CFA Fundamentals

385 Pages·2007·4 MB·English

by  Ahmad Rizki

CFA Fundamentals—The Schweser Study Guide to Getting Started
Chapter One: Quantitative Methods
Overview
In financial analysis, you will need to determine the present and future value of predicted future cash flows.
You will need an understanding of probability distributions and measures of central tendency and
dispersion. This chapter is designed to prepare you for those challenges.
In Section 1 of this chapter, you will learn fundamental concepts regarding the time value of money. In
Section 2 you will learn the meaning and use of many of the most important terms in statistics. In Section 3
you will learn the principles of regression analysis.

Chapter Objective: Review the properties of negative numbers.
Chapter Objective: Discuss the basic form of an equation.
Chapter Objective: Multiply and divide both sides of an equation by a constant.
Chapter Objective: Add and subtract a constant from both sides of an equation.
Chapter Objective: Solve an equation with parentheses.
Chapter Objective: Solve equations containing terms with exponents.
Chapter Objective: Solve two equations with two unknowns.

Section 2: Time Value Of Money
One of the most important tools the financial services professional has is the ability to calculate present and
future values. In addition, the competent financial services professional is very comfortable calculating the
amortization of a loan, the payout from an insurance annuity, or the annual investments necessary to
achieve the desired funds available at retirement.
In this section we introduce and explore the tools necessary for making such calculations.
An annuity is a finite number of equal cash flows occurrring at equal intervals over a defined period of time.
Those intervals could be single days, weeks, months, years, etc. A perpetuity is an infinite annuity (i.e., an
annuity that continues indefinitely).

Terminology
In this section, we will utilize timelines to calculate the present and future values of lump sums and
annuities. Timelines will help you keep cash flows organized and allow you to see the timing of one cash
flow in relation to the other cash flows and in relation to the present (i.e., today). Although certainly not a
requirement for producing time value of money calculations, timelines are invaluable in visually identifying
the timing of cash flows. Before we set up a timeline, however, let’s look at more of the terms we will utilize
throughout the discussion.
A lump sum is a single cash flow. Lump sum cash flows are one-time events and therefore are not
recurring.

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A very simplistic rule that can be used to keep present and future values separate in your mind is this: The
present value will always fall to the left of its relevant cash flows, and the future value will always fall to the
right.

An annuity is a finite number of equal cash flows occurring at equal intervals over a defined period of time
(e.g., monthly payments of $100 for three years).
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.
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.
A perpetuity is a series of equal cash flows occurring at the same interval forever.

Think of discounting as removing or subtracting value, and think of compounding as increasing or adding
value.

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

Chapter Objective: Calculate and interpret the present and future values of a lump sum.

Lump Sums
Future Value of a Lump Sum
We’ll start our discussion with the future value of a lump sum. Assume you put $100 in an account paying
10 percent 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.

Figure 2: Determining Future Value at t = 1
In one year, you’ll have $110. That $110 will consist of the original $100 plus $10 in interest. To set that up
in an equation, we say the future value in one year, FV1, consists of the original $100 plus the interest, i, it
earns.
FV = $100 (i x $100)
1
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Since $100 is the present value, the original deposit, we can substitute PV for the $100 in the equation.
0
©2006 Schweser Study Program
FV = PV + (i PV )
1 0 0
Factoring PV out of both terms on the right side of the equation we are left with:
0
FV = PV (1 + i) (1)
1 0

The effect of compounding is powerful because it allows an investment to earn interest not only on the
principal but also on the interest earned in previous periods. This is referred to as earning interest on
interest.

The result of these mathematical manipulations is the general equation for finding the future value of a lump
sum invested for one year at a rate of interest i. Had it not been so easy to do the calculation in our heads,
we would have substituted for the variables in the equation and gotten:
FV = $100(1.10) = $110
1
What if you leave the money in the account for two years? After one year you’ll have $110, the original
$100 plus $10 interest (or 10% times 100). At the end of the second year you will have the $110 plus
interest on the $110 during the second year. The interest earned in the second year equals 10% times the
$110 balance with which you began the year. Therefore, the interest earned in the second year consists of
interest on the original $100 plus interest earned in the second year on the interest earned during the first
year but left in the account. When interest is earned or paid on interest, the process is referred to as
compounding. This explains why future values are sometimes referred to as compound values.
Now our timeline expands to include two years. Although the numbering is totally arbitrary and we could
have used any number to indicate today, we are assuming we deposit $100 at time 0 on the timeline. We
already know the value after one year, FV1, so let’s start there.

Figure 3: Determining Future Value at t = 2
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 percent
by using Equation (1) (adjusted for the different points in time), which gives us $121.00.
FV =PV (1 + i)
1 0
FV = FV (1+ i)
2 1
FV = $110(1.10) = $121
2
To take this example a step further, we make some additional adjustments. We know from Equation (1) that
FV1 is equal to PV0(1 + i). Let’s further develop relationships between future and present value.
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We start with: FV = FV (1+ i)
2 1

Substituting, we get: FV = PV (1+ i)(1 + i)
2 0
And we end with: FV = PV (1+ i)2 (2)
2 0
Equation (2) is the general equation for finding the future value of a lump sum invested for two years at
interest rate i. In fact, we have actually discovered the general relationship between the present value of a
lump sum and its future value at the end of any number of periods, as long as the interest rate remains the
same. We can state the general relationship as:
FV = PV (1+ i)2 (3)
2 0

Variable interest rate securities can also benefit from compound interest. The calculation of the compound
interest is more complicated, however, since the interest rate for each period is not known in advance.

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 percent.
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). However, most financial institutions pay and charge interest over much shorter periods. For
instance, if an account pays interest every six months, we say interest is “compounded” semiannually.
Every three months represents quarterly compounding, and every month is monthly compounding. Let’s
look at an example with semiannual compounding.

The nominal rate is stated in the contract and does not include the effects of compounding or fees, such as
closing fees on a mortgage.

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 percent, meaning it will pay 5 percent every six months.
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Figure 4: Future Values With Semi-Annual Compounding

Present value and future value formulas can be adjusted to accommodate any compounding period by
dividing the annual interest rate by the number of compounding periods per year and multiplying the
number of years by the number of compounding periods per year.

In order to find the future value in one year, we must first find the future value in six months. This value,
which includes the original deposit plus interest, will earn interest over the second six-month period. The
value after the first six months is the original deposit plus 5 percent 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.
The similarity to finding the FV in two years as we did in Equation (2) is not coincidental. 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
The process for semiannual compounding is mathematically identical to finding the future value in two years
under annual compounding. In fact, we can modify Equation 2 to describe the relationship of present and
future value for any number of years and compounding periods per year.
(4)

where:
FV = the future value after n years
n
PV = the present value
0
i = the stated annual rate of interest
m = the number of compounding periods per year
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m (cid:215) n = the total number of compounding periods (the number of years times the
compounding periods per year)
For semiannual compounding m = 2; for quarterly compounding m = 4; and for monthly compounding m =
12. If you leave money in an account paying semiannual interest for four years, the total compounding
periods would be 4 × 2 = 8. Interest would be calculated and paid eight times during the four years. Let’s
assume you left your $100 on deposit for four years, and the bank pays 10 percent interest compounded
semiannually. We’ll use Equation 4 to find the amount in the account after four years.

As the number of compounding periods per year increases, future values increase and present values
decrease because of the effect on the effective rate of interest. Effective interest rates are the actual rates
earned or paid. They are determined by the stated rate and the number of compounding periods per year.

FV = $100
FV = $100(1.05)8
FV = $100(1.4775) = $147.75
where:
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)
If the account only paid interest annually, the future value would be:
FV = $100
FV = $100 (1.10)4
FV = $100 (1.4641) = $146.41
The additional $1.34 (i.e., $147.75 – $146.41 = $1.34) is the extra interest earned from the compounding
effect of interest on interest. Although the differences do not seem profound, the effects of compounding
are magnified with larger values, greater number of compounding periods per year, or higher nominal
interest rates. In our example, the extra $1.34 was earned on an initial deposit of $100. Had this been a $1
billion deposit, the extra interest differential from compounding semiannually rather than annually would
have amounted to $13,400,000!

An investor who invests in a security promising an annual rate of 10 percent will earn an effective rate of
return greater than 10 percent if the compounding frequency is greater than annually (i.e., quarterly,
monthly, etc.)

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To demonstrate the effect of increasing the number of compounding periods per year, let’s look at several
alternative future value calculations when $100 is deposited for one year at a 10 percent nominal rate of
interest. In each case, m is the number of compounding periods per year.
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:
(cid:1) For the same present value and interest rate, the future value increases as the number of
compounding periods per year increases.
(cid:1) Each successive increase in future value is less than the preceding increase. (The future value
increases at a decreasing rate.)
Effective Interest Rates. The concept of compounding is associated with the related concept of effective
interest rates. In our semiannual compounding example, we assumed that $100 was deposited for one
year at 10 percent compounded semiannually. We represented it graphically using a timeline as follows:

Figure 5: Effective Interest Rates
The stated (nominal) rate of interest is 10 percent. 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.
(5)
effective return =
where:
V = the total value of the investment at the beginning of the year
0
V = the total value of the investment at the end of the year
1
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.
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Returning to our previous example, let’s substitute our beginning-of-year and end-of-year values into
Equation (5). Because the interest was compounded semiannually instead of annually, we actually earned
10.25 percent on the account rather than the 10 percent stated rate.
(5)
effective return =
effective return = = 0.1025 = 10.25%
Let’s employ algebraic principles and rewrite Equation 5 in the following form:
effective return =
(6)
effective return = - 1
Now let’s restate Equation (6) in terms of FV and PV:

(7)

Remember, we always state effective returns in annual terms. Thus we can set n = 1 in the equation and
the PV0 in the numerator and denominator cancel each other out. Substituting Equation (7) back into
Equation (6) we get:
(8)

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.

Although we don’t demonstrate it in this book, an investment could be compounded continuously. This
compounding frequency would provide the greatest effective rate of return possible for a given annual rate
of interest.

m = 1 (annual compounding) -1 = (1.12)1 -1 = 0.12 = 12%
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m = 2 (semiannual) -1 = (1.06)2 - 1= 0.1236 = 12.36%
m = 4 (quarterly) -1 = (1.03)4 - 1= 0.1255 = 12.55%
m = 12 (monthly) -1 = (1.01)12 - 1= 0.1268 = 12.68%
m = 365 (daily) -1 = (1.0003288)365 - 1= 0.12758 = 12.75%
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 (7)
The interest rate in Equation (7) is the geometric mean or compound average annual growth rate earned on
the investment. Solving Equation (7), gives i = 30 percent.[17]
More formally, the geometric mean is found using the following equation:
(8)

where:
GM = the geometric mean
FV = future value of the lump sum investment
n
PV = present value, or the initial lump sum investment
n = the number of years over which the investment is held
Thus if you invested $500 in an mutual fund[18] five years ago and now the original investment is worth
$901.01, we would find the geometric mean return as follows:

The geometric mean is used for interest rates and growth rates. It is a multiplicative mean.

The mutual fund provided an average annual return of 12.5 percent.
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The geometric mean can also be stated using returns over several periods using the following formula:
(9)

where:
GM = the geometric mean
x = the ith return measurement (the first, second, third, etc.)
i
n = the number of data points (observations)
We add 1 to each observation’s value, which is a percentage expressed as a decimal, multiply all the
observations together, find the nth root[19] of the product, and then subtract one.
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 percent. What
was the geometric mean return for the fund?

GM = 12.5% [20]
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.

The geometric mean is the compound annual growth rate for a multi-period investment.

Present Value of a Lump Sum
Recall that Equation (3) showed us the relationship between the present and future values for a lump sum.
FV = PV(1 + i)n (3)
n
In Equation (3) the future value is determined by multiplying the present value by (1 + i)n. To solve for the
present value, we can divide both sides of the equation by (1 + i)n.
(3)

When an interest rate is used to discount a future cash flow to its present value, it is often referred to as a
discount rate.

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