Ethical and Professional Standards and Quantitative Methods (Book 1)
A 175-day T-bill has an effective annual yield of 3.80%. Its money-market yield is closest to:
(1 + .0380) ^ (175/365) -1 = 1.8% = HPY rMM = HPY * 360/t = 1.8% * (360/175)
Future value factor/future value interest factor for single cash flow at I/Y over N compounding periods:
(1+I/Y)^N
Note: There is an alternative definition of the bond equivalent yield in the Corporate Finance topic review on Working Capital Management:
(HPR X 365/days to maturity)
General relationship that holds
(standard deviation > Mean Absolute Deviation)
The arithmetic mean is the most widely used measure of central tendency and has the following properties:
-All interval and ratio data sets have an arithmetic mean -All data values are considered and included in the arithmetic mean computation -A data set has only one arithmetic mean -The sum of the deviations of each observation in the data set from the mean is always zero; it is the only measure of central tendency for which the sum of the deviations from the mean is zero -The arithmetic mean of a sample from a population is the best estimate of both the true mean of the sample and the value of the next observation
It is important to realize that a yield quoted on a bank discount basis is not representative of the return earned by an investor for the following reasons:
-Bank discount yield annualizes using simple interest and ignores the efforts of compound interest -Bank discount yield is based on the face value of the bond, not its purchase price - investment returns should be evaluated relative to the amount invested -Bank discount yield is annualized based on a 360-day year rather than a 365-day year
Once we have established HPY, EAY, or rMM, we can use one as a basis for calculating the other two. Remember:
-The HPY is the actual return an investor will receive if the money market instrument is held until maturity -The EAY is the annualized HPY on the basis of a 365-day year and incorporates the effects of compounding -The rMM is the annualized yield that is based on price and a 360-day year and does not account for the effects of compounding - it assumes simple interest
How to construct a frequency distribution
1) Define the intervals; the range of values for each interval must have a lower and upper limit and be all-inclusive and nonoverlapping; must be mutually exclusive is a way that each observation can be placed in only one interval, and the total set of intervals should cover the total range of values for the entire population 2) Tally the observations; after the intervals have been defined, the observations must be tallied, or assigned to their appropriate interval 3) Count the observations; having tallied the data set, the number of observations that assigned to each interval are counted; The absolute frequency, or simply the frequency, is the actual number of observations that fall within the interval
Loan Amortization
1) Interest component = beginning balance * periodic interest rate 2) principal component = payment - interest 3) The ending balance in a given period, t, is the period's beginning balance minus the principal component of the payment Note* Once you have solved for the payment, the remaining principal on any payment can be calculated by entering N = number of remaining payments and solving for the PV
What is the IRR from the previous problem?
1,800,000 / IRR = $15,000,000 -->IRR must be greater than 10.5% since the NPV is positive =12%
NPV decision rules summarized
1.) Accept projects with a positive NPV. Positive NPV projects will increase shareholder wealth. 2.) Reject projects with a negative NPV. Negative NPV projects will decrease shareholder wealth. 3.) When two projects are mutually exclusive (only one can be accepted), the project with the higher positive NPV should be accepted.
The annual time-weighted return for an investment may be computed by performing the following steps:
1.) Value the portfolio immediately preceding significant additions or withdrawals 2.) Compute the Holding Period Return (HPR) of the portfolio for each sub period. 3.) Compute the product of (1 + HPR) for each subperiod to obtain a total return for the entire measurement period [i.e., (1 + HPR1) * (1 + HPR2)...] -If the total investment period is greater than one year, you must take the geometric mean of the measurement period return to find the annual time-weighted rate of return -annual compound rate
A firm is considering the purchase of a new material handling system for a cost of $15MM. This system is expected to generate a positive cash flow of $1.8MM per year in perpetuity. What is the NPV of the proposed investment if the appropriate hurdle rate is 10.5%?
= 1,800,000 / 0.105 = $17,142,857 -$15,000,000 = $2,142,857
Required interest rate on a security
= nominal risk-free rate + default risk premium + liquidity premium + maturity risk premium
Allison Rodgers, CFA, makes the following statement "The problems with bank discount yields quoted for T-Bills is that they aren't yields, they ignore compounding, and they are based on a short year"
A) Is she correct in all regards? YES B) Which of these problems is remedied by using the holding period yield? -Makes it a true yield C) Which of these problems is remedied by using MM yield? -Also a true yield (based on % of investment) - 360 days D) Which of the following problems is/are remedied by using effective annual yield? -Solves all 3 -True yield -Factors in annual compounding -Based on a full year (365 days)
Relative frequency
Another useful way to present data; The relative frequency is calculated by dividing the absolute frequency of each return interval by the total number of observations -Simply stated, the relative frequency is the percentage of total observations falling within each interval -It is also possible to compute the cumulative absolute frequency and cumulative relative frequency by summing the absolute or relative frequencies starting at the lowest interval and progressing through the highest
Money-weighted return
Applies the concept of IRR to investment portfolios. -The money-weighted rate of return is defined as the internal rate of return on a portfolio, taking into account all cash inflows and outflows -The beginning value of the account is an inflow, as are all deposits into the account. All withdrawals from the account are outflows, as is the ending value
What is the effective annual rate for a credit card that charges 18% compounded monthly?
EAR = [(1 + (.18/12)] ^12 - 1 = 19.56%
For values that are not equal,
Harmonic mean < Geometric mean < arithmetic mean -This mathematical fact is the basis for the claimed benefit of purchasing the same dollar amount of mutual fund shares each month or each week (dollar cost averaging)
Time Value of Money
It is often good to draw a time line before you start to solve a TVM problem - diagram of the cash flows associated with a TVM problem
Maturity risk
Longer maturity bonds have more maturity risk than shorter-term bonds and require a maturity risk premium
Types of statistical measurement scales in order of precision:
Nominal, Ordinal, Interval, Ratio (NOIR)
Terry preferred stocks are expected to pay a $9 annual dividend forever. If the required rate of return on equivalent investments is 11%, a share of Terry preferred should be worth:
PMT/I% = $9/.11 = $81.82
If you deposit $1,000 in the bank today and at the beginning of each of the next three years, how much will you have in six years from today at 6% interest?
Step 1: Compute the FV of the annuity due at the end of year 4 (FV4) -Set the calculator to BEG mode N = 4; I/Y = 6; PMT = -1000; CPT --> FV = $4,637.09 Step 2: Find the future value of FV4 two years from year 4 N = 2; I/Y = 6; PV = -4,637.09; CPT --> FV = 5,210.23
Assume a 35 year-old investor want to retire in 25 years at the age of 60. She expects to earn 12.5% on her investments prior to her retirement and 10% thereafter. How much must she deposit at the end of each year for the next 25 years in order to be able to withdraw $25,000 per year at the beginning of each year for 30 years?
Step 1: Compute the amount required to meet the desired withdrawals. -BEG Mode N = 30; I/Y = 10; PMT = -$25,000; CPT --> PV = $259,240.14 -Make certain you reset your calculator back to END mode Step 2: The annuity payment that must be made to accumulate the required amount over 25 years can be determined by entering the relevant data and computing PMT
What is the PV of four $100 end-of-year payments if the first payment is to be received three years from today and the appropriate rate of return is 9%?
Step 1: Find the PV of the annuity as of the end of year 2 (PV2) N = 4; I/Y = 9; PMT = 100; FV = 0; CPT --> PV = PV2 = $323.97 Step 2: Find the PV of PV2 N= 2; I/Y = 9; PMT = 0; FV = -323.97; CPT --> PV = PV0 = $272.68 -Although the annuity begins at t = 3, we discounted the result for only two periods to get the present (t = 0 value)
A 3-month loan has a holding period yield of 2%. What is the yield on a bond-equivalent basis?
Step 1: convert the 3-month yield to get an effective semiannual (6 month) yield: 1.02^2 - 1 = 4.04% Step 2: 4.04% * 2 = 8.08%
The effective annual yield on an investment is 8%. What is the yield on a bond equivalent basis?
Step 1: convert the effective annual yield to an effective semiannual yield: 1.08^.05 -1 = 3.923% Step 2: 3.923% * 2 = 7.846%
Bank discount yield
T-bills are quoted on a bank discount basis, which is based on the face value of the instrument instead of the purchase price -Key distinction: it expresses the dollar discount from the face (par) value as a fraction of the face value, not the market price of the instrument. Also, it is annualized by multiplying the discount to par by 360/t, where the market convention is to use a 360 day year vs. a 365 day year -This method assumes no compounding
Professor's Note
The geometric mean is always less than or equal to the arithmetic mean, and the difference increases as the dispersion of the observations increases -The only time the arithmetic and geometric mean are equal is when all observations in the sample are equal
What is the third quartile for the following distribution of returns? 8%, 10%, 12%, 15%, 17%, 17%, 18%, 19%, 23%
The third quartile is the point below which 75% of the observations lie. Recognizing that there are 10 observations in the data set, the third quartile can be identified as: Ly = (10 + 1) * 75/100 = 8.25 When the data is arranged in ascending order, the third quartile is one-fourth (0.25) of the way from the eighth data point (18%) to the ninth data point (19%), or 18.25. This means that 75% of the observations lie below 18.25%
If funds are contributed to a portfolio at a favorable time (just prior to a period of relatively high returns), the money-weighted rate of return will be higher than the time-weighted return:
The use of the time-weighted return removes these distortions and thus provides a better measure of a manager's ability to select investments over the period -If the manager has complete control over money flows into and out of the account, the money-weighted rate of return would be the more appropriate performance measure
Statistical Concepts and Market Returns
Two key areas to focus on: -Central tendency -measures of dispersion
Professor's Note: On the Level I exam, you may be asked to convert from any one of these three yields into one of the others:
You should note that the EAY and rMM are merely annualized versions of the HPY -If you concentrate on converting back and forth between the HPY and the other yield figures, you will be well-prepared to answer these types of questions on the Level I exam.
Parameter
a measure used to describe a characteristic of a population -investment analysis usually utilizes just a few parameters, particularly the mean return and standard deviation of returns
Since the overall goal of the firm is to maximize shareholder wealth, the project that:
adds the most value to the firm should be selected
Population mean
all the observed values in the population are summed and divided by the number of observations in the population, N -A given population only has one mean.
IRR decision rule
analyzing an investment (project) using the IRR method provides the analyst with a result in terms of rate of return The following are decision rules of IRR Analysis: 1.) Accept projects with an IRR that is greater than the firm's (investor's) required rate of return. 2.) Reject projects with an IRR that is less than the firm's (investor's) required rate of return. -Note that for a single project, the IRR and NPV rules lead to exactly the same accept/reject decision. If the IRR is greater than the required rate of return, the NPV is positive, and the IRR is less than the required rate of return, the NPV is negative.
Measures of central tendency
arithmetic mean, geometric mean, weighted mean, median, and mode
Another way to compute the FV of an annuity due is to
calculate the FV of an ordinary annuity, and simply multiply the resulting FV by (1 + periodic compounding rate (I/Y)
Future value of an annuity due
calculator can be used to do this, but must be set to (BGN) mode -It should be mentioned that while annuity due payments are made or received at the beginning of each period, the FV of an annuity due is calculated as of the end of the last period
The limit of shorter and shorter periods is called
continuous compounding. To convert an annual stated rate to the EAR with continuous compounding, we use the formula: e^r - 1 = EAR
Opportunity costs of interest rates
current consumption - if the market rate of interest on 1-year securities is 5%, earning an additional 5% is the opportunity forgone when current consumption is chosen rather than saving (postponing consumption)
Dispersion
defined as the variability around the central tendency, where the central tendency is the measure of the reward and dispersion is a measure of risk
Interest rates are also referred to as:
discount rates and, in fact, the terms are often used interchangeably
The rate of interest that investors actually realize as a result of compounding is know as the
effective annual rate (EAR) -EAR represents the annual rate of return actually being earned after adjustments have been made for different compounding periods -Whenever compound interest is being used, the stated rate and the actual (effective) rate of interest are equal only when interest is compounded annually. Otherwise, the greater the compounding frequency, the greater the EAR will be in comparison to the stated rate.
Money market yield (or CD equivalent yield) is
equal to the annualized holding period yield, assuming a 360-day year -Using the money market yield makes the quoted yield on a T-bill comparable to yield quotes for interest bearing money market instruments that pay interest on a 360-day basis
Measures of central tendency
identify the center, or average, of a data set. This central point can be used to represent the typical, or expected, value in the data set.
sample statistic
in the same manner that a parameter may be used to describe a characteristic of a population, a sample statistic is used to measure a characteristic of a sample
Perpetuity
is a financial instrument that pays a fixed amount of money at set intervals over an infinite period of time - in essence is a perpetual annuity -British consol bonds and most preferred stocks are examples of perpetuities since they promise fixed interest or dividend payments forever
Frequency distribution
is a tabular presentation of statistical data that aids the analysis of large data sets -summarize statistic data by assigning it to specified groups, or intervals -intervals are also known as classes
Effective annual yield (EAY)
is an annualized value, based on a 365-day year, that accounts for compound interest
Population variance
is defined as the average of the squared deviations from the mean -Major problem is interpreting the data -The computed variance, unlike the mean, is in terms of squared units of measurement; squared percent/dollars doesn't make any sense -This is mitigated by calculating the population standard deviation by taking the square root
Internal rate of return (IRR)
is defined as the rate of return that equates the PV of an investment's expected benefits (inflows) with the PV of its costs (outflows) -Equivalently, the IRR may be defined as the discount rate for which the NPV of an investment is zero.
Population
is defined as the set of all possible members of a stated group -A sample is defined as a subset of the population of interest. Once a population has been defined, a sample can be drawn from the population, and the sample's characteristics can be used to describe the population as a whole.
Holding period return
is simply the percentage change in the value of an investment over the period it is held. -If the asset has cash flows, such as dividend or interest payments, we refer to the return, including the value of these interim cash flows, as the total return.
Net Present Value (NPV) of an investment project
is the PV of expected cash inflows associated with the project less the PV of the project's expected cash outflows, discounted at the appropriate cost of capital. -Identify all costs (outflows) and benefits (inflows) associated with the investment -Determine the appropriate discount rate or opportunity cost for the investment -Using the appropriate discount rate, find the PV of each cash flow. Inflows are positive and increase NPV. Outflows are negative and decrease NPV -Compute the NPV, the sum of the DCFs
Mean absolute deviation (MAD)
is the average of the absolute values of the deviations of individual observations from the arithmetic mean -The computation of the MAD uses the absolute values of each deviation from the mean because the sum of the actual deviations from the arithmetic mean is zero.
Histogram
is the graphical presentation of the absolute frequency distribution; simply a bar chart of continuous data that has been classified into a frequency distribution
Sample variance (s2)
is the measure of dispersion that applies when we are evaluating a sample of n observations from a population -Sample size - 1 is used in the denominator because just using n (sample size) as the divisor in the computation will systematically underestimate the population parameter, particularly for small sample sizes -N - 1 provides an unbiased estimator of the population variance
Ordinary annuity
is the most common type of annuity. It is characterized by cash flows that occur at the end of each accounting period -This is a typical cash flow pattern for many investment and business finance applications
Sample mean
is the sum of all the values in a sample population, divided by the number of observations in the sample, n -It is used to make inferences about the population mean -The population mean and sample mean are both examples of arithmetic means -The arithmetic mean is the sum of the observation values divided by the number of observations
Since an increase in the frequency of compounding increases the effective rate of interest,
it also increases the FV of a given cash flow and decreases the PV of a given cash flow
Relative Dispersion
makes a meaningful comparison by measuring the amount of variability in a distribution relative to a reference or benchmark -Relative dispersion is commonly measured with the coefficient of variation (CV); measures the amount of dispersion in a distribution relative to the distribution's mean -It is useful because it enables us to make a direct comparison of dispersion across different sets of data -used to measure the risk per unit of expected return
Interval scale
measurements provide relative ranking, like ordinal scales, plus the assurance that differences between scales are equal -temperature measurement in degrees is a prime example. -The weakness of the interval scale is that a measurement of zero does not necessarily indicate the total absence of what we are measuring. This means that interval-scale ratios are meaningless. For example, 30 degrees is not 3x as hot as 10 degrees
Time-weighted rate of return
measures compound growth -Is the rate at which $1 compounds over a specified time horizon -Time-weighting is the process of averaging a set of values over time
Median
midpoint of a data set when the data is arranged in ascending or descending order. Half the observations lie above the median and half are below; find the middle observation -The median is important because the arithmetic mean can be affected by extremely large or small outliers; When this occurs, the median is a better measure of central tendency than the mean because it is not affected by extreme values that may actually be the result of errors in the data
A project may have:
multiple IRRs or no IRR
Geometric mean
often used when calculating investment returns over multiple periods or when measuring compound growth rates -used in the time-weighted rate of return Note that this equation has a solution only if the product under the radical sign is non-negative
Annuity due
payments and receipts occur at the beginning of each period (i.e. the first payment is today at t = 0)
Inferential statistics
pertain to the procedures used to make forecasts, estimates, or judgments about a large set of data on the basis of the statistical characteristics of a smaller sample.
What is the money market yield for a T-bill that is selling for $99,000 if it has a face value of $100,000 and 95 days until maturity?
rMM = HYP * (360/t) =1.01% * (360/95) = 3.83%
Measures of dispersion
range, mean absolute deviation, and most importantly - variance (variability around the center) -When describing investments, measures of central tendency provide an indication of an investment's expected return.
Weighted mean
recognizes that different observations may have a disproportionate influence on the mean; essentially a weighted average -This illustrates an extremely important investments concept; the return of a portfolio is the weighted average of the returns of the individual assets in the portfolio; asset weights are the market weights, the market value of each asset value relative to the market value of the entire portfolio
Bond equivalent yield
refers to 2 x the semiannual discount rate -This convention stems from the fact that yields on U.S. bonds are quoted as twice the semiannual rate, because the coupon interest is paid in two semiannual payments
Cash flow additivity principle
refers to the fact that PV of any stream of cash flows equals the sum of the PVs of the cash flows -if we have two series of cash flows, the sum of the PVs of the two series is the same as the PVs of the two series taken together, adding cash flows that will be paid at the same point in time
Ordinal Scales
represent a higher level of measurement than nominal scales -Every observation is assigned to one of several categories, then these categories are ordered with respect to a specified characteristic -i.e. the ranking of 1000 small cap growth stocks by performance may be done by assigning the number 1 to the best 100 performing stocks, the number 2 to the next 100 best performing stocks, etc.
Ratio scales
represent the most refined level of measurement -Provide ranking and equal differences between scale vales, and they also have a true zero point as the origin -Order, intervals, and ratios all make sense with a ratio scale. The measurement of money is a good example -If you have zero dollars, you have no purchasing power, but $4 is twice as much purchasing power as $2
range
simple measure of variability, but when used with other measures it provides extremely useful information -The range is the distance between the largest and the smallest value in the data set
Chebyshev's inequality
states that for any set of observations, whether sample or population data and regardless of the shape of the distribution, the percentage of the observations that lie within k standard deviation of the mean is at least 1 - 1/k^2 for all k > 1 -The importance is that it applies to any distribution; if we actually know the underlying distribution is normal, we can be even more precise about the percentage of observations that will fall within 2 or 3 standard deviations of the mean
Annuity
stream of equal cash flows that occurs at equal intervals over a given period -There are two types of annuities: ordinary annuities and annuities due
As with the population standard deviation, the sample standard deviation (s) can be calculated by:
taking the square root of the sample variance
For mutually exclusive projects,
the NPV and IRR methods can give conflicting project rankings -This can happen when the projects' initial costs are of different sizes or when the timing of the cash flows is different
Mathematically speaking,
the NPV method assumes the reinvestment of a project's cash flows at the opportunity cost of capital, while the IRR method assumes that the reinvestment rate is the IRR. -The discount rate used with the NPV approach represents the market-based opportunity cost of capital and is the required rate of return for the shareholders at the firm -Given that shareholder wealth maximization is the ultimate goal of the firm, ALWAYS SELECT THE PROJECT WITH THE GREATEST NPV WHEN THE IRR AND NPV RULES PROVIDE CONFLICTING DECISIONS
Note that questions on the FV of an annuity due refer to:
the amount in the account one period after the last deposit is made
NPV Decision Rule
the basic idea behind NPV analysis is that if a project has a positive NPV, this amount goes to the firm's shareholders. -As such, if a firm undertakes a project with a positive NPV, shareholder wealth is increased.
Quantile
the general term for a value at or below which a stated proportion of the data in a distribution lies -Quartiles - quarters -Quintile -fifths -Decile - tenths -Percentile - hundredths -Quantiles and measures of central tendency are known collectively as measured of location
In the majority of IRR applications to capital budgeting,
the initial cash flow, CF0, represents the initial cost of the investment opportunity, and is therefore given a negative value. -As such, any discount rate less than the IRR will result in a positive NPV, and a discount rate greater than the IRR will result in a negative NPV. This implies that the NPV of an investment is zero when the discount rate equals the IRR.
Modal interval
the interval with the greatest frequency
Nominal Scales
the level of measurement that contains the least information -Observations are classified or counted with no particular order. I.e. assigning the number 1 to a bond fund, 2 to another bond fund, etc.
Frequency polygon
the midpoint of each interval in plotted on the horizontal axis, and the absolute frequency for that interval is plotted on the vertical axis
Loan amortization is
the process of paying off a loan with a series of periodic loan payments, whereby a portion of the outstanding loan amount is paid off, or amortized, with each payment
Equilibrium interest rates
the required rate of return for a particular investment, in the sense that the market rate of return is the return that investors and savers require to get them to willingly lend their funds
The cumulative absolute frequency or cumulative relative frequency for any given interval is:
the sum of the absolute or relative frequencies up to and including the given interval
In the investment management industry,
the time-weighted rate of return is the preferred method of performance measurement, because it is not affected by the timing of cash inflows and outflows.
Real risk-free rate of interest
theoretical rate on a single-period loan that has no expectation of inflation in it -When we speak of a real rate of return, we are referring to an investor's increase in purchasing power (after adjusting for inflation) -T-bill rates are nominal risk-free rates because they contain an inflation premium
Harmonic mean
used for certain computations, such as the average cost of shares purchased over time
Descriptive Statistics
used to summarize the important characteristics of large data sets - focus is on the use to consolidate a mass of numerical data into useful information
Mode
value that occurs the most frequently in a data set -A data set may have more than one mode or even no mode -When a distribution has one value that appears most frequency that appears most frequently, it is said to be unimodal; bimodal or trimodal if it has two or three values that occur most frequently
Constructing a time line showing future cash flows
will help in solving many types of TVM problems -Cash flows occur at the end of the period depicted on the time line. -The end of one period is the same as the beginning of the next period -For example, a cash flow at the beginning of year 3 appears at time = 2 on the time line.
