978-1259709685 Chapter 4 Lecture Note Part 2

Unlock access to all the studying documents.

View Full Document

Slide 4.17 Calculator Keys

Lecture Tip: Texas Instruments offers instructors an “Emulator”

software that allows the face of the calculator to be projected onto

a computer. This enables instructors to demonstrate key stroke

sequences.

Slide 4.18 –

Slide 4.19 Multiple Cash Flows

Slide 4.20 Valuing “Lumpy” Cash Flows

There are two ways to calculate the present value of multiple cash flows:

discount the last amount back one period and add them as

you go, or discount each amount to time zero and then add

them.

Lecture Tip: The present value of a series of cash flows depends heavily

on the choice of the discount rate. Changing the interest

rate in one of the examples and recalculating is a good way

to illustrate this point.

There are two ways to calculate the future value of multiple cash flows:

compound the accumulated balance forward one period at a

time, or calculate the future value of each cash flow and

add them at the ending period.

.A The Algebraic Formula

Alternative forms of the TVM relationship:

Solving for future value: Ct+T = Ct × (1 + r)T

Solving for present value: Ct = Ct+T ÷ (1 + r)T

Solving for the discount rate: r = (Ct+T ÷ Ct) 1/T – 1

Solving for the time period: T = ln(Ct ÷ Ct+T )÷ ln(1 + r)

4.2. Compounding Periods

Slide 4.21 –

Slide 4.22 Compounding Periods

To this point, we have assumed annual interest rates; however, many

projects / investments have different periods. For example, bonds typically

pay interest semi-annually, and house loans are on a monthly payment

schedule. To apply the formulas, we must adjust for compounding periods:

FV = C0 × (1 + r/m)m×T

.A Distinction between Annual Percentage Rate and Effective Annual

Rate

Slide 4.23 –

Slide 4.25 Effective Annual Rates of Interest

Stated or quoted interest rate (also called annual percentage rate, or APR)

– rate before considering any compounding effects, such as

10% compounded quarterly

Effective annual rate of interest – rate on an annual basis that reflects

compounding effects, e.g., 10% compounded quarterly has an

effective rate of 10.38%.

.A Compounding over Many Years

Lecture Tip: It is important to stress that the effective annual rate is the

rate of interest that we effectively earn after accounting for

compounding. That seems simple enough, but students still

have a hard time remembering that the EAR already accounts

for all of the interest on interest during the year. It may be

helpful to point out that the EAR is not used directly in time

value of money calculations, except when we have annual

periods. TVM calculations compound (or discount) the values

every period, but the EAR has already done that. The EAR is

primarily used for comparison purposes, not for calculation

purposes.

EAR = [1 + (quoted rate)/m]m – 1, where m is the number of periods per

year

Example: 18% compounded monthly is [1 + (.18/12)]12 – 1 = 19.56%

Slide 4.26 EAR on a Financial Calculator

You can also do this on most calculators. For the TI-BA-II+,

press 2nd I Conv (above the number 2), enter 18 for the NOM

rate (do not forget to press enter), press the up arrow to go to

C/Y and enter 12, press the up arrow to go to EFF and press

CPT = 19.56%. Most other financial calculators can do this as

well.

Lecture Tip: Here is a way to drive the point of this section home. Ask

how many students have taken out a car loan. Now ask one of

them what annual interest rate s/he is paying on the loan.

Students will typically quote the loan in terms of the APR.

Point out that, since payments are made monthly, the effective

rate is actually more than the rate s/he just quoted, and

demonstrate the calculation of the EAR.

Ethics Note: Rent-to-own agreements and tax refund loans have a lot in

common. Because of the structure of the contracts, they do not

have to provide information on interest rates. However, when

you work out the rates implied in the contracts, they can be

extraordinarily high. It is worthwhile to encourage students to

use caution (and their newfound knowledge of time value!)

when considering these situations.

Example: Suppose you are in a hurry to get your income tax refund. If you

mail your tax return, you will receive your refund in 3 weeks. If

you file the return electronically through a tax service, you can

get the estimated refund tomorrow. The service subtracts a $50

fee and pays you the remaining expected refund. The actual

refund is then mailed to the preparation service. Assume you

expect to get a refund of $978. What is the APR with weekly

compounding? What is the EAR? How large does the refund

have to be for the APR to be 15%?

Using a financial calculator to find the APR: PV = 978 – 50 =

928; FV = -978; N = 3 weeks; CPT I/Y = 1.765% per week;

APR = 1.765 (52 weeks per year) = 91.76%!!!

Compute the EAR = (1.01765)52 – 1 = 148.38%!!!!

You would be better off taking a cash advance on your credit card and

paying it off when the refund check comes, even if you have the

most expensive card available.

Refund (net of $50) needed for a 15% APR:

PV + 50 = PV(1 + (.15/52))3

PV = $5,761.14

Lecture Tip: Another point of confusion for many students is what to do

when the payment period and the compounding period do not

match. It’s important to point out that we cannot adjust the

payment to match the interest rate. Neither can we just divide

the APR by any number we want to get a period rate; we can

only divide it by the number of periods used for compounding.

The EAR can be used as a common denominator to help us find

“equivalent” APRs.

Example: Suppose you are going to have $50 deducted from

your paycheck every two weeks and have it placed in an

account that pays 8% compounded daily. How much will you

have in 35 years?

You are depositing money every two weeks (26 times per year), but

compounding occurs daily. You need a period rate that

corresponds to every two weeks, but you can only divide the

APR given by 365. What can you do?

Find the EAR for the daily compounded rate. This is the rate you will

earn each year after you account for compounding.

EAR = (1 + .08/365)365 – 1 = .08327757179 (Point out that

it is extremely important that you DO NOT round on the

intermediate steps.)

What you need is an APR based on compounding every two weeks that

will pay the same effective rate of interest. So you take the EAR

computed above and convert to an APR based on 26

compounding periods per year.

APR = 26[(1.08327757179)1/26 – 1] = .0801144104

At this point, many students feel as if this is wasted effort, because there

is not that much difference. As we will see, the small difference

in rates can make a difference over long periods of time.

Find the FV: PMT = 50; N = 35(26) = 910; I/Y = 8.01144104 /

26 = .308132348; CPT FV = $250,535.24

If you just use I/Y = 8/26, you would get a FV = $249,829.21;

a difference of $706.03.

.B Continuous Compounding

Slide 4.27 Continuous Compounding

Starting with the general formula:

FV = C0× (1 + r/m)m×T

What would happen if m approached infinity? This is

known as continuous compounding. If we take the limit of

the equation, it reduces to:

FV = C0 × erT

where e is a constant, transcendental number equal to

approximately 2.718.

The EAR of a continuously compounded investment is:

EAR = er – 1

4.3. Simplifications

Slide 4.28 Simplifications

.A Perpetuity

Slide 4.29 Perpetuity

Slide 4.30 Perpetuity: Example

Perpetuity – series of level cash flows forever

PV = C / r

Good examples of perpetuities include preferred stock and British consols.

.B Growing Perpetuity

Slide 4.31 Growing Perpetuity

Slide 4.32 Growing Perpetuity: Example

A stream of cash flows that grows at a constant rate forever

PV = C1 / (r-g)

.C Annuity

Slide 4.33 Annuity

Slide 4.34 –

Slide 4.35 Annuity Example

Ordinary Annuity – multiple, identical cash flows occurring at

the end of each period for a fixed number of periods.

The present value of an annuity of $C per period for T periods at r percent

interest:

PV = C[1 – 1/(1 + r)T] / r

Example: If you are willing to make 36 monthly payments of

$100 at 1.5% per month, what size loan can you obtain?

PV = 100[1 – 1/(1.015)36] / .015 = 100(27.6607) = 2766.07

Or, use the calculator: PMT = -100; N = 36; I/Y = 1.5; CPT PV

= $2,766.07 (Remember that P/Y = 1 when using period rates.)

Finding the future value of an ordinary annuity:

FV = C[(1 + r)T– 1] / r

Example: If you make 20 payments of $1000 at the end of each period at

10% per period, how much will be in your account after the

last payment?

FV = $1,000[(1.1)20 – 1] / .1 = 1,000(57.275) = $57,275

Or, use the calculator: PMT = -1,000; N = 20; I/Y = 10; CPT

FV = $57,275 (Remember to clear the registers before working

each problem.)

In general, we assume that cash flows occur at the end of each time period.

This assumption is implicit in the ordinary annuity formulas

presented. An annuity due has cash flows that occur at the

beginning of the period.

Lecture Tip: It should be emphasized that annuity factor tables (and the

annuity factors in the formulas) assume that the first payment

occurs one period from the present, with the final payment at

the end of the annuity’s life. If the first payment occurs at the

beginning of the period, then FV’s have one additional period

for compounding and PV’s have one less period to be

discounted. Consequently, you can multiply both the future

value and the present value by (1 + r) to account for the

change in timing. The values can also be computed directly by

changing the setting in the financial calculator.

.D Growing Annuity

Slide 4.36 Growing Annuity

Slide 4.37 –

Slide 4.38 Growing Annuity: Example

A growing stream of cash flows with a fixed maturity

PV =







4.4. Loan Amortization

Slide 4.39 Loan Amortization

Slide 4.40 Pure Discount Loans

Pure Discount Loans: Borrower pays a single lump sum (principal and

interest) at maturity. Treasury bills are a common example of pure

discount loans.

Slide 4.41 Interest-Only Loans

Interest-Only Loans: Borrower pays interest only each period and

the entire principal at maturity. Corporate bonds are a common

example of interest-only loans.

Slide 4.42 Amortized Loan with Fixed Principal Payment – Example

Slide 4.43 Amortized Loan with Fixed Payment – Example

Amortized Loans: Borrower repays part or all of the principal over the life of

the loan. Two methods are (1) fixed amount of principal to be

repaid each period, which results in uneven payments, and (2)

fixed payments, which results in uneven principal reduction.

Traditional auto and mortgage loans are examples of the second

type of amortized loans.

Lecture Tip: Consider a $200,000, 30-year loan with monthly

payments of $1330.60 (7% APR with monthly compounding). You

would pay a total of $279,016 in interest over the life of the loan.

Suppose instead you cut the payment in half and pay $665.30

every two weeks (note that this entails paying an extra $1330.60

per year because there are 26 two week periods). You will cut your

loan term to just under 24 years and save almost $70,000 in

interest over the life of the loan.

Calculations on TI-BAII plus

First: PV = 200,000; N=360; I=7; P/Y=C/Y=12; CPT PMT =

1330.60 (interest = 1330.60*360 – 200,000)

Second: PV = 200,000; PMT = -665.30; I = 7; P/Y = 26; C/Y =

12; CPT N = 614 payments / 26 = 23.65 years (interest =

665.30*614 – 200,000)

It may be valuable to point out to students that banks often offer

this “service” for a fee; however, with most loans, additional

principal payments are accepted without charge. Thus, a borrower

can effectively create the outcome without paying any fee.

4.5. What Is a Firm Worth?

Slide 4.44 What Is a Firm Worth?

An investment is worth the present value of its future cash flows.

Since a company is a series of investments, it is worth the total

present value of all cash flows generated by the firm.

Slide 4.45 Quick Quiz