Finance 2400

Lecture Notes for the Spring semester 2018

V.71 of

Bite-size Lectures

on

the Time Value of Money (TVM)

and

the discounting of future cash flows.

© Sven Thommesen 2018

Last updated: 2011-09-05

Generated: 2017-11-02

2

© Sven Thommesen 2018

Lectures on the time value of money (TVM)

and the discounting of future cash flows

We live in a modern economy based on the use of money, in which specialization and the

division of labor play a large role. Also playing a large role are the financial markets, in which

people borrow and lend money in a large variety of financial transactions.

One common feature of all financial transactions is that he who borrows money (whatever form

the loan takes) expects to have to pay interest to the lender. Correspondingly, the lender

expects to be able to earn interest on sums lent.

Sometimes, to the lender, the transaction looks not so much like a loan as like an investment,

where he purchases some kind of financial asset, holds it for a while, and then sells it. He will

still expect to be able to earn some rate of return or yield on his investment.

The following lectures give an introduction to simple financial theory (the time value of money

and discounting) and the math used.

They are called “bite-size” (a) because each lecture is small (most are limited to one page), and

(b) because you are supposed to “chew” each lesson thoroughly before going on to the next

one, in order to digest it properly ;)

These lectures should be read in conjunction with lectures explaining the use of your specific

financial calculator to solve practical problems.

There is also a set of problems with worked-out solutions which you can use to test your

understanding of the material.

(The chapter references in blue refer to Ross, Westerfield and Jordan, Fundamentals of

Corporate Finance, 9/e.)

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V.71

TVM LECTURES: TABLE OF CONTENTS

I: SIMPLE LOANS AND SIMPLE INTEREST

Lecture 101: Simple Loans, Simple Interest

Lecture 102: Simple Interest: calculating FV

Lecture 103: Simple Interest: calculating PV [m]

Lecture 104: Simple Interest: solving for INT and N [m]

II: (3610) SIMPLE LOANS AND COMPOUND INTEREST

Lecture 111: Simple Loans, Compound Interest

5.1 Lecture 112: Compound Interest: calculating FV

5.2 Lecture 113: Compound Interest: calculating PV [m]

5.3 Lecture 114: Compound Interest: calculating INT and N [m]

5.3 Lecture 115: Compound Interest: The Rule of 72 [m]

III: PROBLEM SOLVING: SIMPLE LOANS

Lecture 121: Different ways to solve simple loan problems

Lecture 122: Solving simple loan problems using tables

Lecture 123: Solving simple loan problems using a financial calculator

VI: (3610) PROJECTS WITH MULTIPLE EVEN CASH FLOWS: ANNUITIES

6.1 Lecture 201: Cash Flows and Projects

6.2 Lecture 202: FV and PV for multiple cash flows: manual calculations [m]

6.1 Lecture 203: Even cash flows: Annuities

VII: PROBLEM SOLVING: ANNUITIES

Lecture 211: Different ways to solve annuity problems

6.2 Lecture 212: Solving annuity problems using formulas

6.2 Lecture 213: Solving annuity problems using tables

6.2 Lecture 214: Solving annuity problems using a financial calculator

6.2 Lecture 215: Normal Annuities vs. Annuities Due

Lecture 216: Amortization

4

XI: (3610) INFLATION AND TAXES

7.6 Lecture 331: The Effects of Inflation on Interest Rates: the Fisher Equation

7.4 Lecture 332: The Effects of Taxes on After-Tax Bond Yields

7.4 Lecture 333: The Effects of Taxes on Pre-Tax Bond Yields

7.6 Lecture 334: The Combined Effects of Taxes and Inflation on Net Yields

XV: (3700) MORE SPECIAL TOPICS

Lecture 531: The Effects of Inflation on Purchasing Power: using the CPI

Lecture 532: The Effects of Inflation on Purchasing Power: using average Π

XVII: FORMULAS

Lecture 991: A brief discussion of some calculators with financial functionality

Lecture 999: Mathematical formulas

[m] means: Accompanied by a mathematical explanation

5

PART I: SIMPLE LOANS AND SIMPLE INTEREST

6

LECTURE 101

SIMPLE LOANS, SIMPLE INTEREST

We will start our discussion with the simplest possible type of financial transaction:

A simple loan is a financial transaction where only two payments ever take place: one at the

beginning, and another (in the opposite direction) at the end. A common example would be:

you deposit a sum of money to your savings account today, leave it there for a year, then

withdraw the balance, equal to the original deposit plus the interest you have earned.

The interesting question is then: how much can you withdraw at the end?

The answer depends on how your interest is calculated.

There are two possibilities: simple interest and compound interest.

Simple interest is based on the principle that interest gets paid on the original amount that was

loaned or deposited (the principal). However, if the loan or deposit extends over several

interest periods, you do not earn interest on the accumulating interest.

In real life, simple interest is only used in cases where the interest you earn is in fact paid to the

lender when it is earned, so that she may do something with it then (spend it, or invest it

further). One example of such an arrangement is an interest-only mortgage.

Another example is a coupon bond, where the bond issuer (borrower) makes regular interest

payments to the bond holders (lenders, investors) in the form of “coupon payments”. Say the

Treasury sells a $1000 face value 10-year 6% Treasury Note. You purchase the bond for

$1,000, the Treasury sends you a check for $30 every six months for 10 years, then you get

your $1,000 back. You get your interest sent to you as you earn it, and can do with it what you

want as you receive each payment.

7

LECTURE 102

SIMPLE INTEREST: CALCULATING FV

Let us say that you loan $100 to your brother at 10% per week simple interest. (Maybe you

don’t like you brother much, given the rate you’re charging him!)

After 1 week, he will owe you $10 in interest. After two weeks, he owes $20. After 3 weeks,

$30, and so on. In other words, interest is calculated on the original sum he borrowed, which in

financial theory is called the principal.

The general formula for simple interest is this:

FV = PV * (1 + i*n)

where:

PV = present value = the sum lent (the principal);

i = the rate of interest charged or paid per time period (usually per year);

n = the number of time periods (years) involved; and

FV = future value = the total sum owed or due at the end.

Note that in some cases, interest is paid only at the end, while in other cases, the interest earned

is paid as it accrues (weekly, in the case of your brother.)

In the above example, how much would your brother owe you after a year?

FV = PV * (1+i*n) = $100 * (1 + 0.10*52) = $620. (Ouch!)

NOTE #1:

The formula contains the “1+” part because the principal must also be paid back …

NOTE #2:

In financial formulas we render interest rates as decimal fractions, not as percentages.

Thus, i=12% becomes 0.12 and (1+i) becomes 1.12.

Another example: if you placed $1,000 in an investment yielding 12% simple interest per year,

then after 5 years you would have:

FV = PV * (1+i*n) = $1,000 * (1+0.12*5) = $1,600.

8

LECTURE 103

SIMPLE INTEREST: CALCULATING PV

Staying with the topic of simple interest for a minute, we can of course use the formula given to

calculate the other variables as well.

First, let us ask about PV: if you locate a bank account belonging to a deceased uncle, which

contains $81,360.00 and it has been paying 6% simple interest for 21 years, how much did your

uncle originally deposit?

From the basic formula, we can solve for PV:

PV = FV / (1 + i*n) = $81,360.00 / (1 + 0.06*21) = $36,000.

Another way to look at this example is to ask: if I could earn simple interest at a rate of 6% per

year, and if I planned to leave my money in the bank for a period of 21 years, and if I needed to

have $100,000 available to me at that point, how much would I have to deposit into that

account today? The answer is:

PV = FV / (1+i*n) = $100,000 / (1+0.06*21) = $44,248.

9

LECTURE 103 MATH

Simple Interest: Solving for PV

Our basic equation says that

(1 )FV PV i n

To solve for present value, all we have to do is divide by the interest factor on both sides:

(1 )

(1 ) (1 )

FV PV i n

i n i n

Which we can simplify to

(1 )

FVPV

i n

10

LECTURE 104

SIMPLE INTEREST: SOLVING FOR “i” and “n”

Next, we can solve for the interest rate (i): if the account contains $81,360 and it has been

earning interest for 15 years, what would be the applicable interest rate?

FV = PV * (1 + i*n) FV PV

iPV n

So we get: i = (81,360-36,000)/(15*36,000) = 0.0840 or 8.4%

Finally, we can solve for the number of years (n): if the original deposit has earned interest at

9%, how many years has it been sitting in the bank? We have

FV = PV * (1 + i*n) FV PV

nPV i

So we get: n = (81,360-36,000)/(.09*36,000) = 14 years.

11

LECTURE 104 MATH

Simple interest: Solving for “i”

Our basic equation says that

(1 )FV PV i n

1FV

i nPV

1FV

i nPV

1FV

PV in

1

( )FV PV

in PV PV

FV PV

iPV n

Simple interest: Solving for “n”

Our basic equation says that

(1 )FV PV i n

If you follow a similar process as the one above, you should get

FV PV

nPV i

12

II: SIMPLE LOANS AND COMPOUND INTEREST

13

LECTURE 111

SIMPLE LOANS, COMPOUND INTEREST

Look back at the example used in Lecture 102, where you lend $100 to your brother at 10%

simple interest per week. If this loan were like a coupon bond, he’d have to send you a $10

interest check every week, which you could then spend or loan to someone else. If instead he

pays you back the whole $620 owed after a full year, it’s as if he has paid you an interest rate of

0% on the accumulating interest!

For most financial transactions, therefore, (including the very simple one of a bank savings

deposit), if the borrower does not send you regular interest payments as the interest is earned,

we imagine that it is as if he has borrowed those interest payments as well when they come due,

so that he needs to pay you interest-on-the-interest as we go along.

This arrangement is referred to as compound interest.

14

LECTURE 112

COMPOUND INTEREST: CALCULATING FV

[See also: CALC LECTURE #61]

Let us use our previous example: say your brother borrows $100 from you at 10% per week,

but this time using compound interest. Then:

After one week, he owes you $10 in interest ($100 x 0.10), and his total debt is $110. The

following week he owes you an additional $11 in interest ($110 x 0.10) for a total debt of $121.

After yet another week, the debt is $133.10. And so it goes.

The general formula for compound interest is:

FV = PV * (1+i)n

where:

PV = present value = the sum lent (the principal);

i = the rate of interest charged or paid per time period (usually per year);

n = the number of time periods (years) involved; and

FV = future value = the total sum owed or due at the end.

So how much does your brother owe you after one year?

FV = PV * (1+i)n = $100 * (1 + 0.10)52 = $14,204.29 (double ouch!)

Another example: you put $2,500 in a savings account that pays 4.6% per year, and leave it

there for 12 years. How much is in your account now?

Answer: FV = $2,500 * (1+0.046)12 = $4,288.65.

15

LECTURE 113

COMPOUND INTEREST: CALCULATING PV

Now let us go through the same exercises as in lectures 103-4, but this time assuming

compound interest. First, let us “look backward” and calculate PV:

If FV = PV * (1+i)n then that gives us PV = FV / (1+i)n.

So: if your uncle’s bank account contains $81,360 and it has been earning 6% interest for 21

years, how much did your uncle originally deposit?

Answer: PV = FV / (1+i)n = $81,360/(1.06)21 = $23,932.48

(As you can see, compound interest accumulates money at a much faster rate than simple

interest!)

Another example: You are 22 years old. You want to have $1,000,000 in the bank the day you

retire at 67. The bank will pay you 5% interest on your money over that time. How much would

you have to deposit in the bank today to reach your goal?

Answer: PV = $1,000,000 / (1.05)45 = $111,296.51

If instead you could invest your money in the stock market and earn an average return of 12%

per year, how much would you need to invest today to get there?

Answer: PV = $1,000,000 / (1.12)45 = $6,098.02. (Wow!)

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LECTURE 113 MATH

Recall that 1

xy can be expressed mathematically as xy .

Thus, (1 ) n

FVPV

i

is the same as saying that (1 ) nPV FV i .

So $1,000,000 * (1.12)-45 = $6,098.02. [Do it on your calculator!]

17

LECTURE 114

COMPOUND INTEREST: CALCULATING “i” AND “n”

[See also: CALC LECTURE #62]

Question: you want to double your money over 7 years. What yearly return on your money do

you need to earn to reach this goal? We have:

FV = PV * (1+i)n 1nFV

iPV

We get: 7$2

1$1

i = 0.10409 or 10.4% [Notice: 10.4 * 7 = 72.8]

Question: You want to double your money. You invest in a really rotten project which yields

only a meager 2% return on your investment. How many years does your money have to stay

invested?

FV = PV * (1+i)n

ln( )ln( ) ln( )

ln(1 ) ln(1 )

FV

FV PVPVni i

We get:

n = (ln(2.0) – ln(1.0)) / ln(1.02) = 35.003 years [Notice: 2.0 * 35 = 70]

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LECTURE 114 MATH

Compound interest: calculating “i”

Our basic equation states that (1 )nFV PV i from which we get these steps:

(1 )nFVi

PV

(1 ) (1 )nnnFV

i iPV

1nFV

iPV

Compound interest: calculating “n”

Our basic equation states that (1 )nFV PV i from which we get these steps:

(1 )nFVi

PV

ln( ) ln((1 ) )nFVi

PV [Rule: if a=b then ln(a)=ln(b)]

ln( ) ln(1 )FV

n iPV

[Rule: ln( ) ln( )ax a x ]

ln( )ln( ) ln( )

ln(1 ) ln(1 )

FV

FV PVPVni i

[Rule:ln( ) ln( ) ln( )x y x y ]

[Rule: ln( ) ln( ) ln( )x

x yy

]

The natural logarithm ln(x)

The natural log function ln(x) is the inverse of the ex function this way:

If xy e then ln( )x y . [Note that

0 1e and ln( ) 1e and ln(1) 0 .]

19

LECTURE 115

COMPOUND INTEREST: THE RULE OF 72 (A SHORTCUT)

Notice from the two examples in Lecture 114 that if you multiply the number of years by the

yearly interest rate, you get approximately 72 in both cases.

Financial practitioners have come to use this as a quick approximation:

to double your money,

(# periods required) x (period interest rate in %) = 72

OR: IF FV = 2*PV, then N * %INT ~ 72

So to figure out how many years it will take you to double your money, divide 72 by the

expected yearly return.

Or, to figure out what % monthly return you need, divide 72 by the number of months you have

to do it.

ADVANCED:

What if you want to more than double your money?

Let us say you can earn 8% interest. By the rule of 72, it will take you 9 years to double your

money. After another 9 years, you will have doubled it again – i.e. it takes 9x2=18 years to

quadruple your money, 27 years to grow your investment to 8 times its original size, and so on.

20

LECTURE 115 MATH

THE RULE OF 72: WHY IT WORKS

If we are using continuous compounding (see Lecture 143), we have that:

inFV PV e

Since we want to double our money, we get:

2 inFVe

PV

From which we get (using natural logs):

ln(2) 0.693147i n

And if we measure interest rates in percent, that gives us:

100*ln(2) 69.3i n

which we can solve for either “i” or “n”.

(We use 72 instead of 69.3 in our shortcut formula, since 72 is easily divisible.)

For yearly compounding, we get:

(1 )nFV PV i

From which we get:

ln(2)

2 (1 ) ln(2) ln(1 )ln(1 )

nFVi n i n

PV i

Not easy to solve directly for the interest rate, but we can easily construct a little table showing

combinations of “i” and “n”. Examples: i=8% -> n=9.01; i=12%->n=6.12.

If you are dealing with compounding more often than once a year (‘m’ times per year) and you

want to be as accurate as possible, the above expression is modified as follows, where i/m is the

periodic interest rate, and m*n is the total number of compounding periods:

ln(2)

ln(1 )

m ni

m

21

III: PROBLEM SOLVING: SIMPLE LOANS

22

LECTURE 121

DIFFERENT WAYS TO SOLVE SIMPLE LOAN PROBLEMS

There are several different ways to solve financial problems involving simple loans or similar

types of projects. Some of them are:

1. You can solve them “manually” using the mathematical formulas directly. Advantage:

you don’t need a special financial calculator to do this, just a reasonably competent $15

scientific one. We discussed this way of doing things in Lectures 112-114 above.

[See Lecture 999 for relevant formulas.]

2. You can use pre-computed tables of compounded-interest factors [ (1+i)n and its

inverse]. Advantage: as long as you have access to such tables, you only need a very

simple calculator. Disadvantage: some tables give only a limited number of decimals,

thus reducing the accuracy of your answers. (In finance, we sometimes need to be

accurate to the nearest penny!) This solution method is discussed below in Lecture 122.

[Your textbook, RWJ, has such tables in Appendix A.]

3. You can use a financial calculator. There are several such calculators on the market,

by Hewlett-Packard, Texas Instruments, and others. (See Lecture 991 for a list of some

available models.) The use of a financial calculator to solve simple loan problems is

discussed below in Lecture 123. There is a separate set of Lectures available discussing

the operation of specific calculator models (for now, only for the HP-10BII).

The following methods are also available, though they are not discussed in these Lectures:

4. You can use a software emulator: a program that runs on a PC that looks like and acts

like a specific financial calculator. For example, there is an excellent emulator for the

HP-12C available for the Android platform.

5. You can use a spreadsheet such as Microsoft Excel or OpenOffice/LibreOffice Calc,

which have built-in financial functions.

6. You can use one of the many specialized online calculator programs available on

various web sites; these are usually tailored to solving some specific class of problems.

It is to your advantage to become proficient in all these methods if you are going to need

to solve a lot of financial problems in your career.

23

LECTURE 122

SOLVING SIMPLE LOAN PROBLEMS USING TABLES

CALCULATING FV:

You deposit $2,350 in the bank at 7% interest and leave it there for 15 years. How much is in

your account now?

The mathematical formula is FV = PV * (1+i)N = $2,350 * (1.07)15

To use the table, we can look up the value of the interest factor (1.07)15 = 2.7590.

(Find the number at the intersection of the 7% column and the 15-year row.)

Thus, the answer to our problem is $2,350 * 2.7590 = $6,483.65.

CALCULATING PV:

How much do you have to invest today if you want to have $100,000 available 25 years from

now, and you can earn 12% interest?

The math says that FV = PV * (1+I)N so PV = FV / (1+I)N or PV = FV * (1/(1+I)N)

So we can use the same table as above: at N=25, i=12% we find the factor 17.000.

Then: PV = FV / int-factor = $100,000 / 17.000 = $5,882.35.

Or, we can use the table for the inverse factor: at N=25, i=12% we find 0.0588.

Then: PV = FV * (inverse-int-factor) = $100,000 * 0.0588 = $5,880.00.

We see that this second method is a bit less accurate, since the tables give us fewer effective

decimals for the inverse interest factor.

CALCULATING 'i' OR 'n':

We do not have tables to calculate either N or i.

24

LECTURE 123

SOLVING SIMPLE LOAN PROBLEMS USING A FINANCIAL CALCULATOR

Lecture 991 provides a list of some financial calculator models currently available, with their

approximate prices.

Accompanying this set of Lectures on the theory of TVM and cash flow problems is a separate

set of Lectures describing the operations of specific financial calculators. (Currently limited to

the HP-10BII.)

Lecture 214 below describes how to use a financial calculator to solve annuity problems.

Problems involving simple loans and compound interest are solved on the financial calculator

just like annuity problems, with the exception that PMT is always set equal to zero.

Thus, go read Lecture 214.

Or, continue through these lectures until you have completed Lecture 216, then come back

here.

Solving simple loan problems should now seem trivial to you.

25

LECTURE 146

TERMINOLOGY: DIFFERENT CONCEPTS OF “YIELD” and “RETURN”

APR: the annualized yearly interest rate on a loan, found by taking the periodic interest rate

and multiplying by the number of periods in a year. Example: a credit card account charging

1.5% interest per month would report an APR of 1.5% * 12 = 18%.

EFFECTIVE ANNUAL RETURN (EAR): the true compounded yearly return to some

project or investment which compounds interest more often than once a year. As opposed to the

APR, the EAR = (1+i/m)m – 1, or (1.015)12 – 1 = 19.56% for the same credit card.

REQUIRED RETURN: the rate of return we use as the discount rate in computing the NPV of

some financial instrument or project. It is intended to represent the opportunity cost of the

funds used. Thus, in most cases we use the market return for comparable investments as our

required return. However, in special cases when we have access to above-market returns in an

alternate investment, we may use that project’s return as our required return.

YIELD TO MATURITY: the total yield you would earn from purchasing a bond or other debt

instrument at today’s price, and holding it until it matures (when you receive the face value as

your last payment.)

YIELD TO CALL: the total yield you would earn from purchasing a bond or other debt

instrument at today’s price, and holding it until the earliest date the issuer can “call” (pay off)

the bond. The bond indenture specifies what price you would receive when/if the bond is called

(which may not be the same as face value.)

MARKET RATE (MARKET RETURN): the yield or interest rate currently earned by

market participants on the type of financial instrument you are contemplating. This depends,

among other factors, on the current risk-free rate, the risk and liquidity premiums for your

specific instrument, and the time to maturity. The market rate for a given type of financial

instrument is a function of supply and demand, and fluctuates daily.

COUPON RATE: the interest rate used by a bond issuer to determine the yearly coupon

payment, which is fixed from then on. Yearly coupon payments = face value * coupon rate.

TOTAL RETURN: the return you have earned or will earn from a given investment, including

all cash flows from coupons or dividends, as well as the purchase and selling prices. It is found

by calculating the project’s IRR. Sometimes approximated by Current Yield + capital gain.

APY: the yearly yield on a savings account or investment. Equals the EAR (see above.)

QUOTED RATE: the yearly rate quoted on a loan or other debt. Equals the APR (see above.)

CURRENT YIELD: a simplified yield calculation for coupon bonds. See Lecture 314.

YIELD ON A DISCOUNT BASIS: a simplified yield for discount bonds. See Lecture 312.

BOND-EQUIVALENT YIELD: a simplified yield for discount bonds. See Lecture 312.

26

PART VI: PROJECTS WITH MULTIPLE EVEN CASH FLOWS

27

LECTURE 201

CASH FLOWS AND PROJECTS

So far we have been discussing simple loans, that is, financial transactions involving only

two payments. From here on, we discuss the more general concept of projects consisting

of an arbitrary number of cash flows.

CASH FLOWS

A “cash flow” is a single payment that:

(a) has a size (in dollars),

(b) has a direction (you either receive the money, or pay it out), and

(c) takes place at a specific point in time (usually at the beginning or end of a given time

period, which can be a month, a year, or something else).

In our putting-money-in-the-bank example, my deposit today is one cash flow, and my

withdrawal a year later is a second cash flow.

PROJECTS

In finance we will often need to deal with projects which involve multiple cash flows.

A set of cash flows may consist of a number of different size payments (flows), or it may

consist of a sequence of identically sized payments. The cash flows may be spaced out at equal

intervals, or may happen at varying intervals.

A construction project might be an example of a project with cash flows of varying sizes and

intervals, while a mortgage or a coupon bond would be examples of equal-size cash flows

spaced out equally over time.

CASH FLOW DIAGRAMS

The cash flows associated with a given financial transaction or project may be illustrated with a

cash flow diagram. See Lecture 304 for some examples.

THE CASH FLOW SIGN CONVENTION

When we do computations with cash flows, we need to observe the cash flow sign convention:

Amounts which you have received or will receive are represented as positive numbers;

Amounts which you have paid or will pay out are represented as negative numbers.

We see that the signs depend on whether we are looking at the project from the point of view of

the borrower or the lender/investor. On the other hand, the sizes and timings of the cash flows

are the same in either case.

28

LECTURE 202

FV AND PV FOR MULTIPLE CASH FLOWS: MANUAL CALCULATIONS

We have defined a financial project as a set of current and future cash flows. We have seen

above, in Lecture 113, how to compute the present value PV of a single future cash flow.

The Net Present Value (NPV) for a financial project as a whole is equal to the sum of the

present values (PV) of each participating cash flow, discounted at an appropriate discount

rate (interest rate) “i”.

Example: a project promises to pay us $1,000 per year for 3 years, at the end of each year. The

total present value of these three future payments is (assuming a discount rate of 8%):

PV of CF1: $1,000 / (1.08)1 = $925.93

PV of CF2: $1,000 / (1.08)2 = $857.34

PV of CF3: $1,000 / (1.08)3 = $793.83

------------

TOTAL $2,577.10

If this project required an initial investment (cash flow CF0, at time t=0) of $1,200, the net

present value (NPV) of the whole project is:

NPV = $2,577.10 - $1,200.00 = $1,377.10

Similarly, the Net Future Value (NFV) of this project is equal to the sum of the future values

(FV) of all the participating cash flows, at a given point in time, and given the applicable rate of

interest.

For the above project, if the applicable interest rate i=8% then the NFV at time t=3 (at the same

time as we receive the last payment) is:

FV of CF0: -$1,200 * (1.08)3 = $1,511.65-

FV of CF1: $1,000 * (1.08)2 = $1,166.40

FV of CF2: $1,000 * (1.08)1 = $1,080.00

FV of CF3: $1,000 * (1.08)0 = $1,000.00

------------

TOTAL $1,734.75

(Negative cash flows retain their signs in the process.)

We note that for a financial project as a whole,

NFV = NPV * (1+i)n

Verify for yourself that 1,377.10 * (1.08)3 = 1,734.75

[See also Lecture 541 on Net Future Value.]

29

LECTURE 202 MATH

FV AND PV FOR MULTIPLE CASH FLOWS

The net present value (NPV) for a financial project can be expressed in two different ways.

First, if we think of it as the sum of the present values of all the participating cash flows, we

have:

0 (1 )

n t

tt

CFNPV

i

where “i” is the appropriate discount rate.

Second, we can think of it as the present value of all future cash flows, less the initial

investment (CF0):

01 (1 )

n t

tt

CFNPV CF

i

(notice that t goes from 1 to n here).

We use the form that best suits the problem at hand. (See Lectures 431, 435 for an example of

when the second form is used, in calculating the Profitability Index.)

The net future value (NFV) for a project is equal to the sum of the future values of all the

participating cash flows, at a given future point in time, given an applicable interest rate “i”:

0

(1 )n n t

ttNFV CF i

As noted above, for a given project this means that

(1 )nNFV NPV i

30

LECTURE 203

EVEN CASH FLOWS: ANNUITIES

The manual calculations shown in the preceding Lecture works for any arbitrary combination

of cash flows in a project.

However, if we impose certain restrictions on the cash flow profile of the project, we have

available to us special formulas and calculator procedures we can use.

The restrictions are:

(a) the cash flows of the project must all be of the same SIZE and DIRECTION, and they

must be SPACED OUT EVENLY over time;

(b) there may be a single cash flow of a different size and/or direction that happens at the

beginning of time (t = 0);

(c) there may be a single cash flow of a different size and/or direction that happens at the

end of time (t = n), along with the last regular payment.

Examples of financial projects that fit such a profile would be: car loans, mortgages with

balloon payments, and coupon bonds.

Financial projects that fit this cash flow profile are referred to as annuities.

A cautionary note: insurance companies sell a financial product that they call “annuities”,

which involve regular payments by the insurance company to some beneficiary. These products

do fit the profile of an ‘annuity’ in the present sense, but so do various other financial

instruments and relationships.

31

VII: PROBLEM SOLVING: ANNUITIES

32

LECTURE 211

DIFFERENT WAYS TO SOLVE ANNUITY PROBLEMS

There are several different ways to solve financial problems involving projects that take the

form of annuities. Some of them are:

1. You can solve for NPV or NFV manually, cash flow by cash flow, since the NPV for

an annuity is equal to the sum of the PV’s for each cash flow taken by itself. And we

have already seen how to calculate the PV for a single future payment (lump sum). The

same goes for the NFV of an annuity. Lecture 202 above showed how to do this.

2. You can solve them using the mathematical formulas directly. Advantage: you don’t

need a special financial calculator to do this. This procedure is described below in

Lecture 212. [See Lecture 999 for the relevant formulas for FVA, PVA, PMT, SFP.]

3. You can use pre-computed tables of annuity factors [for FVA and PVA]. Advantage:

as long as you have access to such tables, you only need a very simple calculator.

Disadvantage: some tables give only a limited number of decimals, thus reducing the

accuracy of your answers. (In finance, we sometimes need to be accurate to the nearest

penny!) This solution method is described below in Lecture 213. [Your textbook, RWJ,

has such tables in Appendix A.]

4. You can use a financial calculator. There are several such calculators on the market,

by Hewlett-Packard, Texas Instruments, and others. (See Lecture 991 for a list of some

available models.) The use of a financial calculator to solve annuity problems is

discussed below in Lecture 214. There is a separate set of Lectures available discussing

the operation of specific calculator models (for now, limited to the HP-10BII).

The following methods are also available, though they are not discussed in these Lectures:

5. You can use a software emulator: a program that runs on a PC that looks like and acts

like a specific financial calculator. There is an excellent emulator for the HP-12C

financial calculator available for the Android platform, for example.

6. You can use a spreadsheet such as Microsoft Excel or OpenOffice/LibreOffice Calc.

7. You can use one of the many specialized online calculator programs available on

various web sites; these are usually tailored to solving some specific class of problems.

It is to your advantage to become proficient in all these methods if you are going to solve a

lot of financial problems in the future.

33

LECTURE 212

SOLVING ANNUITY PROBLEMS USING MATHEMATICAL FORMULAS

Lecture 999 contains mathematical formulas which you can use to calculate certain annuity

variables, using only a simple scientific calculator.

The variables we can calculate are:

PVA: the present value of an annuity, i.e. the present value of a sequence of future equal size

payments. You only need to supply the size of the recurring payment (PMT), the periodic

interest rate (i) and the number of periods (n).

FVA: the future value of an annuity, i.e. the value of the annuity at the same time as the very

last payment is made or received. Again, you supply the size of the payment (PMT), the

periodic interest rate (i), and the number of periods (n).

SFP: the required size of a sinking fund payment, i.e. the amount of money you have to

contribute each period for n periods, earning a periodic interest rate of “i”, in order to have the

amount FVA available at the end. You supply FVA, n, and i.

PMT: the required size of an annuity loan payment, i.e. the amount of money you have to

contribute each period for n periods, paying a periodic interest rate of “i”, in order to

completely amortize (pay off) an original loan principal of PVA. You supply PVA, n, and i.

WARNING: you have to know the operating characteristics of your calculator to calculate

these formulas correctly! In particular, you may need to enter parentheses at appropriate places

to let the calculator understand which operations go with which number. Some calculators are

smarter than others about ‘understanding’ what you mean when you skimp on the parentheses.

Suggestion: in the beginning, obtain an answer by one of the other methods discussed here as a

check, until you are comfortable with the proper operation of your calculator.

34

LECTURE 213

SOLVING ANNUITY PROBLEMS USING TABLES OF ANNUITY VALUES

There are many sources available [such as Appendix A of your RWJ textbook] where you can

find tables of annuity factors, such as:

PVAF: the present value of a $1 annuity, given the number of periods and the interest rate; and

FVAF: the future value of a $1 annuity, given the number of periods and the interest rate.

Using such tables, we can calculate the following variables:

PVA: the present value of an annuity, given the size of the payment, the number of periods, and

the periodic interest rate.

Example: a particular financial investment promises to pay you $50,000 per year for 12 years.

Your required rate of return is 9%. What is the present value of this stream of payments?

Answer: look in the table for (n=12, I=9%). You find the factor PVAF = 7.1607. Multiply this

factor by the size of the payment, and we get: PVA = $50,000 * 7.1607 = $358,035.00.

FVA: the future value of an annuity, given the size of the payment, the number of periods, and

the periodic interest rate. In the above example (n=12 and I/YR=9%) we find in the table an

FVAF of 20.1407, which gives us FVA = PMT * FVAF = 50,000 * 20.1407 = $1,007,036.00.

(If you refer to the formulas in Lecture 999, you see that the textbook tables supply the values

for the expression in brackets, which you multiply by PMT.)

35

LECTURE 214

SOLVING ANNUITY PROBLEMS USING A FINANCIAL CALCULATOR

Every financial calculator will have a set of TVM functions which can be used to calculate one

of the following variables related to an annuity-type financial project:

PV: the present value of the annuity

FV: the future value of the annuity

PMT: the recurring payment

N: the number of payments or compounding periods

I/YR: the yearly interest rate or yield

You can give the calculator the values for 4 of these variables, then ask it to calculate the value

for the missing 5th variable.

In order for this to work properly, you need to first set these variables, which will be different

from problem to problem:

BEG/END: normally set to END; set to BEGIN if your project is an “annuity due”

(see Lecture 215 below);

P/YR: number of payments or compounding periods per year.

Example: For the simplest type of annuity problem, calculating the required monthly payment

on a mortgage loan, you would enter:

CLEAR ALL : clear the calculator of data from the previous problem

(how to do this will vary from calculator to calculator);

BEG/END = END (since a mortgage is a regular annuity);

P/YR = 12 (since we will be making monthly payments);

N = 360 payments (30 years x 12 payments/year)

I/YR = 5.65 (or whatever yearly interest rate you will be charged);

PV = 300,000.00 (the amount of the loan);

FV = 0 (since we plan to pay off the loan completely);

Then tell the calculator to calculate:

PMT => -1,731.71 (negative, since it’s money you’ll be paying out).

To calculate a savings (“sinking fund”) problem, PV would be zero and FV would be the target

amount you’re saving towards.

NOTE that when entering dollar values for FV, PV, or PMT, you need to obey the cash flow

sign convention: positive for amounts you receive, negative for amounts you pay out.

36

LECTURE 215

NORMAL ANNUITIES VS ANNUITIES DUE

[See also CALC LECTURE 12]

An “annuity” in this context is any stream of future cash flows of even size, spaced out equally

across time. (Not necessarily the same thing as the products of the same name sold by

insurance companies.)

The majority of cash flow projects are such that cash flows take place at the end of the

corresponding time period. We call such projects normal annuities.

For example, your basic home mortgage or car loan is such that each monthly payment is due at

the end of the month. Coupon bonds pay coupon payments at the end of the 6-month period

during which the interest was earned. And so on.

In some cases, however, cash flows happen (are due) at the beginning of the corresponding

time period. We refer to such projects as annuities due.

A typical example of an annuity due would be a lease, where payment for a given month is due

at the beginning of that month. (Remember your apartment lease: rent for the first month was

due in advance. Car leases work the same way.) Another example of an annuity due is the

scenario where you have won some huge sum in a state lottery, and you are offered either a

lump sum or a number of yearly payments. The yearly payments would be in the form of an

annuity due, since the first payment will be received right away (at the beginning of year 1.)

USING FORMULAS

The formulas given in Lecture 999 for FV and PV for an annuity assumes normal annuities.

USING TABLES

The annuity factors given in the textbook tables assume normal annuities.

A NOTE ABOUT CALCULATOR USAGE

Your calculator needs to know whether a particular problem refers to a normal annuity or an

annuity due, since the results will be different. Every financial calculator will have a function

key that toggles between ‘END’ mode (normal annuities) and ‘BEGIN’ mode (annuities due).

Make sure this toggle is set correctly for the problem you are working on.

HINT: Since the majority of financial calculator problems deal with normal annuities, it is

customary to leave the calculator set for ‘END’ mode unless the problem specifically gives you

reason to think it is an annuity due.

37

LECTURE 216

AMORTIZATION

[See also CALC lectures 26 and 29.]

The key characteristic of an annuity such as a mortgage loan is that the recurring payments are

all of the same size. (Also, under normal circumstances we plan to fully amortize the loan over

the given number of payments, so that FV = 0.)

What this means is that part of each payment will go to pay interest on the previous period’s

ending balance, while the rest goes towards reducing the outstanding balance (the principal).

Since we are continually reducing the principal, the interest part of the payments will be

shrinking as we proceed through paying off the loan.

If we want to know, for each payment of the loan, how much goes to pay interest and how

much to reduce the principal, we need to create an amortization schedule.

This is done most efficiently using a spreadsheet such as Microsoft Excel, since doing it

manually for a 360-payment loan would be very tedious.

We can also use the financial calculator to calculate, for a given range of payments,

INT total amount of interest paid with those payments;

PRIN total amount by which the principal has been reduced; and

BAL the remaining principal balance after the last payment.

One use for this would be to calculate how much interest you can deduct from your tax return

for each tax year.

See the separate set of Lectures for your specific calculator to see how to use the amortization

functions.

38

XI: INFLATION AND TAXES

39

LECTURE 331

THE EFFECTS OF INFLATION ON INTEREST RATES: THE FISHER EQUATION

REAL VS. NOMINAL INTEREST RATES

Inflation and the Fisher effect

We distinguish between nominal and real interest rates. The nominal rate is the observed

market rate of interest. The “real” rate is the nominal rate, adjusted for inflation. According

to (American economist Irving) Fisher, the relationship is this:

People’s demand for, and supply of, loanable funds depends on the real (inflation-adjusted)

interest rate r (interest measured in terms of purchasing power). If the economy experiences

inflation, this reduces the purchasing power of the money the borrower pays back to the lender

later, and if the inflation is expected when a loan is granted the lender is going to want to be

compensated for this. The market rate of interest therefore incorporates compensation for

expected inflation.

Ex ante [before the loan is made] we have [using the Greek letter π to stand for inflation]:

(1+i) = (1+r)*(1+E(π) ) -> Nominal rate i = (1+r)(1+E(π)) - 1

Ex post [after the loan has been paid back] we have

(1+r) = (1+i) / (1+ π) -> Real rate r = (1+i)/(1+ π) - 1

In daily use, we often simplify the above equations this way (for small values of i, r, π):

i = r + E(π) and r = i – π

[The nominal or market interest rate = the real interest rate + the expected rate of inflation;

The real interest rate or return = the nominal or market rate – the actual inflation rate ]

Question: Can we know what “the market” expects future inflation to be?

A: Not normally; the interest rates we actually observe in financial markets are nominal interest

rates, which incorporate the different inflation expectations of all borrowers and lenders.

However, a few years back (around 1997) the Treasury started selling so-called TIPS bonds,

which pay a fixed interest rate plus after-the-fact compensation for inflation. These bonds will

therefore carry a nominal market rate equal to the “real” interest rate r, since buyers know

they’ll get compensated for inflation if we have any.

Thus, if we subtract the interest rate on TIPS bonds (r) from the market rate on comparable

standard bonds (i) we get an estimate of what “the market” currently expects the inflation rate

to be over the given time horizon. (I suspect this is exactly why the Treasury started offering

TIPS in the first place – to make economists happy!)

40

LECTURE 332

THE EFFECTS OF TAXES ON AFTER-TAX BOND YIELDS

When you invest in financial instruments, you are subjected to (at least) two forms of taxation:

a) Any interest, dividends or coupon payments you receive are subject to income taxes at

the rates applicable to the individual investor. (As you know, we have a ‘progressive’

graduated income tax system in the United States.)

b) If you buy a financial instrument today and sell it at a higher price later, you are subject

to the capital gains tax. (However, if you hold the investment for less than a year, you

pay regular income taxes on the gain instead.)

Here we discuss the income tax only.

Fact 1A: federal income taxes can amount to up to 35% in the top income bracket (2008).

Fact 1B: state income taxes vary greatly, from zero in some states (Florida, Nevada) up to as

much as 9.9% in high-tax states (CA, IA, NJ, OR, RI, VT, DC).

Fact 2A: income from federal (Treasury and agency) bonds is not taxed by the states.

Fact 2B: income from municipal bonds is not taxed by the federal government (which is why

they are sometimes referred to as “tax-free” bonds.)

Fact 2C: income from corporate bonds is taxed by everybody!

Result: for the same nominal yield (bond coupon rate), the after-tax yield is highest for

municipal bonds, lower for T-Bills, and lowest for corporate bonds.

If the effective tax rate for a given investment is tEFF, then the after-tax yield you earn is:

iAT = iNOM * (1 – tEFF)

So for federal bonds, we have:

iAT = iNOM * (1 – tFed) [ using the taxpayer’s relevant marginal rate ]

For municipal bonds, we have:

iAT = iNOM * (1 – tState) [ using the applicable state tax rate, if any ]

For corporate bonds, we have:

iAT = iNOM * (1 – tFed) * (1 – tState) [ using both ]

41

LECTURE 333

THE EFFECTS OF TAXES ON PRE-TAX BOND YIELDS

Rational investors care about the net after-tax yield they will earn from their investments,

rather than the nominal pre-tax yield. This means that arbitrage activity in financial markets

will see to it that the after-tax yield for bonds of the same quality is equalized.

By ‘same quality’ we mean similar maturity and degrees of liquidity and risk.

We can use the above equations to solve for iNOM, given the common iAT and the applicable tax

rates.

What this means is, for example, that municipal bonds can offer a lower nominal yield than T-

Bills and still be competitive -- even though municipal bonds are both riskier and less liquid…

Example

If you want an after-tax yield of 6%, and your marginal federal income tax rate is 33%, and

your applicable state income tax rate is 10%, what nominal yields do you have to be offered on

federal bonds, municipal bonds, and corporate bonds, respectively?

Municipal bonds: 6% / (1 – 0.1) = 6.67%

Federal bonds: 6% / (1 – 0.33) = 9.00%

Corporate bonds: 6% / (1 – 0.1)*(1-0.33) = 10.00%

42

LECTURE 334

COMBINED EFFECTS OF TAXES AND INFLATION ON NET YIELDS

If you need to know the after-tax real return to an investment, the rule is: first subtract the

taxes (since you pay taxes on the nominal yield), and then adjust the remainder for inflation.

Using simplified math, we get:

(1 )AT NOMr i t

Example: If your fabulous investment scheme yielded a nominal return of 25%, and your

marginal income tax rate is 35%, your nominal after-tax yield is:

25% (1 0.35) 16.25%ATi

And if inflation that year ran 12% (must have been under President Carter, this project ;) your

real net return was:

16.25% 12% 4.25%AT ATr i

which is not quite as impressive as 25%!

NOTE: Referring back to Lecture 331, the more correct way of dealing with inflation is

multiplicative rather than additive:

(1 )*(1 ) (1 (1 ))AT NOMr i t

In this formulation, the correct answer to the above example becomes:

[(1 0.25 (1 0.35)) /(1 0.12)] 1 3.79%ATr

43

LECTURE 531

THE EFFECTS OF INFLATION ON PURCHASING POWER: USING THE CPI

We know from Macroeconomic theory that over the long run, the average price level in the

economy will rise roughly at the same rate as the increase in the money supply less the rate of

real growth. [MV=PQ so dP = dM – dQ when dV=0.]

The federal government, in the guise of the Bureau of Labor Statistics (BLS) tries to measure

this ‘average price level’ P through the calculation of various price indexes such as the

Consumer Price Index (CPI), the Producer Price Index (PPI), and the GDP Deflator. They also

calculate different versions of the CPI, such as … [CPI-U, CPI-W, C-CPI-U, “core” CPI].

Each such index sets the price level in a specific year, called the base year, equal to 100, and

the index for subsequent years is calculated from this basis. If for example prices increase by

5% from the base year till the next year, next year’s index will be 105.

If inflation in 3 subsequent years is 5%, 6%, and 8.5%, the index will increase by:

(1.05)*(1.06)*(1.085) = 1.2076

So if we started at an index of 134.7, after those 3 years the index would be

134.7 * 1.2076 = 162.7.

To compare prices in two different years, we have the following rule:

1 2

1 2

Year Year

Year Year

P P

CPI CPI

From this relationship, if we know 3 values we can calculate the fourth.

Example 1: A given basket of goods cost $1,244.50 in 1977. The CPI in 1977 was 181.8 and

the CPI in 1985 was 322.3 (1967=100). What should this basket of goods cost in 1985?

Answer: $1,244.50 * (322.3 / 181.8) = $1,244.5 * 1.7728 = $2,206.25!

Example 2: A given basket of goods cost $2,500.00 in 1980 and $3,928.75 in 1990. The CPI in

1980 was 247.6. What must the CPI have been in 1990?

Answer: 247.6 * ($3,928.75 / $2,500.00) = 247.6 * 1.5715 = 389.1.

44

LECTURE 532

THE EFFECTS OF INFLATION ON PURCHASING POWER:

USING AN AVERAGE INFLATION RATE

Example 3: Using the data in the lecture above, what was the average rate of inflation from

1977 till 1985? From 1980 till 1990?

Answer: We see that from 1977 until 1985, prices increased by a factor of 1.7728, or by

77.28%. This is over a period of 8 years. The yearly compounded rate, then, is:

8 1.7728 1 0.0742 or 7.42%

From 1980 until 1990, the average inflation rate was:

101.5715 1 0.0462 or 4.62%

Now, we can use these average inflation rates to calculate price changes:

Example 4 (forward): If you spent $375 per month on groceries in 1978, what could you

expect to spend on average for the same groceries in 1979?

Answer: $375.00 * (1 + Π) = 375.00 * 1.0742 = $402.83.

Example 5 (backward): If you purchased a car for $15,000 in 1990, how much would that car

have cost back in 1985 [assuming car prices follow the average inflation rate]?

Answer: $ 15,000 / (1 + Π)5 = $15,000 / (1.0462)5 = $11,968.

We can also use them to calculate real rates of return from nominal rates [see Lecture 331]:

Example 6: Real investment yields. If you invested in some shares of stock in 1985 and sold

them again in 1990, and you earned a nominal annual return of 12.7% on those shares for that

time period, what was your real return?

Answer: Using the Fisher equation,

(1+r) = (1+i)/(1+p) = 1.12 / 1.0462 = 1.077232 i.e. r = 7.72%

The simplified shortcut solution is: r = i – Π = 12.7% - 4.62% = 8.08%.

Note: The above discussion uses geometric averaging for the yearly inflation rate. See lecture

323 for a discussion of geometric vs. arithmetic averages.

45

XVII: FORMULAS

46

LECTURE 991

A BRIEF DISCUSSION OF CALCULATORS WITH FINANCIAL FUNCTIONS

The following is a list of calculator models available as of May 2007 that are advertised as

“financial calculators”, with an indication of their approximate price. To be classified as a

financial calculator, a model must have both TVM functions and CFLO functions. In addition,

some of these models also have specialized bond functions or other functions.

Any of these calculators is usable for FINC-2400, FINC-3610 and FINC-3700, except the ones

from CASIO. We base our lectures on the HP-10BII.

Hewlett-Packard HP-10BII $30 (entry level model)

Hewlett-Packard HP-12C $70 (popular with financial professionals)

Hewlett-Packard HP-12C Platinum $80 (faster than the 12C and more functions)

Hewlett-Packard HP-17BII Plus $100 (menu driven)

Hewlett-Packard HP-19BII (discontinued; about $130 on eBay)

Hewlett-Packard HP-30B $50 (latest model)

Texas Instruments TI-BAII Plus $35 (entry level model)

Texas Instruments TI-BAII Plus Pro $45 (more functions)

Sharp EL-733A $25 (entry level)

CASIO FC-200V $32 (entry level)

Other scientific or graphing calculators sometimes have financial functions (most often TVM

functions, sometimes also CFLO functions). Some of these are:

CASIO CFX-9850GC Plus $117 TVM, CFLO

Sharp EL-9900C $99 TVM

Hewlett-Packard HP-39GS $80 TVM (CFLO by download)

Hewlett-Packard HP-48GII $110 TVM ( “ )

Hewlett-Packard HP-49G+ TVM ( “ )

Hewlett-Packard HP-50G $150 TVM ( “ )

Texas Instruments TI-83,84 series TVM

Texas Instruments TI-89, 92, Voyage200 TVM, CFLO (by download)

Additionally:

- Some general calculators will take plug-in program cards and/or downloadable

programs, which may contain financial functions.

- Some general calculators (esp. Hewlett-Packard) have a generic SOLVER function

where you can enter an equation or set of equations, and have the calculator solve for

unknown variables; this functionality can be used for financial functions.

47

The following is a brief description of the different financial calculator models, with some of

their strengths and weaknesses noted.

HP-10BII

This is HP’s entry level financial calculator. Unusual for HP, it uses standard algebraic data

entry only (no RPN). This calculator has no bond functions or depreciation. The keyboards can

be flaky on some calculators. A new version with a slightly different layout was released in

2008.

Some quirks:

• The cash flow functions arbitrarily limit the repetition factor Nj to 99.

• The TVM functions enter and calculate I/YR. They assume C/YR = P/YR.

• This is the only calculator whose cash flow functions calculate IRR as a yearly rate!

HP-30B

This is a new calculator which is a step up from the HP-10BII in many respects: greater

calculating precision (15 decimals, 10^499), more display digits (12), no limit to P/YR, 50 cash

flow groups (rather than 14) with no limit to repetitions (Nj), NFV and NUS, bond functions,

depreciation, and PB plus DPB (=duration). There are also some additional statistics functions.

The calculator allows for RPN plus 2 different algebraic entry modes.

The downside is: in order to pack the additional functionality into a small form-factor

calculator, HP chose to employ a ‘hidden menu’ system that may take a bit to get used to.

HP-12C

Venerable financial calculator greatly preferred by real estate professionals over the last 20

years, for its small form factor (shirt pocket size), sturdy build, and very reliable keyboard. The

calculator is all RPN all the time, requiring a first time user to “re-wire” his brain first. (But

after that, it’s a cinch to use!) The calculator has some limited programmability.

Benefits over the 10BII:

• Depreciation functions (SL, DB, SOYD)

• Limited bond functions

Notable operating feature:

• The calculator does not know the concept of P/YR: interest rates (I, IRR) are all

periodic, both in the TVM and the CFLO functions.

• Like the 10BII, cash flow groups are limited to 99 repetitions (Nj)

48

HP-12C Platinum

This is a recent update to the older model. It has a bit more memory (80 cash flow registers

instead of 20), and is much faster. Algebraic data entry has been added, to appeal to a broader

audience. A ‘backspace’ function has been added for use during data entry. The limitation on

Nj remains.

HP-19BII

This is HP’s former top-of-the-line financial calculator. It is no longer being made, but used

ones go for about $130 on eBay. It has the form of a two-part fold-out ‘book’. The right-hand

part has the standard calculator set of functions, and the left part has keys for alphabetic data

entry. (You can store text in the calculator, as well as numerical data in named lists.) The

calculator has limited plotting capability. It allows for either RPN or Algebraic operation. It has

an IR port which may be used to talk to HP’s IR printer. (You can print amortization schedules,

for example.) It has a general SOLVER capability: you can enter a single equation or set of

equations, and have the calculator solve for one of the variables as the unknown. The calculator

employs ‘soft menus’: the functions of the top keys change according to the application being

run. The calculator has limited PIM functionality: clock, timer, calendar, appointment book.

Benefits over the 12C:

• Full set of bond functions

• Depreciation (including ACRS, but not MACRS)

• You can plot NPV against the discount rate to find roots (IRR values)

HP-17BII Plus

This is a newer version of the 19BII, where the two-part fold-out format has been abandoned in

favor of a more standard one-piece construction. Other than that, this calculator’s functionality

mirrors that of the older 19BII. It is a bit faster.

49

TI-BAII

This is Texas Instruments’ entry level financial calculator. While costing about the same as the

HP-10BII, it is more capable in some respects. It has bond functions, and depreciation (SL, DB,

SOYD). Operation is algebraic only. This calculator relies heavily on a ‘worksheet’ metaphor

for data entry. Its operations are a bit confusing to understand in the beginning. This calculator

seems well constructed, and the keyboard is reliable.

TI-BAII Professional

This model is a step up from the BAII. For $10 more, you get:

• 32 cash flow registers instead of 24

• Duration calculations (MDUR for bonds, DPBP for TVM calculations)

• Additional cash flow functions: NFV, MIRR

Sharp EL-733A

Another entry level calculator, all algebraic data entry. Construction is kind of ugly and boring,

and the keys can be a bit confusing. This calculator has no bond functions or depreciation

(hence it is a close match to the HP-10BII.) Like the HP-12C, it operates on periodic interest

rates only, not annual ones.

CASIO FC-200V

This is Casio’s entry level bid. Casio has better designers than Sharp: the calculator itself is

very pleasing, constructed in sturdy brushed aluminum. In addition to battery power, it has a

small solar cell. Like the other non-HP calculators, it is algebraic entry only. And like the other

calculators, it has TVM and CFLO functions as well as statistical functions.

Features that set this calculator apart:

• It has special functions for simple interest

• It has bond and basic depreciation functions

• It calculates Duration (in the form of Discounted Payback)

• However, the manual is a really cheap and dinky little booklet.

Really, really stupid limitation in this calculator:

• It does not allow for groups of cash flows! Although the data entry module allows

you to put in frequency data, the calculation routines ignore this information. Grrrr!

50

OTHER HP CALCULATORS

Hewlett Packard makes several advanced graphing scientific calculators, such as the HP-39G,

HP-48GII, and HP-50. These calculators are programmable, and have a general SOLVER

application which permits the entry of a user-defined equation or set of equations, with the

calculator solving for unknown variables. These calculators all come equipped with a basic

TVM module (including amortization), but no cash flows, depreciation, or bonds. They can be

updated with better financial functions via program downloads from a PC.

OTHER TI CALCULATORS

Texas Instruments has a whole range of scientific graphing calculators. Of these, the TI-

83+/Silver and TI-84+/Silver come equipped with a basic TVM module (including

amortization and date calculations), but no cash flows, depreciation, or bonds. Cash flow

routines (NPV, IRR) are available for programs, but not with a graphical interface. The TI-86

has no financial functions. The TI-89, 92, and Voyager200 can be given financial functions via

a download.

CASIO CFX-9850GC Plus

This is a very nice scientific calculator which sports color graphics and soft menus. It has been

given TVM and CFLO functions (but not depreciation or bonds). Unfortunately, the CFLO

functions suffer the same stupid limitation as the FC-200V: no groups! Nj = 1!

SHARP EL-9900C

This is a new graphing scientific calculator with limited financial functionality built in (TVM

only, it appears). While the display is large (8 lines), internally the calculator seems to be

limited: it handles a maximum of 10 significant digits, and the exponent is limited to ±99.

51

LECTURE 999 MATH

SELECTED FORMULAS FOR CERTAIN FINANCIAL RELATIONSHIPS

[From Floyd and Allen’s Real Estate Principles]

FUTURE VALUE OF A SINGLE PAYMENT [LUMP SUM]

(1) (1 ) nFV PV i Calculator: enter n, i, PMT=0, PV; calculate FV

PRESENT VALUE OF A SINGLE PAYMENT [LUMP SUM]

(2) (1 )n

FVPV

i

Calculator: enter n, i, PMT=0, FV; calculate PV

PRESENT VALUE OF AN ORDINARY ANNUITY

(3)

11

(1 )niPVA PMT

i

Calculator: enter n, i, PMT, FV=0; calculate PV

FUTURE VALUE OF AN ORDINARY ANNUITY

(4) (1 ) 1ni

FVA PMTi

Calculator: enter n, i, PV=0, PMT; calculate FV

SINKING FUND PAYMENT [SAVING]

(5) (1 ) 1n

iSFP FVA

i

Calculator: enter n, i, PV=0, FV; calculate PMT

MORTGAGE PAYMENTS [LOANS]

(6) 1

1(1 )n

iPMT PVA

i

Calculator: enter n, i, PV, FV=0; calculate PMT

EFFECTIVE ANNUAL RATE (EAR) where “i” = APR

(7) (1 ) 1miEAR

m

NOTE: formulas 3-6 assume normal annuities (calculator set to END mode).

SEE ALSO: the document "Expanded Financial Formulas."