CFA Fundamentals - Chapter 1 Quantitative Methods

Question 1

PEMDAS =

Answer 1

P

Parentheses first

E

Exponents (ie Powers and Square Roots, etc.)

MD

Multiplication and Division (left-to-right)

AS

Addition and Subtraction (left-to-right)

Question 2

When adding a positive and a negative number…

Answer 2

the sign of the sum is the same as the sign of the number with the largest absolute value

Consider the equation x = (–4) + (+3). Since the absolute value of negative four is greater than the absolute value of positive three, the sum has a negative sign. The answer to the equation is x = (–4) + (3) = –1.

Question 3

When subtracting one number from another…

Answer 3

the sign of the second number is changed, and the two numbers are added together.

Thus, 3 – (+4) = –1 can be rewritten as 3 + (–4) = –1. Similarly, 3 – (–4) = 7 can be rewritten as 3 + 4 = 7.

Question 4

1. Multiplying an even amount of negative numbers yields…
2. Multiplying an odd number of negative numbers yields

Answer 4

1. Multiplying an even amount of negative numbers yields a product with a positive sign.
2. Multiplying an odd number of negative numbers yields a product with a negative sign.

(–1) × (–1) = +1

(–1) × (–1) × (–1) = –1

(–1) × (–1) × (–1) × (–1) = +1

(–1) × (–1) × (–1) × (–1) × (–1) = –1

Notice the number of negative signs in each equation. You will see that when there is an even number, the product has a positive sign, and when there is an odd number, the product has a negative sign.

Question 5

Absolute Value?

Answer 5

Absolute value is the value of the number, ignoring the sign (e.g., the absolute value of +3 is 3, and the absolute value of –3 is also 3).

Question 6

We can multiply or divide all terms on both sides of an equation by the same letter or number…

Answer 6

We can multiply or divide all terms on both sides of an equation by the same letter or number without changing the relationship expressed by the equation or the value of the unknown variables.

a = 6

Let’s multiply both sides of Equation above by the number 5.

5 × a = 5 × 6 ⇒

5a = 30 ⇒

a = 6

By multiplying each side of Equation 1 by five, we did not alter the basic relationship between the left-hand and right-hand side of the equation.6 In fact, we could have multiplied every term in the equation by any number or any letter without changing the relationship expressed by the equation. In every case, the unknown value, signified by the letter a in Equation 1, would still equal 6.

We can also divide both sides of the equation by a number or letter without altering the relationship expressed by the equation.

Question 7

We can add (or subtract) the same letter or number from both sides of an equation…

Answer 7

We can add (or subtract) the same letter or number from both sides of an equation without changing the relationship expressed by the equation or the value of the unknown. Every time you move a variable or number from one side of the equation to the other, its sign changes (i.e., a negative becomes positive or a positive becomes negative).

a = 6

Let’s look again at Equation 1, and this time we will add and subtract a number from both sides.

Adding and subtracting one,

• a + 1 = 6 + 1

or

• a – 1 = 6 – 1

a + 1 = 6 + 1 ⇒

a = 6 + 1 – 1 ⇒

a = 6

a – 1 = 6 – 1 ⇒

a = 6 – 1 + 1 ⇒

a = 6

Question 8

We can formalize the relationship between the amount of money available to purchase apples, the cost of apples, and the number of apples purchased in the following algebraic equation:

Answer 8

1. A = (M/C)
2. Where
   
   1. A = the number of apples you can buy (the unknown)
   2. M = the maximum number of money you can spend ($1.00)
   3. C = the cost per apple ($0.25)
3. We could state the problem another way. How much would four apples cost if each apple costs $0.25. If apples cost $0.25 each, four apples will cost $1.00. This modification can be shown by rearranging Equation 1:
   
   1. M = AC ⇒ M = 4 × 0.25 = $1.00

Question 9

Here’s another algebra example related to apples. You have a total of $5.00 to spend. If the apples cost $0.79 per pound, how many pounds can you buy?

Answer 9

Letting:

1. P = the number of pounds you can buy (unknown)
2. C = the cost per pound ($0.79)
3. M = the total amount of money you can spend ($5.00)

P = (M/C) ⇒

P = ($5.00/$0.79) = $6.33

Now let’s alter this problem to determine the maximum amount of apples you can buy for $5.00. We know you cannot spend more than $5.00, but you can certainly spend less if you prefer. We also know that the number of pounds you buy, P, multiplied by the cost per pound, C, is the total amount you spend. Let’s set up that equation:

P × C ≤ M

Question 10

Multiplying or dividing an inequality by a negative number changes

Answer 10

the direction of the inequality.

Question 11

Parentheses.

Answer 11

Parentheses are used to group together variables and numbers that should be considered together in the equation.

Consider the following equation:

3(x + 4) = 15

That means we can divide both sides of the equation by three (using Rule 1) and solve for the unknown.

(x+4) = 5

x=1

Question 12

3 = 15 / (x + 4)

Answer 12

(x+4) x 3 = ((x + 4) x 15) / (x + 4) is the same as

3(x+4) = 15

Question 13

Distributing

Answer 13

3(x + 4) = 15 ⇒

3x + 12 = 15 ⇒ 3x = 15 – 12

3x = 3 ⇒ x = 1

OR

3x + 12 = 15 ⇒ 3(x + 4) = 15

3/3 (x+4) = 15/3 ⇒

(x+4) = 5

x + 4 = 5 ⇒ x = 5 – 4 ⇒ x = 1

Question 14

Exponents

Answer 14

Algebraic equations become slightly more complicated when an exponent affects one or more of the variables as in Equation 5:

x² = 9

Squaring a number simply means multiplying it by itself. For instance 2 squared is
2 × 2 = 4.

OR

Another way to designate square root is with the exponent ½. Let’s try solving the equation using that method. Instead of using the square root sign, or radical, we’ll use ½ as an exponent.

(x²)^1/2 – (9)^1/2

Question 15

The next logical step is to solve an equation in which we have a set of parentheses with an exponent

Answer 15

Remember that quantities within parentheses can be treated like a single term.

Consider the following equation:

(x + 5)³ – 8

Keep in mind that in order to solve for a variable, its exponent must equal one. In this example we can accomplish this by taking the cube root (third root) of both sides. Remember, we’re going to treat the quantity inside the parentheses just like a single term and multiply the exponents, as we did before.

[(x + 5)³]^1/3 =8^1/3 ⇒

x + 5 = 8^1/3 ⇒

x + 5 = 2 ⇒

x = –3

Question 16

System of Equations

Answer 16

A somewhat more advanced use for algebra is finding the value of two unknown variable terms. When dealing with two unknowns, we will use two equations, and we will solve them “simultaneously.”

Let’s say you are faced with the following two equations:

1. 3x + 4y = 20
2. x + 3y = –4

Even though it might seem like quite a daunting task, solving for these two unknowns, x and y, is actually just a matter of following a very logical series of steps while utilizing the rules of algebra.

Step 1: Start by setting up the equations as if you were going to add them together like two numbers.

3x + 4y = 20
+ x + 3y = –4
= ?

Step 2: Using Rule 1, we multiply all the terms in one or both of the equations so that adding or subtracting them will eliminate one of the variables. In our case we multiply the bottom equation by minus three:

3x + 4y = 20
-3 * (x + 3y) = –4
= ?

⇒

3x + 4y = 20

-3x - 9y = 12

=?

Step 3: Add the two equations together by adding the x terms, the y terms, and the numbers.

3x + 4y = 20
–3x – 9y = 12
0x – 5y = 32

Multiplying both sides of the resulting equation by negative one according to Rule 1, we get the following:
5y = –32
y = –32 / 5 = –6.4

Step 4: Having determined the value for one of the unknowns, we can insert it back into either of the original equations to solve for the second unknown. Using the first of the two original equations, we have:

3x + 4y = 20 ⇒ 3x + 4(–6.4) = 20 ⇒ 3x – 25.6 = 20
3x = 45.6
x = 15.2

Step 5: Plug both values back into the original equations to be sure your answers are correct. Does (3x = 4y = 20) and (x + 3y = –4)?

3x + 4y ⇒ 3(15.2) + 4 (–6.4) = 45.6 – 25.6 = 20
x + 3y = 15.2 + 3(–6.4) = 15.2 – 19.2 = –4

Question 17

Lump Sum

Answer 17

A lump sum is a single cash flow. Lump sum cash flows are one-time events and therefore are not recurring.

Question 18

Annuity

Answer 18

An annuity is a finite number of equal cash flows occurring at fixed intervals of equal length over a defined period of time (e.g., monthly payments of $100 for three years).

Question 19

Present Value

Answer 19

Present value is the value today of a cash flow to be received or paid in the future. On a timeline, present values occur before (to the left of) their relevant cash flows.

Question 20

Future Value

Answer 20

Future value is the value in the future of a cash flow received or paid today. On a timeline, future values occur after (to the right of) their relevant cash flows.

Question 21

Perpetuity

Answer 21

A perpetuity is a series of equal cash flows occurring at fixed intervals of equal length forever.

Question 22

Discount and Compounding Rate

Answer 22

The discount rate and compounding rate are the rates of interest used to find the present and future values, respectively.

Question 23

Future Value of Lump Sum for One Year

Answer 23

We’ll start our discussion with the future value of a lump sum. Assume you put $100 in an account paying 10% and leave it there for one year. How much will be in the account at the end of that year? The following timeline represents the one-year time period.

• FV₁ = $100 + (i × $100)

Since $100 is the present value, the original deposit, we can substitute PV0 for the $100 in the equation.

• FV₁ = PV₀ + (i × PV₀)

Factoring PV0 out of both terms on the right side of the equation we are left with:

• FV₁ = PV₀ + (i × PV₀)
• FV₁ = PV₀ + iPV₀)
• FV₁ = PV₀ (i + 1)

Question 24

Future Value of Lump Sum for Two Years

Answer 24

We know from Equation 1 that the future value of a lump sum invested for one year at interest rate i is the lump sum multiplied by (1 + i ). We simply find the future value of $110 invested for one year at 10% by using Equation 1 (adjusted for the different points in time), which gives us $121.00.

1. FV₁ = PV₀(1 + i)
2. FV₂ = FV₁(1 + i)
3. FV₂ = $110(1.10) = $121
4. We know from Equation 1 that FV1 is equal to PV1(1 + i ).
5. We start with: FV₂ = FV₁(1 + i)
6. FV₂ = PV₀(1 + i)(1+i)
7. FV₂ = PV₀(1 + i)²

Question 25

Equation 3 says the future value of a lump sum invested for n years at interest rate i is the lump sum multiplied by (1 + i)ⁿ. Let’s look at some examples. We’ll assume an initial investment today of $100 and an interest rate of 5%.

Answer 25

Future value in 1 year: $100(1.05) = $105

Future value in 5 years: $100(1.05)⁵ = $100(1.2763) = $127.63

Future value in 15 years: $100(1.05)¹⁵ = $100(2.0789) = $207.89

Future value in 51 years: $100(1.05)⁵¹ = $100(12.0408) = $1.204.08

Regardless of the number of years, as long as the interest rate remains the same, the relationship in Equation 3 holds.

Up to this point we have assumed interest was paid annually (i.e., annual compounding).

Question 26

Again, let’s assume that we deposit $100 at time zero, and it remains in the account for one year. This time, however, we’ll assume the financial institution pays interest semiannually. We will also assume a stated or nominal rate of 10%, meaning it will pay 5% every six months.

Answer 26

The value after the first six months is the original deposit plus 5% interest, or $105. The value after another six months (one year from deposit) is the $105 plus interest of $5.25 for a total of $110.25.

Equation 2 is actually the format for finding the FV of a lump sum after any two periods at any interest rate, as long as there is no compounding within the periods. The periods could be days, weeks, months, quarters, or years. To find the value after one year when interest is paid every six months, we multiplied by 1.05 twice. Mathematically this is represented by:

FV = $100(1.05)(1.05) = $100(1.05)²

FV = $100(1.1025) = $110.25

Question 27

Mutliple Years and Multiple Compounding Periods Future Value

Answer 27

FV_n = PV₀ (1 + (i/M)) ^{m x n}

FVn = the future value after n years
PV0 = the present value
i = the stated annual rate of interest
m = the number of compounding periods per year
m × n = the total number of compounding periods
(the number of years times the compounding periods per year)

EXAMPLE

n = 4, because you will leave the money in the account for four years
m = 2, because the bank pays interest semiannually
i = 10% (the annual stated or nominal rate of interest)

FV = $100 (1 + (0.10/2))^{4 x 2}

FV = $100 (1.05)⁸

FV = $100(1.4775) = $147.75

Question 28

Increasing Number of Compounding Periods

Answer 28

m = 1 (annually) FV = $100(1.10) = $110
m = 2 (every 6 months) FV = $100(1.05)2 = $110.25
m = 4 (quarterly) FV = $100(1.025)4 = $110.38
m = 6 (every 2 months) FV = $100(1.0167)6 = $110.43
m = 12 (monthly) FV = $100(1.008333)12 = $110.47
m = 52 (weekly) FV = $100(1.001923)52 = $110.51
m = 365 (daily) FV = $100(1.000274)365 = $110.52

You will notice two very important characteristics of compounding:

1. For the same present value and interest rate, the future value increases as the number of compounding periods per year increases.
2. Each successive increase in future value is less than the preceding increase. (The future value increases at a decreasing rate.)

Question 29

Effective Interest Rates.

Answer 29

The stated (nominal) rate of interest is 10%. However, determining the actual rate we earned involves comparing the ending value with the beginning value using Equation 5. You can determine the actual or “effective” rate of return by taking into consideration the impact of compounding. Equation 5 measures the change in value as a percentage of the beginning value.

Effective Return =

V₁ - V₀

V₀

_{where:<br></br>V0 = the total value of the investment at the beginning of the year<br></br>V1 = the total value of the investment at the end of the year}

You will notice Equation 5 stresses using the values at the beginning and the end of the year (actually, any twelve month period). By convention, we always state effective interest rates in terms of one year.

10% Annual Rate Compounded Semi-Annually

effective return = ($110.25 - $100) / $100 = 0.1025 = 10.25%

OR

(V₁ / V₀) - 1

($110.25 / $100) - 1 = 1.1025 - 1 = .1025 * 100 = 10.25%

OR

(FV₁ / PV₀) - 1

By substituting System of Equations, we get

Effective Return = (PV₀ (1 + (i/m)) ^{m x 1}) / PV₀) - 1

Effective Return = (1 + (i/m)^{<span>m</span>}

We have arrived at the general equation to determine any effective interest rate in terms of its stated or nominal rate and the number of compounding periods per year. Let’s investigate a few examples of calculating effective interest rates for the same stated interest at varying compounding assumptions. Notice that the effective rate increases as the number of compounding periods increase.

• m = 1 (annual compounding)
  
  • (1 + (.12/1))¹ - 1 =
  • (1.12)¹ - 1 =
  • 12%
• m = 2 (semiannual)
  
  • (1 + (.12/2))² - 1 =
  • (1.06)² - 1 =
  • 12.36%
• m = 4 (quarterly)
  
  • (1 + (.12/4))⁴ - 1 =
  • (1.03)⁴ - 1 =
  • 12.55%
• m = 12 (monthly)
  
  • (1 + (.12/12))¹² - 1 =
  • (1.01)¹² - 1 =
  • 12.68%
• m = 365 (daily)
  
  • (1 + (.12/365))³⁶⁵ - 1 =
  • (1.0003288)^{<span>365</span>} - 1 =
  • 12.75%

Question 30

Geometric Mean Return.

Answer 30

Geometric Mean Return. The geometric mean return is a compound annual growth rate for an investment. For instance, assume you invested $100 at time 0 and that the investment value grew to $220 in 3 years. What annual return did you earn, on average? Using our future value formula from Equation 3:

100(1+i)³ = 220

More formally, the geometric mean is found using the following equation:

Geometric Mean = (FV_n/PV)^(1/n) - 1

1. GM = the geometric mean
2. FV_n = future value of the lump sum investment
3. PV = present value, or the initial lump sum investment
4. n = the number of years over which the investment is held

to solve this problem, divide both sides of the equation by 100, then take the third root and subtract 1;

• i = [220/100]^(1/3) –1

Thus, if you invested $500 in an mutual fund five years ago and now the original investment is worth $901.01, we would find the geometric mean return as follows:

• ($901.01/$500)^(1/5) - 1 =
• (1.80202)^(1/5) - 1 =
• 1.125 - 1 =
• 12.5%

Geometric Mean = The geometric mean can also be stated using returns over several periods of equal length using the following formula:

1. ((1 + x₁)(1 + x₂)(1 + x₃)…(1 + x_n))^(1/n) - 1

Where:

GM = the geometric mean
x<sub>i</sub> = the i<sup>th</sup> return measurement (the first, second, third, etc.)
n = the number of data points (observations)

Let’s return to our mutual fund example. This time we will calculate the geometric mean return differently. Assume that over the last five years the fund has provided returns of 15%, 12%, 14%, 16%, and 6%. What was the geometric mean return for the fund?

GM = ((1 + 0.15)(1 + 0.12)(1 + 0.14)(1 + 0.16)(1 + 0.06))^(1/5) - 1

GM = (1.80544)^1/5 - 1 = 0.125

GM = 12.5%

The geometric mean shows the average annual growth in your cumulative investment for the five years, assuming no funds are withdrawn. In other words, the geometric mean assumes compounding. In fact, when evaluating investment returns, the geometric mean is often referred to as the compound mean.

Question 31

Discounting Value

Answer 31

Finding a present value is actually deducting interest from the future value, which we refer to as discounting. The present value can be viewed as the amount that must be invested today in order to accumulate a desired amount in the future. The “desired amount in the future” is known as the “future value.” Returning to the future value examples used earlier, we can demonstrate how to calculate present values. We will assume the same discount rate of 5%.

• (FV) The value in 1 year of $100 deposited today:
  
  • $100(1.05) = $105
• (PV) The value today of $105 to be received in 1 year
  
  • $100 = $105/(1.05)
• (FV) The value in 5 years of $100 deposited today:
  
  • $100(1.05)⁵ = $105
• (PV) The value today of $127.63 to be received in 5 years:
  
  • $100 = $127.63/(1.05)⁵
• (FV) The value in 15 years of $100 deposited today:
  
  • $100(1.05)¹⁵ = $207.89
• (PV) The value today of $207.89 to be received in 15 years:
  
  • $100 = $207.89/(1.05)¹⁵
• (FV) The value in 51 years of $100 deposited today:
  
  • $100(1.05)⁵¹ = $1,204.08
• (PV) The value today of $1,204.08 to be received in 51 years:
  
  • $100 = $1,204.08/(1.05)⁵¹

Question 32

Annuities

Annuity Due?

Ordinary Annuity?

Fully Amortized?

Answer 32

Recall that an annuity is a series of equal payments that occur at fixed intervals of equal length through time.

Annuity Due

1. When cash flows come at the beginning of the period, the annuity is known as an annuity due
2. An annuity due is typically associated with leases or other situations where payments are made in advance of services or products received or rendered. Our goal is to determine the amount we will have in the account at point five (i.e., the end of year five).
3. Consider an equipment lease. The user of the equipment (the lessee) pays the owner of the equipment (the lessor) a fee for the right to use the equipment over the next period. That is, leases are prepaid.

Ordinary Annuity

1. An example of an ordinary (or deferred) annuity is a mortgage loan (or any debt) with end of period payments.
2. When the money is borrowed, the borrower makes payments at the end of set periods, usually every month or semi-annually.

Fully Amortized

1. When a mortgage is fully amortized, each of the fixed payments includes interest on the outstanding principal as well as a partial repayment of principal. Since the outstanding principal decreases with each payment, the proportion of each successive payment representing interest decreases, and the proportion representing principal increases

Question 33

Summation of Annuities:

Future Value vs. Present Value

Answer 33

Future Value

1. The process of finding the future value of an annuity in this manner is equivalent to summing the future value of each individual cash flow.
2. The cash flow stream is illustrated below. Each cash flow is assumed to earn a 10% return for each of the indicated number of years. For example, The sum of the Ordinary Anuity:
   
   1. Deposit at End of Year 5 = $100.00
   2. Deposit at End of Year 4 = (1 + i)¹ ​= $110.10
   3. Deposit at End of Year 3 = (1 + i)² ​= $121.00
   4. Deposit at End of Year 2 = (1 + i)³ ​= $133.10
   5. Deposit at End of Year 1 = (1 + i)⁴ ​= $146.41
   6. TOTAL = $610.51
3. Future value of the ordinary annuity is less than the future value of the annuity due, although the number and amount of the deposits is the same. The difference results from the fact that the deposits for the annuity due are all received exactly one period earlier than the corresponding deposits for the ordinary annuity.

Present Value

1. We’ll start with the annuity due. The cash flows are assumed to be paid/received at the beginning of each year, and we want to find the aggregate present value of the five cash flows at point zero on the timeline. Again we assume an interest rate of 10%. The process of finding the present value of an annuity due is equivalent to summing the present value of each individual cash flow. The cash flow stream is illustrated in Figure 8. Each cash flow is discounted at a 10% rate for each of the indicated number of years.
   
   1. Beginning of 0 = $100.00
   2. Beginning of 1 = (1 + i) ​= $121.00
   3. Beginning of 2 = (1 + i)³ ​= $133.10
   4. Beginning of 3 = (1 + i)⁵ ​= $161.05
   5. Beginning of 4 = (1 + i)⁵ ​= $161.05
   6. TOTAL = $671.56

Question 34

Financial Calculator for Ordinary Annuity and Annuity Due

Answer 34

Ordinary annuity: Your calculator should be set to end of period payments and one payment per year. To set to end of period payments, press 2nd ⇒ BGN and press 2nd ⇒ SET until END is displayed, then 2nd ⇒ QUIT. (Since end is default, the display will not indicate end of period payments.) To set to one payment per year, press 2nd ⇒ P/Y ⇒ 1 ⇒ ENTER, 2nd ⇒ QUIT. The keystrokes to find the future value are the following:

PMT = -100

• [The calculator assumes one of the payments is an outflow (from the perspective of the investor) and one is an inflow. The negative sign indicates an outflow (the deposit)].

N = 5
I/Y = 10
FV = $610.51

Annuity due. For an annuity due, set the financial calculator for beginning-of-period payments and one payment per year. To set to one payment per year, press 2nd ⇒ P/Y ⇒ 1 ⇒ ENTER, 2nd ⇒ QUIT. To set to end-of-period payments, press 2nd ⇒ BGN and press 2nd ⇒ SET until BGN is displayed, then 2nd ⇒ QUIT. (BGN will show in the calculator display.) The keystrokes to find the future value are the following:

PMT = -100

• [The calculator assumes one of the payments is an outflow (from the perspective of the investor) and one is an inflow. The negative sign indicates an outflow (the deposit)].

N = 5
I/Y = 10
FV = $671.56

Question 35

Interpretations of Annuities

Answer 35

BGN

PMT = -100
N = 5
I/Y = 10
PV = $416.99

There are several ways of interpreting the $416.99. The $416.99 is the present value of the five $100 payments/receipts, but what does “present value” really mean?

1. A simple interpretation is that if you put $416.99 in an account paying 10% interest, you will be able to withdraw $100 per year for five years
2. Another somewhat more sophisticated interpretation is that $416.99 is the maximum you would pay for an investment paying $100 per year with a required return of 10%.
3. A third interpretation is that if you borrow $416.99 to be paid in five equal annual payments, you will pay $100.00 per payment. Regardless, assuming a 10% interest rate, $416.99 today is equivalent to five annual $100 cash flows, the first cash flow occurring today.

Question 36

Amortized Loan

Answer 36

When both principal and interest are included in a loan payment (rather than a series of interest payments followed by the repayment of principal at the maturity of the loan), we say the loan is fully amortized. Let’s use the ordinary annuity of $379.08 as an example.

Let’s assume you have borrowed $379.08 for the purchase of a household item and have agreed to pay for it in five equal payments at 10% interest. We know from the previous example that the payments will be $100 each.

Mortgage Loan (Ordinary or Deferred Annuity) = END

1. The keystrokes to find the present value of an ordinary annuity are [be sure to turn BGN mode off (i.e., you are in END mode)]:

Find the Present Value since that is the ammount you can borrow today:

1. End of Year 1 = $100 / (1 + i) = 90.91
2. End of Year 2 = $100 / (1 + i)² = 82.64
3. End of Year 3 = $100 / (1 + i)³ = 75.13
4. End of Year 4 = $100 / (1 + i)⁴ = 68.60
5. End of Year 5 = $100 / (1 + i)⁵ = 62.90
6. Present Value = $379.08

Amortized Loan

1. Each of the five payments repays a portion of the principal borrowed and pays interest on the balance remaining after the previous payment. The first payment includes 10% interest on the entire loan amount of $379.08, or $37.91
2. The remainder of the payment ($100 – $37.91 = $62.09) is applied to the principal, leaving a balance of $316.99. The second payment includes 10% interest on the new balance of $316.99, or $31.70
3. Again, the remainder of the payment, $68.30, is applied to the principal. This process continues until the loan is fully paid. You can see in Figure 5 that the interest in each payment decreases while the principal increases.
4. Payment #1 = $100
   
   1. Interest = $37.91
   2. Principal = $62.09
   3. Remaining Balance = $316.99
5. Payment #2 = $100
   
   1. Interest = $31.70
   2. Principal = $68.30
   3. Remaining Balance = $248.69
6. Payment #3 = $100
   
   1. Interest = $24.87
   2. Principal = $75.13
   3. Remaining Balance = $173.56
7. Payment #4 = $100
   
   1. Interest = $17.36
   2. Principal = $82.64
   3. Remaining Balance = $90.92
8. Payment #5 = $100
   
   1. Interest = $9.09
   2. Principal = $90.91
   3. Remaining Balance = $0.0

Question 37

Present Value of a Perpetuity

Answer 37

As explained earlier, a perpetuity is a series of equal cash flows occurring at the same interval forever. The present value of a perpetuity equals the periodic cash amount divided by the discount (interest) rate. For example, assume you own a stock paying $2 dividends forever. Further, assume the appropriate discount rate on the stock is 10%. The present value of the perpetuity equals $2 divided by 10%, which equals $20. In other words, you should be willing to pay $20 for a stock expected to pay $2 dividends per year forever, using a discount rate of 10%. More formally, we can state the formula for finding the present value of a perpetuity as follows:

• Present Value of Perpetuity = CF / i

Were:

• CF = the periodic cash flows of the perpetuity
• i = the discount rate

Question 38

A population

What is the important characteristic of Population?

Sample Population?

What is the important characteristic of Sample Population?

Answer 38

Population

is the collection of all possible individuals, objects, measurements, or other items; (e.g., the population of the U.S. is all people who call the U.S. their home country). The population of carp in the world is not just those that live in beautifully maintained and landscaped pools in Japan. The carp population would include every single carp, no matter where it lives.

Important Characteristic of Population?

An important characteristic of a population is its size. For instance, there are approximately 300 million people living in the U.S. That means there are about 300 million members of that particular population. If you wanted to estimate the average height of an American, you would not want to go around to every single person and measure his or her height. Even if you could afford the extreme cost in both time and money, it would be a logistical nightmare. How about weighing every carp in the world to find the average weight of an adult carp?

Sample?

A sample is a portion or subset of a population that is used to estimate characteristics of (i.e., make inferences about) the population. If we were interested in the average height of a U.S. citizen, we could select people from all over the U.S. (a sample of people), measure them, and find the average height. The average height of the individuals in the sample is then used to infer the average height of all people in the United States.

Important Characterisitc of Sample Population?

1. An important characteristic of samples is randomness. You don’t want to force the sample to yield statistics that are biased because of the way the sample is taken. For example, if you are trying to estimate the percentage of people in the U.S. who are over age 65, you would not take your sample observations24 from a retirement community. If you were to do that, you could estimate that nearly 100% of the U.S. population is over 65! An appropriate sample would be drawn from many different areas across the country in a random, unbiased way.
2. Another characteristic important to the sample is size. When an extremely large sample is drawn, the costs can be very high, and the inferences not significantly stronger than those of a somewhat smaller sample. However, if the sample is too small, the inferences drawn from the sample may not be trustworthy. Even though we will not pursue ideal sample size in this chapter, it is important to remember that sample size is very important to the value (the confidence) you can place on the inferences you make about a population.

Question 39

Variables

Answer 39

A variable is an unknown quantity (measurement) that can have different values. For example, if you were estimating average height and measured every person in your sample,

1. the first value of the variable “height” would be the height of the first person measured.
2. The second value would be the height of the second person;
3. the third value would be the height of the third person, and so on.

The variable “height” would have as many values (observations) as there are people in your sample. For example, we might use the letter x to denote height. x1 is the notation used to denote the height of the first person sampled. x2 is the notation used to denote the height of the second person sampled, etc. For instance, if the first person is 70 inches tall, then x1 = 70. If the second person is 71 inches tall, then x2 = 71.

Question 40

Qualitative and Quantitative Variables

Quantiative Discrete vs. Continuous?

Answer 40

A qualitative variable measures attributes. These could include gender, religious preference, eye color, type of running shoe preferred, and place of birth. In other words, qualitative variables do not use numbers

Conversely, quantitative variables are expressed numerically. These could include the average number of children in the typical household, the average height of American females, the percentage of people in the population with false teeth, or the average number of computers sold daily.

Quantitative variables can be categorized as discrete or continuous.

1. Think of discrete as meaning that the variable can only have a countable number of easily identified values. If the variable can only take on a whole number value from 1 to 10, it would be considered discrete. Its only possible values are 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10. You’ll notice that you can easily count each possible value.
2. Now let’s say the variable can have any incremental value between 1 and 10. In this case the variable can assume an infinite number of possible values and is called a continuous variable. When the variable was discrete, two of its possible values were three and four, and it could not have a value between three and four. As a continuous variable, it can have the values three and four, but it can also take on any of the infinite values between three and four.

EXAMPLES:

1. An example of a discrete variable is the outcome of the roll of a die (1, 2, 3, 4, 5, or 6).
2. An example of a continuous variable is the amount of rainfall during July in a city.

Question 41

Frequency Distribution

Answer 41

Frequency Distribution

Often, we do not want to see all the data points (especially if the sample size is large) but rather are interested in seeing merely a summary of the data. One popular way to summarize the data is to tabulate the frequency of observations falling in various categories. The tally of observations falling in equally spaced intervals is called the frequency distribution of the data. The frequency distribution shows how the data are scattered. For instance, consider the following frequency distribution.

Height Interval (inches) and Frequency Tally
64 up to 66 = 10
66 up to 68 = 35
68 up to 70 = 40
70 up to 72 = 10
72 up to 74 = 5

Question 42

Frequency Distribution Histogram

Answer 42

Question 43

Frequency Distribution Histogram and SKEW

Answer 43

A graph of a frequency distribution is called a histogram. Notice how the histogram illustrates how the data are scattered. There is a center (at the 3rd category) with a pattern around the center. In this example, the distribution is skewed, meaning that there are not an equal number of observations on either side of the middle interval. Alternatively, a symmetric distribution is one in which there are an equal number of observations on either side of the middle interval (i.e., follows a bell-shape).

To describe the data further, we often want to find the center point of the data. The center point of the data is known as the central tendency of the distribution. There are a few statistical measures of central tendency, the most popular of which are the mean, median, and mode.

Question 44

Mean

Answer 44

Mean is just another word for average, or the center of the data. For instance, when we previously mentioned estimating the average height, we could have used the expression “mean height.” The most common measure of central tendency is the arithmetic mean.

Earlier, to estimate the average height of the U.S. population, we took a sample of people and measured their heights. By substituting the data (individual measurements) into the equation to find the sample mean, we get the following:

• 70, 71, 73, 66, 62, 70 =
• Arithmetic Mean = 68.7

Question 45

Median

Why do we need so many measures of the center of the distribution in the first place?

Answer 45

The median is the middle observation of the ranked data. If we ranked our height observations from the smallest to the largest, the same number of observations will fall above and below the median value. Our observations are 62, 66, 70, 70, 71, and 73. The median of the six observations falls between the third- and fourth-ranked observations. Thus, the simple average of the third and fourth observations, 70, represents the median of our sample. There are two observations greater than 70 (71 and 73), and there are two observations less than 70 (62 and 66).

Question 46

Why do we need so many measures of the center of the distribution in the first place?

Answer 46

This question deserves some attention. The median finds the center of the distribution by number of observations. There are an equal number of observations above and below the median, regardless of their values. The mean, on the other hand, actually adds all the observations together and divides by the number of observations to find the mathematical center

The reason we calculate both the mean and median of a data set is to get an idea of where the true center lies.

Since the median and mean find the center in different ways, a more accurate estimation of where the center is and what is influencing its location can be gained by observing both the mean and median.

Question 47

Mode

Answer 47

The mode is the observation that appears most often. In our case both the median and the mode are 70. The mean, median, and mode are all measures of central tendency. They all locate the center of the observations or population.

Question 48

Measures Of Dispersion

Answer 48

1. range
2. mean
3. absolute deviation
4. standard deviatio

Question 49

Range

Question 50

Mean Absolute Deviation

Answer 50

The mean absolute deviation is a measure of the dispersion of the sample observations around the center of the distribution. It measures the average deviation from the mathematical mean. A deviation is measured as the distance from the mean to each observation.

With a mean of 68.7”, the deviations for our sample are:

70. 0” – 68.7” = 1.3”
71. 0” – 68.7” = 2.3”
72. 0” – 68.7” = 4.3”
73. 0” – 68.7” = –2.7”
74. 0” – 68.7” = –6.7”
75. 0” – 68.7” = 1.3”

Mean Absolute Deviation =

(1.3 + 2.3 + 4.3 + 2.7 + 6.7 + 1.3) / 6 = 3.1

The mean absolute deviation for our sample is 3.1 inches. Therefore, on average, the sample observations fall 3.1 inches from the sample mean. If the mean absolute deviation had been 1.0 inch, the observations would be much more closely grouped around the mean, or less dispersed. Had the mean absolute deviation been 6 inches, the observations would be more spread out or dispersed.

Question 51

1. Variance
2. Standard Deviation
3. Mean Absolute Deviation

Answer 51

Both the variance and standard deviation are measures of dispersion of the data around the mean of a distribution. The calculation of the variance is similar to the mean absolute deviation calculation. Once again we begin with the deviations of each data point from the mean. Instead of averaging the deviations, however, we average the squared deviations. This also gives us all positive numbers because squaring a negative number results in a positive product.

After we calculate the variance for the population, we can find the standard deviation. The standard deviation, σ, is simply the square root of the variance, σ².

If we assume that our earlier data set of heights for six people included the entire population, the standard deviation would be calculated as follows:

(((70-68.7)² + (71 - 68.7)² + (73 - 68.7)²​ + (66 - 68.7)²​ + (62 - 68.7)²​ + (70 - 68.7)² ​)^(1/2) ) / 6

σ = (1.69 + 5.29 + 18.49 + 7.29 + 44.89 + 1.69)^(1/2) / 6

σ = 3.64

Thus, the standard deviation of our population is 3.64 inches.

Question 52

Sample vs. Population Variance and Standard Deviation

Answer 52

The equations for variance and standard deviation of a sample are slightly different than those for the population. Since a sample is smaller than the population, finding the arithmetic mean of the squared deviations tends to underestimate the true value. We have to adjust the formula slightly to account for this effect. The variance for a sample is as follows:

S² = (Σ (x - Avg x)²) / (n - 1)

where:
s2 = the sample variance
x = an individual observation in the sample
Avg x = the sample mean
(n – 1) = the number of sample observations minus 1

(((70-68.7)² + (71 - 68.7)² + (73 - 68.7)^2​ + (66 - 68.7)²​ + (62 - 68.7)²​ + (70 - 68.7)² ​)(1/2) ) / (6 -1)

σ = (1.69 + 5.29 + 18.49 + 7.29 + 44.89 + 1.69)(1/2) / 5

σ = 3.98

Intiuition behind unbiased estimate

• Dividing by a smaller number gives us a better estimate of the true population variance because it creates a bigger variance
• Think of a number line where you have 14 numbers that means Population is N
• Sample Size is 3
• True Population Mean is in the middle of the total 14 numbers.
• If you take the mean of the sample size, there is a possibility that your sample mean is equal to the True Population Mean
• BUT
• Distinct possibility that you sample three low numbers and we aren’t close
• Your population mean could be outside of the sample.
  *

Question 53

The Normal Distribution and the Empirical Rule for Probabilities

Answer 53

Used extensively in financial analysis, the normal distribution is the famous “bell-shaped curve” of which statisticians and others speak.

In a normal distribution, the mean, median, and mode are equal.

Although the normal distribution may not precisely describe the pattern of returns for common stocks, for example, it is commonly used because of its statistical properties, especially symmetry.

Perfect symmetry implies:

1. The center point of the distribution is the mean, median, and mode
2. 50% of all possible returns are to the right of (above) the mean
3. 50% of all possible returns are to the left of (below) the mean

Of particular interest are the symmetry of the distribution and the counting of standard deviations from the mean. To the right of (above) the mean, lines are drawn that would approximate the returns within one and two standard deviations of the mean. Symmetry implies we can include lines on the other side of (left of) the mean that are the same distance from the mean as their counterparts to the right.

Question 54

Normal Distribution - likelihood of certain events

Question:

For instance, assume you have been hired to manage a portfolio for a client. Assume your portfolio has an expected mean annual return of 10% and a standard deviation of 5%. Your client asks you to estimate the probability that she will lose money next year on your portfolio.

Answer 54

In many business settings, we wish to estimate the likelihood of certain events (e.g., likelihood of positive earnings, of debt rating change, etc). A helpful rule for estimating the likelihood of events using the normal distribution is that

1. approximately 68% of the data lie within 1 standard deviation of the mean, and
2. 95% of the data lie within 2 standard deviations of the mean.

Question/Ansnwer:

You can use the empirical rule to answer the question. For instance, the empirical rule states that there is approximately a 95% chance that next year’s portfolio return will lie within 2 standard deviations of the mean. Since the mean is 10% and the standard deviation is 5%, then 95% of the time the annual return will range from 0% to 20% (i.e., the mean return plus and minus 2 standard deviations). The remaining 5% of the time, the portfolio return will lie either below 0% or above 20%. If we assume the portfolio returns are symmetric (equal probabilities for returns above and below the mean), there is a 2.5% chance the return will lie below 0% and another 2.5% chance the return will lie above 20%. This client should feel pretty safe about her investment: There is only a 2.5% chance she will lose money next year.