Formulas

Question 1

Name this formula and use:

V=D1/ (r−g)

Answer 1

Constant Growth Dividend Discount Model (DDM):

• Calculates the value of a dividend-paying security (with constant dividend growth) in dollar terms.

Question 2

What are the determinants of stock purchase decision using the Constant Growth Dividend Discount model formula

Answer 2

When IV (intrinsic value) < MV (market value) the stock is overvalued and Er (expected return) < k; (required return) the investor should avoid the stock

When IV > MV, the stock is undervalued and Er > k; the investor should buy the stock

When IV = MV, the stock is fairly valued and Er = k; the investor should buy the stock

Question 3

Name this formula and use

r =D1/P + g

Answer 3

Expected Rate of Return
The rate an investor should expect based on the price paid for a security.

Where:
D1= next year’s dividend
P = market price paid for a security
g = the dividend growth rate

Question 4

What are the determinants of stock purchase decision using the expected rate of return model formula

Answer 4

When Er>k, the stock is undervalued and IV > MV; the investor should buy the stock
When Er

Question 5

Name this formula and use

COVij=ρijσiσj

Answer 5

Covariance
Measures how one security behaves as a direct result of another.

Where:
ρij = correlation between securities i and j
σi = standard deviation of security i
σj = standard deviation of security j

Question 6

How is covariance applied to investment decision making

Answer 6

Since Covariance helps us understand the directional relationship between investments, a positive covariance would indicate that two investments would tend to move in the same direction. A negative covariance would indicate that two investments would tend to move inversely.

Covariance, however, doesn’t tend to allow the investor to understand the severity of that co-movement.

Question 7

Name this formula and use
‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾
σp=√W2i σ21+W2jσ2j+2WiWjCOVij

Answer 7

Standard Deviation of a Two Asset Portfolio

Provides the weighted standard deviation for a two-stock portfolio. It helps understand how much an outcome can vary from the expected outcome. With investing standard deviation is used to understand how much an investment’s return can vary from the expected return.

Where:
Wi = weight of stock ‘i’
Wj = weight of stock ‘j’
σi = standard deviation of stock ‘i’
σj = standard deviation of stock ‘j’
COVij = covariance between ‘i’ and ‘j’

Question 8

Name this formula and use

βi= COVim / σ2m = (Pim)(σi) / σm

Answer 8

Beta

Provides risk as a measure of volatility relative to that of the market. Therefore, an investment with a beta of around 1.0 will have a similar performance to the market in a given time period. An investment with a beta of less than 1.0 will be less volatile than the market. This means when the market is down, the investment will be down less. Conversely, if the market is up, the market is up less.

With a beta greater than 1.0, the investment will be more volatile than the market. This means when the market is up, the investment will be up more. When the market is down, the investment will be down more.

Where:
σi = standard deviation of the individual security
ρim= correlation between an individual security and the market
COVim= covariance between an individual security and the market
σm = standard deviation of the market

Question 9

Name this formula and use
‾‾‾‾‾‾‾‾
σr√ ∑t=1n(rt−r⎯⎯)2 / n

Answer 9

Standard Deviation of a Population

Identifies the deviation of a single security over a series of periods of return.

Where:
σr = standard deviation of results from the expected return
Σ = summation of all terms
n = number of periods being considered
rt = actual return
r⎯⎯ = average return

Question 10

Name this formula and use
‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾
Sr √ ∑t=1n(rt−r-)2 / (n−1)

Answer 10

Standard Deviation of a Sample

Identifies the deviation of a single security over a series of periods of return.

Where:
Sr = standard deviation of results from the expected return
Σ = summation of all terms
n = number of periods being considered
rt = actual return
r⎯⎯ = average return

Calculate using: ∑+ = each period return; g, x (zero key) = mean); g, s (decimal key) = standard deviation

Question 11

How do you apply standard deviation of sample results

Answer 11

Investors can use average (or expected) return and standard deviation to give them an idea of how variable a stock’s performance can be. The greater the standard deviation, the less predictable a stock’s performance will be in a given time period.

Question 12

Name this formula and use

ri = rf+(rm−rf)βi

Answer 12

Capital Asset Pricing Model (CAPM)

Used to determine a theoretically appropriate required rate of return of an asset. CAPM helps an investor quantify what return they should expect from an investment, or how much return should the investor require from the investment given an expected market return and the risk of the investment relative to that market (Beta).

ri = the investor’s required rate of return
rf = risk-free rate (T-Bill rate serves this end)
rm = return of the market (S&P 500 or some broad index)
βi = beta of the security being measured for the required return

Market risk premium is the same as (rm−rf);

Question 13

Name this formula and use

αp=rp−[rf+(rm−rf)βp]

Answer 13

Jensen’s Performance Index (Alpha)

Measures the performance of a portfolio manager relative to the performance of the market. The purpose of alpha is to quantify the risk-adjusted performance of an investment manager relative to the risk they exposed the portfolio to in terms of beta
One of the functions of the capital asset pricing model (CAPM) is to draw the security market line (SML). The SML plots out all of the expected returns of portfolios relative to their Betas (a measure of market risk) given a market return and a risk-free rate of return.

αp = difference of return from the amount required by investors
rp = return of the portfolio
rf = risk-free rate of return
rm = return of the market
βp = Beta of the portfolio being measured
(calculation following rp- = CAPM)

Question 14

How do you interpret alpha results relative to stock portfolio performance

Answer 14

If alpha is positive, the portfolio manager provided a greater amount of return than what was expected given the amount of risk the portfolio was exposed to in terms of beta.

If alpha is negative, the portfolio manager provided less return than what was expected given the amount of risk the portfolio was exposed to in terms of beta.

If alpha is zero, the portfolio manager provided the exact amount of return that was expected given the amount of risk the portfolio was exposed to in terms of beta. This would plot right on that security market line (SML).

Question 15

Name this formula and use

Tp=(rp−rf) / βp

Answer 15

Treynor Ratio

Measures the risk-adjusted performance of a portfolio manager, when R2 is substantially high enough (R2 > .7). Otherwise if the R2 is not substantially high enough (R2 < .7), use the Sharpe Ratio.

Treynor Ratio is useful as a comparative value, meaning that you need to compare the Treynor Ratios of two or more investments to get use out of it.

When interpreting the results of a Treynor Ratio comparison, you always want to choose the investment with the higher Treynor Ratio as it will provide a greater risk-adjusted rate of return.

Tp = Treynor Index equals:
rp = return of the portfolio
rf = risk free rate of return
βp = beta of the portfolio being measured

Question 16

Name this formula and use

D = [(1+y) / y] − { (1+y)+t(c−y) / c [(1+y)t−1]+y }

Answer 16

Duration

Identifies the length of time the discounted cash flow of a bond remains outstanding.

Where:
c = rate of interest paid on the coupon
t = number of periods to maturity
y = Yield to Maturity (as a %)

Question 17

Name this formula and use

ΔP / P= −D[Δy/(1+y)]

Answer 17

Change of Bond Price

States the change of price that will occur in a bond as interest rates change. Given the inverse relationship that bond prices and yields have, when yields go up, prices go down; when yields go down, prices go up. The change in bond price formula estimates the damage, or benefit, to the fixed income portfolio from changes in rates.

Where:
ΔP = the dollar change in price
P = price of a bond
ΔPP= % price change of bond
(-D) = the duration in terms of years used as a negative value
∆y = the % change in interest rates. If they go down this number should be negative
1+y = 1 + yield to maturity

Question 18

Name this formula and use

IR = (Rp−RB) / σA

Answer 18

Information Ratio

Measures return above benchmark divided by standard deviation, returning a risk-adjusted relative (or comparative) value. When comparing multiple managers based on tracking error, the manager with the highest information ratio provided the highest risk-adjusted rate of return.

Where:
RP = return of a portfolio
RB = return of a benchmark
σA = tracking error of active return / standard deviation

Question 19

Name this formula and use

EAR= (1+i/n)n −1

Answer 19

Effective Annual Rate

Accounts for intra-year compounding. The effective annual rate is higher than the stated nominal rate due to the more frequent compounding. The more frequent compounding provided a higher overall return or greater borrowing cost

Where:
i = interest rate
n = number of periods

Question 20

Name this formula and use

TEY= r/ (1−t)

Answer 20

Taxable Equivalent Yield

Provides the return that is required on a taxable investment to make it equal to the return on a tax-exempt investment. TEY helps an investor choose between a taxable corporate bond and a tax-free municipal bond

Where:
r = nominal rate of return
t = investor’s marginal tax rate (as a decimal)

Question 21

Name this formula and use

SP = ( rp−rf ) / σp

Answer 21

Sharpe Ratio

Measures the risk-adjusted performance of a portfolio in terms of standard deviation, and s useful as a comparative value, meaning that you need to compare the Sharpe Ratios of two or more investments to get practical use out of it.

If the R2 is substantially high enough (R2 > .7), use the Treynor Ratio. If the R2 is not substantially high enough (R2 < .7), use the Sharpe Ratio.

Where:
Sp = sharpe Index =
rp = return of the portfolio
rf= risk free rate of return
σp = standard deviation of the portfolio being measured

Question 22

Name this formula and use

1RN = [(1+R11)(1+E(r12))…(1+E(r1N))]1/N −1

Answer 22

Unbiased Expectations Theory

Implies long-term investors will choose to purchase debt instruments on whether forward interest rates are more or less favorable than current short-term interest rates.

Where:
N = Term to maturity
1R1 = Actual current one-year rate today
E(2r1) = Expected one-year rate for period 2
E(Nr1) = expected one-year rate for years, i = 1 to N

Please note: It is unlikely that you will need to use the Unbiased Expectations Theory formula on the Exam.

Question 23

Name this formula and use

HPR = [(1+r1)×(1+r2)×…(1+rn)] − 1

Answer 23

Holding Period Return

Provides the total return received from holding an asset or portfolio of assets over a period of time.

Where:
r = rate of return for given period
n = number of periods

Question 24

Name these formulas and their use

(1) AM=(a1+a2+a3+…+an) / n

(2) ‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾
n√(1+r1)×(1+r2)×…(1+rn) −1

Answer 24

(1) Arithmetic Mean

Offers the average of a set of numerical values, calculated by adding them together and dividing by the number of terms in the set.

Where:
a = rate of return for given period
n = number of periods

(2) Geometric Mean
What does it do? States the central number in a geometric progression, also calculable as the nth root of a product of n numbers.

Where:
n = number of periods
r = rate of return for given period

Question 25

What is difference between Alpha and Beta and how each is used

Answer 25

Alpha - Jensen’s Performance Index (Alpha)

Measures the performance of a portfolio manager relative to the performance of the market. The purpose of alpha is to quantify the risk-adjusted performance of an investment manager relative to the risk they exposed the portfolio to in terms of beta One of the functions of the capital asset pricing model (CAPM) is to draw the security market line (SML). The SML plots out all of the expected returns of portfolios relative to their Betas (a measure of market risk) given a market return and a risk-free rate of return.

Beta

Provides risk as a measure of volatility relative to that of the market. Therefore, an investment with a beta of around 1.0 will have a similar performance to the market in a given time period. An investment with a beta of less than 1.0 will be less volatile than the market. This means when the market is down, the investment will be down less. Conversely, if the market is up, the market is up less.

With a beta greater than 1.0, the investment will be more volatile than the market. This means when the market is up, the investment will be up more. When the market is down, the investment will be down more.

Question 26

How do you calculate geometric mean

Answer 26

Multiply 1+return (1-return) for each period return, then multiply by number of periods, [1/x], then [yX], then subtract 1-