Ch. 4 - The Time Value of Money

Ch. 4 - The Time Value of
Money
Topics Covered
Future Values
Present Values
Multiple Cash Flows
Perpetuities and Annuities
Effective Annual Interest Rate
Inflation & Time Value
The Time Value of Money
Compounding and
Discounting Single
Sums
Future Values
Future Value - Amount to which an
investment will grow after earning interest.
Compound Interest - Interest earned on
interest.
Simple Interest - Interest earned only on the
original investment.
Future Values
Example - Simple Interest
Interest earned at a rate of 6% for five years on
a principal balance of $100.
Interest Earned Per Year = 100 x .06 = $ 6
Future Values
Example - Simple Interest
Interest earned at a rate of 6% for three years
on a principal balance of $100.
Today
Interest Earned
Value
100
1
6
106
Future Years
2
3
6
6
112 118
Value at the end of Year 3 = $118
Future Values
Example - Compound Interest
Interest earned at a rate of 6% for three years on
the previous year’s balance.
Interest Earned Per Year =Prior Year Balance x
.06
Future Values
Example - Compound Interest
Interest earned at a rate of 6% for three years
on the previous year’s balance.
Today
Interest Earned
Value
100
Future Years
1
2
3
6.00
6.36 6.74
106.00 112.36 119.10
Future Value of $100 compounded at 6% for three
years = $119.10
Future Value of Single Cash Flow
FV  PV  (1  r )
t
Future Values
FV  $100  (1  r )
t
Example - FV
What is the future value of $100 if interest is
compounded annually at a rate of 6% for three years?
FV  $100  (1  .06)  $119.10
3
Future Values with Compounding
Interest Rates
7000
0%
6000
5000
10%
4000
15%
3000
2000
1000
Number of Years
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
0
FV of $100
5%
Example: Mutual Fund Fees and
Retirement Savings
Prof. Finance moves to a new university and has
$100,000 in retirement savings to invest (rollover)
into a new retirement account.
Prof. Finance wants to invest this money for 25
years into an indexed stock fund, which is
expected to return 9% annually.
Prof. has two choices: Vanguard Total Equity
Fund with a 0.4% annual expense fee and
Onguard Total Fencing Fund with an 1.2% annual
expense fee.
What is the difference in Prof. Finance’s expected
future retirement savings between the two funds?
Present Values
Present Value
Discount Factor
Value today of a
future cash
flow.
Present value of
a $1 future
payment.
Discount Rate
Interest rate used
to compute
present values of
future cash flows.
Present Values
Present Value = PV
PV =
Future Value after t periods
(1+r) t
Example: Paying for Baby’s
MBA
Just had a baby. You think the baby will
take after you and earn academic
scholarships to attend college to earn a
Bachelor’s degree. However, you want
send your baby to a top-notch 2-year MBA
program when baby is 25. You have
estimated the future cost of the MBA at
$85,000 for year 1 and $89,000 for year 2.
Example: Paying for Baby’s
MBA
Today, you want to finance both years of
baby’s MBA program with one payment
(deposit) into an account paying 8%
interest compounded annually.
How large must this deposit be?
Time Value of Money
(applications)
Value of Free Credit
Implied Interest Rates
Internal Rate of Return
Time necessary to accumulate funds
Example : Finding Rate of
Return or Interest Rate
A broker offers you an investment (a zero
coupon bond) that pays you $5,000 five
years from now for the cost of $3,700
today.
What is your annual rate of return?
Important Time Value
Relationships
Increasing interest rate and time
increases future value. POSITIVE
RELATIONSHIP.
Increasing interest rate and time
decreases present value. INVERSE
RELATIONSHIP.
The Time Value of Money
Compounding and
Discounting
Cash Flow Streams
0
1
2
3
4
Perpetuities
Suppose you will receive a fixed payment
every period (month, year, etc.) forever.
This is an example of a perpetuity.
Perpetuities
PV of Perpetuity Formula
PV 
C = cash payment
r = interest rate
C
r
Perpetuities & Annuities
Example - Perpetuity
You want to create an endowment to fund a
football scholarship, which pays $15,000 per
year, forever, how much money must be set
aside today in the rate of interest is 5%?
PV 
15,PV
000

.05
15, 000
.05
 $300,000
 $300,000
Perpetuities & Annuities
Example - continued
If the first perpetuity payment will not be
received until three years from today, how much
money needs to be set aside today?
PV 
300, 000
(1.05) 3
 $259,151
Annuities
Annuity: a sequence of equal cash flows,
occurring at the end of each period. This
is known as an ordinary annuity.
0
PV
1
2
3
4
FV
Examples of Ordinary Annuities:
If you buy a bond, you will receive equal
semi-annual coupon interest payments
over the life of the bond.
If you borrow money to buy a house or a
car, you will pay a stream of equal
payments.
Annuity-due
A sequence of periodic cash flows
occurring at the beginning of each period.
0
PV
1
2
3
4
FV
Examples of Annuities-due
Monthly Rent payments: due at the
beginning of each month.
Car lease payments.
Cable TV and most internet service bills.
Perpetuities & Annuities
PV of Ordinary Annuity Formula
PV  C

1
r

1
t
r ( 1 r )

C = cash payment
r = interest rate
t = Number of years cash payment is
received
Perpetuities & Annuities
PV Annuity Factor (PVAF) - The present
value of $1 a year for each of t years.
PVAF 

1
r

1
r ( 1 r ) t

Perpetuities & Annuities
Applications
Value of payments
Implied interest rate for an annuity
Calculation of periodic payments



Mortgage payment
Annual income from an investment payout
Future Value of annual payments
FV   C  PVAF   (1  r )
t
Perpetuities & Annuities
PV (and FV) of Annuity-dues = PV (or FV) of
ordinary annuity x (1 + r) or BGN mode on
financial calculator.

PV  C 
1
r
1
t
r (1 r )
(1 r)
C = cash payment
r = interest rate
t = Number of years cash payment is
received
Example: Invest Early in an IRA
How much would you have at age 65 if
you deposit $2,500 at the end of each
year in an account paying 9% annually
starting at:


(A) age 41?
(B) age 22?
Why an IRA?
Imagine in the last example, you didn’t take
advantage of the tax-sheltered environment of
an IRA.
Your annual investment return would be taxed!
With a 28% tax rate, our annual after-tax return
would fall from 9% to 6.48% (=9%(1-.28)).
At age 65: I would have $135,519 vs. $191,975.
You would have $535,392 vs. $1,102,114: 52%
less!! The IRS killed Kenny,…!
Example: Enjoying your
Retirement
You go ahead and make the contributions
starting at age 22 in the last example, giving you
$1,102,114 at age 65.
You expect to live to age 85. So, you want to
make 20 annual withdrawals from your IRA
paying 9% at the beginning of each year starting
at age 65.
How large can this annual withdrawal be?
More annuity fun, enjoying your
release from baseball
Bob B. is released from the last year of his
guaranteed contract from a New York
baseball team. He is due $5.9 million from
the last year of this contract. Bob and the
team agree to defer the $5.9 million at 8%
interest for 15 years. At this time (15 years
from today), the team will begin the first of 15
equal annual payments at 8% interest.
The press is reporting the payments will total
$30 million. Are they correct?
Non-Annual Interest
Compounding
When interest is compounded more frequently than once
a year.
Important non-annual compounding terms and things to
know:



Quoted Annual Rate, or Annual Percentage Rate (APR):
Stated nominal annual rate before compounding.
Effective Annual Rate (EAR): the actual (effective) annual
interest rate earned or paid.
Periodic Interest Rate: the interest rate paid or charged
each interest compounding period = quoted rate/m, where
m = number of compounding periods per year.
Effective Interest Rates
example
Given a monthly rate of 1%, what is the Effective
Annual Rate(EAR)? What is the Annual
Percentage Rate (APR)?
12
EAR = (1 + .01) - 1 = r
EAR = (1 + .01)12 - 1 = .1268 or 12.68%
APR = .01 x 12 = .12 or 12.00%
FV and PV with non-annual
interest compounding
n = number of years
m = number of times interest is paid per
year
APR = nominal annual rate (APR)
APR/m = periodic rate
Single CF
FVnm = PV(1+ARR/m)nm
PV = FVnm/(1+APR/m)nm
Non-annual annuities
Ordinary:
PV = C(PVAFAPR/m,nm)
FVnm = C(PVAFAPR/m,nm)(1+APR/m)nm
Annuity-Due:
PV = C(PVAFAPR/m,nm)(1+APR/m)
FVnm = C(PVAFAPR/m,nm)(1+APR/m)nm+1
Example: Low Rate or Rebate?
The Frontier family want to buy a sport ut (SUV). They decide
on a 4-wheel drive Jeep Grand Cherokee. The purchase price
with tax of the vehicle is $32,500. The Frontiers have $4,000
as a down payment.
Jeep offers the choice of two incentives on the 4-door Grand
Cherokee.



0% APR Financing for 60 months, or
$3,000 rebate which would be applied toward the purchase price.
If the Frontiers elect to take the rebate, they can get 4.49% APR
financing for 60 months.
Question: Which incentive would give the Frontiers the lowest
monthly payment?
Example: The $200 national ISP
signup credit: good deal for whom?
A national ISP all provide $200 for new customers to use
at a particular electronics store chain if they sign-up for a
2-year internet service contract at $21.95/month.
What interest rate (APR) are you paying on this “free
money” if you wanted internet service and could get it for
free? (200 PV, -21.95 PMT, 24 N 0 FV, CPT I/Y =
9.8%/month x 12= 117.8% APR!!)
What interest rate (APR) are you paying on this “free
money” if you wanted internet service and could get it for
$9.95/month?(-12 PMT CPT I/Y = 3.15%/mo x 12 =
37.8%! Thanks, but no thanks!
Inflation
Inflation - Rate at which prices as a whole
are increasing.
Nominal Interest Rate - Rate at which
money invested grows.
Real Interest Rate - Rate at which the
purchasing power of an investment
increases.
Inflation
1+ nominal interest rate
1  real interest rate =
1+inflation rate
approximation formula
Real int. rate  nominal int. rate - inflation rate
Inflation
Example
If the interest rate on one year govt. bonds is
5.0% and the inflation rate is 2.2%, what is the
real interest rate?
1+.050
1  real int erestrat e = 1+.022
Savings
1  real int erestrat e = 1.027
Bond
real int erestrat e = .027 or 2.7%
Approximat
ion = .050 - .022 = .028or 2.8%
Example: Real retirement income
Going back to your retirement in 43 years,
you expect 3% inflation along with your
9% nominal investment rate annually and
want to withdraw $32,000 in real terms at
the beginning of each year for 20 years
once you retire.
How will this change your retirement
saving plans?