow at time t N = the investment’s projected life
r = the discount rate or appropriate cost of capital
Net Present Value
The difference between the present value of cash inflows and the present value of cash outflows. NPV is used in capital budgeting to analyze the profitability of an investment or project.
NPV analysis is sensitive to the reliability of future cash inflows that an investment or project will yield.
Bank Discount Yield
rBD = (D/F) x (360/t)
where: rBD = the annualized yield on a bank discount basis. D = the dollar discount (face value – purchase price) F = the face value of the bill t = number of days remaining until maturity
Bank Discount Yield
Discount yield is a measure of a bond’s percentage return. Discount yield is most frequently used to calculate the yield on short-term bonds and treasury bills sold at a discount. This yield calculation uses a 30-day month and 360-day year to simplify calculations. Discount yield is calculated by the following formula: Discount Yield = [(par value - purchase price)/par value] * [360/days to maturity]
Bank Discount Yield
Discount yield is a measure of a bond’s percentage return. Discount yield is most frequently used to calculate the yield on short-term bonds and treasury bills sold at a discount. This yield calculation uses a 30-day month and 360-day year to simplify calculations. Discount yield is calculated by the following formula: Discount Yield = [(par value - purchase price)/par value] * [360/days to maturity]
Holding Period Yield
The total return received from holding an asset or portfolio of assets. Holding period return/yield is calculated as the sum of all income and capital growth divided by the value at the beginning of the period being measured. Holding period return is a very basic way to measure how much return you have obtained on a particular investment. This calculation is on a per-dollar-invested basis, rather than a time basis, which makes it difficult to compare returns on different investments with different time frames. When making comparisons such as this, the annualized calculation shown above should be used.
Holding Period Yield
The total return received from holding an asset or portfolio of assets. Holding period return/yield is calculated as the sum of all income and capital growth divided by the value at the beginning of the period being measured. Holding period return is a very basic way to measure how much return you have obtained on a particular investment. This calculation is on a per-dollar-invested basis, rather than a time basis, which makes it difficult to compare returns on different investments with different time frames. When making comparisons such as this, the annualized calculation shown above should be used.
Effective Annual Yield
The yield after taking into account the consequences of compounding. It is computed as [1 + (stated interest/n)]n - 1; where n is the number of payments within the year. For instance, a bond’s return is 5% and is to be paid semi-annually, thus, the effective annual yield is calculated as: [1 + (.05/2)2 - 1 = 5.062%.
Money Market Yield
RMM = (360 x rBD) / 360 - (t x rBD) RMM = HPY X (360/t)
Money Market Yield
The interest rate earned by investing in securities with high liquidity and maturities of less than one year such as negotiable certificates of deposit, U.S. Treasury bills and municipal notes. Money market yield is calculated by taking the holding period yield and multiplying it by a 360-day bank year divided by days to maturity. It can also be calculated using bank discount yield. Also known as “CD-equivalent yield”. To earn a money market yield, it is necessary to have a money market account. Banks offer money market accounts because they need to borrow funds on a short-term basis to meet reserve requires and to participate in interbank lending. The money market yield will be lower than the yield on stocks and bonds because of the low risk associated with money market investments.
Money Market Yield
The interest rate earned by investing in securities with high liquidity and maturities of less than one year such as negotiable certificates of deposit, U.S. Treasury bills and municipal notes. Money market yield is calculated by taking the holding period yield and multiplying it by a 360-day bank year divided by days to maturity. It can also be calculated using bank discount yield. Also known as “CD-equivalent yield”. To earn a money market yield, it is necessary to have a money market account. Banks offer money market accounts because they need to borrow funds on a short-term basis to meet reserve requires and to participate in interbank lending. The money market yield will be lower than the yield on stocks and bonds because of the low risk associated with money market investments.
Bond Equalent Yield
A calculation for restating semi-annual, quarterly, or monthly discount-bond or note yields into an annual yield. For a fixed income security with a par value of $1000, the calculation is as follows: The BEY allows fixed-income securities whose payments are not annual to be compared with securities with annual yields. The BEY is the yield that is quoted in newspapers. Alternatively, if the semi-annual or quarterly yield to maturity of a bond is known, the APR calculation may be used.
Population Mean
u= x/ N
Population Mean
u= x/ N
Sample Mean
X (bar) = x / n
Percentiles
Ly= (n+1) y / 100
where y= percentage point at which we are dividing the distribution; Ly= location (L) of the pecentile (Py) in the data set sorted in accending order
Percentiles
Ly= (n+1) y / 100
where y= percentage point at which we are dividing the distribution; Ly= location (L) of the pecentile (Py) in the data set sorted in accending order
Population Variance
o^2= (X1-u) ^ 2 / N
where: Xi = observation i u = population mean
N = size of the population
Sample Variance
Sample Variance= s^2= (X1- X(bar) ^2 / (n-1)
where: n = sample size.
Coefficient of Variation
Coefficient of variation= s/ X (bar)
where: s = sample standard deviation
X (bar)= the sample mean
Sharpe Ratio
Sharpe Ratio= r(bar) p - rf / sp
where: r (bar) p= mean portfolio return
rf= risk-free return sp= standard deviation of portfolio returns
Sharpe Ratio
Sharpe Ratio= r(bar) p - rf / sp
where: r (bar) p= mean portfolio return
rf= risk-free return sp= standard deviation of portfolio returns
Sample skewness, also known as sample relative skewness, is calculated as:
SK = [ n / (n-1) (n-2) ] (X1- X(bar)^ 3) / s^ 3
Odds for an event
P (E) = b / (a+b)
Where the odds against are given as ‘a to b’, then:
Conditional Probabilities
P (AB) = P (A|B) x P (B)
Multiplication Rule for Probabilities
P (AB)= P (A | B ) x P (B)
For Independant Events
P(A|B) = P(A), or equivalently, P(B|A) = P(B)
P(A or B) = P(A) + P(B) - P(AB)
P(A and B) = P(A) P(B)
The Total Probability Rule
P(A) = P(AS) + P(AS^ c) P(A) = P(A|S) x P(S) + P(A|S ^ c)) X P(S ^ c)
The Total Probability Rule
P(A) = P(AS) + P(AS^ c) P(A) = P(A|S) x P(S) + P(A|S ^ c)) X P(S ^ c)
Variance and Standard Deviation
o^2 (X) = E{ [X - E(X)]^ 2 }
The Total Probability Rule for Expected Value
- E(X) = E(X|S)P(S) + E(X|S ^ c ) P(S ^ c )
- E(X) = E(X|S1) x P(S1) + E(X|S2) x P(S2
) + …+ E(X|Sn) x P(Sn)
Where:
E(X) = the unconditional expected value of X E(X|S1) = the expected value of X given Scenario 1 P(S1) = the probability of Scenario 1 occurring The set of events {S1
, S2 ,…, Sn} is mutually exclusive and exhaustive.
Covariance
Cov (XY) = E{[X - E(X)][Y - E(Y)]}
Cov (Ra,Rb) = E{[Ra - E(Ra)][Rb - E(Rb)]}
The demand function captures the effect of all these factors on demand for a good
Demand function: QDx = f(Px, I, Py, . . .)
Equation 1 is read as “the quantity demanded of Good X (QDX ) depends on the price of Good X (PX), consumers’ incomes (I) and the price of Good Y (PY), etc.”
The supply function can be expressed as:
Supply function: QSx = f(Px, W, . . .)
The own-price elasticity of demand is calculated as:
ED Px= % change QDx / % change Px
Arc elasticity is calculated as:
EP =% change in quantity demanded/ % change in price
Income Elasticity of Demand
Income elasticity of demand measures the responsiveness of demand for a particular good to a change in income, holding all other things constant.
Income Elasticity of Demand
ED I= % change QDx/ % change I
Income Elasticity of Demand
ED I= % change QDx/ % change I
Cross-Price Elasticity of Demand
EC =% change in quantity demanded / % change inpriceofsubstituteor complement
The Utility Function
U = f(Qx1, Qx2,…, Qxn)
Accounting Profit
Accounting profit (loss) = Total revenue – Total accounting costs
Economic profit (also known as abnormal profit or supernormal profit)
Economic profit = Total revenue – Total economic costs
Economic profit = Total revenue – (Explicit costs + Implicit costs)
Economic profit = Accounting profit – Total implicit opportunity costs
Economic profit (also known as abnormal profit or supernormal profit)
Economic profit = Total revenue – Total economic costs
Economic profit = Total revenue – (Explicit costs + Implicit costs)
Economic profit = Accounting profit – Total implicit opportunity costs
Normal Profit
Normal profit = Accounting profit - Economic profit
Total Revenue
Total revenue (TR) Price times quantity (P x Q), or the sum of individual units sold times their respective prices; Pi x Qi)
Marginal revenue (MR)
Change in total revenue divided by change in quantity; ( changeTR / change Q)
Marginal revenue (MR)
Change in total revenue divided by change in quantity; ( changeTR / change Q)
Total fixed cost (TFC)
Sum of all fixed expenses; here defined to include all opportunity costs
Total costs (TC)
Total fixed cost plus total variable cost; (TFC + TVC)
Total costs (TC)
Total fixed cost plus total variable cost; (TFC + TVC)
Average fixed cost (AFC )
Total fixed cost divided by quantity; (TFC / Q)
Average variable cost (AVC)
Total variable cost divided by quantity; (TVC / Q)
Marginal cost (MC)
Change in total cost divided by change in quantity;( changeTC / change Q)
Marginal revenue product (MRP) of labor is calculated as:
MRP of labor = Change in total revenue / Change in quantity of labor
For a firm in perfect competition, MRP of labor equals the MP of the last unit of labor times the price of the output unit
MRP
MRP = Marginal product * Product price
A profit-maximizing firm will hire more labor until: MRPLabor = PriceLabor
Profits are maximized when:
MRP1/Price of input 1 = MRPn /Price of input n
The relationship between MR and price elasticity can be expressed as:
MR = P[1 – (1/EP)]
The relationship between MR and price elasticity can be expressed as:
MR = P[1 – (1/EP)]
In a monopoly, MC = MR so:
P[1 – (1/EP)] = MC
N-firm concentration ratio:
Simply computes the aggregate market share of the N largest firms in the industry. The ratio will equal 0 for perfect competition and 100 for a monopoly
Nominal GDP
refers to the value of goods and services included in GDP measured at current
prices.
Nominal GDP = Quantity produced in Year t x Prices in Year t
Nominal GDP
refers to the value of goods and services included in GDP measured at current
prices.
Nominal GDP = Quantity produced in Year t x Prices in Year t
Real GDP
refers to the value of goods and services included in GDP measured at base-year prices
Real GDP = Quantity produced in Year t x Base-year prices
GDP Deflator
GDP deflator = [ Real GDP/ Nominal GDP ] x 100
The Components of GDP
Based on the expenditure approach, GDP may be calculated as:
GDP = C + I + G + (X - M)
C = Consumer spending on final goods and services I = Gross private domestic investment, which includes business investment in capital goods (e.g. plant and equipment) and changes in inventory (inventory investment) G = Government spending on final goods and services X = Exports M = Imports
Expenditure Approach
Under the expenditure approach, GDP at market prices may be calculated as:
GDP = Consumer spending on goods and services+ Business gross fixed investment
+ Change in inventories
+ Government spending on goods and services + Government gross fixed nvestment
+ Exports – Imports + Statistical discrepancy
