Formulas Flashcards
Dividend Discount Model
V = intrinsic value of stock
D1 = expected dividend per share one year from now
r = required rate of return (derived from CAPM)
g = growth rate in perpetuity expected for the dividends
Gordon Constant Growth Model is a stock valuation model that assumes there are constant growth in dividends. May be referred to as the Dividend Discount Model (DDM). It is used to determine the intrinsic value of a stock based on a future series of dividend payments that grow at a constant rate over time.
Covariance
Ρij = the correlation coefficient between the assets
σ = the standard deviation of each asset
COVij = the covariance between two assets
This is the formula for the covariance between two assets: “i” and “j”. It is equal to the correlation coefficient between the two assets (represented by the Greek letter “rho” or “ρ”), multiplied by their respective standard deviations (σ). Covariance measures how two assets move together or apart: if they move in the same direction they have a positive covariance; if they move in opposite directions they have a negative covariance. Covariance is a critical input in calculating the standard deviation of a portfolio.
Beta
Βi = beta of the individual asset
σ = the standard deviation of both the asset (i) and the market (m)
ρim = the correlation coefficient between the individual asset and the market
σ 2 = the variance of returns for the market (m)
Alpha
rp = rate of portfolio return actually received
rf = risk-free rate of return
rm = return of the market
ßp = beta of the portfolio
The Jensen Index (or most commonly known as alpha “α”) compares a portfolio’s expected return with its actual return. The difference between the two is called alpha (α) and it indicates the additional return earned by an investment or portfolio after adjusting for risk. Positive alpha means the manager outperformed the market on a risk-adjusted basis. A negative alpha does NOT mean negative performance, but rather indicates that the manager’s performance was less than expected given the risk they took to achieve that performance. Jensen’s alpha is an absolute measure of risk-adjusted performance.
Duration
D = duration for the bond in years
y = yield-to-maturity per period expressed as a decimal
c = coupon rate per period expressed as a decimal
t = number of periods until maturity
The bond duration calculation uses just three inputs. If the compounding period is annual, then all numbers reflect annual rates. If the compounding period is semiannual, then the number of periods is twice the number of years, and the coupon rate and YTM are one- half of the annual rates.
Information Ratio
IR = Information Ratio
RP = the return of the portfolio
RB = the return of the index or benchmark
σA = the standard deviation of the difference between returns of the portfolio and the returns of the index (in other words, the tracking error)
The Information Ratio (IR) is a ratio of portfolio returns that are above the returns of a benchmark—usually an index—to the volatility of those returns. The IR is ultimately used to determine a portfolio manager’s ability to generate excess returns relative to a benchmark but also attempts to identify the consistency of the investor.
The Taxable Equivalent Yield (TEY) is the pretax yield that a taxable bond needs to have for its yield to be equal to that of a tax-free municipal bond. This calculation can be used to fairly compare the yield of a tax-free bond to that of a taxable bond in order to determine which bond has a higher applicable yield. Note that the CFP Board’s formula (TEY = r/(1-t) is a truncated version of the one shown here.
Sharpe Index
rp = expected portfolio return
rf = risk-free rate of return
σp= portfolio standard deviation
The Sharpe Index (or Ratio) relates the return on an investment portfolio to the degree of total risk (σ) taken. The higher the index, the greater the return for each unit of risk. This is a relative measurement of risk, as in it must be used in comparing one fund’s risk-adjusted- return to another.
Holding Period Return
HPR = holding period return
rn = % return per period
The holding period return is the total return that is received after holding an asset or portfolio of assets over a period of time and is generally expressed as a percentage. It is particularly useful for comparing returns between investments that are held for different periods of time.
Dividend Discount Model II
r = expected rate of return
D1 = expected dividend per share one year from now
P = the actual price of the stock
g = growth rate in perpetuity expected for the dividends
This is a version of the Dividend Discount Model (DDM) that solves for the expected rate of return on a particular stock. Note that “V” in the Gordon Constant Growth Model represents the intrinsic value of the stock, whereas “P” represents the current market price or value of the stock.
Standard Deviation of a Two-Asset Portfolio
It is a measure of variability of the expected returns from a portfolio.
W = asset weight within the portfolio σ = standard deviation
COV = covariance between two assets
rf = risk-free rate of return
rm = return of the market
ßi = beta of the security (or fund)
The Capital Asset Pricing Model (CAPM) equation determines the rate of return an investor should require on an investment. The required return is a function of three factors: the risk-free rate (rf), the expected premium for risk taking (rm – rf), and the level of systematic risk versus the benchmark (ß). The bigger the risk, the bigger the market risk premium.
rp = expected portfolio return rf = risk-free rate of return
βp = beta of the portfolio
The Treynor Index (or Ratio) relates the return of an investment or portfolio to the degree of systematic risk (β) taken. The higher the index, the greater the return for each unit of risk. This is a relative measurement of risk, so it is helpful in comparing diversified funds to one another. Since relative risk is measured with beta (β) in this formula, the index should be used to evaluate the performance of diversified portfolios only—those with a coefficient of determination (R2) of 70 or higher. It can also be used to calculate the risk- adjusted return for the broader market.
% Change in Bond Price due to Duration
ΔP = change in price
P = current price
-D = duration of the bond (expressed as negative) Δy = expected change in interest rates
y = current yield-to-maturity (current interest rate)
Formula for the approximate percentage price change in a bond for a given change in yield using duration. Duration is always negative to express the inverse relationship between bond prices and the change in interest rates.
EAR = effective annual rate
i = stated annual interest rate
n = number of compounding periods
The effective annual rate (EAR) is the interest rate that is actually earned or paid on an investment, loan, or other financial product due to the result of compounding over a given time period. Additionally, it is also referred to as the effective interest rate, the effective rate, or the annual equivalent rate. The EAR is an important financial concept because it is used to compare different products that calculate compounded interest differently.
1Rn = Actual N-period rate today
N = term to maturity
1R1 = actual one year current rate today
E(ir1) = expected one-year rates for now
The Unbiased Expectations Theory (UET) formula states that long-term interest rates hold a forecast for short-term interest rates in the future. The formula assumes that an investor earns the same amount of interest by investing in a one-year bond in the present and rolling the investment into a different one-year bond after one year as compared to purchasing a two-year bond in the present.
Geometric Mean
rn= % return per period
n = number of compounding periods, which is then used as the Nth root when making the calculation.
The geometric return is the compound return over the period being measured. The periodic returns are first multiplied by one another and the Nth root is then applied to the resulting figure. The Nth root is the reciprocal of the number of periods being measured.
