CFA Level 1 – Quantitative Methods
CFA Level 1 - Quantitative Methods
SESSION 2
Reading 1.A: Time Value of Money
LOS a: Calculate the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, and an annuity due.
Future Value Example:
Using a financial calculator, with FV of a $300 investment (PV), given you earn a compound rate of return (I/Y) of 8% over a 10-year (N) period of time:
N = 10, I/Y = 8, PV = 300; CPT FV = $647.68 (ignore the sign).
Present Value Example:
Using a financial calculator, with PV of a $1,000 cash flow (FV) to be received in 5 (N) years, given a discount rate of 9% (I/Y).
N = 5, I/Y = 9, FV = 1,000; CPT PV = $649.93 (ignore the sign).
Calculate the FV of an ordinary annuity: Find the FV of an ordinary annuity that will pay $150 per year at the end of each of the next 15 years, given the investment is expected to earn a 7% rate of return.
N = 15, I/Y = 7%, PMT = $150; CPT FV = $3,769.35 (ignore the sign).
The time line for the cash flows in this problem is depicted below.
Calculate the FV of an annuity due: Find the FV of an annuity due that will pay $100 per year for each of the next three years, given the cash flows can be invested at an annual rate of 10%.
Note: you MUST put your calculator in the beginning of year mode (BGN)
N = 3, I/Y = 10%, PMT = $100; CPT FV = $364.10 (ignore the sign).
Calculate the PV of an ordinary annuity: Find the PV of an annuity that will pay $200 per year at the end of each of the next 13 years, given a 6% rate of return.
N = 13, I/Y = 6, PMT = 200; CPT PV = $1,770.54
Calculate the PV of an annuity due: Find the PV of a 3-year annuity due that will make a series of $100 beginning of year payments, given a 10% discount rate.
BGN mode: N = 3, I/Y = 10, PMT = 100; CPT PV = $273.55
LOS b: Calculate the PV of a perpetuity.
Example: Assume a certain preferred stock pays $4.50 per year in annual dividends (and they're expected to continue indefinitely). Given an 8% discount rate, what's the PV of this stock?
PVperpetuity = PMT / I/Y
PVperpetuity = 4.50 / .08 = $56.25
This means that if the investor wants to earn an 8% rate of return, she should be willing to pay $56.25 for each share of this preferred stock.
LOS c: Calculate an unknown variable, given the other relevant variables, in single-sum problems, annuity problems, and perpetuity problems.
Example 1: Solving for I/Y
In this example, you want to find the rate of return (I/Y) that you'll have to earn on a $500 investment (PV) in order for it to grow to $2,000 (FV) in 15 years (N).
N = 15, PV = -500, FV = 2,000; CPT I/Y = 9.68%
Example 2: Solving for PMT
In this example, you want to find out the dollar amount of payments (PMT) it will take at 7% (I/Y) to achieve $3,000 (FV) in 15 years (N).
N = 15, I/Y = 7, FV = 3,000; CPT PMT = -$119.38 (ignore sign)
LOS d: Calculate the FV and PV of a series of uneven cash flows.
FV Example: Given a 10% discount rate and cash flows of (starting with year 1) -1,000, -500, 0, 4,000, 3,500, and 2,000, compute the FV.
PV = -1,000, I/Y = 10, N = 5, CPT FV = -1,610.51
PV = -500, I/Y = 10, N = 4, CPT FV = -732.05
PV = 0, I/Y = 10, N = 3, CPT FV = 0
PV = 4,000, I/Y = 10, N = 2, CPT FV = 4,840.00
PV = 3,500, I/Y = 10, N =1 , CPT FV = 3,850.00
PV = 2,000, I/Y = 10, N = 0, CPT FV = 2,000.00
Total cash flow stream = $8,347.44
PV Example: Given a 10% discount rate and cash flows of (starting with year 1) -1,000, -500, 0, 4,000, 3,500, and 2,000, compute the PV.
FV = -1,000, I/Y = 10, N = 1, CPT PV = -909.09
FV = -500, I/Y = 10, N = 2, CPT PV = -413.22
FV = 0, I/Y = 10, N = 3, CPT PV = 0
FV = 4,000, I/Y = 10, N = 4, CPT PV = 2,732.05
FV = 3,500, I/Y = 10, N =5 , CPT PV = 2,173.22
FV = 2,000, I/Y = 10, N = 6, CPT PV = 1,128.95
Total cash flow stream = $4,711.91
LOS e: Solve time value of money problems when compounding periods are other than annual.
Example: PV = $100, N = 1 year, I = 12%. Find the FV for various compounding periods.
Annual: / N = 1 / I = 12% / PV = $100 / CPT FV = $112.00Semi-annual: / N = 2 / I = 6% / PV = $100 / CPT FV = $112.36
Quarterly: / N= 4 / I = 3% / PV = $100 / CPT FV = $112.55
Monthly: / N = 12 / I = 1% / PV = $100 / CPT FV = $112.68
Daily: / N = 365 / I = .03287 / PV = $100 / CPT FV = $112.74
Continuous: / FV = (PV) / e(i rate)(n) / =100e(.12)(1) / CPT FV = $112.75
In the continuous compounding equation, the interest rate is the stated or nominal annual rate.
Example: Given a 10% annual rate paid quarterly; PV = 500; time is 5 years; CPT FV.
Solve: I/Y = 10/4 = 2.5; N = 5*4 = 20; PV = 500: CPT FV = $819.31
LOS f: Distinguish between the stated annual interest rate and the effective annual rate.
The stated rate of interest is known as the nominal rate, and represents the contractual rate. The periodic rate, in contrast, is the rate of interest earned over a single compound period - e.g., a stated (nominal) rate of 12%, compounded quarterly, is equivalent to a periodic rate of 12/4 = 3%. The true rate of interest is known as the effective rate and represents the rate of return actually being earned, after adjustments have been made for different compounding periods.
LOS g: Calculate the effective annual rate, given the stated annual interest rate and the frequency of compounding.
Example: Compute the effective rate of 12%, compounded quarterly. Given m = 4, and periodic rate = 12/4 = 3%.
Effective rate = (1 + periodic rate)m - 1
Where m = the number of compounding periods in a year.
(1 + .03)4 - 1 = 1.1255 - 1 = 12.55%
LOS h: Draw a time line, specify a time index, and solve problems involving the time value of money as applied to mortgages, credit card loans, and saving for college tuition or retirement.
Example: Paying off a Loan (or Mortgage)
A company wants to borrow $50,000 for five years. The bank will lend the money at a 9% rate of interest and will require that the loan be paid off in five equal, annual (end-of-year) installment payments. What are the annual loan payments that this company will have to make in order to pay off this loan?
N = 5, I/Y = 9, PV = 50,000; CPT PMT = $12,854.62
Example: Loan Amortization
An individual borrows $10,000 at 10% today amortized over 5 years. What are his payments?
PV = 10,000, N = 5, I/Y = 10; CPT PMT = $2,637.97
Example: Funding a Retirement Program
A 35-year old investor wants to retire in 25 years at age 60. Given he expects to earn 12.5% on his investments prior to his retirement, and then 10% thereafter, how much must he deposit annually (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 the next 30 years?
This is a two-part problem. First, use PV to compute the present value of the 30-year, $25,000 annuity due and second, use FV to find the amount of the fixed annual deposits that must be made at the end of the first 25-year period to come up with the needed funds.
Step 1: N = 29, I/Y = 10, PMT = 25,000; CPT PV = 234,240 + 25,000 = $259,240
Step 2: N = 25, I/Y = 12.5, FV =259,240; CPT PMT = $1,800.02
SESSION 2
Reading 1.B: Statistical Concepts and Market Returns
LOS a: Differentiate between a population and a sample.
A population is defined as all members of a specified group. Any descriptive measure of a population characteristic is called a parameter.
A sample is defined as a portion, or subset of the population of interest. Once the population has been defined, we can take a sample of the population with the view of describing the population as a whole.
LOS b: Explain the concept of a parameter.
Any descriptive measure of a population characteristic is called a parameter.
LOS c: Explain the differences among the types of measurement scales.
§ Nominal scale: Observations are classified or counted with no particular order.
§ Ordinal scale: All observations are placed into separate categories and the categories are placed in order with respect to some characteristic.
§ Interval scale: This scale provides ranking and assurance that differences between scale values are equal.
§ Ratio scale: These represent the strongest level of measurement. In addition to providing ranking and equal differences between scale values, ratio scales have a true zero point as the origin.
LOS d: Define and interpret a frequency distribution.
A frequency distribution is a grouping of raw data into categories (called classes) so that the number of observations in each of the nonoverlapping classes can be seen and tallied. The purpose of constructing a frequency distribution is to group raw data into a useable visual framework for analysis and presentation.
LOS e: Define, calculate, and interpret a holding period return.
Holding period return (HPR) measures the total return for holding an investment over a certain period of time, and can be calculated using the following formula:
HPR = [Pt - Pt - 1 + Dt ] / P t - 1
Where: Pt = price per share at the end of time period t, and Dt = cash distributions received during time period t.
Example: A stock is currently worth $60. If you purchased the stock exactly one year ago for $50 and received a $2 dividend over the course of the year, what is your HPR?
(60 - 50 + 2) / 50 = 24%
LOS f: Define and explain the use of intervals to summarize data.
§ Define the intervals. An interval (classes) is the set of return values within which an observation falls. Each observation falls into only one interval, and the total number of intervals covers the entire population. Intervals must be all-inclusive and non-overlapping.
§ Tally the observations.
§ Count the number of observations.
LOS g: Calculate relative frequencies, given a frequency distribution.
Relative frequency is calculated by dividing the frequency of each return interval by the total number of observations. Simply, relative frequency is the percentage of total observations falling within each interval.
LOS h: Describe the properties of data presented as a histogram or a frequency polygon.
A histogram is the graphical equivalent of a frequency distribution. It is a bar chart of continuous data that has been grouped into a frequency distribution. The advantage of a histogram is that we can quickly see where most of the observations lie.
To construct a frequency polygon, we plot the midpoint of each interval on the horizontal axis and the absolute frequency for that interval on the vertical axis. Each point is then connected with a straight line.
LOS i: Define, calculate, and interpret measures of central tendency, including the population mean, sample mean, arithmetic mean, geometric mean, weighted mean, median, and mode.
A population mean is the entire group of objects that are being studied. To find the population's mean, sum up all the observed values in the population (sum X) and divide this sum by the number of observations (N) in the population.
A sample mean is sum of all the values in a sample of a population divided by the number of values in the sample. The sample mean is used to make inferences about the population mean.
Arithmetic mean is the sum of the observation values divided by the number of observations. It is the most widely used measure of central tendency, and is the only measure where the sum of the deviations of each value from the mean is always zero.
Geometric mean is often used when calculating investment returns over multiple periods, or to find a compound growth rate. It is computed by taking the nth root of the product of n values.
Weighted mean is a special case of the mean that allows different weights on different observations.
The median is the mid-point of the data when the data is arranged from the largest to the smallest values. Half the observations are above the median and half are below the median. To determine the median, arrange the data from highest to the lowest and find the middle observation.
The mode of a data set is the value of the observation that appears most frequently.
LOS j: Describe and interpret quartiles, quintiles, deciles, and percentiles.
§ Quartiles: The distribution is divided into quarters.
§ Quintiles: The distribution is divided into fifths.
§ Decile: The distribution is divided into tenths.
§ Percentile: The distribution is divided into hundredths (percents).
LOS k: Define, calculate, and interpret (1) a portfolio return as a weighted mean, (2) a weighted average or mean, (3) a range and mean absolute deviation, and (4) a sample and a population variance and standard deviation.
Refer to LOS 1.B.i for a review of weighted mean and weighted average.
§ Range is the distance between the largest and the smallest value in the data set.
§ Mean absolute deviation (MAD) is the average of the absolute values of the deviations of individual observations from the arithmetic mean.
§ Population variance is the mean of the squared deviations from the mean. The population variance is computed using all members of a population.
§ Population standard deviation is the square root of the population variance.