HFT 4464
Chapter 5
Time Value of Money
3/23/2016
1
Chapter 5
Introduction


This chapter introduces the topic of
financial mathematics also known as the
time value of money.
This is a foundation topic relevant to many
finance decisions for a hospitality firm:
Capital budgeting decisions
 Cost of capital estimation
 Pricing a bond issuance

5-2
Organization of Chapter


Financial math will be presented in the context of
personal finance. Later chapters will apply financial
math to various finance applications for a hospitality
firm.
Financial math topics covered include:



Computation of future values, present values, annuity
payments, interest rates, etc.
Perpetuities and non-constants cash flows
Effective annual rates and compounding periods other
than annual
5-3
Future Value of a Lump Sum

The future value in 2 years of $1,000 earning
5% annually is an example of computing the
future value of a lump sum. We can
compute this in any one of three ways:
Using a calculator programmed for financial
math
 Solve the mathematical equation
 Using financial math tables (Table 5.1) p. 91

5-4
Solve for the Future Value

The general equation for future value is:


FVn = PV x (1+i)n
Computing the future value in the example:

FV2 = $1,000 x (1+5%)2 = $1,102.50
5-5
Present Value of a Lump Sum

How much do you need to invest today so
you can make a single payment of $30,000
in 18 years if the interest rate is 8%? This is
an example of the present value of a lump
sum.

Again we can solve it using a programmed
calculator, solving the math or using Table
5.2. p.93
5-6
Solve for the Present Value

The general equation for present value is:
FVn
PV 
n
1  i 

Computing the present value in the example:
$30,000
PV 
 $7,507.47
18
1  8%
5-7
Annuities

Two or more periodic payments

All payments are equal in size.

Periods between each payment are equal in
length.
5-8
PV Versus FV of an Annuity

The value of an annuity can be expressed
as an equivalent lump sum value.

The PV of an annuity is the lump sum value
of an annuity at a point in time earlier than
the payments.

The FV of an annuity is the lump sum value
of an annuity at a point in time later than the
payments.
5-9
Ordinary Annuity Versus Annuity Due

The PV of an ordinary annuity is located one period before the
first annuity payment. (Paid at the end of each period)

The PV of an annuity due is located on the same date as the first
annuity payment. (Paid at the beginning of each period)

The FV of an ordinary annuity is located on the same date as
the last annuity payment. (Last annuity payment is at the end of
the last period)

The FV of an annuity due is located one period after the last
annuity payment. ( Last annuity payment is at the beginning of
the last period)
5-10
Future Value of an Annuity

Suppose you plan to deposit $1,000 annually
into an account at the end of each of the next
5 years. If the account pays 12% annually,
what is the value of the account at the end of 5
years? This is a future value of an annuity
example.

We can solve this problem using a
programmed calculator, solving the math, or
using Table 5.4. (p. 96)
5-11
Solve for the Future Value of an Annuity

The general equation for a FV of an annuity is:
 1  i n  1
FVA n  PMT x 

i



The FV of the annuity in the example is:
 1  12% 5  1
FVA 5  $1,000 x 
  $6,352.85
12%


5-12
Present Value of an Annuity

You plan to withdraw $1,000 annually from an
account at the end of each of the next 5 years.
If the account pays 12% annually, what must
you deposit in the account today? This is an
example of a present value of an annuity.

We can solve this problem using a
programmed calculator, solving the math, or
using Table 5.5. ( p. 102)
5-13
Solve for the Present Value of an Annuity

The general equation for PV of an annuity is:
1

1 - 1  i n
PVA n  PMT x 
i








The PV of the annuity in the example is:
1


1
 1  12%5 
  $3,604.78
PVA 5  $1,000 x 
12%




5-14
Perpetuity—An Infinite Annuity



A perpetuity is essentially an infinite annuity.
An example is an investment which costs you
$1,000 today and promises to return to you $100
at the end of each forever!
What is your rate of return or the interest rate?
PMT
$100
i

 10%
PV
$1,000
5-15
The Present Value of a Perpetuity

Another investment pays $90 at the end of
each year forever. If 10% is the relevant
interest rate, what is the value of this
investment to you today? We need to solve
for the present value of the perpetuity.
PMT $90
PV 

 $900
i
10%
5-16
Present Value of a Deferred Annuity

There are 3 different PV of annuity computations:

The payments on an ordinary annuity begin one period
after the PV.

The payments on an annuity due begin on the same date
as the PV.

The payments on a deferred annuity begin 2 or more
periods after the PV. Thus it is called a deferred annuity
since the payments are deferred more than one period
from the present.
5-17
Computing the PV of a Deferred Annuity

An investment promises to pay $100 annually
beginning at the end of 5 years and continuing
until the end of 10 years. What is the value of
this investment today at a 7% interest rate?
Because the payments are deferred 5 years, this
is a PV of deferred annuity problem.

1st step:Compute the PV of an ordinary annuity.
1


1
 1  7% 6 
  $476.65
PVA  $100 x 
7%




5-18
Computing the PV of a Deferred Annuity

2nd step : Discount the PV of the ordinary
annuity through deferral period.
1


1 - 1  7%6 
1
 x
PV  $100 x 
4
7%
1  7%




1
PV  $476.65 x
 $363.63
4
1  7%
5-19
General Formula for PV
of a Deferred Annuity
1

1
 1  i n
PV  PMT x 
i








1
 x
m


1

i


PMT = $ amount of the perpetuity payment
i = interest rate
n = the number of perpetuity payments
m = the deferral period minus 1
5-20
PV of a Series
of Non-Constant Cash Flows

The PV of a series of non-constant cash flows is
just the sum of the individual PV equations for
each cash flow.
CF1
CF2
CFn
PV 

 .............
1
2
1  i  1  i 
1  i n

Where the Cfi’s are a series of non-constant cash
flows from year 1 to year n.
5-21
PV of a Series
of Non-Constant Cash Flows

Suppose some new kitchen equipment for your
restaurant is expected to save you $1,000 in 1
year, $750 in 2 years, and $500 in 3 years.
What is the PV of these cost savings today if
10% is the relevant interest rate?
$1,000
$750
$500
PV 


 $1,904.58
1
2
3
1  10% 1  10% 1  10% 
5-22
Compounding Periods
Other Than Annual

Future value of a lump sum.
 i nom 
FVn  PV x 1 

m 

 inom
mx n
= nominal annual interest rate
 m = number of compounding periods per year
 n = number of years
5-23
Compounding Periods
Other Than Annual

A $1,000 investment earns 6% annually
compounded monthly for 2 years.
12 x 2
 6% 
FV2  $1,000 x 1 

12 

FV2  $1,000 x 1  0.5%   $1,127.16
24
5-24
Compounding Periods
Other Than Annual

PV of a lump sum uses a similar adjustment to the basic
equation for non-annual compounding.
PV 



FVn
 i nom 
1 

m 

mx n
inom = nominal annual interest rate
m = number of compounding periods per year
n = number of years
5-25
Compounding Periods
Other Than Annual

Annuity computations require the annuity period and the
compounding period to be the same.

For example, suppose a car loan for $12,000 required 20
equal monthly payments and uses a 12% annual rate
compounded monthly.



The annuity payments and the compounding periods are
both monthly.
The interest rate needs to be expressed as a monthly rate:
I = 12% / 12 = 1%
5-26
Compounding Periods
Other Than Annual

The car loan payment can be computed with the
following equation:
1


1
 1  1.00% 20 

$12,000  PMT x 
1.00%





And the car loan payment = $664.98.
5-27
Effective Annual Rate

An effective annual rate is an annual
compounding rate. When compounding periods
are not annual, the rate can still be expressed as
an effective annual rate using the following:
m
 i nom 
Effective Annual Rate  1 
 1
m 

 inom
= nominal annual rate
 m = number of compounding periods in 1 year
5-28
Effective Annual Rate

A bank offers a certificate of deposit rate of 6%
annually compounded monthly. What is the
equivalent effective annual rate?
12
 6% 
12
1 
  1  1  0.5%  - 1  6.17%
12 

5-29
Amortized Loans

Amortized loans are paid off in equal payments
over a set period of time and can be viewed as
the PV of an ordinary annuity.

An amortization schedule follows for a $120,000
mortgage to be paid of with 360 monthly
payments of $965.55 each over 30 years. The
interest rate is 9% annually compounded
monthly or 0.75% per month.
5-30
Loan Amortization Schedule
Month
Payment
Interest
Principal
Balance
0
1
2
3
4
$965.55
965.55
965.55
965.55
$900.00
899.51
899.01
898.51
$65.55
66.04
66.53
67.03
$120,000.00
119,934.45
119,868.41
119,801.88
119,734.85
357
358
359
360
965.55
965.55
965.55
965.55
28.43
21.40
14.32
7.19
937.12
944.14
951.23
958.36
2,853.73
1,909.58
958.36
0.00
5-31
Loan Amortization Schedule

A loan amortization schedule shows:

The amount of each payment apportioned to pay
interest. The amount paid towards interest
declines since the principal balance is declining.

The amount of each payment apportioned to pay
principal balance. The amount paid towards
principal balance increases as the interest amount
declines.

The remaining balance after each payment.
5-32
Summary

FV (p. 91)& PV of a lump sum (p. 93)

FV (p 96) & PV (p 102) of an annuity and PV of a
perpetuity

PV of a series of non-constant cash flows

Compounding other than annual and effective
annual rates

Loan amortization schedule
5-33
Homework
Problems 1,2,3,4,13
and Problem Sheet
on the website
5-34