唯學(xué)網(wǎng)小編提醒：金融分析師考試定量投資在國(guó)內(nèi)才剛起步，一提起定量投資，很多人首先想到的是“程序化交易”、“高頻”這樣的字眼。這樣的理解是否正確?定量投資分很多門類，程序化交易或高頻交易是其中的一部分。

二、Investment Tools: Quantitative Methods

1.A.: Time Value of Money

a: Calculate the future value (FV) and present value (PV) of a single sum of money.

Future Value:

FV = PV(1 + I/Y)N

Where PV = the amount of money invested today, I/Y = the rate of return, and N = the length of the holding period.

Example: Using a financial calculator, here's an example of how you would find the 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:

PV = FV / (1 + I/Y)N

Example: Using a financial calculator, here's an example of how you'd find the 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).

b: Calculate an unknown variable, given the other relevant variables, in single-sum 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). This very same problem could also be set up in terms of growth rates - e.g., what rate of growth (I/Y) is necessary for a company's sales to grow from $500 per year (PV) to $2,000 per year (FV) in 15 years (N).?

N = 15, PV = -500, FV = 2,000; CPT I/Y = 9.68%

Example 2: Solving for N

In this example, you want to find out how many years (N) it will take for a $500 investment (PV) to grow to $1,000 (FV), given that we can earn 7% annually (I/Y) on your money.

I/Y = 7, PV = -500, FV = 1,000; CPT N = 10.24 years.

c: Calculate the FV and PV of an regular annuity and an annuity due.

Calculate the FV of an ordinary annuity:

Example: 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).

Calculate the FV of an annuity due:

Example: 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: When solving for a FV of an annuity due, you MUST put your calculator in the beginning of year mode (BGN), otherwise you'll end up with the wrong answer.

N = 3, I/Y = 10%, PMT = $100; CPT FV = $364.10 (ignore the sign).

Calculate the PV of an ordinary annuity:

Example: 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:

Example: 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.

Note: There are two ways to approach this question. The first is to put your calculator in BGN mode and then input all the variables as you normally would. The second is to shorten the annuity by one year (N - 1) and find the PV of that shortened annuity as if it were an ordinary annuity, then add the first annuity payment (PMT0) to it to come up with the PV of this annuity due. In this second alternative, you will leave your calculator in the END mode.

1. BGN mode: N = 3, I/Y = 10, PMT = 100; CPT PV = $273.55

2. END mode: N = 2, I/Y = 10, PMT = 100; CPT PV = $173.55 + 100 = PV = $273.55

d: Calculate an unknown variable, given the other relevant variables, in annuity problems.

Example: Find the PMT required to fund a retirement program of $3,000 at the end of 15 years, given a rate of return of 7%.

N = 15, I/Y = 7%, FV = 3,000; CPT PMT = $119.38 (ignore the sign).

Example: Suppose that you will deposit $100 at the end of each year for 5 years into an investment account. At the end of 5 years, the account will be worth $600. What is the rate of return?

N = 5, FV = 600, PMT = 100; CPT I/Y = 7 years.

Example: Solve for the PMT given a 13-year annuity with a discount rate of 6%, and a PV of $2,000.

N = 13, I/Y = 6, PV = 2,000; CPT PMT = $225.92.

Example: Suppose that you have $1,000 in the bank today. If the interest rate is 8%, how many annual, end-of-year payments of $150 can you withdraw?

I/Y = 8, PMT = 150, PV = -1,000; CPT N = 9.9 years.

Example: What rate of return will you earn on an annuity that costs $700 today and promises to pay you $100 per year for each of the next 10 years?

N = 10, PV = 700, PMT = 100; CPT I/Y = 7.07%.

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e: 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.

f: Calculate an unknown variable, given the other relevant variables, in perpetuity problems.

Example: Continuing with our example from LOS 1.A.e, what rate of return would the investor make if she paid $75.00 per share for the stock?

I/Y = PMT / PVperpetuity

4.50 / 75.00 = 6.0%

g: Calculate the FV and PV of a series of uneven cash flows.

FV Example: Given: I = 9%; PMT1 is $100; PMT2 is $500; and PMT3 is $900. How much is this future stream worth at the end of the 3rd year?

Solve:	enter PMT1	=	$100 as PV;	I=9%	n=2:	Compute FV1	=	$118.81
	enter PMT2	=	$500 as PV;	I=9%	n=1:	Compute FV2	=	$545.00
	enter PMT3	=	$900 as PV;	I=9%	n=0:	Compute FV3	=	$900.00
						Sum of FVs	=	$1,563.81

PV Example: Given: I = 10%; PMT1 is $100; PMT2 is $200; and PMT3 is $300. Solve for the PV of this cash flow stream.

Solve:	enter PMT1	=	$100 as PV;	I=10%	n=1:	Compute PV1	=	$90.91
	enter PMT2	=	$200 as PV;	I=10%	n=2:	Compute PV2	=	$165.29
	enter PMT3	=	$300 as PV;	I=10%	n=3:	Compute PV3	=	$225.39
						Sum of PVs	=	$481.59

h: Calculate 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	compute FV = $112.00
Semi-annual:	N = 2	I = 6%	PV = $100	compute FV = $112.36
Quarterly:	N= 4	I = 3%	PV = $100	compute FV = $112.55
Monthly:	N = 12	I = 1%	PV = $100	compute FV = $112.68
Daily:	N = 365	I = .03287	PV = $100	compute FV = $112.747
Continuous:	FV = (PV)	(e(i rate)(n))	=100(e)(.12)(1)	App6A compute 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; compute FV.

Solve: I = 10/4 = 2.5; N = 5 * 4 = 20; PV = 500: compute FV = 819.31.

i: 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%. Finally, 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.

j: 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%

k: 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

This loan can be paid off in five equal annual payments of $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

He will pay $2,637.97 at the end of each of the next 5 years.

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

The investor will need a nest egg of $259,240. He will then have to put away $1,800 per year at the end of each of the next 25 years in order to accumulate a nest egg worth $259,240 - which will enable him to withdraw $25,000 per year for the following 30 years.

1.B: Statistical Concepts and Market Returns

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. Populations can have many parameters, but investment analysts are usually only concerned with a few, such as the mean return, or the standard deviation of returns.

A sample is defined as a portion, or subset of the population of interest. Even if it is possible to observe all members of a population, it is often too expensive or time consuming to do so. Once the population has been defined, we can take a sample of the population with the view of describing the population as a whole.

b: Explain the concept of a parameter.

Any descriptive measure of a population characteristic is called a parameter.

c: Explain the differences among the types of measurement scales.

Nominal scale: this represents the weakest level of measurement. Observations are classified or counted with no particular order. An example would be assigning the number one to a large cap value fund, the number two to a large cap growth fund, etc.

Ordinal scale: this is a higher level of measurement. All observations are placed into separate categories and the categories are placed in order with respect to some characteristic. An example would be ranking 100 large cap growth mutual funds by performance and assigning the number one to be the 10 best performing funds and the number ten to the 10 worst performing funds.

Interval scale: this scale provides ranking and assurance that differences between scale values are equal. Measuring temperature is a prime example.

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. Money is a good example.

d: Define and interpret a frequency distributions.

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.

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 / Pt - 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%

f: Define and explain the use of intervals to summarize data.

An interval 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. It is important to consider the number of intervals to be used. If too few intervals are used, too much data may be summarized and we may lose important characteristics; if too many intervals are used, we may not summarize enough. Each interval has a lower limit and an upper limit. Intervals must be all-inclusive and non-overlapping.

After intervals have been defined, you must tally the observations and assign each observation to its respective interval.

Once the data set has been tallied, you should count the number of observations that were placed in each interval. The actual number of observations in a given interval is called the absolute frequency, or simply the frequency.

g: Calculate relative frequencies, given a frequency distribution.

Another useful way to present data is the relative frequency. 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.

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 histogram, the class intervals are scaled on the horizontal axis and the absolute frequencies are scaled on the vertical axis.

A second graphical tool used for displaying data is the frequency polygon. 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.

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.

Example: A stock you and your research partner are analyzing has 12 years of annualized return data. The returns are 12%, 25%, 34%, 15%, 19%, 44%, 54%, 33%, 22%, 28%, 17%, and 24%. Your research partner is exceedingly lazy and has decided to collect data based on only five years of returns. Given this data, calculate the population mean and calculate the sample mean. (Your partner's data set is shown above as bold).

Population mean = 12 + 25 + 34 + 15 + 19 + 44 + 54 + 33 + 22 + 28 + 17 + 24 / 12 = 27.25%

Sample mean = 25 + 34 + 19 + 54 + 17 / 5 = 29.8%

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.

Example: A data set contains the following numbers: 5, 9, 4, and 10. The mean of these numbers is: ( 5 + 9 + 4 + 10) / 4 = 7. The sum of the deviations from the mean is: (5 - 7) + (9 - 7) + (4 - 7) + (10 - 7) = -2 + 2 - 3 + 3 = 0.

Geometric mean is often used when calculating investment returns over multiple periods, or to find a compound growth rate.

Example: For the last three years the return for Acme Corporation common stock have been -9.34%, 23.45%, and 8.92%. Find the geometric mean.

Take the cube root of (-.0934 + 1)(.2345 + 1)(.0892 + 1) = The cube root of 1.21903. On your TI calculator, enter 1.21903 and hit the yx key, then enter .3333 = to get of 1.06825. Now you must subtract this number from one to get an answer of 6.825%.

Weighted mean is a special case of the mean that allows different weights on different observations.?

Example: A portfolio consists of 50% common stocks, 40% bonds, and 10% cash. If the return on common stocks is 12%, the return on bonds is 7%, and the return on cash is 3%, what is the return to the portfolio?

Weighted mean = [(0.50 *0.12) + (0.40 * 0.07) + (0.10 * 0.03)] = 0.091, or 9.1%

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.

Example: The five-year annualized total returns for five investment managers are 30%, 15%, 25%, 21%, and 23%. Find the median return for the managers.

First, arrange the returns from hi to lo: 30, 25, 23, 21, 15.

The return observation half way down from the top is 23%.

The mode of a data set is the value of the observation that appears most frequently.

Example: In the following set of numbers, 30%, 28%, 25%, 23%, 28%, 15%, and 5%, 28% is the most frequently occurring value.

j: Distinguish between arithmetic mean and geometric means.

The value for the arithmetic mean is higher. The geometric mean will always be less than or equal to the arithmetic mean. In general, the difference between the two means increases with the variability between period-by-period observations. The only time when the two means will be equal is when there is no variability in the observations (e.g., all observations are 10%).

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.

Example: The five-year annualized total returns for five investment managers are 30%, 12%, 25%, 20%, and 23%. What is the range of the data? Range = 30 - 12 = 18%.

Mean absolute deviation (MAD) is the average of the absolute values of the deviations of individual observations from the arithmetic mean. Remember that the sum of all of the deviations from the mean is equal to zero. To get around this zeroing out problem, the mean deviation uses the absolute values of each deviation.

Example: Continuing from above, what is the mean deviation of investment returns and how is it interpreted?

MAD = {I (30 - 22) I + I (12 - 22) I + I (25 - 22) I + I (20 - 22) I + I (23 - 22) I } / 5

MAD = [ 8 + 10 + 3 + 2 + 1] / 5 = 4.8%

Population variance is the mean of the squared deviations from the mean. The population variance is computed using all members of a population.

Example: Assume the five-year annualized total returns for the five investment managers used in the earlier example represent all of the managers at a small investment firm. What is the population variance?

μ = {30 + 12 + 25 + 20 + 23} / 5 = 22%

ó = { (30 - 22)2 + (12 - 22)2 + (25 - 22)2 + (20 - 22)2 + (23 - 22)2 } / 5 = 35.60%2

Population standard deviation is the square root of the population variance.

Example: Continuing with our example, take the square root of 35.60 = 5.97%

Sample variance applies when we are dealing with a subset, or sample of the total population.

Example: Assume the five-year annualized total returns for the five investment managers used in the earlier example represent only a sample of the managers at a large investment firm. What is the sample variance?

sample mean = {30 + 12 + 25 + 20 + 23} / 5 = 22%

s2 = { (30 - 22)2 + (12 - 22)2 + (25 - 22)2 + (20 - 22)2 + (23 - 22)2 } / 5 - 1 = 44.5%2

Sample standard deviation can be found by taking the positive square root of the sample variance.

Example: Continuing with our example, take the square root of 44.50 = 6.67%

l: Calculate the proportion of items falling within a specified number of standard deviaitons of the mean, using Chebyshev's inequality.

Chebyshev's inequality states that for any set of observations (sample or population, regardless of the shape of the distribution), the proportion of the observations within k standard deviations of the mean is at least 1 - 1/k2 for all k > 1. If we know the standard deviation, we can use Chebyshev's inequality to measure the minimum amount of dispersion, regardless of the shape of the distribution.

Chebyshev's inequality states that for any distribution, approximately:

36% of observations lie within 1.25 standard deviations of the mean

56% of observations lie within 1.50 standard deviations of the mean

75% of observations lie within 2 standard deviations of the mean

89% of observations lie within 3 standard deviations of the mean

94 of observations lie within 4 standard deviations of the mean

m: Define, calculate, and interpret the coefficient of variation.

The coefficient of variation expresses how much dispersion exists relative to the mean of a distribution and allows for direct comparison of dispersion across different data sets.

CV = [standard deviation of returns]/[Expected rate of return]

Example:

Investment A has an ER of 7% and a s of .05.

Investment B has an ER of 12% and a s of .07.

Which is riskier?

A’s CV is .05/.07 = .714

B’s CV is .07/.12 = .583

A has .714 units of risk for each unit of return while B has .583 units of risk for each unit of return. A is riskier, it has more risk per unit of return.

n: Define, calculate, and interpret the Sharpe measure of risk-adjusted performance.

The Sharpe measure seeks to measure excess return per unit of risk. The numerator of the Sharpe measure recognizes the existence of a risk-free return. Portfolios with large Sharpe ratios are preferred to portfolios with smaller ratios because it is assumed that rational investors prefer return and dislike risk. The Sharpe ratio is also called the reward-to-variability ratio.

Example: The mean monthly return on T-bills is 0.25%. The mean monthly return on the S&P 500 is 1.30% with a standard deviation of 7.30%. Calculate the Sharpe measure for the S&P 500 and interpret the results.