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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 =
T
r
g
gr
C
1
1
1
1
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
