Chpt 5 - Quantitative Concepts

(CS) Dispersion

- Measures of *dispersion* are important for describing the *spread of the data*, or its *variation around a central value*. Two common measures of dispersion are range and standard deviation. - Dispersion (also called variability, scatter, or spread) denotes how stretched or squeezed is a distribution (the relative distribution or arrangement of its individuals within a given amount of space). - Common examples of measures of statistical dispersion are the variance, standard deviation and interquartile range. - Dispersion is contrasted with location or central tendency, and together they are the most used properties of distributions.

(CS) 3 considerations when comparing investments

1) [reward] The cash flows each investment will generate in the future 2) [timing] The timing of these cash flows 3) [risk] The risk associated with each investment. Note: The discount rate reflects the riskiness of the cash flows. Using present value as the criterion is appropriate for comparing investments when the initial outflow for each investment is the same. However, *investments may not have the same initial cash outflow*, and *outflows may occur at times other than Time Zero*. The net present value (NPV) of an investment is the present value of future cash flows or returns less the present value of the cost of the investment (which often—but not always—occurs only in the initial period).

(CS) Valuing Financial Instruments...

Financial instruments can be valued as the present value of their expected future cash flows

quantitative

numerical data / mathematically based

Annuities and Mortgages

(p122 examples) Two time value of money applications that require the final balance of money to be zero are annuities (the investment products) and mortgages. In fact, most consumer loans result in a final balance of money equal to zero. An annuity involves the initial payment of a sum, usually to an insurance company, in exchange for a fixed number of future payments of a certain amount. Each period, the insurance company makes payments to the annuity holder; these payments are equivalent to the annuity holder making withdrawals. These withdrawals can be viewed as negative cash flows because they reduce the annuity balance. The number of withdrawals and the timing of the withdrawals are fixed. In this scenario, the *initial payment (to the insurer) is termed the "value of the annuity"* and the *final value is to equal zero*. The initial amount paid, by the annuity buyer to the insurance company, funds the later withdrawals. A repayment (amortizing) mortgage involves a loan followed by a series of fixed payments. The initial amount of the loan is referred to as the principal. Although the payment amount is fixed, the portion of each payment that is interest is based on the remaining principal at the beginning of the period. As some principal is repaid each period, the amount of interest decreases over time, and thus the amount of principal paid increases with each successive payment until the value to zero when the loan matures (the last payment is made).

(CS) Net Present Value

*The present value of future cash flows net of the investment required to obtain them*. It is useful when comparing alternatives that require different initial investments. Note that the concepts of present value and net present value have widespread applications in the valuation of both business projects and financial assets and products because they incorporate future cash flows. For example, equities may pay dividends and/or be sold in the future, bonds may pay interest and principal in the future, and insurance may lead to future payouts.

Learning Outcomes

*Time Value of Money* a Define the concept of *interest*; b Compare *simple* and *compound interest*; c Describe effects of *time and discount rate* on *value*; d Explain the relevance of the *net present value* in valuing financial investments; *Descriptive Statistics* e Explain uses of *mean, median, mode, range, percentile, and standard deviation*; f Describe and interpret the characteristics of a *normal distribution*; g Describe and interpret *correlation*.

(CS) Effective annual rate (EAR)

- The effective annual rate (EAR) of interest is calculated by annualising a rate that is stated on a less than annual basis. EAR = (1 + APR/ Number of time periods per year)^ Number of periods per year −1

(CS) Correlation

- The strength of a relationship between two variables can be measured using correlation. - Correlation is measured by the correlation coefficient on a scale from -1 to +1. - When the two variables move exactly in harmony with each other, the variables are said to be perfectly positively correlated; The correlation coefficient is +1. -When the two variables move exactly in step in opposite directions, they are perfectly negatively correlated. The correlation coefficient is -1. - Variables with no relationship to each other will have a correlation coefficient close to 0. - It is important to realize that correlation does not imply causation.

(CS) Normal Distribution (common)

A distribution is simply the values that a variable can take showing its observed or theoretical frequency of occurrence. *A common distribution is the normal distribution*, a bell-shaped curve that is *represented by its mean and standard deviation*. *For a perfectly symmetrical distribution, for example a normal distribution, the mean, median, and mode will be identical.*

Discount rate

Before you can calculate present or future values, you must decide on the appropriate interest or discount rates to apply to these future sums. The rate chosen will usually depend on the overall level of *interest rates in the economy*, a measure of *opportunity cost*, and the *particular riskiness* of the investments under consideration. Future value = Present value × (1 + Discount rate)^Number of periods Present value = Future value/ (1+ Discount rate)^Number of Periods 1) The interest rate used in discounted cash flow analysis to determine the present value of future cash flows. The discount rate takes into account the time value of money (the idea that money available now is worth more than the same amount of money available in the future because it could be earning interest) and the risk or uncertainty of the anticipated future cash flows (which might be less than expected). 2) The interest rate that an eligible depository institution is charged to borrow short-term funds directly from a Federal Reserve Bank. Different types of loans are available from Federal Reserve Banks and each corresponding type of credit has its own discount rate. This type of borrowing from the Fed is fairly limited. Institutions will often seek other means of meeting short-term liquidity needs. The Federal Funds Discount Rate is determined by the average rate which banks are willing to charge each other for overnight funds.

Bell-Shaped Distributions w. Fat & Thin Tails

Even using two standard deviations, the chance of the return being in the left tail more than two standard deviations from the mean (which would be an extreme loss under typical circumstances) is 2.5%. Out of 200 days, 5 days are expected to have observations that are more than two standard deviations from the mean. During the financial crisis of 2008, the losses that were incurred by some banks over several days in a row were 25 standard deviations below the mean. To put this in perspective, *if returns are normally distributed, a return that is 7.26 standard deviations below the mean would be expected to occur once every 13.7 billion years*. That is approximately the age of the universe. The frequency of extreme events during the financial crisis of 2008 was, therefore, much higher than predicted by the normal distribution. This inconsistency is often referred to as the distribution having "fat tails", meaning the probability of observing extreme outcomes is higher than that predicted by a normal distribution that has thin tails.

Symmetrical/asymmetrical Distribution characteristics

For a perfectly symmetrical distribution, such as a normal distribution, the mean, median, and mode will be identical. If the distribution is skewed, these three measures of central tendency may differ. - There may be no identifiable mode. - The mean may be larger or smaller than the median because it is more affected by extreme values. If a distribution is skewed to the right...the mean is dragged toward the extreme positive values. The reverse is true for distributions that are negatively skewed - in this case, the mean is smaller than the median because the mean is pulled left in the direction of the skew.

Future Value (with compounded interest)

Future value = Original principal × (1 + Simple interest rate)^Number of periods = principal x (1+r), where "r%" = return

Measures of Dispersion (+ central tendency)

Important for describing the spread of the data, or its variation around a central value. Two distinct data sets may have the same mean or median, but completely different levels of variability, or vice versa. A proper description of a data set should include both a measure of central tendency, such as the mean, and a measure of dispersion. Investment risk is often measured by using some measure of variability. Two common measures of dispersion of a data set are the range and the standard deviation.

(CS) Annuity

Income from capital investment paid in a series of regular payments - also, known as a yearly allowance, payment, or income. An annuity, a common structure for financial instruments, *has multiple cash flows*, each of an equal amount. *Mortgages are a typical example of annuities*; the periodic payment is fixed and each period, some of the payment covers the interest on the loan and the rest of the payment pays off some of the principal.

Interest

Interest can be defined as payment for the use of borrowed money. Interest is paid by a borrower and earned by the lender to compensate for opportunity cost and risk. From the borrower's perspective, interest is the cost of having access to funds that they would not otherwise have. *An interest rate is determined by two factors: Opportunity cost (compensation for deferred spending) and risk (chances of default and expected inflation)*. Even if a loan is viewed as riskless (zero likelihood of default), there would still have to be compensation for the lender's opportunity cost (compensation for deferred spending or consumption) and for expected inflation.

(CS) Interest - simple & compound

Interest is a return earned by a lender that compensates for opportunity cost and risk. For the borrower, it is the cost of borrowing. - The simple interest rate is the cost to the borrower or the rate of return to the lender, per period, on the original principal borrowed. *Simple interest* = Simple interest rate × Principal × Number of periods, where the number of periods here is assumed to be 1 (because the interest rate is assumed to be an annual rate). However, in reality, the number of periods could be any multiple or fraction of 1. *Compound interest* is the return to the lender or the cost to the borrower when *interest is reinvested and added to the original principal*. Compound interest is often referred to as *"interest on interest"*.

Standard Deviation

It measures the variability or volatility of a data set around the average value (the arithmetic mean) of that data set [Xi - E(X)] = difference between value of observation Xi and the mean value of X *The differences between the observed values of X and the mean value of X capture the variability of X*. These differences are squared and summed. Note that because the differences are squared, what matters is the size of the difference not the sign of the difference. The sum is then divided by the number of observations. Finally, the square root of this value is taken to get the standard deviation. *The value before the square root is taken is known as the variance*. Variance is another measure of dispersion. *The standard deviation is the square root of the variance*. The standard deviation and the variance capture the same thing—how far away from the mean the observations are.*The advantage of the standard deviation is that it is expressed in the same unit as the mean*.

Implications from Standard Deviation

Larger values of standard deviation relative to the mean indicate greater variation in a data set. Also, using standard deviation, you can determine how likely any given observation will occur based on its distance from the mean.

Overview of Future Value

Money value fluctuates over time: $100 today is not worth $100 in five years. This is because one can invest $100 today in a bank account or any other investment, and that money will grow/shrink due to interest. Also, if $100 today allows the purchase of an item, it is possible that $100 will not be enough to purchase the same item in five years, because of inflation (increase in purchase price). An investor who has some money has two options: to spend it right now or to invest it. The financial compensation for saving it (and not spending it) is that the money value will accrue through the interests that he will receive from a borrower (the bank account on which he has the money deposited). Therefore, to evaluate the real worthiness of an amount of money today after a given period of time, economic agents compound the amount of money at a given interest rate. Most actuarial calculations use the risk-free interest rate which corresponds the minimum guaranteed rate provided the bank's saving account, for example. If one wants to compare their change in purchasing power, then they should use the real interest rate (nominal interest rate minus inflation rate).

Opportunity cost

Opportunity cost, in general, is the value of the next best alternative (opportunity) that has been given up to pursue the chosen course of action. In lending, opportunity cost is the cost of not having cash to invest, spend, or hold—that is, the cost of giving up opportunities to use cash. This cost includes any other use to which the money could have been put, including lending to others, investing elsewhere, or simply spending the funds. Opportunity cost can also be seen as compensation for deferring consumption. The lender could use the funds for consumption today. Lending delays that consumption by the term of the loan (the time over which the loan is repaid). The longer the consumption is deferred, the more compensation (higher interest) the lender will demand.

(CS) Present Value

The amount of money you would need to deposit now (given interest rates) in order to attain a desired amount in the future. - The present value of *a future sum of money* is found by discounting the future sum by an appropriate *discount rate*. - The present value of *multiple cash flows* is the *sum of the present value of each cash flow*. The concept of present value is important in financial markets because expected cash flows from investments occur at different points in time. Calculating present values is what allows investors and analysts to translate cash flows of different amounts and at different points in the future into sums in the present that can be compared to each other. Alternatively, the cash flows could be translated into the values they would be equivalent to at a common future point.

(CS) Autocorrelation / Serial Correlation

The correlation of a time series with its own past values. For example, a measure of the correlation between successive returns for a single security or portfolio over time. When a variable's value at one point in time is correlated with its values at prior points in time, it can often be a problem in regression models and must be fixed (as this can inflate the value of the inferential statistics (e.g. t or F), thereby resulting in an increased probability of a Type I error).

(CS) Arithmetic Mean

The arithmetic mean is the most commonly used measure. It represents the sum of all the observations divided by the number of observations. The mean has one main disadvantage: It is particularly susceptible to the influence of outliers. When there are one or more outliers in a set of data in one direction, the data are said to be skewed in that direction (right, positive; left, negative)

(CS) Purpose of Measuring "Average" or "Central Tendency"

The purpose of measuring average or central tendency is to *describe a group of individual data scores with a single measurement*. This measure can be represented by an *arithmetic mean*, *median*, *mode*, or *geometric mean*. Different measures are appropriate for different types of data.

Range

The range is the difference between the highest and lowest values in a data set. It is the easiest measure of dispersion to calculate and understand. However, it is very sensitive to outliers. The 50th percentile is, in fact, the median.

present value

The current worth of a future sum of money or stream of cash flows given a specified rate of return. Future cash flows are discounted at the discount rate, and the higher the discount rate, the lower the present value of the future cash flows. Determining the appropriate discount rate is the key to properly valuing future cash flows, whether they be earnings or obligations. Also referred to as "discounted value". Investopedia explains 'Present Value - PV' This sounds a bit confusing, but it really isn't. The basis is that receiving $1,000 now is worth more than $1,000 five years from now, because if you got the money now, you could invest it and receive an additional return over the five years. The calculation of discounted or present value is extremely important in many financial calculations. For example, net present value, bond yields, spot rates, and pension obligations all rely on the principle of discounted or present value. Learning how to use a financial calculator to make present value calculations can help you decide whether you should accept a cash rebate, 0% financing on the purchase of a car or to pay points on a mortgage.

Geometric Mean/Average Return

The geometric mean return formula is used to calculate the average rate per period on an investment that is compounded over multiple periods. The geometric mean return may also be referred to as the geometric average return; measures the status of an investment over time. Geometric mean return (3 years, assuming compounding) = [(1 + 8%) × (1 + 3%) × (1 + 7%)]1/3 - 1 ≈ 5.98% The formula to arrive at the geometric mean return is shown below: (Return 1 x Return 2 x Return t) 1/t -1 where "Return 1" = 1 + ri ri = the return in period i expressed using decimals t = the number of periods (or "n")

(CS) Geometric Mean

The geometric mean return is the average compounded return for each period—that is, the average return for each period assuming that returns are compounding. [the mean of n numbers expressed as the n-th root of their product; for 2 positive numbers found by solving a/x = x/b so x²=ab] In mathematics, the geometric mean is a type of mean or average, which indicates the central tendency or typical value of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the nth root (where n is the count of numbers) of the product of the numbers. For instance, the geometric mean of two numbers, say 2 and 8, is just the square root of their product; that is 2√2 × 8 = 4. As another example, the geometric mean of the three numbers 4, 1, and 1/32 is the cube root of their product (1/8), which is 1/2; that is 3√4 × 1 × 1/32 = ½ . A geometric mean is often used when comparing different items - finding a single "figure of merit" for these items - when each item has multiple properties that have different numeric ranges.[1] For example, the geometric mean can give a meaningful "average" to compare two companies which are each rated at 0 to 5 for their environmental sustainability, and are rated at 0 to 100 for their financial viability. If an arithmetic mean was used instead of a geometric mean, the financial viability is given more weight because its numeric range is larger- so a small percentage change in the financial rating (e.g. going from 80 to 90) makes a much larger difference in the arithmetic mean than a large percentage change in environmental sustainability (e.g. going from 2 to 5). The use of a geometric mean "normalizes" the ranges being averaged, so that no range dominates the weighting, and a given percentage change in any of the properties has the same effect on the geometric mean. So, a 20% change in environmental sustainability from 4 to 4.8 has the same effect on the geometric mean as a 20% change in financial viability from 60 to 72. The geometric mean can be understood in terms of geometry. The geometric mean of two numbers, a and b, is the length of one side of a square whose area is equal to the area of a rectangle with sides of lengths a and b. Similarly, the geometric mean of three numbers, a, b, and c, is the length of one side of a cube whose volume is the same as that of a cuboid with sides whose lengths are equal to the three given numbers. The geometric mean applies only to positive numbers.[2] It is also often used for a set of numbers whose values are meant to be multiplied together or are exponential in nature, such as data on the growth of the human population or interest rates of a financial investment. The geometric mean is also one of the three classical Pythagorean means, together with the aforementioned arithmetic mean and the harmonic mean. For all positive data sets containing at least one pair of unequal values, the harmonic mean is always the least of the three means, while the arithmetic mean is always the greatest of the three and the geometric mean is always in between (see Inequality of arithmetic and geometric means.)

"Lying in the Tails"

The observations more than a specified number of standard deviations from the mean may be described as lying in the tails of the distribution. Assuming that returns on a portfolio of stocks are normally distributed, the chance of extreme losses (a return more than three standard deviations lower than the mean return) is relatively small.

Present to Future (Capitalization), Future to Present (Discounting)

The operation of *evaluating a present value into the future value is called a capitalization (how much will $100 today be worth in 5 years?)*. *The reverse operation* which consists in *evaluating the present value of a future amount of money is called a discounting (how much $100 that will be received in 5 years- at a lottery, for example -are worth today?*). It follows that if one has to choose between receiving $100 today and $100 in one year, the rational decision is to cash the $100 today. If the money is to be received in one year and assuming the savings account interest rate is 5%, the person has to be offered at least $105 in one year so that two options are equivalent (either receiving $100 today or receiving $105 in one year). This is because if you cash $100 today and deposit in your savings account, you will have $105 in one year.

Amortization

The paying off of debt in regular installments over a period of time

(CS) Role of Descriptive Statistics

The role of descriptive statistics is to summarise the information given in large quantities of data for the purpose of making comparisons, predicting future values, and better understanding the data [used to organize, compare, summarize and better understand data]

Normal Distribution

The shape of a normal distribution depends on two measures: The mean and the standard deviation. For a perfectly symmetrical distribution, such as a normal distribution, the mean, median, and mode will be identical. A normal distribution is represented in a graph by a bell curve; the shape of the curve is symmetrical with a single central peak at the mean of the data and the graph falling off evenly on either side of the mean; 50% of the distribution lies to the left of the mean, and 50% lies to the right of the mean. SD 1 (-1 to 1) = 68.26% (a little more than 2/3) SD 2 (-2 to 2) = 95.44% (a little more than 95%) SD 3 (-3 to 3) = ~99% A normal distribution has special importance in statistics because many variables have the approximate shape of a normal distribution—for example, height, blood pressure, lengths of objects produced by machines. This distribution is often useful as a description of data when there are a large number of observations. A normal distribution is a distribution of a continuous random variable (i.e., a vari- able that can take on an infinite number of values). The vertical axis for the normal distribution is the probability or likelihood of occurrence. (In contrast, on the histogram shown earlier [with company employees' salaries], the vertical axis was frequency of occurrence.)

Annualize

To convert a rate of any length into a rate that reflects the rate on an annual (yearly) basis. This is most often done on rates of less than one year, and usually does not take into account the effects of compounding. The annualized rate is not a guarantee but only an estimate, and its accuracy depends on the variance of the rate. This rate is also known as "annualized return" and is similar to "run rate".

Time Value of Money - example

Two basic time-value-of-money problems are finding the value of a set of cash flows now (present value problems) and as of a point of time in the future (future value problems)."How much money is needed today to produce a certain sum in the future given the rate of interest, r?"

Key aspects of Geometric Mean Return

Usually the geometric average/mean is lower than the calculated arithmetic average, even though the annual returns are identical. This is because the returns are compounded when calculating the geometric average return. Recall that compounding will result in a higher value, so a lower rate of return is required to get the final accumulation if compounding occurs. In fact, using the same set of numbers, the geometric average return is never greater than the arithmetic average return and is normally lower.

(CS) Median

[The value below which 50% of the cases fall; the middle number in a set of numbers that are listed in order] The median is the value such that there are as many observations that are greater than the median as there are observations that are less than the median. When observations are ranked in ascending order of size from the smallest to the largest, the median is the middle value. One advantage of the median over the arithmetic mean is that it is not sensitive to outliers; the median is usually a better measure of central tendency than the arithmetic mean when the data are skewed

(CS) Mode

[the most frequent value of a random variable] The mode is the most frequently occurring value in a data set. Benefits: The mode may be used as a measure of central tendency for data that have been sorted into categories or groups. For example, if all the employees in a company were asked what form of transport they used to get to work each day, it would be possible to group the answers into categories, such as car, bus, train, bicycle, and walking. The category with the highest number would be the mode. Problems with the mode: 1) it is often not unique, in which case there is no mode/agreed upon method to choose the representative value 2) The mode may also be difficult to compute if the data are continuous (data that can take on an infinite number of values between whole numbers—for example, weights of people) 3) The most frequently occurring observation may be far away from the rest of the observations, and thus does not meaningfully represent them.

qualitative

descriptive data / relating to or involving comparisons based on qualities

initial investment (outflow)

interest rate = coupon rate