CFA level 1 study guide sample

Study Session 2: Quantitative Methods—
Basic Concepts

© 2016 Wiley

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The Time Value of Money

Reading 5: The Time Value of Money
LESSON 1: INTRODUCTION, INTEREST RATES, FUTURE VALUE, AND
PRESNT VALUE
The Financial Calculator
It is very important for you to be able to use a financial calculator when working with
TVM problems. CFA Institute allows only two types of calculators for the exam—the TI
BA II Plus™ (including the TI BA II Plus™ Professional) and the HP 12C (including the
HP 12C Platinum). We highly recommend that you choose the TI BA II Plus™ or the TI
BA II Plus™ Professional, and the keystrokes defined in our readings cater exclusively to
TI BA II Plus™ users. However, if you already own an HP 12C and would like to use it,
by all means continue to do so.
The TI BA II Plus™ comes preloaded from the factory with the periods per year (P/Y)
function set to 12. This feature is not appropriate for most TVM problems, so before
moving ahead please set the “P/Y” setting of your calculator to “1” by using the following
keystrokes:
[2nd] [I/Y] “1” [ENTER] [2nd] [CPT]

Your calculator’s P/Y setting will remain at 1 even when you switch it off. However, if you
replace its batteries you will have to reset the P/Y setting to “1”. If you wish to check this

setting at any time, simply press [2nd] [I/Y] and the display should read “P/Y = 1.”
With these setting in place, you can think of “I/Y” as the interest rate per compounding
period, and of “N” as the number of compounding periods. Please take the time to
familiarize yourself with the following keys on your TI Calculator:
N = Number of compounding periods
I/Y = Periodic interest rate
PV = Present Value
FV = Future Value
PMT = Constant periodic payment
CPT = Compute
Timelines
To illustrate some examples, we will use timelines to present the information more clearly.
It is very important for you to recognize that the cash flows occur at the end of the period
depicted on the timeline. Further, the end of one period is the same as the beginning of the
next period. For example, a cash flow that occurs at the beginning of Year 4 is equivalent to
cash flow that occurs at the end of Year 3, and will appear at t = 3 on the timeline.
Sign Convention
Finally, pay attention to the signs when working through TVM questions. Think of inflows
as positive numbers and outflows as negative numbers. We will continue to emphasize this
point through the first few examples in this reading so that you get the hang of it.

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The Time Value of Money

Suppose you were offered a choice between receiving $100 today or $110 a year from
today. If you are indifferent between the two options, you are attaching the same value to
receiving $110 a year from today as you are to receiving $100 today. It is obvious that the
cash flow that will be received in the future must be discounted to account for the passage
of time. An interest rate, r, is the rate of return that shows the relationship between two
differently dated cash flows. The interest rate implied in the tradeoff above is 10%.
Present value (PV) is the current worth of sum of money or stream of cash flows that
will be received in the future, given the interest rate. For example, given an interest rate of
10%, the PV of $110 that will be received in one year is $100.

0

PV = $100
  0

1
$110

Future value (FV) is the value of a sum of money or a stream of cash flows at a specified
date in the future. For example, assuming a 10% interest rate, the FV of $100 received
today is $110.

0
$100
 

1

FV1 = $110

LOS 5a: Interpret interest rates as required rates of return, discount rates,
or opportunity costs. Vol 1, pp 278–279
Interest rates can be thought of in three ways:
1.The minimum rate of return that you require to accept a payment at a later date.
2.The discount rate that must be applied to a future cash flow in order to determine
its present value.
3.The opportunity cost of spending the money today as opposed to saving it for a
certain period and earning a return on it.

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© 2016 Wiley

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The Time Value of Money

LOS 5b: Explain an interest rate as the sum of a real risk‐free rate and
premiums that compensate investors for bearing distinct types of risk.
Vol 1, p 279
Interest rates are determined by the demand and supply of funds. They are composed of
the real risk‐free rate plus compensation for bearing different types of risks:

• The real risk‐free rate is the single‐period return on a risk‐free security assuming
zero inflation. With no inflation, every dollar holds on to its purchasing power,

so this rate purely reflects individuals’ preferences for current versus future
consumption.
• An inflation premium is added to the real risk‐free rate to reflect the expected
loss in purchasing power over the term of a loan. The real risk‐free rate plus the
inflation premium equals the nominal risk‐free rate.
• The default risk premium compensates investors for the risk that the borrower
might fail to make promised payments in full in a timely manner.
• The liquidity premium compensates investors for any difficulty that they might
face in converting their holdings readily into cash at their fair value. Securities that
trade infrequently or with low volumes require a higher liquidity premium than
those that trade frequently with high volumes.
• The maturity premium compensates investors for the higher sensitivity of the
market values of longer term debt instruments to changes in interest rates.

LOS 5e: Calculate and interpret the future value (FV) and present value (PV)
of a single sum of money, an ordinary annuity, an annuity due, a perpetuity
(PV only), and a series of unequal cash flows. Vol 1, pp 280–284, 289–303
The Future Value of a Single Cash Flow

We shall go through
LOS 5c after LOS
5e.

Let’s start off with a relatively simple concept. If you had $100 in your pocket right now,
and interest rates were 6%, what would be the future value of your money in one year, and
in two years?
FVN = PV(1 + r) N

In one year the value of $100 will be:
$100 × (1 + 0.06)1 = $106

Or using your calculator:
PV = −100; I/Y = 6; N = 1; CPT FV → FV = $106.
TI calculator keystrokes:
[2nd ] [FV] “−100” [PV] “6” [I/Y] “1” [N] [CPT] [FV]

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We have shown the
PV as a negative
number so that the
resulting FV is
positive. Basically,
an investment
(outflow) of $100
today at 6% would
result in the receipt
(inflow) of $106 in
one year.

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The Time Value of Money

In solving time
value of money

problems remember
that the stated
interest rate, I/Y,
and the number
of compounding
periods, N, should
be compatible.
For example if N
is stated in days,
I/Y must be the
unannualized daily
interest rate.

In two years the value of $100 will be:
100 × (1 + 0.06)2 = $112.36.
Or using your calculator:
PV = − 100; I/Y = 6; N = 2; CPT FV → FV = $112.36
TI calculator keystrokes:
[2nd ][FV] “−100” [PV] “6” [I/Y] “2” [N] [CPT] [FV]
On your investment of $100 you earn 0.06 × 100 = $6 in simple interest each year. In the
second year, the $6 simple interest earned in Year 1 also earns interest in addition to the
principal. This $6 × 0.06 = $0.36 of additional interest earned is compound interest. Over
the two years, total interest earned equals $6 + $6 + $0.36 = $12.36
Drawing up timelines will help you avoid careless mistakes when handling TVM
questions. A general timeline for the future value concept looks like this:

N–1

N

FVN = PV (1+ r)N
If we wanted to determine the future value after 15 periods, the PV and FV would
be separated by a future value factor of (1 + r)15, where r would be the interest rate
corresponding to the length of each period.
Since PV and FV are separated in time, remember the following:

• We can add sums of money only if they are being valued at exactly the same point
in time.
• For a given interest rate, the future value increases as the number of periods
increases.
• For a given number of periods, the future value increases as the interest rate
increases.
An investment
(outflow) of
$750 today at 7%
would result in the
receipt (inflow) of
$1,689.14 in 12
years.

Example 1-1: Calculate the FV of $750 at the end of 12 years if the annual interest rate
is 7%.

Important: After
each problem, get
into the habit of
clearing your TI
calculator’s memory
by pressing [2nd]
[FV] and [2nd]

[CE|E].

Example 1-2: Calculate the value after 20 years of an investment of $500, which will be
made after 7 years. The expected annual rate of return is 8%.

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Solution
PV = −$750; N = 12; I/Y = 7; CPT FV → FV = $1,689.14

Solution
PV = −$500; N = 13; I/Y = 8; CPT FV → FV = $1,359.81
Note: The investment will earn interest for 13 periods.

© 2016 Wiley

8 October 2015 8:51 PM